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Theses and Dissertations
2003
Preservice secondary school mathematics teachers'
knowledge of trigonometry : subject matter content
knowledge, pedagogical content knowledge and
envisioned pedagogy
Cos Dabiri Fi
Copyright © 2003 Cos Dabiri Fi. Posted with permission of the author.
This dissertation is available at Iowa Research Online: http://ir.uiowa.edu/etd/4936
Recommended Citation
Fi, Cos Dabiri. "Preservice secondary school mathematics teachers' knowledge of trigonometry : subject matter content knowledge,
pedagogical content knowledge and envisioned pedagogy." PhD (Doctor of Philosophy) thesis, University of Iowa, 2003.
http://ir.uiowa.edu/etd/4936.
Follow this and additional works at: http://ir.uiowa.edu/etd
Part of the Science and Mathematics Education Commons
PRESERVICE SECONDARY SCHOOL MATHEMATICS TEACHERS'
KNOWLEDGE OF TRIGONOMETRY: SUBJECT MATTER CONTENT
KNOWLEDGE, PEDAGOGICAL CONTENT KNOWLEDGE AND ENVISIONED
PEDAGOGY
by
Cos Dabiri Fi
A thesis submitted in partial fulfillment
of the requirements for the Doctor o f
Philosophy degree in Education
in the Graduate College o f
The University o f Iowa
August 2003
Thesis Supervisor: Professor Douglas A. Grouws
Copyright by
COS DABIRI FI
2003
All Rights Reserved
Graduate College
The University o f Iowa
Iowa City, Iowa
CERTIFICATE OF APPROVAL
PH.D. THESIS
This is to certify that the Ph.D. thesis o f
Cos Dabiri Fi
has been approved by the Examining Committee
for the thesis requirement for the Doctor o f Philosophy
degree in Education at th4 August 2003 graduation.
Thesis Committee:
Douglas A. Grouws, Thesis Supervisor
Flarold Schoen
William Nibbelink
Tim Ansley
W a l t e r Seaman
To the spirit o f perseverance and the desire for knowledge, knowing, understanding and
advancement o f knowledge, and to the evolution o f peace.
ii
ACKNOWLEDGMENTS
I would like to express my profound heartfelt thanks for the tutelage and direction
of my thesis committee that made this thesis possible. A very special thank you goes to
my academic advisor, Douglas A. Grouws, for his incubation of my fledgling ideas for a
proposal o f study. I also want to express my profound gratitude to my wonderful mother
Dr. Blessing Mboma Fubara, my loved ones, friends, and colleagues. I could not have
completed this dissertation without your support, advice and vigilance. Thank you.
iii
ABSTRACT
The education community recognizes that subject matter content knowledge and
pedagogical content knowledge form the bases for effective teaching. The purpose o f this
study was to assess the subject matter content knowledge, pedagogical content
knowledge, and envisioned practice of preservice secondary school mathematics teachers
in the area of trigonometry.
Data was collected in two phases. Phase 1 involved 14 preservice secondary
mathematics teachers who had completed at least two methods courses in mathematics
education and a practicum course. All participants in phase one o f the study completed
(1) a test measuring knowledge o f trigonometry, (2) a measure o f pedagogical content
knowledge with respect to trigonometry via two card-sorting activities, and (3) two
concept maps.
Five case studies formed the basis for phase two o f the study. The five cases were
selected after a preliminary analysis o f the data from phase one. Individuals were chosen
to achieve distinct profiles: (1) high subject matter content knowledge and (2) low subject
matter content knowledge. Each case study participant was interviewed twice using semi­
structured interviews designed to delve deeply into their trigonometric knowledge and
envisioned teaching practice. Interview data were transcribed and analyzed using the
qualitative methods o f constant comparison and content analysis.
Subject matter content knowledge items were scored for correctness with credit
given for partially correct solutions. The card sorting activities and concept maps, and the
interview data were analyzed using primarily qualitative methods.
Results indicate that these preservice secondary school mathematics teachers have
poorly developed understanding in such areas as: radian measure o f angles, inverse
trigonometric functions, reciprocal functions, periodicity, and co-functions. Many o f the
scores on the test o f trigonometric knowledge were below the 50 percent correct level.
iv
These findings agree with prior research findings that preservice teachers' knowledge of
many areas o f school mathematics is weak and considerably below what is usually
expected. The preservice teachers' concept maps o f trigonometric ideas generally focused
on either right triangles or notions of function. The sequencing task revealed that these
preservice teachers seldom considered prerequisite skills in planning lessons.
v
TABLE OF CONTENTS
LIST OF TABLES
............................................................................................................... XI
LIST OF FIGURES
............................................................................................................... xii
CHAPTER 1: PROBLEM STATEMENT AND SIGNIFICANCE..................................
1
Purposes o f the study...............................................................................
Background for the stu d y .......................................................................
Rationale and theoretical basis for the stu d y.......................................
Why study trigonometry?.............................................................
Why study preservice high school mathematics
teachers?...............................................................................
Research questions..................................................................................
Summary o f chapter I ..............................................................................
1
1
n
n
16
70
23
CHAPTER II: REVIEW OF THE LITERATURE............................................................... 25
Introduction and organizational structure o f chapter II......................
Teacher know ledge.................................................................................
Epistemology...........................................................................................
Literature dealing with trigonom etry...................................................
Concept m apping....................................................................................
Card sorting...............................................................................................
Summary o f chapter II.............................................................................
75
75
31
37
37
36
40
CHAPTER III: DESIGN OF THE STU DY .......................................................................... 41
Methodology.............................................................................................
Introduction (review of the purpose o f the stu d y )...................
Methodological foundations........................................................
Description of methodology..................................................................
Participants.....................................................................................
Instrumentation..............................................................................
Test of trigonometric knowledge......................................
Card sorting ta sk s ...............................................................
Interviews.............................................................................
Concept m a p s......................................................................
Consent fo rm .......................................................................
Procedures and data collection...............................................................
Pilot study.......................................................................................
The main study: Phase O n e .........................................................
Concept m a p s......................................................................
Card sorts..............................................................................
Test o f trigonometric knowledge......................................
The main study: Phase tw o..........................................................
Interview 1............................................................................
Interview 2 ............................................................................
Data analysis.............................................................................................
Data analysis framework...............................................................
Concept m apping...........................................................................
vi
41
41
41
43
43
45
45
47
48
46
50
50
51
55
58
56
60
60
67
63
63
64
65
Meaning of quantitative scores........................................
Card sorts.......................................................................................
Test o f trigonometric knowledge...............................................
The interview data........................................................................
The research questions and the related analyses.....................
68
69
71
73
74
CHAPTER IV: PRESENTATION AND DISCUSSION OF RESU LTS........................ 77
Pilot study results...................................................................................
Concept m a p .................................................................................
Card sorts.......................................................................................
Test o f trigonometric knowledge...............................................
Instrument modifications based on pilot testing.....................
Phase one results: Concept maps, card sorts, and test o f
trigonometric knowledge.............................................................
Concept maps (CM1 and CM 2).................................................
Further analysis o f concept map 1 and concept map 2 ..........
Correct definitions, examples, and relationships..........
Mention o f sinusoids and their transformations...........
Mention o f applications o f trigonom etry.......................
Mention o f radian measure...............................................
Unused concepts and ideas...............................................
Misconceptions in concept map 1 and concept
map 2 ........................................................................
Card sort 1 .....................................................................................
Emergent themes for card sort 1......................................
Card sort 2 .....................................................................................
Test o f trigonometric Knowledge..............................................
Phase two results: Interviews and case studies..................................
Interview 1.....................................................................................
Concept map 1...................................................................
Concept map 2 ...................................................................
Comparison o f concept maps one and two....................
Student task number o n e ..................................................
Without the graphing calculator...........................
With calculator........................................................
Summary o f student’s task number o n e.........................
Student task number tw o ..................................................
Without the graphing calculator...........................
With the graphing calculator.................................
Card sort 2 ..........................................................................
Card sort 1 ..........................................................................
Technology.........................................................................
Interview 1 Summary........................................................
Interview 2 .....................................................................................
Problem solving, proof and justification........................
Interview question one: What is the radian
m easure?..................................................................
Interview question two: Prove that there are
360° in one revolution.............................................
Interview question three: Ferris wheel problem ...........
Interview question four: A proof question....................
Interview question five: Find domain values for
a given set o f range values.....................................
vii
77
78
81
8?
84
85
85
88
89
89
89
90
90
90
91
97
97
99
132
134
134
135
137
137
138
141
142
142
142
147
149
157
163
168
169
169
170
172
172
174
178
Five profiles o f understanding..............................................................
A X ..................................................................................................
N M .................................................................................................
E S ...................................................................................................
L N ..................................................................................................
A B ..................................................................................................
Summary o f cases........................................................................
Summary o f results.................................................................................
Knowledge of basic ideas, essential features, and
basic repertoire of trigonometry......................................
Multiple perspectives, different representations, and
alternative ways of approaching trigonometry
(including problem solving ideas)..................................
Connectedness, strength o f trigonometric concepts,
knowledge and understanding o f trigonometric
concepts...............................................................................
Longitudinal coherence and knowledge o f
trigonometry.......................................................................
Summary........................................................................................
180
180
184
186
189
191
193
194
196
196
197
198
199
CHAPTER V: CONCLUSIONS, DISCUSSION, LIMITATIONS AND
IMPLICATIONS............................................................................... 200
Overview and discussion of the results...............................................
What content knowledge o f trigonometry do
preservice secondary school mathematics
teachers possess?...............................................................
Definitions and terminology.............................................
Degree and radian measures............................................
Co-functions.......................................................................
Angles of rotation, coterminal angles, and
reference angles.......................................................
Special angles (30°, 45°, 60°), their triangles,
and their use to simplify com putation.................
Trigonometric functions and their graphs.....................
Domain and range..............................................................
Transformation of trigonometric functions...................
Even and odd functions....................................................
Laws o f cosines and sin es................................................
Trigonometric identities...................................................
Algebra and calculus o f trigonometry............................
The use of trigonometry in solving and
modeling mathematical and real-world
situations..................................................................
What pedagogical content knowledge o f trigonometry
do preservice secondary school mathematics
teachers possess?...............................................................
What prerequisite knowledge is necessary for
the learning o f trigonometry?................................
How do the preservice secondary mathematics
teachers understand multiple
representations that will prove useful to
unpacking the content of trigonometry for
students?...................................................................
viii
200
201
201
201
202
202
203
203
204
204
204
204
205
205
205
206
206
207
How do preservice secondary mathematics
teachers sequence and organize
trigonometric concepts for teaching?................... 208
Do the sequence and organization o f the
concepts anticipate both students’
preconceptions and misconceptions, and
possible approaches to help students
overcome such misunderstanding?....................... 208
If preservice secondary mathematics teachers
were presented with difficulties that
students might encounter, how would they
help students get better conceptualizations
of trigonometry?...................................................... 208
How are preservice secondary school mathematics
teachers’ content and pedagogical content
knowledge of trigonometry organized?......................... 209
How do preservice secondary school mathematics
teachers envision teaching trigonometry?..................... 209
How will they develop the six basic
trigonometric ratios?............................................... 209
What pedagogical approaches (didactic or
heuristic) will the preservice secondary
mathematics teachers employ?.............................. 209
How are preservice secondary school mathematics
teachers’ content and pedagogical content
knowledge of trigonometry related to their
envisioned application o f their content and
pedagogical content knowledge in mathematics
classroom s?........................................................................ 210
Limitations............................................................................................... 211
Im plications............................................................................................ 212
High school trigonometry (teaching and learning)................. 212
Preservice teacher education...................................................... 213
Concluding remarks and suggestions for further research............... 213
APPENDIX A: CONSENT F O R M ....................................................................................... 216
APPENDIX B: TEST OF TRIGONOMETRIC KNOWLEDGE...................................... 221
APPENDIX C: CARD SORT TASK 1................................................................................. 235
APPENDIX D: CARD SORT TASK 2................................................................................. 238
APPENDIX E: CONCEPT M A P S ........................................................................................ 241
APPENDIX F: INTERVIEW 1 .............................................................................................. 245
APPENDIX G: INTERVIEW 2 .............................................................................................. 250
APPENDIX H: INFORMATION AND INVITATION TO PARTICIPATE................. 254
APPENDIX I: INTRODUCTION TO CONCEPT M A PPIN G ........................................ 258
ix
APPENDIX J: PILOT VERSIONS OF THE TEST OF TRIGONOMETRIC
KNOWLEDGE.................................................................................. 264
APPENDIX K: REPRESENTATIVE SAMPLE CONCEPT M APS............................... 286
APPENDIX L: COMPARISON OF CM 1 & CM2, AND REASONS GIVEN
IN CARD SORT 1............................................................................ 297
APPENDIX M: V ITAE........................................................................................................... 304
REFERENCES.......................................................................................................................... 308
X
LIST OF TABLES
Table 1. Post-secondary course taking history o f the 14 participants in Phase 1 of
the study...........................................................................................................................
46
Table 2. Item Importance Rating by a mathematics professor o f the 25-item pre­
version of the test o f trigonometry............................................................................... 56
Table 3. The analytic framework for the concept mapping activities showing the
scales used in the analysis of the data..........................................................................
66
Table 4. Categorization o f trigonometric topics in card sort 2 into early,
intermediate, and advanced concepts and ideas.........................................................
70
Table 5. The results o f the first pilot concept mapping in which the participants
generated the terms used in the map............................................................................
79
Table 6. Difficulty Ratings o f the items on a pre-version of the test o f
trigonometric knowledge by 7 experienced teachers and one student teacher......
83
Table 7. Phase one participants’ performance scores from CM1 and CM 2..................... 87
Table 8. Correct classification for each o f the fifteen propositions o f card sort 1,
participant responses, and the number o f correct responses by participant...........
92
Table 9. The placements o f the propositions into the three piles (AT, TS, NT) by
the 14 participants...........................................................................................................
93
Table 10. Number o f participants that provided valid reasons for selected
propositions...................................................................................................................... 96
Table 11. Participants’ scores on the 17 items of the test o f trigonometric
knowledge........................................................................................................................ 100
Table 12. Test o f trigonometric knowledge item statistics................................................. 101
Table 13. Rating o f participants’ knowledge o f trigonometry........................................... 133
Table L - 1. Comparison of CM1 and CM2 including the terms/ideas/relations
used in both activities and some salient features o f both concept maps................. 298
Table L - 2. Reasons provided for placing the propositions for which the
participants had the most difficulties in card sort one............................................... 300
XI
LIST OF FIGURES
Figure 1. Concept Map Schematic........................................................................................
39
Figure 2. Concept map scoring model o f Novak and Gowin (1984, p. 3 7 ) ...................
67
Figure 3. A holistic model for scoring some of the free-response items on the test
of trigonometric knowledge.........................................................................................
73
Figure 4. Comparison between participants’ item scores in concept map 1 and
concept map 2 ................................................................................................................
85
Figure 5. Comparison o f the stream scores from concept map 1 and concept map
2 ........................................................................................................................................
86
Figure 6. Comparison of the ratio scores from concept 1 and concept map 2 ...............
87
Figure 7. Comparison o f composite scores from concept map 1 and concept map 2....
88
Figure 8. Sequence agreement with Hirsch & Schoen (1990) and Senk et al.
(1998)...............................................................................................................................
98
Figure 9. A holistic model for scoring free-response items on the test o f
trigonometric knowledge............................................................................................... 100
Figure 10. Description o f item 1 ............................................................................................. 102
Figure 11. The only incorrect response to item 1 ................................................................ 103
Figure 12. Description o f item 2 ............................................................................................. 104
Figure 13. Sample response to item 2 ................................................................................... 105
Figure 14. Second sample response to item 2 ....................................................................... 105
Figure 15. Description o f item 3 ............................................................................................. 106
Figure 16. Sample response to item 3 ................................................................................... 107
Figure 17. Second sample response to item 3 ....................................................................... 107
Figure 18. Description o f item 4 .............................................................................................
108
Figure 19. Sample response to item 4 ................................................................................... 109
Figure 20. Second sample response to item 4 .......................................................................
110
Figure 21. Description o f item 5 ............................................................................................. 11 1
Figure 22. Sample response to item 5 ................................................................................... 1 11
Figure 23. Second sample response to item 5 ....................................................................... 112
xii
Figure 24. Description o f item 6 ............................................................................................ 113
Figure 25. Sample response to item 6 ................................................................................... 114
Figure 26. Second sample response to item 6 ....................................................................... 114
Figure 27. Description o f item 7 ............................................................................................ 115
Figure 28. Sample response to item 7 ................................................................................... 116
Figure 29. Second sample response to item 7 ....................................................................... 116
Figure 30. Description o f item 8 ............................................................................................ 117
Figure 31. Sample response to item 8 ................................................................................... 118
Figure 32. Second sample response to item 8 ....................................................................... 119
Figure 33. Description o f item 9 ............................................................................................ 120
Figure 34. Description o f item 1 0 .......................................................................................... 121
Figure 35. Sample response to item 1 0 ................................................................................. 122
Figure 36. Second sample response to item 10.................................................................... 122
Figure 37. Description o f item 1 1 .......................................................................................... 123
Figure 38. Description o f item 1 2 .......................................................................................... 124
Figure 39. Description o f item 1 3 ......................................................................................... 125
Figure 40. Description o f item 1 4 ......................................................................................... 126
Figure 41. Sample response to item 1 4 ................................................................................ 126
Figure 42. Second sample response to item 14................................................................... 127
Figure 43. Description o f item 1 5 ......................................................................................... 127
Figure 44. Sample response to item 1 5 ................................................................................ 128
Figure 45. Second sample response to item 15................................................................... 129
Figure 46. Description o f item 1 6 ......................................................................................... 130
Figure 47. Sample response to item 1 6 ................................................................................ 130
Figure 48. Description o f item 1 7 ......................................................................................... 132
Figure I - 1. Hierarchical Concept Map Schematic............................................................ 260
Figure I - 2. A concept map o f how to construct a hierarchical concept map ACES....... 763
xiii
Figure K - 1. Concept map 1 produced by LN in phase one o f the study........................ 287
Figure K - 2. Concept map 2 produced by LN in phase one o f the study........................ 288
Figure K - 3. Concept map 1 produced by AB in phase one o f the study........................ 289
Figure K - 4. Concept map 2 produced by AB in phase one o f the study........................ 290
Figure K - 5. Concept map 1 produced by ES in phase one o f the study......................... 291
Figure K - 6. Concept map 2 produced by ES in phase one of the study......................... 292
Figure K - 7. Concept map 1 produced by NM in phase one o f the study....................... 293
Figure K - 8. Concept map 2 produced by NM in phase one o f the study....................... 294
Figure K - 9. Concept map 1 produced by AX in phase one o f the study........................ 295
Figure K - 10. Concept map 2 produced by AX in phase one o f the study..................... 796
xiv
1
CHAPTER 1
PROBLEM STATEMENT AND SIGNIFICANCE
Purposes of the study
The purpose o f this study was to assess the content knowledge, pedagogical
content knowledge, and envisioned practice o f preservice secondary school mathematics
teachers in the area o f trigonometry. More precisely, the study attempted to accomplish
the following:
Characterize the depth o f preservice secondary school mathematics teachers’
content and pedagogical content knowledge o f trigonometry in the school mathematics
curriculum.
Explore how preservice secondary school mathematics teachers envision applying
their content and pedagogical content knowledge o f trigonometry in teaching situations.
Provide a description o f the relationships among preservice secondary school
mathematics teachers’ content knowledge, pedagogical content knowledge, and
envisioned practice.
The study was intended to complement prior research on teacher knowledge and
to enhance our understanding of teachers’ knowledge o f school mathematics. Educators
have recognized the nature of teacher knowledge as an important component in planning,
orchestrating instruction, and reflecting on instruction and students’ learning. The
literature review in chapter II details and explains what we know about the relationship
between what teachers know and classroom pedagogy.
Background for the study
In this section, I situate conceptions of teacher knowledge within the greater
domain o f mathematics education. Teacher knowledge has meant different things during
different eras in mathematics education in the United States. As I present the different
2
eras in mathematics education, I invite the reader to reflect on what effective teaching
would have looked like during the eras.
To assist the reader in better understanding the fundamental assumptions o f this
study, a definition of effective teaching is presented next. A major goal o f mathematics
education is the improvement o f students’ understanding of mathematics. The main factor
in achieving this goal is the mathematics teacher. Therefore effective teaching of
mathematics is a primary concern for mathematics education.
Adding It Up (Kilpatrick, Swafford, & Findell, 2001) refers to effective teaching as
proficient teaching. They define proficient teaching as teaching for:
•
conceptual understanding;
•
fluency;
•
strategic competence (problem solving);
•
adaptive reasoning (proof and justification); and
•
productive disposition (perseverance and thinking mathematically) (p. 380).
Mathematics education has undergone several re-evaluations and re-organizations
of itself since the beginning o f the fledgling profession late in the 19th century. In the US,
the period before 1888 was considered the Era o f the Greek Mind1. That era emphasized
deductive reasoning, the nature o f mathematics, and logical transfers. According to
Mayer (1992), beginning theories on how humans think and learn were traceable to the
Greek philosopher Aristotle. Aristotle claimed that learning and memory occur by
contiguity, similarity, and contrast. The doctrine of contiguity posits that events or
objects that occur in the same time or space are associated in memory, so that thinking of
one will cause thinking of the other. The doctrine o f similarity claims that events or
objects that are similar tend to be associated in memory. And finally, the doctrine of
1 Professor William H. Nibbelink of the University of Iowa in the Foundation of
Mathematics Education course that he teaches supplied era demarcation and terminology.
3
contrast stipulates that events and objects that are opposites tend to be associated in
memory (Mayer, 1992).
Locke reformulated the ideas o f Aristotle in the eighteenth century (Mayer, 1992).
Locke categorized mental processes into atomism, mechanization, empiricism, and
imagery. Atomism is the notion that all ideas and the association between any two of
them are specific. Mechanization is the idea that the movement from one idea to another
is automatic and based solely on strength o f association. Empiricism is the theory that all
knowledge comes from sensory experience. Within empiricism the mind at its inception
is a blank slate. Imagery is the idea that thinking is merely the automatic movement from
point to point along mental paths established through learning.
The period froml888 to 1923 was the Era o f the Practical. During this period
Thorndike developed the theory o f identical elements transfer, which served as the basis
of theorizing in that era. The theory of identical elements transfer states, “positive
transfer occurs because some o f the elements in the to-be-learned task (B) are identical to
elements that the learner has already learned from previous task (A)” (Mayer, 1992, p.
37). John Dewey was the major influence on the Era of the Child (1923 - 1940) that
followed the era o f the practical. The era o f the child emphasized the nature o f the child,
inductive reasoning, and the humanization o f mathematics and other disciplines. The
Period of Confusion ensued after the Era o f the Child. It lasted from 1940 through 1950.
The confusion may have been the result o f Second World War and its aftermath. The
years from 1950 to 1957 was the Period o f the Underground Movements & Curricula
Experimentations. This was the period of university lab schools.
In 1957, the Soviet Union sent an unmanned spacecraft Sputnik into space. In
response the educational communities in the US and the US public marshaled resources
to train students to become the next cadre o f scientists and engineers. School mathematics
became more rigorous and reflected greater symbolism. Understanding o f mathematics
structure became the cornerstone o f mathematics education. This was the era of the New
4
Math, with Jerome Bruner as the arch-theorist. The new math era saw spiraling o f the
curriculum and emphases on deductive reasoning, formalism, symbolism, algebra, and
even topology in school mathematics. Educators viewed kids as miniature scientists. By
1971, teachers, parents, and policymakers had become disillusioned with the miniature
scientist paradigm. Reasons for the disillusionment vary, but lack o f teacher
preparedness, and ill fashioned and conceived curricula materials have been blamed for
the failure o f the massive educational reform called the New Math.
The country for the better part of five years experimented with individualized
instructional strategies during the After Math Era. By 1975, drill and computation had
replaced individualized instruction as the agenda for mathematics education. Brownell
(1987)2 anticipated this seesawing when he cogently explicated the mutuality o f both
conceptual and procedural understanding o f mathematics. However, the wisdom o f
Brownell was not enough to persuade the field and mathematics education ushered in the
Back to Basics Movement. The Back to Basics movement redesigned school mathematics
to first teach the “basics.” According to proponents of the back to basics movement, once
students have gotten the basics, then they will be free o f any encumbrances with
fundamental details when they problem-solve.
According to Brownell, meaning and computational competence are valuable. He
observed that some school districts wanting to improve their students’ mathematical
understanding, during the new-math era, had embarked on programs for meaningful
arithmetic, only to see their students' scores on standardized tests decline. The response to
the decline in scores has been to disparage the tests as ineffective at assessing
understanding of arithmetical ideas and procedures. Furthermore, the tests were
disparaged as measuring outcomes o f little significance, thus in essence arguing that
2 Originally printed in 1956 in the Arithmetic Teacher, by the NCTM
5
computational skills are unimportant. Brownell argued that such disparagements were not
acceptable to stakeholders in the mathematics education o f students.
Sources o f the errors in thinking that computational competence is useless or at
the least should be minimized in our schools came from experts' advice and
recommendations to teach for "relational" understanding and their comparative silence on
the role or place for computational competence. The experts' advice caused teachers to
de-emphasize practice. And "in fleeing from over reliance on practice, we may have fled
too far" (Brownell, 1987, p. 19). Another source o f error came from misinterpretation or
misapplication o f psychological theories. Conditioning theory (S —R) has been
misunderstood to mean that once an instance o f a desired response is made to an event
(stimulus), then there exist a connection that ties the stimulus to the response. For
example, once a student respond correctly to 5+2 = 7, it is assumed that they understand
single-digit addition.
Another psychological theory that is often misunderstood is field theory. "It is
often said that one experience o f "insight" or "hindsight" —before, during, or after
success —is enough" (Brownell, 1987, p. 19). But that is not so, argued Brownell. A third
source o f error exists in educational theories. When educational theories are
misunderstood, dogmatism may set in. For example, dogmatism in beliefs and practices
as it related to the nature o f the child resulted in the view that practice was evil and
should be avoided in favor o f conceptual or meaningful learning. "Attempts to guide and
direct learning and to organize learning experiences were frowned on as 'violating child
nature' and as almost certainly productive of serious derangement o f child personality" (p.
19). Teachers’ misunderstanding o f pedagogical and cognitive actions needed to teach for
understanding in arithmetic is another source of error. Teachers accept memorized
responses to questions. Students may not understand the base-ten system, and they may
not understand the reasons for the many operations and properties of operations on
different sets o f numbers.
6
To situate both practice and meaningful learning centrally in the education of
mathematics, Brownell argued that we should conceptualize meaning as a continuum and
not an "all or nothing" proposition. Furthermore, "there are degrees or levels of
understanding ... and not all forms o f practice are alike" (p. 21). Brownell exhorted
teachers to teach meaningfully by "directing learning in such a way that children ascend,
as it were, a stairway o f levels o f thinking arithmetically [mathematically] to the level of
meaningful habituation in those aspects o f arithmetic [mathematics] that should be
thoroughly mastered, among them the basic computational skills" (p. 22). Brownell
defined meaningful habituation as "the almost automatic way in which the required
response is invariably made; meaningful implies that the seemingly simple behavior has a
firm basis in understanding" (p. 22).
Children need to move progressively along the continuum. Moreover, teachers
should not require students to perform at levels that they have not achieved because the
children may refuse to learn (I won't, I can't, I don't care), they may become silent, yet
perform, and if not carefully monitored could come across as having attained a level that
they have not. Or they may begin to guess and rely on recall o f memorized facts without
any understanding. Teachers should also use "varied practice" that incorporates different
approaches to help students move upwards from where they are towards meaningful
habituation. Additionally, teachers should use repetitive practice for memorization,
develop competence, and save time. They should also include practices that span the
whole spectrum from the repetitive practice to varied practice.
If we are to achieve the dream set forth by Brownell and countless educators that
have proposed meaningful learning and teaching for understanding, our teachers will
have to be well educated in the subject matter, and the pedagogical methodologies.
Furthermore, our teachers ought to experience mathematics meaningfully so that they can
teach meaningfully.
7
The back to Basics movement disregarded Brownell’s call for both conceptual
and procedural learning. Rather it focused almost exclusively on procedural learning. The
Back to Basics failed and by 1979, Problem Solving presented a clear and unequivocal
alternative to the drudgery and bore o f drill and computation. The phenomenon of
problem solving took hold and lasted through the 1990’s. However, not all problem
solving was problem solving. The set of practices that were encapsulated into problem
solving ranged from drill and practice exercises, to Polya-type problem solving
(Schoenfeld, 1992; Lester, 1994).
The problem-solving era built upon the earlier ideas o f Polya (1945). Due to
events outside o f mathematics education, such as the Second World War and the space
race with the Soviet Union, the ideas o f Gestalt Psychologists (later made rigorous by
Cognitive Scientists), and Polya (1945) remained underutilized until the late 1970’s. The
publication o f a series o f seminal documents: (1) The Agenda for Action (National
Council o f Teachers o f Mathematics, 1980); (2) Everybody Counts (National Research
Council, 1989); (3) and the Standards (NCTM, 1989, 1991, 1995, 2000) emphasized the
need to stay the course towards problem-solving and meaningful mathematics.
By the late 1980s and early 1990s, a reformulation o f cognitive science that
incorporated Piaget (cognitive constructivism), Vygotsky (zone o f proximal learning),
and situated cognition (Lave, Greeno, Brown, Collin, Duguid, and Cobb) began to
germinate. The Era of Constructivism was bom. The major theoretical frameworks
emerged from the works o f Piaget (1954), Vygotsky (1978) and Lave (1988).
Serendipitously, the notion o f progressive formalization undergirds the refocusing
on teaching and learning mathematics meaningfully (Bransford, Brown, & Cocking,
2000; Hiebert, Carpenter, Fennema, Fuson, Weame, Murray, Olivier, & Human, 1997).
Progressive formalization, which is akin to Brownell’ habituation, argues for beginning
with informal approaches to teaching and learning for understanding. Formalization and
abstraction are then built from the prior understanding of the concepts. The resurgence of
8
meaningful learning and teaching o f school mathematics in the late 1990s has refocused
mathematics educators on the types o f knowledge that teachers need to teach
mathematics effectively. Although a focus on teachers’ know-how about teaching is not
new, the accumulation o f data and theories (see for example, von Glasersfeld, 1996;
Hiebert et al., 1997; Hiebert & Carpenter, 1992; Shulman, 1986, 1987; Leinhardt &
Smith, 1985; Fennema & Franke, 1992; Thompson, 1992; Koehler & Grouws, 1992; and
Dossey, 1992) have helped mathematics educators in better understanding teacher
knowledge through articulation o f the connections among belief, affective domain,
pedagogical content knowledge, subjectivity o f knowing, and teachers’ subject matter
knowledge.
In other words, for teachers to teach effectively, they need to acquire a “profound
understanding o f fundamental mathematics [PUFM]” (Ma, 1999). But how do we help
teachers gain such understanding o f mathematics? Should we develop PUFM in
preservice education or is it only possible through in-service experience? Preservice
teacher education provides students with the tools to be effective teachers in the various
subject domains. However, cognitive science has found that transfer does not occur as
easily and often as we would want it (Holyoak & Koh, 1987; Brown, 1989). But studies
by Lave, Wenger, and others, dubbed everyday thinking studies, have also shown that
when people learn in their natural settings, engaged in meaningful activities, they tend to
transfer their knowledge to novel situations better than if they learned in decontextualized situations.
The implication o f the everyday thinking studies, and studies on teachers’
pedagogical content knowledge in mathematics is that preservice teachers ought to be
conversant with the subject matter they intend to teach; in all its manifestations (Ball,
1988, 1991; Ma, 1999; CBMS, 2001). This includes a thorough understanding o f the
content, to an extent that enables the teachers to unpack the content during their teaching
practice (Kilpatrick, et al, 2001).
9
Syntheses o f the literature on teacher knowledge paint a picture o f three major
components that are necessary for effective teaching: knowledge o f mathematics,
knowledge of students, and knowledge of instructional practices (Kilpatrick, et al., 2001).
Preservice teachers have limited knowledge o f students and knowledge of instructional
practices. Arguably, their recency with advanced mathematics in institutions o f higher
learning ought to make them experts o f school mathematics. But the truth cannot be
farther from that assumption (Ball, 1990, 1991).
The education community considers content knowledge of mathematics the basis
for effective mathematics teaching. As such, efforts and energies are being spent on
providing preservice teachers with a thorough understanding of school mathematics. The
new conception in the profession is to re-organize the preservice experience so that the
mathematics needed for school mathematics is thoroughly elaborated, explicated and
their connections made transparent; including the “horizons o f that mathematics - where
it can lead and where their students are headed with it” (Kilpatrick, et al., 2001, p. 369).
In the past, mathematics training of teachers has focused on giving teachers the
procedural skill and the content o f mathematics without much integration and connection.
The result o f studies that have investigated formal course-taking by teachers and its
impact on student achievement have shown that, at best, 5 advanced courses is the
threshold for any impact (Begle, 1979; Monk, 1994). And that beyond five courses, there
is no appreciable impact on student achievement. However, these studies have been
critiqued on the grounds that they used an inaccurate predictive measure of course-taking
to assess teachers’ understanding o f mathematics. The critics posit that qualitative
descriptions of teachers’ knowledge o f mathematics that address the connectedness of
teachers’ mathematical knowledge, ability o f teachers to flexibly use the knowledge and
a grasp o f the fundamental basics, and an understanding o f the longitudinal coherence o f
school mathematics are better measures (Fennema & Franke, 1992).
10
One problem with preservice teachers’ knowledge o f mathematics is that they
learn increasingly abstract concepts in institutions o f higher learning. However, what is
required o f them is an ability to unpack the content “in ways that make the basic
underlying concepts visible” (Kilpatrick, et al., 2001, p. 376). In fact, when teachers are
given the opportunities to learn in ways that address the needs o f students and the ways of
unpacking the content o f mathematics that agree with the ways students think, the
teachers have been shown to improve students’ mathematical achievements (Hiebert, et
al., 1997).
Knowledge o f students and instructional practices are tied to inservice
experiences in the classroom. Although preservice teachers receive instruction in these
areas in educational psychology and methods courses, the experiential realities o f actual
teaching in the classroom and dealing with diverse learning preferences and abilities of
students are more powerful means o f attaining expertise in these areas.
•
Historical aspects o f pedagogy o f trigonometry in High School
•
Trigonometry was introduced into the school mathematics curriculum around
1890 (Alspaugh, Kerr, and Reys, 1970; Allen, 1977). Presently trigonometry
is either integrated into Geometry, Algebra II or Precalculus; or it is offered as
a stand-alone semester course after algebra II. When the pedagogy of
trigonometry is conducted in the former approach the subject seems to be
relegated to the end o f the course or treated superficially (Markel, 1982). Thus
the integration approach trivializes trigonometry in the school mathematics
curriculum. Such trivialization is counter-productive to success o f students in
higher mathematics in which knowledge o f trigonometry is assumed.
•
Allen (1977) presents a history o f trigonometry in US schools from 1890 to
1970. In 1890, trigonometry was introduced as college freshman course in US
colleges and universities and also as a high school terminal elective course. In
1903, The American Mathematical Society (AMS) argued for requiring
11
trigonometry as a college entrance requirement. The AMS requirements for
trigonometry were as follows:
•
Definitions and relations o f the six trigonometric functions as ratios and
circular measurement o f angles;
•
Proofs o f principal formulas, in particular for the sine, cosine, and the tangent
o f the sum and difference o f two angles, o f the double angle and the halfangle, the product expressions for the sum or the difference of two sines or of
two cosines, etc.; the transformation o f trigonometric expressions by means of
these formulas;
• Solution o f trigonometric equations of a simple character;
•
Theory and use o f logarithms (without the introduction o f work on infinite
series); and
•
The solution o f right and oblique triangles, and practical applications,
including the solutions o f right spherical triangles (Allen, 1977, p. 91).
In the early years o f the twentieth century, three arch-theorists: Moore (USA),
Perry (UK), and Klein (Germany) re-conceptualized the organization o f school
mathematics both in Europe and the US. They argued for formalization and rigor in
school mathematics through the scientific (laboratory) approach to learning mathematics.
Moreover, Klein argued for the unification o f school mathematics away from
disconnected strands and towards a connected and holistic arrangement o f the content of
school mathematics.
During these early years of trigonometry in school mathematics, the subject was
justified as a high school course by virtue of its utility in the vocations o f surveying and
navigation. However, by the late 1950s, the justification has moved from vocational to an
academic one. The study o f trigonometry in the late 1950s was justified as a high school
course as a prerequisite knowledge for higher mathematical and scientific training. In
12
fact, The Committee o f Ten (1893) had alluded to such conceptualization o f trigonometry
as a high school subject (Allen, 1977).
During the early twentieth century through 1923, “trigonometric topics remained
predominantly upper-year subject matter (on an elective basis) throughout the period,
with possible introduction o f right triangle solution at the junior high school level”
(Allen, 1977, p. 127). In general, high school mathematics was seen as college
preparatory courses. The period from 1923 through 1939 saw the number o f students in
high school mathematics courses drop and “mathematics lost ground as a graduation
requirement and as an elective” (p. 139). Trigonometry was still being “taught primarily
as an upper-year elective, with stress on plane triangle solutions and its useful
applications - primarily surveying and navigation” (p. 139).
The period o f the Second World War, Cold War, and pre-Sputnik, from 1940
through 1957 involved discussions and recommendations on school mathematics. But the
discussions were not focused. The launch o f Sputnik by the Soviet Union changed all of
that and brought focus to the discussions and gave birth to the “new-mathematics”
movement. The treatment o f trigonometry as a separate elective course for the higher
grades was de-emphasized during the “new-mathematics” era. Rather, a three-level
organization o f trigonometry that presented “rudimentary right-triangle trigonometry in
junior high school, angle trigonometry, with Algebra, in Grade XI, and circular function
study in Elementary Functions in Grade XII” (Allen, 1977, p. 180) was proposed. And
the integration o f trigonometry into algebra and geometry courses has continued to today.
In today’s high school mathematics curriculum, students are most likely to encounter
trigonometric ideas in geometry, Algebra II, or Pre-Calculus.
13
Rationale and theoretical basis for the study
Why study trigonometry?
Trigonometry is a fascinating content strand in school mathematics. It is
conceptually rich and contains connections to several other mathematical ideas and
structures (Burch, 1981). Trigonometry brings together algebra, function ideas, domain,
range, inverse, algebra o f functions, and recursive reasoning in terms o f periodicity. In
1950, the Mathematical Association [MA] o f England recognized the unifying role o f
trigonometry in school mathematics, in addition to its ancillary and amplifying roles.
According to the MA, “trigonometry fuses together arithmetic, algebra, geometry and
mechanics” (MA, 1950, p. 3). Moreover, trigonometry provides students with the
mathematical power needed to resolve important mathematical questions and gives
students a coherent picture o f mathematics (MA, 1950). Trigonometry is rich in visual
representations o f phenomena. It is applicable in navigation, astronomy, motion,
rotations, elevations, and bearing, to name a few o f the areas in which trigonometry is
indispensable. Trigonometry has rich problem solving opportunities and involves acute
reasoning and proof capacities. It could be used to foster mathematical discourse and
reasoning and proof capacities in students. Moreover, trigonometry is indispensable in
modeling periodic phenomena (Thomas & Finney, 1996; Hirsch, Weinhold & Nichols,
1991; Maor, 1998).
In their 2001 text on trigonometry, Israel M. Gelfand and Mark Saul stated:
Trigonometry sits at the center o f high school mathematics. It
originates in the study o f geometry when we investigate the ratios of sides
in similar right triangles, or when we look at the relationship between a
chord o f a circle and its arc. It leads to a much deeper study of periodic
functions, and o f the so-called transcendental functions, which cannot be
described using finite algebraic processes. It also has many applications to
physics, astronomy, and other branches o f science. ... Trigonometry is an
important introduction to calculus, where one studies what mathematicians
call analytic properties o f functions (p. ix).
Research on trigonometry at the college and high school levels has focused on the
relative impacts o f methods o f teaching trigonometry: right triangle (geometric in focus),
14
unit-circle or wrapping function, transformational, and vector approaches to teaching
trigonometry on students’ achievement on trigonometric tests. The studies have all found
that the different pedagogical approaches to teaching trigonometry do not produce
significant differences in students’ performance on tests that address trigonometry
concepts and skills (Burch, 1981; Evanovich, 1974; Palmer, 1980; Huber, 1977). The
main point I take away from the research findings is that trigonometry is a coherent body
of knowledge, it is versatile, it is malleable, and thus can be approached from multiple
perspectives.
The National Assessment o f Education Progress (NAEP) and the National
Council o f Teachers o f Mathematics (NCTM) acknowledge that more US students are
taking more advanced school mathematics courses such as trigonometry or courses that
include trigonometry. According to Dossey & Usiskin (2000), in 1996 fifty percent o f
seventeen-year-old students reported to have completed or were currently taking Algebra
II; that was a significant increase from 1978 when 37 percent of the same age group
reported to have completed or were currently taking Algebra II. Similarly the percentage
o f 17 year old students that have completed or were currently taking Precalculus or
Calculus jumped from 6% in 1978 to 13% in 1996; a significant increase. Trigonometry
is an integral part o f college preparatory courses like Geometry, Algebra II (second year
algebra), Precalculus (mathematics analysis) and Calculus (Alspaugh, Kerr, and Reys,
1970; Hirsch, Weinhold and Nichols, 1991). Since more students are taking these courses
that include trigonometry than ever before, more and more teachers will invariably teach
trigonometry because trigonometry makes up a significant portion o f these courses.
Therefore, understanding preservice secondary mathematics teachers’ understanding of
trigonometry would enable colleges o f education and mathematics departments to better
educate the future cadre o f secondary school mathematics teachers to be effective
teachers o f mathematics.
15
Seven percent of the 60 multiple-choice questions that make up the ACT
mathematics test are trigonometry questions (http://www.act.org). Although trigonometry
forms the smallest portion o f the ACT mathematics test, it is nonetheless an integral part
o f the test. And if students are to succeed on the ACT mathematics test they must be well
prepared in trigonometry. That requires teachers competent in their knowledge of
trigonometry to guide the students. Moreover, the College Board has stated a willingness
to include trigonometry in the SAT test starting in 2005. For students to succeed in
trigonometry, they need to have teachers who can unpack the trigonometric concepts so
that they are understandable.
Furthermore, Bolte (1993) and Howald (1998) in their study o f preservice
teachers and experienced mathematics teachers, respectively, found that trigonometric
functions were the least understood by the participants. Both Bolte (1993) and Howald
(1998) explored the depth of understanding o f functions by both preservice and
experienced teachers, respectively. However, trigonometric functions constituted a
minute portion o f Bolte’s (1993) and Howald’s (1998) works. Some o f the most difficult
concepts and ideas were domain, range, periodicity and inverses o f the trigonometric
functions. The present study builds on the methodology o f both Bolte (1993) and Howald
(1998) and focus only on trigonometry in its broadest sense as it is presented in school
mathematics.
Searches o f the ERIC database, professional journals, reports, conference
proceedings, and handbooks did not turn up any studies that looked specifically at
preservice teachers’ knowledge o f trigonometry, their pedagogical content knowledge of
trigonometry or their teaching practice in relation to high school trigonometry. Even the
Conference Board o f the Mathematical Sciences’ (CBMS) 2001 issue on the
mathematical education o f teachers barely covered the topic o f trigonometry. It
recommends that preservice high school mathematics teachers develop an “understanding
of trigonometry from a geometric perspective and skill in using trigonometry to solve
16
problems” (pp. 41, 129). More precisely, preservice teachers o f high school mathematics
are to have command o f the “law o f sines, law o f cosines, Pythagorean theorem, the
addition formulas, and the general notion of identity - and to make or reinforce
connections with geometry” (p. 132).
The point is, although professional organizations like The National Council of
Teachers of Mathematics, and the CBMS; education researchers (for example, Ruhama
Even (1990), Ball, Lubienski, & Mewbom (2001), and others); and mathematicians (for
example, Richard Askey, H. Wu) and mathematics teachers argue that preservice teachers
ought to understand the content of trigonometry, there has been little exploration o f the
kind or depth o f understanding o f trigonometry that would prove adequate for high
school mathematics teaching. Wu (2002) ventured to stipulate some parameters for a
meaningful understanding of trigonometry that teachers ought to possess for effective
high school mathematics teaching. He proposed that teachers o f high school mathematics
ought to understand the foundational ideas behind such concepts and facts as radian
measure, the Pythagorean theorem, the sine addition formula, inverse trigonometric
functions, graphs of the trigonometric functions and their inverses, De Moivre’s theorem,
and the usefulness o f trigonometry beyond the geometric realm; for example, a
qualitative understanding o f the fact that all periodic phenomena are expressible using
sine and cosine functions, thus understandable via the trigonometric functions, will serve
“to disabuse teachers o f the possible misconception that trigonometry is a purely
geometric subject” (p. 27).
Why study preservice high school mathematics teachers?
There is a consensus in mathematics education that what preservice mathematics
teachers know impacts what they will eventually do in the mathematics classroom.
Furthermore, for the sake of the continued professionalism and effectiveness o f
mathematics teaching, understanding the knowledge of preservice mathematics teachers
17
serves to support and improve, and perhaps reform the education o f future teachers of
mathematics. “Analyzing the images o f mathematics and mathematics teaching held by
preservice teachers is important because these teachers will significantly impact upon the
nature o f mathematics that will transpire in the future classrooms” (Wilson, 1992, p. 1).
Furthermore, preservice teachers come to mathematics education programs with
little or inadequate understanding and notions o f mathematics. They seldom are prepared
to view mathematics education as a profession imbued with both content and pedagogy.
Mathematics education should help preservice teachers gain a better appreciation of
mathematics and help them integrate the content, pedagogy, and beliefs necessary for
proper teaching (Cooney, 1999). Moreover, we should help preservice teachers move
from, what Cooney (1999) calls, isolationist perspective to a reflective connectionist
perspective. According to Cooney (1999), there are four ways to view teachers. (1) They
can be viewed as isolationists: Teachers think they know the right way to teach and
bothers minimally to incorporate new ideas. (2) They can be viewed as naive idealists
(received knowing): Teachers integrate outside knowledge without much reflection. (3)
They can also be viewed as native connectionists: Teachers engage in reflection, but
compartmentalize contradictions or conflicts that arise among theories without any
attempt to resolve the apparent contradictions. (4) Finally, teachers can also be viewed as
reflective connectionists: Teachers at this stage are reflective and they endeavor to
resolve conflicts among theoretical perspectives.
However, there is a bifurcation o f opinions as to how to prepare preservice
teachers of school mathematics. There are those that want preservice education to prepare
teachers to be competent and fully developed at the completion o f teacher training
programs. On the other hand, there are those who are willing to accept the proposition
that teacher education programs cannot possibly achieve the aforementioned goal. If
nothing else, the component o f pedagogical content knowledge is experiential and cannot
18
be taught but experienced through actual practice, reflection on the practice, and re­
negotiation of practice over an extended time.
Within the second conception of teacher education, the best that preservice
teacher preparation can hope to achieve is provide preservice teachers with the
conceptual framework and methodological framework so that they can anticipate,
recognize, articulate, and incorporate facets o f teaching that improve their teaching into
their repertoire.
Cross-national studies that have compared US teaching and teachers with their
Asian counterparts have found evidence that being an effective teacher is a life-long
journey (TIMSS, 1995, 1999). But the camp that argues for preservice teachers to be
expert teachers by the time they enter the profession continue to worry that “unqualified”
teachers are dooming US educational system.
Teaching is a complex enterprise. It is neither a hard science nor an abstract art.
The possession o f some knowledge structures is correlated with being an effective
teacher. According to Shulman (1987):
Teaching necessarily begins with a teacher's understanding o f what
is to be learned and how it is to be taught. It proceeds through a series of
activities during which the students are provided specific instruction and
opportunities for learning, though the learning itself ultimately remains the
responsibility o f the students. Teaching ends with new comprehension by
both the teacher and the student (p. 7)
The categories o f knowledge necessary for effective teaching according to
Shulman (1987, p. 8) are (1) Content knowledge; (2) general pedagogical knowledge,
with special reference to those broad principles and strategies o f classroom management
and organization that appears to transcend subject matter; (3) curricula knowledge, with
particular group of the materials and programs that serve as "tools o f the trade" for
teachers; (4) pedagogical content knowledge, that special amalgam of content and
pedagogy that is uniquely the province o f teachers, their own special form o f professional
understanding; (5) knowledge o f learners and their characteristics; (6) knowledge o f
19
educational contexts, ranging from the workings of the group or classroom, the
governance and financing o f school districts, to the character o f communities and
cultures; (7) and knowledge o f educational ends, purposes, and values, and their
philosophical and historical grounds.
To be effective at teaching, the teacher should first and foremost comprehend the
subject matter content knowledge with degrees o f flexibility and adaptability that enables
the teacher to transform that knowledge into "forms that are pedagogically powerful and
yet adaptive to the variations in ability and background presented by the students" (p. 15).
Preservice teachers have to develop the aforementioned competencies as well. However,
the development o f such expertise is difficult and is not a straightforward event.
Preservice teachers’ “development from students to teachers, from a state o f expertise as
learners through novitiate as teachers, exposes and highlights the complex bodies of
knowledge and skill needed to function effectively as a teacher” (p. 4).
Furthermore, preservice teachers’ recency with college mathematics courses in
which trigonometric ideas were explored and the future impact o f the preservice teachers’
content knowledge on what transpires in mathematics classrooms makes a study of
preservice teachers a reasonable and worthwhile endeavor. The recency with college
mathematics courses needs to be qualified if we are to better understand the
understanding that preservice teachers possess.
In the case o f trigonometry, most preservice teachers would have only their high
school experience to draw from. According to Wu (2002) this is problematic because
high school textbooks and teachers do not necessarily present trigonometry with its
logical progression from similar triangles. Moreover, neither high school teachers nor
high school textbooks explicate the necessity o f the radian measure, the fundamentalism
of the Pythagorean theorem sin2 jc + cos2x = \, the sine and cosine addition theorems, or
Fourier series to proper understanding o f trigonometry.
20
Preservice teachers need to understand the content they want to teach. But they
need to also understand how to unpack and present the content so that students can learn
with understanding (Kilpatrick, et al., 2001). Teachers’ knowledge is dynamic, and is
dialecticized by content knowledge, knowledge o f pedagogy, knowledge o f students’
cognition, and teachers’ beliefs; and is situated in practice. Teachers’ content knowledge
influences their instructional practices but its influences on students’ mathematical
achievements are not clear from the research literature.
The implication is that the more connected and broad the content knowledge of
the teacher, the richer the learning environment facilitated by the teacher can be. “The
important factor in a positive relationship between content knowledge and classroom
instruction appears to be the mental organization of the knowledge that the teacher
possesses” (Fennema & Franke, 1992, p. 153). Expert teachers have better connected
schemata o f content and pedagogy. “This knowledge of subject matter [has] an impact in
several ways: (1) on agendas, because teachers with more knowledge had richer mental
plans than did teachers with less knowledge; (2) on scripts, because more knowledgeable
teachers were able to use more representations and richer explanations; and (3) on
teachers’ response to students’ comments and questions during instruction” (Leinhardt et
al., 1991, quoted in Fennema & Franke, 1992, p. 161). Therefore, teachers’ content
knowledge affects the teachers’ actions, and thus impacts students’ opportunity to learn
(OTL). OTL is believed to be the most important variable in students’ success in
mathematics (Kilpatrick, et al., 2001)
Research questions
This section states the research questions explored in the present study. The
questions arose from reading the related literature on trigonometry in the school
curriculum, and from my personal interest in this area. Finding answers to the questions,
even partial answers, will provide a better understanding o f preservice secondary
21
mathematics teachers’ subject matter knowledge o f trigonometry, pedagogical content
knowledge of trigonometry, and their envisioned pedagogy o f trigonometry. The
questions are first presented and then elaborated on.
•
What content knowledge o f trigonometry do preservice secondary school
mathematics teachers possess?
•
What pedagogical content knowledge of trigonometry do preservice
secondary school mathematics teachers possess?
•
How are preservice secondary school mathematics teachers’ content and
pedagogical content knowledge of trigonometry organized?
•
How do preservice secondary school mathematics teachers envision teaching
trigonometry?
•
How are preservice secondary school mathematics teachers’ content and
pedagogical content knowledge of trigonometry related to their envisioned
application o f their content and pedagogical content knowledge in
mathematics classrooms?
To study question 1, the following areas o f knowledge were explored.
•
Definitions o f the six basic trigonometric functions; ability to motivate the
definitions using ratios of similar right triangles
•
Periodicity/Domain/Range/Frequency. Functions
approach/Domain/Range/Restricted domain/Transformations on the
trigonometric functions/Sinusoidal functions
•
Representations: Graphical/Tabular/Symbolic representations. Graphs of
inverses/Symbolic representations o f inverses/use o f rt. Triangle for inverses,
•
Knowledge o f fundamental identities and ability to generate new identities
•
Manipulations/Operations (Sum, difference, product, quotient, power)
22
•
Applications (where and how can basic trigonometric function be used to
solve real-world problems?) Applications to both geometric (right triangles,
non-right triangles, and other situations amenable to triangular interpretations)
•
Rt. Triangle, and Wrapping Function Approaches (circle representation,
radian measure, ratio definitions)
•
Inverses (domain, range, D j = Rj- 1 ; Rf = D^-i, note that the domains o f the
trigonometric functions are restricted to make them one-to-one)
•
Proofs in trigonometry (identities, sum and difference equations, and ability to
use these in other proofs and reasoning problems)
To gain a better understanding o f preservice secondary mathematics teachers’
pedagogical content knowledge, as stated in research question 2, the following questions
were explored. What prerequisite knowledge is necessary for the learning of
trigonometry? How do the preservice secondary mathematics teachers understand
multiple representations that will prove useful to unpacking the content for students?
How would the preservice secondary mathematics teachers sequence and organize
trigonometric concepts for teaching? Do the sequence and organization o f the concepts
anticipate both students’ preconceptions and misconceptions, and possible approaches to
help students overcome such misunderstanding? If preservice secondary mathematics
teachers were presented with difficulties that students might encounter, how would they
go about helping students to get more valid conceptualizations o f trigonometry?
Research question 3 is intended to reveal the depth o f the understanding of
trigonometry held by the preservice secondary mathematics teachers. Analysis of both a
survey o f trigonometry concepts and facts, and the concept maps o f the domain of
trigonometry were used to address this particular question. Part o f the analysis explored
preservice secondary mathematics teachers’ integration o f and translation amongst the
different representations that are possible in trigonometry (Bolte, 1993).
23
Research question 4 is at heart o f the pre-active phase o f teaching. Planning for
content delivery, the sequence, pre-requisite knowledge, ways o f explaining difficult
concepts, important examples, students activities during instruction, student practice, and
student questions are important ingredients for effective teaching. For this study, the
following questions were addressed as they relate to the planning phase or the pre-active
phase o f teaching. How will they develop the six basic trigonometric ratios? Will they
approach the ratios from the perspective o f right triangles, or unit circle, or as functions?
What is the depth o f the preservice secondary mathematics teachers’ understanding of
curricula knowledge as per trigonometry (sequencing, prerequisite knowledge, core
components o f trigonometry, application o f trigonometry)? What strategies will the
preservice secondary mathematics teachers use in their teaching o f trigonometry? Will
they use multiple representations, manipulatives, and electronic technologies? What
pedagogical approaches (didactic or heuristic) will the preservice secondary mathematics
teachers employ? Will the preservice secondary mathematics teachers use justification
and proof as part o f their method for validating claims and ascertaining truth?
Central to research question 5 are the connections, patterns and systems o f
relations that might exist between the preservice teachers knowledge base and what they
intend to do with that knowledge in the classroom. Questions that were explored in
relation to research question 5 are (1) what are the relationships between the preservice
teachers’ depth o f knowledge and what they intend to do in the classroom with their
knowledge; and (2) how does subject matter content knowledge impact pedagogical
content knowledge, or vice versa?
Summary o f chapter I
This chapter laid out the foundation for the study. I discussed the purpose of the
study, background for the study, rationale and significance o f the study, and the research
questions. What follows in chapter II is the review o f related literature. The literature
24
review presents related literatures on teacher knowledge, epistemological foundations for
the study, and review of the literature on trigonometry. Chapter III explicates the design
o f the study. It includes the methodology, procedure and data collection, description of
the pilot study, and instrumentation. Chapter IV presents the results o f the study and
discussions. Chapter V summarizes the study with further discussions o f implications,
and limitations.
25
CHAPTER II
REVIEW OF THE LITERATURE
“Before you begin to rock the boat, be sure you are in it”
- Wolcott (2001, p. 71).
Introduction and organizational structure o f chapter II
Wolcott (2001) guided and shaped the following review o f the literature. The
review relates method, theory and prior research to the present descriptive study. The
review presents a selection o f the available literature that bear upon the knowledge of
preservice teachers and the methodology o f studying and cataloging such knowledge. It
was not intended to be a “dump” o f the literature but rather it presents evidence to justify
and motivate the present study. First teacher knowledge is discussed. Then the
epistemological foundation o f the study is presented. That is followed by a review of the
literature on school trigonometry. Discussions o f concept mapping and card sorting
conclude this chapter’s review o f relevant literature.
Teacher knowledge
The knowledge that teachers possess and use in their classrooms is the instrument
of change in students’ learning. A synthesis and furtherance o f the available research on
teacher knowledge led Fennema and Franke (1992) to propose a five-component
framework for teachers’ knowledge; the framework includes content knowledge,
pedagogical knowledge, beliefs, knowledge o f students’ cognition, and knowledge of the
context o f classrooms. This study explored two o f these five domains o f teacher
knowledge; namely content knowledge and pedagogical [content] knowledge. In
addition, envisioned practice served as a proxy for teaching in real classrooms with real
students. Furthermore, the envisioned practice portion o f the study explored the level and
depth o f preservice secondary mathematics teachers’ ability to reflect on the interaction
of their understanding o f mathematical content and the pedagogy o f that content.
26
Fennema and Franke (1992) relied heavily on Shulman (1986; 1987). According
to Shulman (1987) the categories o f teacher-knowledge are (1) Content knowledge; (2)
general pedagogical knowledge, with special reference to those broad principles and
strategies o f classroom management and organization that appear to transcend subject
matter; (3) curricula knowledge, with particular group o f the materials and programs that
serve as "tools o f the trade" for teachers; (4) pedagogical content knowledge, that special
amalgam o f content and pedagogy that is uniquely the province o f teachers, their own
special form o f professional understanding; (5) knowledge o f learners and their
characteristics; (6) knowledge o f educational contexts, ranging from the workings o f the
group or classroom, the governance and financing o f school districts, to the character of
communities and cultures; and (7) knowledge o f educational ends, purposes, and values,
and their philosophical and historical grounds.
Furthermore, Shulman (1987) argued that teachers' understanding o f content is
critical and paramount, irrespective o f the pedagogy that is employed by the pedagogue.
However, "among these categories, pedagogical content knowledge is o f special interest
because it identifies the distinctive bodies o f knowledge for teaching. It represents the
blending o f content and pedagogy into an understanding o f how particular topics,
problems, or issues are organized, represented, and adapted to diverse interests and
abilities o f learners, and presented for instruction. Pedagogical content knowledge is the
category most likely to distinguish the understanding o f the content specialist from that of
the pedagogue" (p. 8). In essence it is the teacher that plans the “learning trajectories” of
the students. Thus the teacher is central and inextricable from the learning episodes that
occur in the classroom. Moreover, "the manner in which that understanding is
communicated conveys to students what is essential about a subject and what is
peripheral. In the face o f student diversity, the teacher must have a flexible and
multifaceted comprehension, adequate to impart alternative explanations o f the same
concepts or principles" (p. 9).
27
Similarly, Lappan & Theule-Lubienski (1994) presented a model o f the domain of
teachers’ knowledge that incorporates three spheres o f knowing: Pedagogy o f
mathematics, Students, and Mathematics. The three spheres o f teacher knowledge
represent knowledge o f the mathematics content; knowledge o f students’ cognition,
knowledge o f students’ difficulties with concept domains, and how to motivate and
facilitate learning; and finally knowledge of how to orchestrate pedagogy o f mathematics
that empowers learning and students involvement. Lappan and Theule-Lubienski (1994)
characterization fits within the model expounded by Shulman (1986, 1987).
Leinhardt’s model o f agenda, scripts, and routines makes claims about the forms
o f teacher knowledge as perceived through observation o f classroom practice. Agenda is
the master plan that teachers impose on the mathematical content to facilitate pedagogy.
Scripts are specific plans for dealing with specific topics that allow teachers to unpack the
mathematical content for pedagogy. Routines are “scripted sets o f behaviors that allow
teachers to carry out some activities in a relatively automated manner and with minimum
cognitive load” (Sherin, Sherin, & Madanes, 1999, p. 361).
Leinhardt’s model is a cognitive one that explains teachers’ behaviors from data
gathered, perhaps, from videotaping o f teachers teaching in classrooms, viewing the
videotapes, and inferring from the videotapes teachers’ cognitions and then using the
inferred cognitive structures (forms) o f teacher knowledge to explain the scripts and
routines of the teachers (Sherin, Sherin, & Madanes, 1999). Moreover, the model is more
appropriate for analyzing teachers’ pre-active, active, and post-active phases of
instruction; always remaining focused on actual classroom practice.
Yet another model o f teacher knowledge presented by Schoenfeld and the
Teacher Model Group at Berkeley University is an attempt to ’’explain why a teacher
does what he or she does during the moment o f instruction ... to be able to account for
different teaching styles and different types o f lessons” (Sherin, Sherin, & Madanes,
1999, pp. 362 - 363). Precisely, the goals for the Teacher Model Group are to construct a
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model of teaching that
(1) accommodate all teaching in its architecture; (2) works at all
levels o f grain size, from planning curricula to planning lessons to
utterance-by-utterance interactions; and (3) provides a fine-grained
explanation o f how and why any teacher does what he or she does, in the
midst of learning interactions (Schoenfeld, 1999, p. 244).
The focus o f the present study is not on the practices o f teachers, rather it
categorizes and catalogues the level o f preservice secondary mathematics teachers’
understanding o f trigonometry and the pedagogical content knowledge that is associated
with it. As explained in the section that addressed why study preservice teachers,
understanding preservice teachers’ knowledge o f mathematics and pedagogy with respect
to particular mathematical strands is useful knowledge for planning effective education of
the future cadre o f mathematics teachers. The model from the Teacher Model Group does
not quite address the interests o f the present study because as a model, it is primarily
concerned with teacher problem solving while teachers are in the act o f teaching.
The model o f Shulman was chosen over other models of teacher knowledge (for
example, Leinhardt and colleagues (Leinhardt & Smith, 1985); and Schoenfeld (1999))
because it explains the phenomena of teacher knowledge; methodologically, Shulman’s
model is mostly ethnographic supporting the use o f interviews as means to elaborate the
observations and inquiry; and finally it does not attempt to explain behavior, rather it is a
model for analyzing content o f teachers’ knowledge (Sherin, Sherin, & Madanes, 1999).
Ma (1999) stressed the importance of the culture o f teacher education on the
efficacy o f teachers. For example she posits that the differences between Chinese
elementary teachers’ and US elementary teachers’ understanding o f elementary
mathematics are partly explainable by their different educational experiences. According
to Ma (1999), “teachers’ subject matter knowledge develops in a cyclic process o f
schooling, teacher preparation, and teaching” (p. 144). However,
in China, the cycle spirals upward. When teachers are still
students, they attain mathematical competence. During teacher education
programs, their mathematical competence starts to be connected to a
29
primary concern about teaching and learning school mathematics. Finally,
during their teaching careers, they develop a teacher’s subject matter
knowledge, which I call in its highest form PUFM [Profound
Understanding of Fundamental Mathematics]. Unfortunately, this is not
the case in the United States. It seems that low-quality school mathematics
and low-quality teacher knowledge of school mathematics reinforce each
other (Ma, 1999, p. 145).
To make matters worse, the purposes o f teacher education are bifurcated in the
United States. On one hand, graduates o f teachers’ education programs are considered
experts in their fields that will not need serious additional support and structure for
improvement. In other words, within this perspective, new teachers are expected to be
independent and effective in their practice without any serious expectation for further
development. On the other hand, new teacher education is viewed as a stepping-stone for
further elaboration of practice and study of content.
The latter conception finds needed support in cross-national studies between US
and Asian educational system and practices. Pedagogical content knowledge is
particularly situated in this conception since studies have shown that it is predicated on
experience. However, experience does not equal number o f years o f teaching. If that were
the case, the 11 experienced US teachers with an average teaching experience of 11 years
that participated in Ma (1999) would have developed the necessary pedagogical content
knowledge o f elementary mathematics. But Ma (1999) found that none o f the US
teachers possessed PUFM, defined as a deep, vast, and thorough understanding of
mathematics.
What is central to attaining PUFM is the quality o f the experience vis-a-vis
focused reflection on practice and the subject matter ala Dewey. Teachers with PUFM
understand mathematics with breadth and depth. They show connectedness among
mathematical concepts and procedures; are flexible in their approach to solving
problems; have a grasp of the fundamental basics; and understand the longitudinal
coherence o f the mathematics that they teach. That is, they understand the pre and post
implications and ramifications for the mathematical concepts that they teach.
30
Ma (1999) posits that it takes both internal and external factors to develop PUFM.
The internal factors are the teacher’s disposition (to know how and why), knowledge of
mathematical structure, and attitude (to want to solve problems in multiple ways and
provide justification for ones reasoning as well). The external factors include use of
teaching materials (textbooks, National framework (curriculum or teaching standards),
and teacher manuals), learning from colleagues and students, learning mathematics by
doing it, and teaching round-by-round (teaching different grades and hence teaching at
different levels of mathematical sophistication).
Theoretically, Ma (1999) fits within Shulman (1986, 1987) framework. However,
her categorization o f teachers’ knowledge along the dimensions o f connectedness,
multiple perspectives, basic ideas, and longitudinal coherence presents an appropriate
additional way to categorize preservice teachers’ understanding o f trigonometry, its place
in school mathematics, and its pedagogical implications. Therefore, this study analyzed
the preservice teachers’ depth and breadth o f trigonometry understanding using the
scheme o f Ma (1999).
This study looked at the understanding preservice secondary mathematics
teachers have about the content of trigonometry, the necessary pedagogy that would
foster meaningful student learning, and how the preservice secondary mathematics
teachers intend to use their knowledge structures to facilitate student learning. Because to
be effective at teaching, the teacher should first and foremost comprehend the subject
matter knowledge with degrees o f flexibility and adaptability that enables the teacher to
transform that knowledge into "forms that are pedagogically powerful and yet adaptive to
the variations in ability and background presented by the students" (Shulman, 1987, p.
15). ,
31
Epistemology
The present study is an investigation o f individual preservice teachers’
construction of knowledge about trigonometry. The knowledge is idiosyncratic to each
preservice teacher. The epistemological foundation for the present study is rooted in
radical constructivism o f von Glasersfeld. Radical constructivism explains the
assumptions that gird the study o f the preservice teachers’ understanding o f trigonometry,
since such understanding is resident in each individual preservice teacher. The only social
aspect o f this study o f preservice teachers understanding o f trigonometry has to do with
the reality that they are part o f a social organization o f people engaged in advanced study.
Radical constructivism is an epistemology that posits,
“real world, in the sense o f ontological reality, is inaccessible to
human reason” (von Glasersfeld, 1996, p. 309). Furthermore the
“cognizing activity is instrumental and neither does nor can concern
anything but the experiential world of the knower. The experiential world
is constituted and structured by the knower’s own ways and means of
perceiving and conceiving, and in this elementary sense it is always and
irrevocably subjective (that is, construed by the cognizing subject). It is
the knower who segments the manifold of experience into raw elementary
particles, combines these to form viable things, abstracts concepts from
them, relates them by means o f conceptual relations, and thus constructs a
relatively stable experiential reality (p. 308).
Knowledge is fallible, according to radical constructivism. In addition it argues
that knowledge is not unique. Radical constructivism argues that since reality is not
accessible to human reason, the cognizing agent structures and constructs her knowledge.
The constructed knowledge is considered true by virtue o f its viability (that is, its fit) into
the world o f the knower’s experience, the only reality accessible to human reason.
However, such fit cannot be considered unique because there are other cognizing agents
that construct other viable knowledge. The present study will investigate the viability of
preservice secondary school mathematics teachers’ knowledge o f trigonometry in lieu of
expert construction o f trigonometry in school mathematics.
According to radical constructivism, knowledge is resident in individuals. It does
not argue that the world does not exist, but that “reality is unknowable and that it makes
32
no sense to speak o f a representation o f something that is inherently inaccessible” (von
Glasersfeld, 1996, p. 309). Moreover, society and others are not discounted. Rather they
are viewed as environmental structures (realities) that help the individual test the viability
of her knowledge constructs.
Consider, for a moment, a cognizing individual engaged in an inter-agent social
interaction mediated through a common shared language. At the end o f the interaction, at
the point where the agents dissociate into intra-agent reasoning, at the point when the
social interaction is no more; if knowledge was constructed, then it must reside in the
individual cognizing agents, albeit, each individual carries with him or her idiosyncratic
versions o f the knowledge. In other words, the locus o f knowledge is in individuals and
not in the social interaction. However, social interaction is not rejected or deemed useless
by radical constructivism; rather it is considered epiphenomenal to individual
understanding.
Social constructivism, on the other hand, argues that learning and knowing is
social (Ernest, 1991, 1994, 1996). Social constructivism tries to account for both
subjective knowledge (radical constructivism) and objective knowledge. Objective
knowledge is knowledge that is socially agreed upon having been critiqued and
reformulated (Ernest, 1991, 1996). Radical constructivism does not neglect this socially
constructed knowing. Rather it would argue that the socially constructed knowing is
ultimately subsumed into individual constructions and as such the social becomes
“subjectified” by the individuals in the social. As a study of individuals, in contexts that
do not lend themselves to social negotiation o f knowledge, the present study rested on
radical constructivism and not social constructivism.
Literature dealing with trigonometry
The following review o f the literature on trigonometry represents a product of
searches for pertinent literatures on school trigonometry and preservice secondary
33
mathematics teachers understanding of the subject. The search covered the following
literature sources: (1) School science and mathematics', (2) Issues in mathematics
education (CBMS); (3) Educational studies in mathematics', (4) Journal fo r research in
mathematics education', (5) Journal o f mathematical behavior, and (6) Hiroshima journal
o f mathematics education. The search yielded very few studies that discussed
trigonometry. There were no studies that investigated teachers’ understanding o f
trigonometry. At best, trigonometry was subsumed under the study o f functions (Even,
1990). However, the space given to trigonometry in studies o f function knowledge was
very minimal (Even, 1989, 1990; Bolte, 1993; Howald, 1998). The discussions o f the
results o f the search that follow are organized by the sources from which the data were
pulled.
The search of School Science and Mathematics from 1990 to July 8, 2002 yielded
the following studies on trigonometry: O ’Shea (1993), Flores (1993), Bidwell (1993,
1994), and Doerr (1996). O ’Shea’s (1993) research grew out o f a personal conversation
the author had with two Canadian mathematics educators on the lack understanding of
trigonometry o f preservice secondary mathematics teachers who were student-teaching at
the time. The author developed an activity-based approach to teaching the law o f cosines
after an examination o f leading textbooks on trigonometry revealed that the books either
presented convoluted or cumbersome approaches to the law o f cosines. The author
claimed that preservice teachers who went through the intervention activity were able to
use “geometry, trigonometry, algebra, and arithmetic” to understand and explain the law
of cosines (p. 74).
Flores (1993) presented a way to operationalize the true intent o f spiraling the
curriculum that is intended to create and develop deeper understanding o f concepts over
time. The author used the case o f the Pythagorean theorem via the Van Hiele levels to
illustrate how depth o f understanding o f a concept can be achieved through spiraling. At
the 0-Level - the recognition level, students are familiarized with the Pythagorean
34
theorem using pictures and perhaps measurements. At the 1-Level - the analysis level,
students are given instances o f the theorem via puzzles and manipulative problem
solving. The idea is to allow students to come to the truth o f the theorem through
exploration. At the 2-Level - the informal deduction level, students begin to justify the
theorem diagrammatically with the aide o f algebra. At the 3-Level —the axiomatic
deduction level, students use the ideas of similar triangles and properties, and
axioms/theorems o f geometry to construct proofs o f the Pythagorean theorem. At the 4Level - the rigor level, the students begin to examine the theorem in Euclidean and nonEuclidean systems. In the Euclidean system, both analytic geometry and vectors are
explored. The geometry o f the sphere is used to support explorations in non-Euclidean
systems.
Bidwell (1993) used Ptolemy’s theorem to derive the sum and difference, double,
half-angle, and the triple angle trigonometric identities. Bidwell (1994) presented the
recursion approach to approximating the value o f Jt used by Archimedes o f Syracuse (287
- 212 B.C.). The only link to trigonometry in Bidwell (1994) comes in the form of a
connection o f Archimedes’ method to later invention o f infinite expansion of arctan x by
James Gregory in 1671. Doerr (1996) is a case study that investigated the construction of
understanding o f motion of an object down an inclined plane in an integrated algebra,
trigonometry, and physics class at an alternative public school. The students were 17
average-ability students in Grade 9 - 1 2 . The only link to trigonometry is the use of
trigonometry to provide both horizontal and vertical component analysis o f force and
velocity of the body in motion on the inclined plane.
Search o f the Conference Board of Mathematical Sciences’ (CBMS) publication
Issues in Mathematics Education from 1990 to July 8, 2002 yielded a mention of
trigonometry in the eleventh volume on the mathematical education o f teachers. The
report recommended that preservice high school mathematics teachers develop
“understanding of trigonometry from a geometric perspective and skill in using
35
trigonometry to solve problems” (p. 14, and p. 129). The report also makes the argument
that preservice teachers ought to have a command o f the “law o f sines, law of cosines,
Pythagorean theorem, the addition formulas, and the general notion o f identity - and to
make or reinforce connections with geometry” (p. 132).
Search o f Educational Studies in Mathematics from 1990 to July 8, 2002 yielded
the following studies related to trigonometry: (1) Even (1990), and (2) Shama (1998).
Even’s (1990) study was a discussion o f the author’s dissertation study o f preservice
teachers understanding o f function that was conducted in 1989. She proposed a
framework for analyzing teachers’ subject matter knowledge. The analytic components of
the framework are: essential features, different representations, alternative ways o f
approaching, strength o f the concept, basic repertoire, knowledge and understanding o f a
concept, and knowledge about mathematics.
Even argued that teachers’ mathematical knowledge can be divided into these
categories o f knowledge. She indicated that teachers with adequate knowledge o f school
mathematics ought to possess the aforementioned seven categories o f knowledge.
Teachers understand the essential features o f school mathematics if they have knowledge
of the critical attributes and prototypes o f any given school mathematics concept. An
understanding o f the different ways that mathematical concepts can be manifested and the
ability to navigate amongst the varied representational systems encapsulates the different
representations categories. Alternative ways o f approaching involves the teacher’s ability
to apply mathematics both to mathematical and non-mathematical situations. If teachers
grasp the scope o f utility and limitations of given mathematical concepts, and are able to
apply such knowledge to render mathematics useful and applicable, then they would have
manifested the knowledge o f the strength of the concept.
Even argued further that teachers ought to have a basic repertoire o f routinized
essential and fundamental mathematics. An indication o f knowledge and understanding
of a concept is the teacher’s ability to integrate both conceptual and procedural
36
knowledge and use both readily in problem solving situations. Finally, teachers who
understand school mathematics have knowledge about mathematics: They have an
understanding o f the nature o f mathematics, its truth structures, and understand
mathematics’ progression, accretion, and development. In using the seven categorical
analytical framework, she found that preservice teachers understanding o f functions is
fragile and weak. The types of functions studied involved few trigonometric functions.
However, trigonometric functions formed a minute portion o f the families o f functions
that were studies by Ruhama Even. Thus the link o f Even to trigonometry is at best
tangential. Nonetheless, the author argued that trigonometry ought to be part of school
mathematics teachers’ repertoire. Even’s vote for trigonometry to form part of
mathematics teachers’ repertoire provides further support for the present study of
preservice secondary mathematics teachers’ knowledge o f trigonometry.
Shama (1998) was an integrated qualitative and quantitative study of students in
Grades 3, 6, 9, and 11. The purpose o f the study was to assess students’ understanding o f
periodicity. The link to trigonometry lies in the fundamental notion o f periodic
phenomena that undergird the whole domain o f trigonometry. In the qualitative study,
classroom teachers were interviewed, then their classes were observed, and finally 28
students were interviewed. The theoretical framework of the study was predicated on the
field-grounded theory o f Glazer and Strauss because prior theory on students’
understanding of periodicity did not exist at the time the author (Gilli Shama) conducted
her dissertation. The quantitative portion o f the study involved 895 eleventh graders
learning advanced mathematics. A 45-minute questionnaire survey was the instrument of
choice for data collection in the quantitative portion o f the study.
Shama found that 52% o f the 11th graders defined periodicity as only dependent
on time scale, and 93% of the 28 interviewees gave examples that used time as the
independent variable. The interviewees tended to over-generalize periodicity onto non­
periodic phenomena, and they tended to associate a period with only the fundamental
37
period (the period o f minimal length). Shama also found that 62% o f the students tended
to view a period as a closed interval with only 30% accurately identifying a period as a
half-closed interval. In concluding, the author claimed that since students tended to tie
periodicity with time, they were procedurally driven. And that this may be a function of
classroom instruction that ties periodicity to completion o f cycles on a certain interval of
time.
Concept mapping
A concept map is an external visual “schematic device for representing concept
meanings embedded in a framework o f propositions” (Novak & Gowin, 1984, p. 15). A
concept can be defined as “regularity in events or objects designated by some label” (p.
4). Propositions are “two or more concept labels linked by words in a semantic unit”
(Novak & Gowin, 1984, p. 15). Both hierarchical (Novak & Gowin, 1984) and web-like
models (Bolte, 1993; Howald, 1998) have been used to assess students’ learning and
understanding, and also as learning tools. In this study, concept mapping was used to
assess preservice secondary school mathematics teachers’ knowledge o f trigonometry.
A hierarchical model presents a definite structure o f a knowledge domain. The
structure entails the superordinate and subordinate concepts or ideas in that knowledge
domain. These superordinate-subordinate pairs form levels of abstractions inherent in the
knowledge domain. Thus the hierarchical model affords the developer o f the model a
means of encapsulating his or her depth of understanding o f the conceptual coherence of
the knowledge domain. In contrast, the web-like model cannot be used to articulate levels
of abstraction, nor argue for conceptual coherence o f a knowledge domain. It is useful for
displaying how concepts or ideas are connected and dispersed. Furthermore, the web-like
model is not suited for displaying superordinate and subordinate relationships between
concepts or ideas.
38
As an assessment tool, concept mapping was used to assess participants’
performance levels on the Bloom’s Taxonomy: Knowledge, Comprehension,
Application, Analysis, Synthesis, and Evaluation (Novak & Gowin, 1984). By
constructing concept maps o f trigonometry, participants recalled trigonometric facts
(knowledge), comprehended and gave examples o f the scope o f applicability of
trigonometry. Furthermore, the participants were challenged to analyze the relationships
among concepts and synthesize their analyses into a schematic representation (concept
map). Finally, the participants evaluated the accuracy and veracity o f their semantic units.
In this study o f preservice teachers’ understanding o f school mathematics, the
participants were challenged to construct hierarchical concept maps because the study
was interested in the participants’ breadth, depth, and knowledge o f the structure of the
domain of trigonometry. However, knowledge webs were accepted as valid
demonstrations o f the study participants’ knowledge o f trigonometry. According to
Novak and Gowin (1984), the hierarchical model allows for such organizing of
knowledge domains with the more inclusive and general concepts at the top and the more
concrete and less inclusive ones included at lower and lower levels in the hierarchy (see
figure l) 3. The coherence o f the hierarchy is supremely fundamental to the integrity of
the model. Therefore, the validity o f the hierarchy receives utmost importance. The
validity o f the propositions is only secondary to the hierarchy. Cross-link propositions
that connect concepts in different clusters are viewed as more important than regular
propositions within clusters. This is because cross-links represent meaningful
connectedness of the constructor’s knowledge. Moreover, cross-links can represent
creativity and ingenuity in schematizing a knowledge domain.
For the purposes o f this study, the contents o f the concept maps were valued more
than the format in which the contents were organized. Moreover, relationships between
3 The concept map schematic was adapted from Novak and Gowin (1984, p. 37).
39
items in the concept maps and mathematical claims were carefully analyzed to generate
the study participants’ status, depth, and organization o f trigonometric knowledge.
Figure 1. Concept Map Schematic
Card sorting
Card sorting was employed by Even (1989), Wilson (1992), Bolte (1993), and
Howald (1998) to measure preservice teachers’ and in-service teachers’ knowledge o f the
development o f conceptual domains. Card sorting activities involve participants
arranging index cards of concepts or ideas into a predefined classification structure. For
example, the aforementioned studies gave preservice teachers (Even, 1989; Wilson,
1992; Bolte, 1993), and experienced teachers (Howald, 1998) a set o f index cards with a
function on each one card and asked them to sort the cards into piles with labels such as
40
True for all, and True for some. In the present study, participants were given cards and
the classification structure o f Always True, Sometimes True, and Never True to use in
their card sorting. This form o f card sort was a way o f assessing the participants’
understanding o f trigonometric knowledge. Moreover, the participants also arranged
cards of trigonometric and mathematical ideas into a pedagogical sequence. This
sequencing exercise assessed the participants’ pedagogical knowledge within the domain
of trigonometry. In a sense, the study explored the pedagogical pow er (Cooney, 1994) of
the preservice teachers in the area of trigonometry.
Summary o f chapter II
The review of the literature revealed that very little is known about what
preservice teachers understand about trigonometry and how preservice teachers
understand trigonometry that obtains at the high school level. What we know about
preservice or inservice teachers’ knowledge o f trigonometry is inferred from studies on
function knowledge. However, studies on functions have explored very little
trigonometric concepts. Thus a study that focused on the breadth and depth of
trigonometry is warranted and would contribute still to our understanding o f preservice
teachers’ knowledge o f high school mathematics.
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CHAPTER III
DESIGN OF THE STUDY
Methodology
Introduction (review o f the purpose o f the study)
The purpose o f the present study was to assess the subject matter content
knowledge, pedagogical content knowledge, and envisioned practice o f preservice
secondary school mathematics teachers in the area o f trigonometry. The study was
divided into two phases: phase one involved 14 the participants. In this phase each
participant completed two concept maps, two card sorts, and a test o f trigonometric
knowledge. Phase two involved in-depth interviewing o f 5 of the 14 participants to
investigate their cognition, reasoning and problem-solving processes with respect to
trigonometry. Furthermore, the interviews provided the researcher with data for an indepth analysis o f the preservice teachers’ understanding and conception o f trigonometry.
This chapter discusses the methodological foundations, description of
methodologies, instrumentation, procedures and data collection, the main study, phase 1
and phase 2 of the main study, data analysis, and the research questions and the related
analyses.
Methodological foundations
Methodologically, this study was a descriptive one and built on previous research by
Even (1989), Bolte (1993), and Howald (1998). Even, Bolte, and Howald were all
conducted to investigate the knowledge o f functions held by either preservice or
experienced mathematics teachers. However, trigonometric functions formed a minute
part o f the three studies. The study used descriptive statistics to convey some o f the
results from the first phase o f the study. Case study methodology (Merriam (2001), Yin
(1994), and Stake (1995, 2000)) was employed in the second phase o f the study. The
42
choice o f case study for the second phase of the study makes sense because the present
study investigated a well-defined population: preservice secondary mathematics teachers
and a specific content strand o f mathematics. Taken together they form a closed system
for which qualitative case study is appropriate. Moreover, case study is compatible with
many methodologies (Merriam, 2001) and “virtually any phenomenon can be studied by
means of the case study methodology” (Gall, Borg, & Gall, 1996, p. 544).
Moreover, the use o f case study as the theoretical framework for the methodology
of phase two o f the present study is justified because this study was “interested in insight,
discovery, and interpretation rather than hypothesis testing” (Merriam, 2001, pp. 28 29). Furthermore, the use of case study design allowed for the study to focus on a
particular phenomenon (in this case, preservice secondary mathematics teachers
knowledge of trigonometry); allowed for thick description o f that knowledge via analyses
o f the status o f preservice teachers’ knowledge and organization of trigonometry, and
pedagogical ramifications o f their content knowledge; and also provides the reader with
heuristic understanding o f the phenomenon o f preservice secondary mathematics
teachers’ understanding o f trigonometry.
Interviews present the best method for rendering transparent the reasons and
motivation o f research participants’ actions and choices o f explanations. According to
Merriam, “interviewing is necessary when we cannot observe behavior, feelings, or how
people interpret the world [in this case the mathematical world o f trigonometry] around
them” (p. 72). For the present study, the interview was conducted to find out how
preservice secondary mathematics teachers organize and categorize the domain of
trigonometry, and how they envision applying their knowledge to the pedagogy of
trigonometry.
Furthermore, since the present study sought to contribute to theory and knowledge
of preservice teachers’ knowledge, a semi- structured in-depth interview suited the
purposes o f this study (Wengraf, 2001). Semi-structured interviews are made up o f two
43
parts: one is structured and the other is open-ended (Merriam, 2001). The structured
component o f the interviews in the present study sought to collect a standardized set of
information from all five cases. The open-ended portion anticipated and utilized the
different approaches the five cases used to categorize and organize the domain of
trigonometry. According to Strauss, Schatzman, Bucher, and Sabshin (1981), four types
of interview questions are helpful in getting at participants’ knowledge structures:
hypothetical, devil’s advocate, ideal position, and interpretive (Strauss, Schatzman,
Bucher, and Sabshin, 1981, in Merriam, 2001). The present study utilized both
hypothetical and d e vil’s advocate interview questions.
Description o f methodology
Participants
The targeted population for this study was preservice teachers who were nearing
the completion o f a mathematics teaching certification program but had not yet student
taught. Participation was solicited from the 17 students in a high school mathematics
methods course for preservice mathematics teachers offered at a large Midwestern
university. Fourteen o f the 17 students signed a consent form to have their data analyzed
and released in published form. The results reported in this study are from the 14
consenting participants. Twelve o f the participants were seniors, one had an
undergraduate degree and was completing his teaching certification, and one was a
graduate student. They had completed a practicum course, an elementary methods course,
a course in educational psychology, a course in special education, and a course in the
history o f education in the US, a course in Human Relations for the classroom teacher.
These students had also completed some field experiences in schools. They had observed
mathematics teaching from a vantage point other than that o f a student being taught, in
addition to their experiences as students. They had also constructed lesson plans in
previous courses. Through lesson planning experiences, they would have invariably
44
reflected on mathematics, students’ misconceptions and preconceptions, possible
difficulties that students may have with specific topics, and how to motivate student
learning. Moreover, the participants had taught some lessons to K-12 students in real
classroom settings during their field experiences. Thus they had developed some
rudiments o f pedagogical content knowledge through classroom experience and had
experienced the three phases o f teaching: Pre-active (planning), Active (instruction), and
Post-active (reflecting on the lesson). In summary, the participants possessed the typical
attributes and experiences of preservice secondary school mathematics teachers late in
their education prior to student teaching.
Furthermore, the participants had developed an appreciation o f the K-12
mathematics curriculum through their work with curricula materials in their methods
courses. Their experiences with K-12 mathematics as students played a complementary
role in the development o f their appreciation o f the K-12 mathematics curriculum. The
participants had formed notions about mathematics as students who took school
mathematics, as students in college mathematics courses, and as preservice teachers
learning to develop their craft of teaching. Such beliefs about mathematics will impact
their teaching practice (Fennema & Franke, 1992). These preservice teachers would have
also developed an understanding of mathematics that goes beyond trigonometry. Their
understanding o f trigonometry should be “a mile deep” since they have successfully
completed calculus, which is predicated on a thorough understanding o f trigonometry.
Their beliefs and their conceptions o f themselves as teachers and in particular as
mathematics teachers spanned both personal and social considerations. Six of the 14
participants indicated personal gratification, for example “love o f math”, as their primary
reason for wanting to teach mathematics. Another 6 highlighted the desire to help
students learn mathematics and humanize mathematics as their reasons for entering the
profession. Two participants did not respond to the questions on their beliefs and their
conceptions of themselves mathematics teachers.
45
As per subject matter content knowledge, their transcripts showed that the
participants had completed a variety o f undergraduate mathematics courses that ranged
from introductory Abstract Algebra, introductory Analysis, Foundations of Geometry,
Discrete Mathematics, and introductory Differential Equations (see Table 1). The letter Y
in a cell means that the participant had completed the indicated course at the time of the
study or was currently enrolled in the course in fall o f 2002. Table 1 also shows that the
preservice secondary mathematics teachers have completed an elementary school
mathematics course, an introduction and practicum to secondary mathematics education,
and most o f them have completed or were completing courses in middle school
mathematics methods, high school mathematics methods, and elementary school
mathematics student teaching. Most of them have or were completing foundational
educational courses dealing with history and theory o f education, human relations, and
the exceptional learner. None of the participants had completed nor were enrolled in the
secondary school mathematics student teaching course, which is required for
certification.
Instrumentation
Test o f trigonometric knowledge
A test o f trigonometric knowledge (the test) was developed to assess conceptual
knowledge and meaningful understanding o f trigonometry. The test was designed to
investigate preservice secondary mathematics teachers’ understanding o f the nuanced
complexities o f the fundamental ideas o f trigonometry. The test of trigonometric
knowledge was a non-calculator test, because the researcher wanted to investigate the
study participants’ cognition o f fundamental ideas o f trigonometry. The focus on
conceptual understanding is in accord with present conception o f learning and teaching
mathematics for understanding (NCTM; MAA; NRC; MSEB; CBMS; Hiebert, et al.,
1997; Hiebert & Carpenter, 1992; Ma, 1999).
46
Table 1. Post-secondary course taking history o f the 14 participants in Phase 1 o f the
study
Participants
AB
NM
LN
AX
ES
ZN
PM
Al
El
IA
SY
IB
AD
CT
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Educational Psychology
Educational Psychology and
Measurement
Mathematics
Intro. Abstract Algebra
Space & Functions
Foundations o f Geometry
Transformation Geometry
Discrete Mathematics 1
Discrete Mathematics 2
Elementary Num. Analysis
Intro. Ordinary DiffEq
Y
Y
Y
Y
Y
Y
Y
Y
Y
History o f Mathematics
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Intro. Analysis 1 and 2
Complex Analysis
Y
Abstract Algebra 1
Theory o f Numbers
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
M athematics Education
Elem. School Mathematics
Intro. & Practicum
Middle School Methods
High School Methods
Elem. Mathematics Student
Teaching
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
General Education
Human Relations
Foundations o f Education
Mainstreaming
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
The test addressed the following aspects o f trigonometric knowledge: (1)
definitions, terminology, and conventions; (2) degree and radian measures; (3) co­
functions; (4) angles o f rotation, coterminal angles, and reference angles; (5) special
angles (30°, 45°, 60°), their triangles, and their use to simplify computation; (6)
trigonometric functions and their graphs; (7) domain and range; (8) transformation o f
trigonometric functions; (9) even and odd functions; (10) laws o f cosines and sines; (11)
periodicity; (12) trigonometric identities; (13) algebra and calculus o f trigonometric
functions; and (14) the use of trigonometry in solving and modeling mathematical and
real-world situations.
47
The test was developed after reviewing several textbooks on the subject (Fort,
1963; Gelfand & Saul, 2001; Maor, 1998; Ryan, Doubet, Fabricant, & Rockhill, 1993),
recommendations of experts on what preservice teachers ought to know about the subject
o f trigonometry (Mathematical Association, 1950; Shama, 1998; Wu, 2002; Usiskin,
2002; CBMS, 2001) and instruments used in function knowledge studies (Bolte, 1993;
Quesada & Maxwell, 1994; Howald, 1998). Problems from the aforementioned textbooks
were used as items on the test o f trigonometric knowledge and also as problems in the
interviews.
In particular, the test measured the fundamental notion o f the restriction of the
domains o f the trigonometric functions that produce functions whose inverses are also
functions. The test also measured facility with the algebra o f trigonometric functions and
the nature and behavior o f trigonometric functions through transformations. And finally,
the ability to construct proofs o f trigonometric facts such as identities, and the ability to
use these in solving problems were also assessed. Professors in mathematics education,
mathematics, and educational measurement evaluated the test and made suggestions for
improving the test. The revised test was used in the pilot study. The test was further
refined after the pilot study. The final form o f the test o f trigonometric knowledge (see
appendix B) represents a significantly reduced version o f the initial test. The reduction in
the number and complexity o f the items in the instrument was a result o f feasibility and
levels of difficulty considerations.
Card sorting tasks
The study employed two card-sorting activities. In Card Sort 1 participants
received statements about trigonometric functions (see appendix C) to sort into three
piles: (1) Always true; (2) True sometimes; and (3) Never true. For example, statements
such as trigonometric functions [sin (x), tan (x), cos(x), esc (x), cot (x), and sec (x)~\ are
periodic formed the basis of the first card sorting activity. In Card Sort 2, participants
48
arranged a list o f trigonometric concepts and ideas (see appendix D) in what they
determined to be a logical order in which students should encounter the ideas in a school
mathematics curriculum. The card-sorts gave the participants an opportunity to articulate
their pedagogical content, curricular, and subject matter content knowledge of
trigonometry.
Interviews
Two in-depth, semi-structured interviews were conducted with 5 o f the 14 study
participants. These participants were chosen after an analysis o f the data from the test of
trigonometric knowledge, card sorts, and concept maps. Three o f the five interviewees
were chosen because of their strong content knowledge of trigonometry, and the other
two were because o f their low content knowledge of trigonometry. Originally, six
participants (3 high and 3 low), plus one backup, had been identified as possible
interviewees. However, one o f the six declined to participate, and the backup also
declined to participate in the interview for unknown reasons.
The first set o f interviews (see appendix F) used a semi-structured methodology to
gather information on how the interviewees constructed their concept maps, on what they
tried to convey with the concept maps, and on their knowledge o f trigonometry
terminology (see items 1, 2, and 3). The first interview also provided the interviewees an
opportunity to respond to how they would deal with some common students’
misconceptions about trigonometry in the classroom (see items 4 and 5). The researcher
also used the first interview as follow-up to interviewees’ incorrect responses to items on
the test o f trigonometric knowledge. Furthermore, in the first interview, participants’
articulation and sequencing o f topics in trigonometry (see items 6 and 7) provided
additional sites for analyzing their understanding o f the content o f trigonometry and their
pedagogical content knowledge o f trigonometry. As part o f the first interview, the
interviewees also discussed how they envisioned using technology in the classroom to
49
teach mathematics (in general) and trigonometry (in particular). To complete interview 1,
the interviewees were asked to use the TI-83+ to graph the function f ( x ) = C o f x[x) in
item 8.
The second semi-structured interview (see appendix G) focused on problem
solving in trigonometry. The interviewees also answered questions related to radian
measure (see item 1), proofs (see items 2 and 4), modeling (see item 3), transformations
and periodicity (see item 5). Probing questions (why, how, can you extend the question?)
were used to extend the interviewees’ responses to the problems presented and to gain
insight into the reasoning and justification abilities o f the interviewees. Interviewees’
problem-solving approach, use o f multiple strategies, disposition, use o f multiple
representations, abilities to extend problems to related and new situations were recorded
for analysis.
Concept maps
All participants completed two concept-mapping activities. The terms for the first
concept map came from the participants. The researcher provided the terms for the
second concept map (see appendix E). The concept maps are the outward representations
of the participants’ schemata for the domain of trigonometric knowledge. The process of
concept mapping afforded the participants opportunities to articulate the connections and
relations among trigonometric ideas and concepts. More discussion on concept mapping
can be found in the following sections: Pilot Study, and procedure for concept mapping.
The participants completed two concept maps o f the domain o f trigonometry. In
the first concept-mapping activity, participants constructed conceptual maps of the field
of trigonometry using their own terms and ideas. In the second concept mapping activity,
the researcher provided a list o f 89 terms and phrases that the participants used. The size
of the list of terms that was provided for the second concept mapping activity speaks to
the complexity o f trigonometry and also to the desire to be near exhaustive so that
50
participants are forced to relate and integrate the many ways o f thinking about
trigonometry: right-triangle, function, geometric, symbolic, graphical, periodic
phenomena, application and unit-circle. Furthermore, some o f the terms tease out
participants’ grasp o f mathematical conventions used in trigonometry. The concept maps
so constituted addressed both the emic and etic perspectives o f the participants and the
researcher. The concept mapping activities were open-ended in scope and the participants
were free to create personal conceptions of the field o f trigonometry. However, the emic
situation (in concept map 1) presented an unstructured format for the participants to
express their understanding o f trigonometry in their voice and symbolism. The etic
situation (in concept map 2) imposed some structure via the provided terms. As such, the
etic situation was an attempt to gauge the fit between the participants’ understanding of
trigonometry with that of an expert’s understanding o f the subject o f trigonometry.
Moreover, both the emic and etic perspectives are supported in radical constructivism,
which undergird the epistemological foundations for this study.
Consent form
A consent form was used to solicit participation in the study (see appendix A). It
included a brief description o f the purpose o f the study, the research procedures, contact
information for the Human Subjects’ Office and my academic advisor, and finally, but
most importantly it discussed the ramifications o f the study with respect to harm, adverse
effects, and anonymity. Participants gave their consent to participate and contribute to the
advancement o f the mathematics education field’s understanding o f preservice teachers’
knowledge of school mathematics.
Procedures and data collection
Participation in the study was sought in November o f 2002. The researcher visited
classes o f preservice teachers to recruit participants for the study. Participation in the
study was encouraged as a way to help further the mathematics education field’s
51
understanding of preservice teachers’ knowledge o f school mathematics. The participants
were briefed about their rights as research subjects using a consent form. Furthermore,
the researcher stressed the need for anonymity and assured them that their information
would not be shared with any of their instructors. Moreover, the researcher emphasized
their responsibility to safeguard their anonymity by not conferring with their peers about
their participation status or reveal to their peers any o f the activities that they completed
in the course o f the study should they choose to participate. In addition, interviewees
were told that the audiotapes of the interviews would be kept safe and confidential.
Pilot study
All instruments were piloted in October o f 2002. The test o f trigonometric
knowledge, the researcher-developed concept map activities, the researcher-developed
card sorts, the questions for the semi-structured interviews, and the general data
collection procedure were pilot tested with 4 graduate students enrolled in the fall
semester of the Doctoral Seminar in Mathematics Education in a large Midwestern
university. All four students had a Masters degree in mathematics. One had an additional
Masters degree in mathematics education. The dissertation committee also reviewed the
instruments and the data collection process. The purpose o f the pilot study was to refine
the instruments, gauge the extent of time commitments, validate the questions and terms
used in the instruments, and situate the instruments within the understanding o f experts in
mathematics and mathematics education.
In order to simulate the main study, the pilot study incorporated all aspects of the
intended research procedures and instruments. Permission to use the seminar class was
obtained from the professor in charge o f the seminar. An initial introduction to the
research study was presented to the seminar group and willingness to participate in the
pilot work was obtained from the graduate students in the class. The graduate students
received instruction on how to construct concept maps using functions (see appendix I).
52
The graduate students were asked to use all the terms provided in their concept maps. The
graduate students read Bolte (1999) and portions o f Novak and Gowin (1984) as
homework. Page 79 o f Novak and Gowin explicates the schema for concept mapping (see
figure 1 in chapter II). The assigned readings provided the graduate students with
additional foundational ideas on concept mapping.
On the second meeting o f the seminar class, vis-a-vis the pilot study, the
participants in the pilot study were given 1 hour to complete two concept maps: In the
first concept map, the participants used their own terms, concepts and idea of
trigonometry. The participants received a set o f 107 terms to use in the second concept
mapping activity. The participants were instructed to use all the terms provided to
construct their second concept map. At the end o f the concept mapping activity, the
seminar took a for 5 minute break. When they resumed, the participants were given 30
minutes to complete two card-sorting activities. In the first card sort, the participants
placed 48 trigonometric concepts and ideas (each on an index card) in a logical order that
students should encounter them in a school mathematics curriculum. In the second card
sort, participants arranged 15 index cards with statements about trigonometry into three
piles: Always True, Sometimes True, and Never True. At the conclusion o f the card sorts,
the seminar again took a 5 minute break. The remainder o f the seminar time was used to
discuss and receive feedback about the concept maps and the card sorts. The group
provided reaction on such things as difficulty levels, significance, clarity o f directions
and questions, and usefulness o f the list o f terms provided by the researcher for the
concept map.
Due to the shear volume of items (85 items developed for the item bank for the
test o f trigonometric knowledge), the item bank was subdivided into 4 test forms for the
pilot study. Each test form contained 31 items; thirteen o f which were identical on all
four forms (see appendix J). The items were sorted along the following dimensions:
Content (Function, Unit Circle, Angle-Triangle), Representation (Graphical, Symbolic -
53
algebraic, Geometric/Pictorial), Difficulty {Advanced, Intermediate, Basic), Process
{Proof/Justification, Problem Solving, Fact/Definition/Formula), and Application. After
serious consideration o f the best way to give the pilot group enough time to think about
the problems and provide in-depth and useful feedback such as which questions posed the
most difficulty fo r them, the participants were asked to take the test home and work on it
as homework and return the test along with responses to the following questions.
•
How long did you spend on the test? Itemize the length o f time spent on each
question.
•
Which items were just exercises for you?
•
Which items were problematic for you? Were there questions that you could
not answer?
• State what aspect o f trigonometry you thought each individual question on
your test was assessing.
• Which questions would be good items for a problem-solving test?
• Which items were open-ended, and which items were straightforward?
• Which items were vague and how did you interpret them?
• Are there questions about trigonometry that you think should have been asked
that were not on your test?
Furthermore, the participants in the pilot study were encouraged to take the test
without any support from peers or textbooks, as if under the testing conditions to be used
in the main study. The importance o f completing the test in one seating was also
emphasized. After their initial attempt, they were to go back over the test and answer the
aforementioned questions. Additionally, they were told that the objective of the test was
to assess what they already understood and remembered about trigonometry. The pilot
study was intended to help determine content validity and to assess time requirements
needed for completion o f the test. Moreover, the researcher assumed that the
mathematical ability o f the seminar group (pilot participants) was superior to that o f the
54
participants o f the main study. As such, the researcher was keen on finding out the
cognitive challenge o f the items for this group.
The selection o f the pilot study participants addressed the issue o f possible
contamination o f the potential research group. It was possible that any one o f the
graduate students could have had contacts with the preservice teachers, although none of
the graduate students served in the capacity o f teaching assistants. They were all research
assistants. The need to keep the items, activities, purpose o f the study and all other facets
of the study confidential and secret was communicated to the pilot group o f graduate
students. Moreover, as doctoral students, the importance o f all o f the aforementioned
precautions and procedures were well understood and they cooperated as requested. The
final instruments that were used in the main study reflected changes and improvements
made as a result o f the pilot study and inputs from mathematics and mathematics
education professors.
The pilot study revealed that trigonometric knowledge has a large decay factor
associated with it when it is not used on a regular basis. The graduate students, whose
mathematical knowledge is superior to that o f preservice undergraduate students had
difficulties with organizing, ordering, and categorizing the knowledge domain of
trigonometry. Their concept maps showed limited knowledge o f certain aspects o f high
school trigonometry. In some cases, there was confusion about radian measure,
periodicity (both fundamental and general), and the addition formulas. Application of
trigonometry to resolving mathematical problems and real world problems also provided
serious challenges to some o f the pilot participants. The performance of the graduate
students revealed that many o f the items chosen for the pilot study would have been too
difficult for use with the preservice students. Although, the preservice teachers had
greater recency of use o f trigonometry than that o f the graduate students, the preservice
teachers lacked the mathematical maturity that the graduate students possessed. The pilot
55
study also revealed that the tests and instruments took substantial amounts of time to
complete.
As a result o f the pilot study, a 25-item test o f trigonometry was created for use in
the main study (see appendix B). This 25-item test was submitted to a mathematics
professor for his scrutiny. He agreed to record his time on task and also rate the items on
their importance to high school mathematics teacher preparation. The researcher was
interested in what the mathematics professor thought was important for preservice
teachers to know.
The mathematics professor reported spending about an hour on the 25 questions.
The final test o f trigonometric knowledge reflects the inputs from the mathematics
professor in terms of importance o f the items, difficulty o f the items, and time
requirement needed to complete the test. The professor used a 3-point system to rate the
25 items. A rating o f 1 meant that the professor viewed that item to be of high importance
for preservice teachers to know. A 2 meant that the professor viewed that item to be of
medium importance. And a rating o f 3 meant that item was o f low importance because
either it was too specific or relied heavily on a hard-to-recall factual information for its
resolution. The ratings o f importance (see table 2) o f the items provided by the
mathematics professor show 18 o f the 25 items received ratings of highest importance,
and the remaining 7 items received ratings o f medium importance. As a consequence of
the professor’s reaction to the 25 items, two questions that asked for proofs o f the laws of
sines and cosines were changed to factual questions o f stating the laws. The professor
agreed with the literature on trigonometry that knowledge of these laws is essential to
understanding and using trigonometry.
The main study: Phase One
The study was organized into two phases. The first phase involved both
quantitative and qualitative investigations of subject matter content knowledge and
56
pedagogical content knowledge o f the 14 preservice teachers. Five o f the 14 preservice
teachers participated in the second phase o f the study. The 5 phase two participants were
selected after an initial analysis of the data from phase one o f the study.
Table 2. Item Importance Rating by a mathematics professor o f the 25-item pre-version
of the test o f trigonometry
Items
1
2
3
4
5
Ratings
1
1
2
2
1
Items
6
7
8
9
10
Ratings
Items
Ratings
Items
Ratings
Items
Ratings
1
1
1
1
1
11
12
13
14
15
1
1
1
2
1
16
17
18
19
20
1
1
1
2
2
21
22
23
24
25
1
2
1
2
1
Prior to the collection o f data for this phase o f the study, participants were given
two articles about concept mapping to read. The intent o f the assigned readings was to
prepare the class for a preliminary in-class activity on concept mapping o f functions. This
served as a prelude to and practice for the concept mapping for the main study. See
appendix I for details o f the procedure and the set o f assigned readings.
All participants were assigned to write and provided an academic autobiography
that highlighted their course taking history (from high school to present), courses with
pedagogical components, their experiences in actual classrooms (as preservice trainees)
and details concerning their decision to teach mathematics at the high school level. The
duration o f the assignment was one week. The autobiographies were collected on the
same day the data for phase one were collected. The purpose o f the assignment was to
gather demographic information about the participants. The demographic information
provided information-rich descriptions o f the participants. The assignment also allowed
57
for interview time, in phase 2 of the study, to be devoted primarily to substantive
questions regarding trigonometry rather than the collection o f demographic data.
Fourteen participants, enrolled in a high school methods course, completed a test
on trigonometric knowledge, a survey of trigonometry pedagogical content knowledge
via two card sorting activities and two concept maps. The instruments were administered
in the following order: Concept mapping, cards sorts, and then the test on trigonometric
knowledge. This sequence followed what has been done in previous research (Bolte,
1993; Howald, 1998) and it avoided turning the investigation into a learning episode;
which would have been the case if the test o f trigonometric knowledge had been
administered prior to the concept mapping and card sorting activities.
Moreover, by following the aforementioned sequence, the study was able to first
expose the participants’ conceptions and structure o f the domain o f trigonometry, then
evaluate their pedagogical content knowledge with respect to trigonometry, and finally
assess their understanding o f facts, concepts, application, and ability to problem solve
within the domain o f trigonometry. This resulted in a better measurement o f the
participants’ knowledge as opposed to ‘parroting’ o f the ideas and concepts contained in
the test o f trigonometric knowledge on subsequent instruments, if the test of
trigonometric knowledge had been the first instrument. Moreover, it eliminated possible
review of the concepts and ideas on the test o f trigonometric knowledge that might have
skewed the results o f subsequent instruments. All instruments were completed
individually. There was no group work. The researcher and the instructor of the high
school methods course monitored the participants as they worked.
The instruments in this phase of the study were completed during a single college
class period o f 140 minutes. House keeping activities by the instructor o f the course took
the first 30 minutes o f class time. Thus phase one lasted 110 minutes. As a result o f the
shortened time (I had anticipated using the full 140 minutes), the researcher made the
tactical decision to cut the time allotted for students to complete each instrument and also
58
cut the number o f items to be completed on the test o f trigonometric knowledge. The 25item test o f trigonometric knowledge (see appendix B) was reduced to a 17-item test.
Items 9, 10, 13, 14, 17, 19, 21, and 23 were eliminated from the 25-item test, and portions
o f items 11, 16, 20, and 24 were also eliminated or modified. The changes were
communicated to the participants both verbally and written on the board before they
started the test o f trigonometric knowledge. The final test is shown in appendix B. Two
5-minutes breaks separated the three major components: Concept mapping, card sorting,
and test o f trigonometric knowledge. The concept mapping activities lasted 40 minutes,
with Concept Map 1 and Concept Map 2 consuming 20 minutes each. The card-sorting
activities lasted about 20 minutes (10 minutes for each card sort). And the test of
trigonometry lasted 40 minutes. Electronic devices such as graphing calculators were not
allowed or used in phase one.
Concept maps
The procedures for the concept mapping activities were as follows. For the first
concept mapping activity, the participants were asked to create a concept map of
trigonometry using a similar structure that they had used with functions on a previous
occasion. They were reminded to first write what they knew about trigonometry on sheets
of paper (sheets were provided). After they had done that, they were to sort the
ideas/terms into clusters according to the extent o f relatedness among the terms. The
participants were then to relate and connect the terms and phrases in each cluster. The
participants were reminded that any one term could possible reside in more than one
cluster. After creating the intra-connections within the clusters, the participants were to
create inter-connections among the clusters. Within each cluster, the participants were
told to specify relationships among the terms and the phrases. A relationship was
represented with a line segment (or arc) between two ideas/terms and a descriptive phrase
or word connecting dyadic pair. Each participant received a sheet o f paper detailing all
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the directions (see appendix E). For the second concept mapping activity, the participants
also received the researcher-generated list o f 89 terms and phrases related to trigonometry
to use in the construction o f their concept maps. They were asked to use as many terms as
they deemed necessary to convey their organization and structure o f trigonometry. They
were also free to supplement the list of terms with additional terms. The participants
spent 20 minutes on concept map 1 and 20 minutes on concept map 2
Card sorts
In the first card-sort, participants arranged fifteen 4 inch by 6 inch index cards
with statements about trigonometry (see appendix C) into three piles: always true,
sometimes true, and never true. In addition participants received three label index cards,
each labeled with one of the following: Always True, Sometimes True, or Never True.
The participants used the labeled index cards to separate and organize their piles. In the
second card-sort, each participant received and sequenced thirty-four 4 inch by 6 inch
index cards of mathematical ideas in a logical order in which students should encounter
them in school mathematics curriculum. Ideas such as triangles, Pythagorean theorem,
angle o f rotation, law o f sines, coterminal angles, and quadrants made up the list of ideas
that were presented to the participants on the index cards (see appendix D). The
participants were also provided binder clips to preserve their piles in card-sort 1, and their
sequencing in card-sort 2. The participants placed all completed work in pre-labeled
envelopes for pick up by the researcher. The methodology used in the card-sorting
activities are recognized and utilized in the mathematics education field (Even, 1989;
Bolte, 1993; Howald, 1998). The purpose o f the card-sorting activities was to assess the
participants’ content and pedagogical content knowledge o f trigonometry. The
participants spent 10 minutes on card sort 1 and another 10 minutes on card sort 2.
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Test o f trigonometric knowledge
Each participant took a 17-item test o f trigonometric knowledge. The questions
ranged from definitions, identification of graphs, to problem-solving tasks. They received
the test o f trigonometric knowledge after they had completed the concept maps and card
sorts. They had 40 minutes for this activity. They were told to answer the questions to the
best of their ability. They were told that the items would be graded for correctness but
that partial credit would be given for correct portions o f an otherwise incorrect solution.
The participants were provided blank sheets of paper to use for their work. Completed
work was collected in pre-labeled envelopes.
The main study: Phase two
After an initial analysis of the data from phase one of the study, five individual
one-on-one case studies were conducted. The case studies were conducted during two
interview sessions that were convened a month after phase one o f the study. The profiles
chosen for the case studies were (1) high subject matter content knowledge, and (2) low
subject matter content knowledge. The five participants that met such profiles were
selected for further in-depth one-on-one interviews. The interviews attempted to uncover
nuanced complexities of the relationships among the content knowledge o f trigonometry,
knowledge of pedagogy specific to the teaching o f trigonometric ideas, and the
envisioned practice held by the participants. The interviews followed a semi-structured
interview format and presented participants with problems to solve and proofs to
construct (see appendix F for an outline o f topics addressed). Planned questions and
activities such as describe how you went about constructing your concept map and how
would you change your concept map or card-sorts to fit your present conceptions? were
used to structure the interview.
Furthermore, the interviews presented participants with constructed hypothetical
cases of students’ work (see appendix F). The participants validated or refuted the
students’ work, and discussed extensions and ways to help the students attain better
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understanding of trigonometry. The participants were presented with situations that
required them to advocate an ideal position for trigonometry in school mathematics given
the present dearth o f trigonometry in mathematics education literature. A question such
as, in your opinion, how important is trigonometry in the high school mathematics
curriculum allowed the participants to advocate for trigonometry. Throughout the
interviews the researcher asked questions that informed the interpretation o f participants’
responses. These unplanned probes rendered the interview open-ended to a certain point.
The interview format heeded the advice of Merriam (2001) to avoid multiple questions,
these are questions with multiple parts, because they may confound and confuse the
participants; and thus yield inaccurate information. Other types o f questions that were
avoided were leading questions and yes-or-no questions since they do not generate much
meaningful or rich data. The interview methodology ensured that the interviews did not
degenerate into teaching episodes.
During the interviews, the preservice teachers evaluated and renegotiated the two
card sorts that they had previously constructed in phase one o f the study. The card sorting
activities were geared toward the preservice teachers’ conceptions o f the sequence and
order in which trigonometric concepts should be presented and the veracity o f
propositions about trigonometry. The interviewees also evaluated the earlier concept
maps o f trigonometry. The methodology o f allowing participants to re-synthesize their
earlier concept maps agrees with Novak and Gowin (1984) claim that “a second map
usually shows key relationships more explicitly” (p. 35) when compared with the first
concept map. The final concept maps embodied any learning, reflections and new
understanding since the start o f the research. Thus it also served as a measure o f the
extent to which preservice secondary mathematics teachers reflect and learn to
understand the content that they will be required to teach once they become teachers. The
interviews were recorded on audiotapes and the researcher took notes. Additional data
that were collected included participants’ work that they generated during the interviews.
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Participants in the present study had a quiet place to do their work. The interviewees
spent as much time as was they deemed necessary to resolve questions posed and to also
problem-solve. On an average, interviewees spent anywhere from 3 hours in one seating
to a combined time o f 5 hours at two seatings to complete the interviews. The TI-83+ was
used as a post-analytic tool to check and accept, or check and modify their first attempts
on the interview items.
Interview 1
In the first interview (see appendix F), participants evaluated and renegotiated the
concept maps they created in phase on o f the study. The interviewees had the option of
talking into the microphone while they worked or first reviewing and renegotiating their
concept amps, and then talk into the microphone. All five interviewees used both
approaches. There were times when they talked into the microphone while they worked.
And at other times they waited until they had finished with a process before talking into
the microphone. Interviewees ended their discussions o f the concept maps by comparing
and contrasting the concept maps.
The next activity in interview one involved the interviewees supporting or
refuting two constructed students’ works rife with common misconceptions. The
interviewees completed these tasks first without the aid o f any electronic aid. And then,
they were given the option to review their conclusions to students’ work 1 and students’
work 2 using the TI 81 + graphing calculator.
After they had completed students’ work 1 and students’ work 2, they arranged
the 34 cards of card sort 2 on a large conference table into the sequence they had
produced in phase one o f the study. They discussed reasons why they had put the cards in
the order used to arrange the cards into a pedagogical sequence. They also addressed
questions regarding prerequisite concepts. Then they evaluated their first card sort: The
arrangements of the 15 propositions into 3 piles of Always True, True Sometimes, and
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Never True. In the evaluations o f the concept maps and the card sort the interviewees had
the option to make changes to their creations. The final and eighth activity of the first
interview involved the interviewees discussing the use o f technology in teaching and
learning o f trigonometry. That also addressed issues related to the place o f trigonometry
in a high school mathematics curriculum. To conclude interview one, the interviewees
were asked to graph the function / ( * ) = Cot~l(x) on the TI 83+ graphing calculator.
Interview 2
The interviewees started interview 2 (see appendix G) by discussing how they
begin the problem-solving process and how they persevere while problem solving. They
also discussed the role o f proof and justification in the teaching o f trigonometry. Then
they moved on to defining radian measure. The next activity involved writing a proof for
the claim “there are 360° in one revolution”. For the remainder o f interview 2, the
interviewees completed another proof question, and two problem-solving items without
the use of the graphing calculator. At the completion o f the interview items the
interviewees were given the option to go back and review their conclusions with the aid
of the TI 83+ graphing calculator. They had the option to make changes as they wished.
Phase two ended with the researcher thanking the interviewees for their contributions to
the research study.
Data analysis
This section discusses the analysis o f the data from the two phases of the study.
First, a general framework for the analysis is described. Then, details o f the analysis of
phase one data from the two concept maps, the two card sorts, and the test of
trigonometric knowledge are presented. That is followed by a discussion o f the analysis
of the data from phase two o f the study from the two semi-structured interviews. Finally,
a discussion o f how the data analysis addresses the research questions is presented.
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Data analysis framework
Merriam (2001) provided the framework for analyzing the interview and conceptmapping data. Basically six data analysis methods underlie all data analysis done in
qualitative studies: ethnographic analysis, narrative analysis, phenomenological
analysis, constant comparison method, content analysis, and analytic induction.
Ethnographic analysis seeks to provide thick descriptions of cultures through different
artifacts, both pre-existing and constructed. The artifacts are analyzed using categories
and typologies, and possible or tentative hypotheses and explanations are attempted.
Narrative analysis attempts to understand experiences through analysis o f peoples’ livedstories. Phenomenological analysis “attends to ferreting out the essence or basic
structures o f a phenomenon” (p. 158). Constant comparison method constantly compares
incidents in data to develop explanatory categories towards building a theory. Qualitative
content analysis attends to themes and recurring patterns in the content o f interviews,
field notes and other data collected in a qualitative study. Analytic induction is a
“rigorous process o f successively testing each new incident or case against the most
recently formulated hypothesis or explanation of the phenomenon under study. ... The
process continues until the reformulation covers all cases studied or no negative cases can
be found” (pp. 160-161).
O f the six data analysis methods, constant comparison and content analysis
methods were applied to analyze the data collected in the present study. Furthermore, the
analytic framework employed by Bolte and Howald to analyze their surveys o f function
knowledge and concept maps was employed for the same purpose in the present study.
For the concept maps, the scales o f item score, stream score, ratio score, endpoint, open
chain, closed chain, cross-link, and composite score were employed to gauge the
complexity and integration o f the concept maps. According to Bolte and Howald,
Item Score is the number o f discrete entries included in the map
indicating the degree o f differentiation o f the map; (2) Stream Score is the
number o f lines emanating from the focal concept that lead to one or more
words or phrases indicating the degree o f complexity o f the map; (3) Ratio
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Score is the ratio o f item score to stream score indicating the degree of
integration o f the map; (4) Endpoint is the number o f single concepts
emanating from other terms, that is, only one linking line drawn from the
term; (5) Open Chain represents three or more concepts linked in a single
chain within a cluster; (6) Closed Chain represents concepts that form a
closed system within the cluster; (7) Cross Links are connecting lines
between clusters; and (8) Composite Score is the sum of the endpoint,
open chain, closed chain, and cross-link scores (Howald, 1998, pp. 75 76).
Concept mapping
The concept maps were analyzed both quantitatively and qualitatively. For the
quantitative analysis, the frameworks o f Bolte, Howald, and Novak and Gowin were
employed. For the qualitative analysis both constant comparison and content analysis
were employed.
Following Bolte and Howald, three categories o f tallies, groupings, and
connections were used to organize the data from the two concept maps. Tallies consist of
the item and stream scores. Groupings are made up of endpoint scores, open chain scores,
and closed chain scores. Connections comprise ratio scores, cross-link scores, and
composite scores. Table 3 presents a tabular representation of the quantitative analytic
framework o f tallies, groupings, and connections. A table similar to table 3 was
constructed for the data from each o f the two concept mapping activities. Again,
following Bolte and Howald, distinct items on the concept maps were scored a point
each, streams were scored one point each, endpoints were scored one point each, open
chains were scored two points each, the closed chains were scored three points each, and
cross-links were scored four points each. These scores were then combined to generate
individual participants’ totals for the scales o f item, stream, endpoint, open-chain, closed
chains, and cross-links. The ratio (item to stream) scores were computed for each
participant. Composite (endpoints, open-chains, closed chains, and cross-links) scores
were also computed for each participant.
The item scores were computed by counting the distinct concepts and ideas
presented in each concept map. These did not depend on the veracity o f the items or the
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linkages between the items. The stream scores were computed by counting the branches
emanating from the focal concept o f the concept maps. A focal concept was identified by
its position at the top o f the page or the concept that is centrally placed and has the most
branches emanating from it. The ratio scores were computed from these two scores: item
score to stream score. The endpoint scores were computed by counting the terminal
concepts or ideas in an open chain. Closed chains do not have endpoints due to their
looped nature. In closed chains, concepts and ideas are either directly or indirectly selfreferential.
Table 3. The analytic framework for the concept mapping activities showing the scales
used in the analysis of the data
Item
Tallies
Stream
Endpt
Groupings
Open
Connections
Closed
Cross
Ratio
Composite
AB
NM
LN
AX
ES
ZN
PM
Al
El
IA
SY
IB
AD
CT
Open chains and closed chains occur embedded in clusters. Clusters were
identified as “bunching” o f concepts and ideas off the streams (or branches from the focal
concept). Linear progressions of 3 or more concepts or ideas within clusters were selected
as open chains for the analysis. On the other hand, “looping” o f concepts and ideas were
selected as closed chains and scored accordingly. Having identified clusters, the
connections between any two clusters were identified as cross-links and scored
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accordingly. Each concept map was scored and re-scored on three separate occasions in
order to stabilize the scores for the participants on each item. This process o f re-scoring
was applied to all the instruments in the study.
For this study, both web and hierarchical concept maps were acceptable.
Hierarchical concept mapping allows for an assessment o f the participants’ conceptions
of the levels o f abstractions and the related super-ordinate and subordinate structure of
trigonometric concepts and ideas. Figure 2 shows how a hierarchical concept map could
be scored. Observe that only valid relationships, hierarchies, cross-links, and examples
are counted and credited.
Figure 2. Concept map scoring model o f Novak and Gowin (1984, p. 37)
One of the stated objectives o f the study was to assess preservice secondary
mathematics teachers’ status and organization o f trigonometric knowledge in both the
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emic and etic situations presented in concept map 1 and concept map 2, respectively. As
such, the scores on the two concept maps were compared. The comparisons were along
the scales o f item, stream, ratio, and composite scores. Box plots were used to present the
comparisons visually.
Meaning o f quantitative scores
The focus o f the quantitative analysis was on the item scores, ratio scores, and the
composite scores. The item score offered a window into the amount o f discrete
trigonometric knowledge bits that the preservice teachers were able to recall. The number
of recalled items should be viewed in light o f the list o f 89 items that were provided for
use in the second concept map. The list was conceived as an expert’s conception of
trigonometry in a high school curriculum. Thus higher numbers o f items were interpreted
to mean that the participants have greater flexibility with low order thinking skills. To
qualify for high knowledge o f trigonometry, the participants had to also achieve high
ratio score, high composite scores, high level o f accuracy, completeness, and
organization.
The ratio scores indicated the level o f integration o f the concept map. Thus the
higher the ratio score, the greater the integration o f the concept maps. The composite
scores measure the richness o f the knowledge captured by the concept map. Higher
composite scores might indicate possession of rich knowledge of trigonometry. But
again, that may not be the case. So the composite scores were coupled with the
qualitative analysis along the dimensions o f completeness, accuracy, and organization to
generate finer grain analysis o f the nature o f the knowledge captured by the concept
maps.
The qualitative dimensions of completeness, accuracy, and organization provided
ways o f making sense o f the quantitative scores and the nature o f the knowledge that was
captured by the concept maps. I measured completeness by analyzing the presence and
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nature o f fundamental ideas and relationships, definitions, and applications of
trigonometry in the concept maps. Accuracy had to do with the validity o f the concepts
and ideas proposed by the participants and the relationships among them. And finally, the
process o f constant comparison was employed to generate patterns and categorize the
organizational structures that were employed by the participants.
Card sorts
Quantitative and qualitative analysis were conducted for both card sorts. In card
sort 1, the participants arranged the 15 propositions on index cards into three piles and
provided reasons for the placements. The placement o f the propositions into the piles was
scored for correctness o f the proposed membership. That is, if the proposition belongs in
the pile in which it was placed, then that correct placement was scored one point. If an
incorrect placement was made, that incorrect placement received zero points. Once the
quantitative analysis was completed, the researcher identified the propositions that more
participants placed incorrectly than had placed it correctly. Then the reasons for placing
those propositions were analyzed for correctness and substance. Substantive reasons refer
to mathematical arguments and not “I guessed”. If no reason was given for the placement
of the propositions, that reality was also recorded so that the researcher could discuss the
nature and complexity o f the subject matter content knowledge o f the preservice teachers.
Card-Sort 2 was scored for coherence o f the sequence, with particular attention
paid to the participants’ assumptions vis-a-vis necessary prerequisite knowledge and the
learning trajectory encapsulated in the proposed sequence. To accomplish the
aforementioned analysis, both quantitative and qualitative analyses were conducted. The
quantitative analysis involved counting the cards that agreed with Hirsch and Schoen
(1990), and Senk et al. (1998) sequence o f trigonometric concepts and ideas. The
concepts and ideas on the 34 index cards used in the second card sort were divided into
three categories: early concepts and ideas, intermediate concepts and ideas, and advanced
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concepts and ideas (see table 4). Percentages o f agreement along the dimensions o f early
concepts and ideas, intermediate concepts and ideas, and advanced concepts and ideas
were computed.
Table 4. Categorization o f trigonometric topics in card sort 2 into early, intermediate, and
advanced concepts and ideas
Dimensions
Early
(Cards 1 - 9)
Intermediate
(Cards 11, 12, 13, 15 - 23)
Advanced
(Cards 10, 14, 2 4 - 3 4 )
Trigonometric Concepts and Ideas
D egree
Similar Right Triangles
Coordinate Plane/Angles/Rotations/Quadrantal/Coterminal Angles
Pythagorean Theorem
Six Basic Trigonometric Ratios
Special Angles/Reference Angles/Trigonom etric Ratios o f Special and
Reference A ngles
Radian M easure/Arclength/Angular - Linear V elocity
Trigonometric Identities (reciprocal, Quotient, Pythagorean)
Circular Functions
Unit Circle
Periodicity
Sinusoids
Graphs/Domain/Range/Asymptotes/Frequency/1 - 1/Even-Odd Functions
Inverse Trigonometric Functions/}1 = x line/Principal Values
Trigonometric Equations
Triangles & Trigonometry/Law o f cosines/Law o f sines/Solving Triangles
Sum /D ifference/D ouble/H alf A ngles/Product from Sum/Difference/Sum
from H a lf A ngles
Polar Equations
Vectors & Trigonom etry
C om plex N um bers and Trigonom etry
Circular Functions and Series
Column 1 o f table 4 gives the concepts and ideas o f card sort 2 and how they fit
into the early, intermediate, and advanced scheme. Concepts and ideas are presented in a
prerequisite sequence in column two of the table. For example, degree is a prerequisite
knowledge for understanding similar right triangles. A set of concepts and ideas that are
co-requisites, that is, at the same point in the prerequisite sequence are separated by
slashes (/). Some concepts and ideas in column 2 o f table 4 did not appear in card sort 2,
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but they were included in the table to provide a complete map o f trigonometry at the high
school level.
The analytic framework presented in table 4 was adopted for this study because
the focus o f card sort 2 was on prerequisite integrity, not realities o f curriculum
implementation or classroom practice. For example, knowledge o f inverse trigonometric
functions is one of the prerequisite knowledge needed to solve problems as simple as
“find all the missing data in a right triangle with legs measuring 4 units and 7 units
respectively”. Therefore in arranging cards into a prerequisite and pedagogical sequence,
solving triangles ought to be placed later in the sequence after inverse trigonometric
functions. Use o f the inverse trigonometric functions keys on the calculator for
pedagogical reasons does not negate the stated argument. In fact, by calling on those
inverse functions to solve problems, we argue that they are needed, hence prerequisite to
the problem at hand.
The qualitative analysis o f the data from the second card sort focused on the
pedagogical integrity and prerequisite integrity o f the sequences produced by the
participants. Integrity as used here refers to the coherence o f the sequences. To assess the
pedagogical and prerequisite integrities of the participants’ sequences, close attention was
paid to the placement o f the topics in the sequences. And “out-of-place” placements were
flagged and catalogued. For example, placing angular rotations before mention of
quadrant was categorized as an out-of-place placement. Hence the organization o f the
qualitative analysis was along the dimensions o f misplacements.
Test of trigonometric knowledge
The first order o f analysis was the description and categorization o f the items on
the test o f trigonometric knowledge. Each item was categorized along the dimensions of
levels o f difficulty, representational modality, knowledge structure, process, importance
and the nature o f importance, and projected success rate. Each item was assigned low,
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medium or high level o f difficulty depending on the depth o f knowledge and prerequisite
knowledge required for accurate resolution o f that item. Each item was assigned one of
the following representational modalities: graphical, pictorial, tabular, and
symbolic/equation depending on the primary mode o f the expected response. Each item
was assigned one o f the following processes: factual/recall, or problem-solving. Finally,
each item was ascribed a success rate depending on the level o f difficulty ascertained
from the prerequisite knowledge, representational systems, conceptual depth of
knowledge, and the process (recall or problem solving) associated with the item. The
aforementioned description and categorization served as an interpretive framework for
the discussions o f the results.
The test o f trigonometric knowledge was analyzed both quantitatively and
qualitatively. The quantitative aspects o f the analysis involved scoring the 17 items for
correctness. Each item had a possible score, ranging between 3 and 9 points. A holistic
scoring rubric (see figure 3) was used to score the problem-solving tasks, definitional
tasks, and proofing tasks on the test of trigonometric knowledge. These were items 1, 2,
3,4, 5, 6, 7, 8, 10, 11, 13, 14, 15, and 16. O f these, items 7, 8, and 15 are two-part
questions. However, the rubric was only applied twice to item 15 since it was valued at 3
points per part. Items 7 and 8 were each scored 3 points. Part (a) o f item 7 was scored 2
points while part (b) received a single point. Part (a) o f item 7 required the definition of
the radian measure for angles and part (b) asked for the conversion between radians and
degrees. The depth o f knowledge needed for an accurate definition in part (a) necessitated
valuing part (a) more than part (b). Item 8 part (a) was scored 1 point, if it met the rigor
of the highest level of understanding presented in the holistic rubric while part (b) was
scored 2 points, again, if a response met the highest rigor o f the highest level of
understanding presented in the holistic rubric. Part (a) o f item 8 required the definition of
a unit circle and part (b) required a discussion o f the uses o f the unit circle in
trigonometry. Therefore, part (b) was considered to require more in-depth knowledge
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than part (a). Items 9, 12, and 17 were outside the holistic rubric because they were true false, or matching items. Correct responses to parts o f item 9 were scored a point each.
Correct matches to parts o f 12 and 17 were scored three points each. Items 12 and 17
demand knowledge o f sinusoids and their graphs, and inverse trigonometric functions and
their graphs, respectively. Therefore parts o f item 12 and item 17 were valued more than
the parts of item 9.
1.
All procedures and solutions are accurate, complete and appropriate. Theorem s, definitions, and all
conventions are spelled out correctly and used appropriately
2.
M inor errors in at m ost one part o f the solution process (could be definitional, related to a theorem,
conventions: In a triangle if one o f the angles is labeled ,4, then the side opposite that angle is
labeled a; algebra, or the final answer
3.
Serious and m ajor errors in process, solution and/or multiple algebra m istakes, or understanding
and use o f definitions, theorems, or conventions, but shows an understanding o f the question
4.
Did not understand the problem , solution process is irrelevant to the question asked, or
inappropriate process or solution was provided. Lack o f understanding in use o f theorems,
definitions, or conventions
Figure 3. A holistic model for scoring some o f the free-response items on the test of
trigonometric knowledge
The total score possible for all correct responses in the test o f trigonometry was
64. Descriptive statistics o f mean, range, mode, maximum, and minimum were used to
analyze the scores obtained by the participants. The quantitative analysis was done both
item-wise and case-wise per participant. Items on which 50 percent of the participants
scored 50 percent or better, or scored less than 50 percent were identified. Further
qualitative analysis was conducted to elucidate discemable patterns o f response and error
patterns. These patterns were categorized into emergent themes o f responses o f the
preservice teachers.
The interview data
The audiotapes o f the interviews were transcribed and coded using the NUD*IST
software program to generate themes and patterns o f the preservice content and
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pedagogical content knowledge o f trigonometry. To ameliorate loss o f data and minimize
subjective interpretations o f the interview data during transcription, debriefing notes and
notes taken during the interviews were used to complement the data from the audiotapes
(Wengraf, 2001). The transcripts were also cross referenced with the interviewees’ work.
The data from the interviews were analyzed item-wise. Salient patterns o f responses were
recorded and compared among the interviewees. These patterns were synthesized to yield
a picture o f the preservive secondary mathematics teachers’ knowledge o f trigonometry.
Additional qualitative analyses were carried out using M a’s (1999) categorization
o f teachers’ knowledge along the dimensions of connectedness, multiple perspectives,
basic ideas, and longitudinal coherence. This classification scheme is helpful in assessing
preservice teachers’ understanding of trigonometry, its place in school mathematics, and
its pedagogical implications. Further analyses were conducted using Even’s (1990)
framework for analyzing teachers’ subject matter knowledge along the dimensions of
essential features, different representations, alternative ways o f approaching, strength of
the concept, basic repertoire, knowledge and understanding o f a concept, and knowledge
about mathematics.
The research questions and the related analyses
What content knowledge o f trigonometry do preservice secondary school
mathematics teachers possess?
The test o f trigonometric knowledge was used to help answer this question. The
scores derived from the scoring process previously discussed were used to analyze the
preservice secondary mathematics teachers’ understanding o f (1) definitions and
terminology; (2) degree and radian measures; (3) co-functions; (4) angles o f rotation,
coterminal angles, and reference angles; (5) special angles (30°, 45°, 60°), their triangles,
and their use to simplify computation; (6) trigonometric functions and their graphs; (7)
domain and range; (8) transformation o f trigonometric functions; (9) even and odd
functions; (10) geometric underpinnings o f trigonometry, for example, triangles,
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Ptolemy’s theorem, and figures inscribed in circles; (11) laws o f cosines and sines; (12)
trigonometric identities; (13) algebra and calculus o f trigonometry; and (14) the use of
trigonometry in solving and modeling mathematical and real-world situations. Additional
insights about participants’ subject matter content knowledge were gained from the
analysis o f card sort 1.
What pedagogical content knowledge o f trigonometry do preservice secondary
school mathematics teachers possess?
To assess preservice secondary mathematics teachers’ pedagogical content
knowledge, data from the card sorting activities, the interviews, and concept maps were
analyzed to address the following questions: What prerequisite knowledge is necessary
for the learning o f trigonometry? How do the preservice secondary mathematics teachers
understand multiple representations that will prove useful to unpacking the content of
trigonometry for students? How do preservice secondary mathematics teachers sequence
and organize trigonometric concepts for teaching? Do the sequence and organization of
the concepts anticipate both students’ preconceptions and misconceptions, and possible
approaches to help students overcome such misunderstanding? If preservice secondary
mathematics teachers were presented with difficulties that students might encounter, how
would they go about helping students get to truer conceptualizations o f trigonometry?
How are preservice secondary school mathematics teachers ’ content and
pedagogical content knowledge o f trigonometry organized?
This question is intended to reveal the depth o f understanding of trigonometry
held by the preservice secondary mathematics teachers. Analysis o f both the test of
trigonometric knowledge, interviews, card sorts, and concept maps o f the domain of
trigonometry were used to address this particular question. The analysis also explored
preservice secondary mathematics teachers’ integration o f and translation amongst the
different representations (the rule of four: graphical, tabular, symbolic, and verbal) that
are possible in trigonometry.
76
How do preservice secondary school mathematics teachers envision teaching
trigonometry?
Analyses o f the interview data and the card sorts were used to generate themes
and patterns for the preservice teachers’ envisioned pedagogy. This question is at the
heart o f the pre-active phase o f teaching. Planning for content delivery, the sequencing,
pre-requisite knowledge, ways o f explaining difficult concepts, important examples,
students activities during instruction, student practice, and student questions are
important ingredients for effective teaching. For this study, the following questions were
addressed as they relate to the planning phase or the pre-active phase o f teaching. How
will they develop the six basic trigonometric ratios? Will they approach the ratios from
the perspective o f right triangles, or unit circle, or as functions? What is the depth o f the
preservice secondary mathematics teachers’ understanding o f curricula knowledge as per
trigonometry (sequencing, prerequisite knowledge, core components o f trigonometry,
application of trigonometry)? What strategies will the preservice secondary mathematics
teachers use in their teaching o f trigonometry? Will they use multiple representations,
manipulatives, and electronic technologies? What pedagogical approaches (didactic or
heuristic) will the preservice secondary mathematics teachers employ? Will the
preservice secondary mathematics teachers use justification and proof as part of their
method for validating claims and ascertaining truth?
How are preservice secondary school mathematics teachers ’ content and
pedagogical content knowledge o f trigonometry related to their envisioned application o f
their content and pedagogical content knowledge in mathematics classrooms'?
All data points were analyzed to generate some hypotheses regarding the
flexibility, adaptability, and the robustness o f the preservice teachers’ pedagogical
content knowledge and their pedagogical problem-solving capacities.
77
CHAPTER IV
PRESENTATION AND DISCUSSION OF RESULTS
“Seek simplicity, but distrust it”
—Alfred North Whitehead,
quoted in Novak & Gowin, 1984, p. 1.
This study investigated the status and organization o f preservice secondary school
mathematics teachers’ knowledge o f high school level trigonometry. This chapter
presents the results o f the study. First, the results o f the pilot study are presented. That is
followed by a presentation o f the results o f the activities (two concept maps, two card
sorts, and the test o f trigonometric knowledge) from phase one o f the study. Third, the
results of interview one and interview two from phase two o f the study are presented.
Then Even’s (1990) and M a’s (1999) analytic frameworks are used to organize and
summarize the results. Finally, explorations o f the five cases investigated in phase two of
the study are attempted. The explorations provide profiles o f understanding o f the five
interviewees. The chapter concludes with a brief summary.
Pilot study results
The instruments for the study were piloted with four graduate students in
mathematics education. In addition, seven experienced high school mathematics teachers
with median teaching experience o f 10 years evaluated the test o f trigonometric
knowledge for item difficulty and validity. The teachers provided feedback on the
phrasing of the test items to minimize ambiguity and misinterpretation. A professor of
mathematics, two professors o f mathematics education and one professor o f educational
measurement provided additional feedback on item validity, phrasing o f the items, and
item difficulty.
78
Concept map
The four graduate students (G l, G2, G3, G4) felt that the sequence o f the two
concept maps (emic, then etic) was appropriate and they liked the fact that they were
given the opportunity to come up with their own terms prior to using a generated list.
They felt that the initial direction to use all o f the 107 terms in the second concept map
was not practical. In fact, they were overwhelmed by the size o f the list. They suggested
that the directions for the second concept mapping activity be changed to suggest that
participants in the study use as many terms as possible in their maps.
The quantitative results o f the pilot test o f the first concept map are presented in
table 5. The maximum number o f terms generated by the participants was 26. G l and G4
displayed the most connected and elaborate understanding o f trigonometry. Their higher
item and cross-link scores attest to that fact. Qualitatively, G l used angle, unit circle,
right triangle, and function to organize trigonometry. G l included the six basic
trigonometric functions: sine x, cosine x, tangent x, cosecant x, secant x, and cotangent x
under functions. G l mentioned domain and range as necessary things to know about the
six basic trigonometric functions. G l also mentioned degrees and radians as types of
angles. G l mentioned that the unit circle could be used to determine the values o f sine,
cosine, and tangent o f angle measures. Under right triangle, G l included the properties of
legs, and hypotenuse. G l also connected the legs to opposite/adjacent and mentioned the
two special triangles (45° - 45° - 90°; 30° - 60° - 90°).
G2 used functions as the main and only organizer for trigonometry. G2 mentioned
graphs (waves, amplitude, asymptotes, and periods), degrees, and radians as measures.
Furthermore, G2 stated that functions are part o f identities found on unit circles, which
have something to do with polar coordinates and complex numbers. G2 also stated that
trigonometric functions can be undefined at some angle values and that trigonometric
functions have inverses. Finally, G2 stated that, sine, cosine, and tangent can be
represented by Taylor series.
79
Table 5. The results o f the first pilot concept mapping in which the participants generated
the terms used in the map
Gl
G2
G3
G4
Item
23
15
15
26
Tallies
Stream
12
11
11
19
Endpoint
6
7
7
9
Groupings
Open
3
8
6
5
Closed
10
5
1
4
Cross
5
0
0
6
Connections
Ratio
Composite
23/12
24
15/11
20
14
15/11
24
26/19
G3 used similar triangles as the main organizing concept for trigonometry, with
similar right triangles as subordinate structures to the more general idea o f similar
triangles. G3 related right triangles to the Theorem o f Pythagoras and the unit circle. The
unit circle was then related to trigonometric functions. There was an indication that the
functions were one-to-one. G3 went on to connect the law o f sines and cosines to all
triangles as means o f solving triangles. G 3’s concept map also included “subtract acute
angles and add obtuse angles” without elaboration or explanation.
G4 used the dimensions o f triangles, functions, uses, and fa cts to organize
trigonometry. Triangles were subdivided into right (which was further subdivided into
basic ratios and special triangles), degree and radian measures, and identities (which were
further subdivided into law o f cosines and law o f sines). Functions were subdivided into
form (a + frsin(c + dx)), and cyclic (period, amplitude, and frequency). Uses were
subdivided into solving triangles for angles and sides, environment fo r p ro o f and problem
solving (circular motion). Facts were subdivided into arccos, arcsin, cos2x + sin2x = 1,
and sin(x + y).
As previously mentioned, the graduate students were overwhelmed by the size of
the list o f terms for the second concept mapping activity. After 15 minutes, the researcher
realized the frustration level o f the graduate students had reached a saturation point and
that moving forward with the activity would not have been productive. Thus the
researcher asked the graduate students whether they would like to discuss their
impressions o f the activity, their feelings, and provide constructive feedback to the
80
researcher on how to improve the instrument. Without any hesitation, the graduate
students accepted the offer and discussion of the activity ensued. The direction for the
second concept map had required the use all 107 terms, which the graduate students felt
was too burdensome. An analysis of the graduate students’ concept maps revealed that
the list o f 107 terms, provided for the second concept map, contained all the terms that
the graduate students had used in their first concept map. Thus the list accurately
anticipated the ideas and terms that might be included in a concept map of trigonometry.
However, there were a number o f terms that were underutilized by the graduate students.
Again, this is indicative o f the exhaustive nature of the list o f 107 terms.
The graduate students suggested that the researcher change the direction of the
task to read Use as many o f the trigonometric terms provided as you can and not require
the preservice teachers to use all the terms. The researcher anticipated that the graduate
students would make such a suggestion. However, the researcher wanted to validate his
ideas with the graduate students (his peers) prior to finalizing the directions for the
concept mapping activity. Moreover, the direction, allowing participants to use as many
of the trigonometric terms provided, agrees with the methodologies employed by Bolte
(1993) and Howald (1998). Further changes were made to the list o f terms after careful
and consultative analysis with faculty in mathematics education and mathematics. The
number o f terms was reduced from 107 to 89 terms. The idea was to present only
fundamental notions and to leave out the advanced notions (perhaps for a later study of
experienced teachers o f trigonometry).
Other useful insights that emerged from the pilot o f the concept mapping
activities were: (1) hierarchical concept map o f trigonometry is the more difficult one of
the two forms (web-like and hierarchical) to construct, (2) the arrangement o f the terms
on the paper could potentially influence participants and mislead them to intuit a
viewpoint that was not intended, and (3) participants wanted the freedom to augment the
list with terms and ideas o f their own. The difficulties associated with the construction of
81
hierarchical concept maps for trigonometry, such as difficulties with constructing super­
ordinate and subordinate relationships, are indicative o f the complexity and nebulousness
of advanced high school mathematics content. Furthermore, the difficulty revealed that
the state of knowledge o f advanced high school mathematics topics might not be as
thorough and coherent in participant’s minds as we would like to believe. Similar results
have been shown with elementary school mathematics (Ma, 1999).
Card sorts
The graduate students felt that a group discussion o f the card sort would serve me
better than having them individually complete the task. Moreover, the time constraint
under which we were working necessitated a modification o f the methodology. Thus the
four of them discussed the card sorts and suggested that participants first complete the
task that required sorting the cards into piles, and then complete the sequencing task. This
suggestion was incorporated into the main study. The pilot group also suggested a
reduction in the number o f cards and that suggestion was heeded as well in the main
study.
The graduate students expressed concern about the clarity, scope and verbiage on
some of the cards. To address those concerns the researcher sought inputs from two
mathematics education professors and a mathematics professor. The cards that were used
in the main study reflect the changes and modifications that were suggested for the initial
instruments. For example, the task of sorting propositions into truth piles was restricted to
considerations of the six basic (fundamental) trigonometric functions: sin(x), cos(x),
tan(x), csc(x), sec(x), and cot(x).
The graduate students’ discussions were primarily focused on the logistics of
implementing the card sorts in the main study. Thus the researcher could not ascertain,
one way or the other, the graduate students’ pedagogical content knowledge or curricular
knowledge of trigonometry in a high school mathematics curriculum.
82
Test o f trigonometric knowledge
Two o f the four graduate students (G1 and G4) completed pilot versions (see
appendix J) o f the test o f trigonometric knowledge. G1 and G4 completed forms B and C
of the pilot versions. The other two graduate students reported not having time to take the
test. The following discussions on the results o f the pilot test o f trigonometric knowledge
reflect the results from the two graduate students. G1 and G4 spent about 2 hours each on
31 questions. Correct responses to questions received three points. G1 and G4 found the
questions on the test o f trigonometric knowledge to be difficult and removed from their
recent mathematical experience. G1 scored 51 out o f the 93 possible points. G4 scored 48
out o f the 93 possible points. The graduate students did best on the definitional items and
factual items, and less well on the proof items and the problem-solving items.
The results from the graduate students on the test o f trigonometry necessitated
further piloting o f the instrument because the graduate students minimal contact with
trigonometry presented a confounding issue for any meaningful analysis or claim. A
thirty-one item test was constructed from items from the pilot versions o f the test of
trigonometry. The instrument was given to seven experienced inservice high school
mathematics teachers and one student teacher. The minimum number o f years of teaching
for the group was three years and the maximum was thirty years. The teacher with the
three years of teaching experience had taught Honors Precalculus for two years. The
experienced teachers and the student teacher were asked to use a four-point rating rubric
to predict how well preservice teachers would do on each item o f the test o f trigonometric
knowledge (see table 6). An item rating of zero means that no preservice teachers will be
able to correctly answer that item. A rating of one means that few preservice teachers will
be able to correctly answer that item.
A rating o f two means that many preservice teachers should correctly answer that
item. A rating of three means that most to all preservice teachers should correctly answer
that item. They were also asked to provide feedback on the phrasing o f the items on the
83
test of trigonometric knowledge and to suggest new phrasing if they deemed it necessary
to insure clarity.
Table 6. Difficulty Ratings o f the items on a pre-version o f the test o f trigonometric
knowledge by 7 experienced teachers and one student teacher.
Teachers
TI
T4
T5
T6
T7*
T8
2
3
1
3
2
2
2
0
2
1
1
3
0
3
3
2.5
2
2.5
2
1
2.5
2.5
3
1.5
2
3
3
2.5
3
1.5
1.5
0.5
3
0.5
3
2
2.5
1
2
1
2.5
1.5
1.5
2
3
3
2
2
3
2
3
2
2
3
2
3
3
3
2
3
2
3
2
3
2
3
2
2
2
3
2
2
2
2
2
3
3
2
2
3
2
1
1
2
2
2
2
3
3
2
1
1
2
0
3
1
3
2
2
2
2
2
2
2
0
2
3
3
2
2
3
3
2
2
3
2
2
3
3
2
2
3
2
2
2
3
3
2
3
3
2
3
2
3
3
2
3
3
2
1
2
2
3
2
2
2
2
2
2
1
2
2
2
1
2
1
2
1
2
2
3
2
3
1
2
2
2
2
3
3
2
2
3
3
3
3
3
3
2
3
3
3
3
3
3
2
3
3
2
2
3
2
2
3
2
3
2
2
2
3
2.8
1.9
1.6
2.8
2.4
2
2.2
2.4
2.5
1.8
2.5
2.8
2.6
2.2
2.5
1.6
2.1
1.4
2.9
1.5
2.6
2.3
2.4
1.9
2.3
1.7
2.2
1.9
1.8
1.9
T2
T3
M ean Rating
Items
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
3
3
1
1
3
2
1
2
2
2
1
2
3
3
2
2
1
2
0.5
3
3
2
3
0
3
2
3
3
3
3
2
3
3
2
2
3
1
2
Total
39.5
66
65
75
60
78
60
81
M ean o f
Means**
Mean
j qg***
2.1
2.1
2.42
1.94
2.52
1.94
2.61
2.2
* T7 represents the student teacher , ** The average o f all 8 participants
***T1 did not complete the rating o f items 21 - 3 1 , 1.98 = 39.5 •+ 20
84
Furthermore, the inservice teachers were asked to provide information on the
prerequisite knowledge required for correct resolution o f each item. Three o f the seven
inservice teachers worked out the problems and provided solutions. The remaining four
inservice teachers browsed the items and provided the requested information without
actually providing solutions to the items.
The seven experienced teachers and one student teacher gave an average item
rating of 2.2 (see table 6) to the thirty-one items on the test. That translated into a
consensus that many preservice secondary mathematics teachers ought to be able to
answer most o f the questions posed in the pilot test of trigonometric knowledge. If the
ratings were interpreted as attainable scores on the items o f the test, then the results show
that the minimum predicted score would have been 60 out o f a possible 93 points, and a
maximum o f 81 out o f 93 points.
Instrument modifications based on pilot testing
In conclusion, the pilot testing o f the tasks and instruments resulted in a number
of changes and modifications. The direction of the second concept-mapping task was
modified to allow participants to use as many terms as they could and also to augment the
list provided if they wanted to. Furthermore the number o f terms on the list was reduced
from 107 to 89 by removing ideas and terms such as hyperbolic, angular velocity, linear
velocity, (sin;c)/.x. The order o f the card sorts was switched. So the placement o f
proposition into three piles of always true, sometimes true, and never true was placed
ahead of the sequencing activity. The test of trigonometric knowledge was reduced from
31 items to 19 items. The set of 19 items was further reduced to 17 items in the main
study, due to feasibility issues that arose at the time o f testing. The researcher had
anticipated having more time (about 2.5 hours) for the study participants to complete the
instruments, but housekeeping tasks such as returning exams, assigning required readings
and answering students’ questions consumed some 30 minutes o f the class time.
85
Phase one results: Concept maps, card sorts, and test of
trigonometric knowledge
Concept maps (CM1 and CM2)
The 14 study participants generated the terms and ideas they used in their first
concept maps. This allowed the participants to draw from their native understanding and
to organize trigonometric ideas in a high school mathematics curriculum from their
perspectives. The resulting concept maps were idiosyncratic and followed varied formats.
Some participants used a hierarchical format and others used a web-like format. The
participants emphasized and stressed different aspects o f trigonometry. Some emphasized
the functional approach, some emphasized the right triangle approach and others implied
that trigonometry was about measuring and using angles.
The results o f the first concept map are presented in table 7. The columns marked
by m 1 and m2 represent the scores from concept map one and concept map two,
respectively. In general, the participants’ second concept maps (where a list o f terms was
provided) contained more items (also see figure 4), higher number o f endpoints, more
open chains, higher ratio scores, and higher composite scores.
Figure 4. Comparison between participants’ item scores in concept map 1 and concept
map 2
The stream scores from the second concept maps were slightly lower than the
stream scores obtained in the first concept maps (see figure 5). In concept map one, one
participant had a zero stream score, an outlier when compared to the other 13
86
participants. In concept map two, one participant had nine streams, an outlier when
compared to the other 13 participants. Lower stream scores are logical because the ratio
scores depend on the stream scores. And the ratio scores are higher for the second
concept maps (see figure 6). One participant scored a ratio score o f 16 in concept map
one, which was significantly different from the other 13 ratio scores. Both closed chains
and cross-links stayed about the same. The 14 participants generated a total of 65 terms
and ideas that were used to construct their concept maps. That number represents 73% of
the number o f items (89) provided in the second concept-mapping task.
Two participants produced concept maps that were not integrated as shown by
their ratio scores o f zero (also see appendix K). The second concept maps o f these two
participants were made up o f disjointed clusters of concepts and ideas. Nine o f the 14
participants had cross-link scores that were zero for at least one o f their concept maps. As
discussed in chapter III, cross-links are indicative of connected and rich knowledge of a
trigonometry. So by the foregoing account, these preservice teachers do not seem to
possess a rich and connected knowledge of trigonometry.
Figure 5. Comparison o f the stream scores from concept map 1 and concept map 2
87
Table 7. Phase one participants’ performance scores from CM1 and CM2.
p
item
Connections
Groupings
Tallies
stream
endpoin
t
open
closed
31
2
28
18
12
18
0
2
24
0
14
0
6
31
1
3
9
15
0
15
0
18
3
6
0
0
SI
9
16
10
22
2
5
13
6
16
24
20
8
3
8
18
*AX
*n m
*ES
*LN
*AB
IB
ZN
AD
Al
El
PM
13
2
44
21
48
32
20
20
31
26
30
64
10
31
1
4
0
3
3
3
3
2
3
4
3
4
31
2
4
0
4
2
6
9
4
2
3
0
3
31
1
8
3
16
13
0
12
9
6
0
0
10
CT
28
35
4
3
IA
SY
28
21
46
39
3
3
23
33
3
SI
22
10
25
15
5
16
14
21
16
47
6
31
I
14
4
22
26
0
20
14
16
0
0
0
m
I
26
19
33
26
6
19
31
14
30
22
18
ratio
cross
15
6
21
21
9
0
6
0
0
0
0
31
1
20
0
0
16
0
4
0
0
16
0
12
31
2
8
0
4
20
0
0
0
0
8
28
0
31
i
7
0
11
9
2
6
16
5
8
7
5
21
3
0
0
7
18
9
28
12
Means
14
8
11
33
15
8
16
12
12
9
7
7
composite
m2
mi
m2
45
16
53
55
15
36
41
25
22
0
22
73
34
62
74
14
18
44
21
38
75
12
30
41
9
7
11
0
12
16
3.3
2.2
7.8
13
10
0
3.3
11.
7
23
8
50
42
79
79
7
8.7
32
47
Note: m l and m2 represent scores from concept map 1 and concept map 2, respectively.
Column P contains the pseudonyms o f the participants.
*Phase 2 participants
Figure 6. Comparison o f the ratio scores from concept 1 and concept map 2
88
The composite scores for the second concept maps are higher than those obtained
in concept map 1 (also see figure 7). As previously discussed, the composite scores are
sums of the endpoint scores, open chain scores, closed chain scores and the cross-link
scores. Endpoints and open chain scores increased in the second concept maps, while the
closed chain scores and cross-link scores stayed about the same.
Figure 7. Comparison o f composite scores from concept map 1 and concept map 2
Further analysis o f concept map 1 and concept map 2
The following presentation discusses the emergent themes and dimensions from
the qualitative analysis o f the two concept maps. The themes were generated via constant
comparisons of the participants’ responses and content analysis o f the terms/items
89
included in the participants’ concept maps. Terms and ideas were provided for the study
participants to use in concept map two.
Correct definitions, examples, and relationships
In CM1, the participants generated correct definitions o f sine, cosine, and
tangents, and came up with the Pythagorean identities. There were mentions o f the law o f
sines or the law o f cosines, but they were not defined explicitly. In CM2, participants
used the terms sine and cosine from the provided list, but they did not take the extra step
of generating explicit definitions.
Mention of sinusoids and their transformations
Two participants in CM1 generated and used terms and ideas such as amplitude,
phase-shift, stretch, shrink, and vertical shift. In CM2 on the other hand, 4 participants
used amplitude in their concept maps, six used phase-shift and stretch, 5 used shrink and
frequency, and 7 used periodic in their concept maps. It is obvious that more participants
used these concepts and ideas because they had ready access to them since they were
provided in the list. Qualitative analysis o f the use of these terms and ideas, in both CM 1
and CM2, revealed that they were mentioned in relation to graphs of trigonometric
functions and not in relation to the application o f sinusoids in resolving periodic
phenomena.
Mention of applications o f trigonometry
In CM1, five participants mentioned solving triangles: finding missing angle
measures and side lengths as an application trigonometry. No other application o f
trigonometry was presented. However, in CM2, participants did not use the applications
of trigonometry; not even the use of trigonometry in solving triangles even though these
concepts and ideas were provided in the list that they received.
90
Mention of radian measure
Eleven participants mentioned radian measure in CM1 as did 9 participants in
CM2 as another way o f measuring sizes o f angles different from the degree measure. One
participant gave the conversion formula between radians and degrees in concept map 1.
However, none o f the participants gave a definition o f the radian measure, not even in
CM2. The list provided for CM2 had terms that could have been used to define the radian
measure.
Unused concepts and ideas
It is noteworthy to mention some of the terms that were left mostly unused in
CM2. Recall that participants were provided with list o f 89 terms/ideas in CM2. The
absence o f the following terms from any of the CM2 concept maps may indicate weak
knowledge o f some o f the foundations of high school trigonometry. Some of the
application terms/ideas that were not used were bearing, direction, angle o f elevation,
and angle o f depression. Some angular terms/ideas that were consistently underutilized
were quadrantal angles, reference angles, argument, and initial side. The function ideas
that were underutilized were y = x line, sinusoidal functions, odd and even functions,
domain, rate o f change, image, composition, and range. W heny = x line was mentioned
it was mentioned in terms o f graphing the line y = x line or as the transformation (reflect
overy = x line) but never as the transformation that yields inverses o f functions or
relations.
Misconceptions in concept map 1 and concept map 2
The following misconceptions were noticed in CM1. Some proclaimed false
identities are sin2 x + cos2 a = tan2 x , cos2x = 1/2 + (1/2) sinx cos x , and 1 - sec2x = tan2x .
Ten of the fourteen participants wrote incorrect inverse relations or linked functions
without any stated relationships. Five of those ten participants claimed that reciprocal
functions are inverses o f one another. Five other participants connected sine, cosine, and
91
tangent with the functions cosecant, secant, and cotangent but did not specify the
pertinent relationship. In CM2 one participant included sine, cosine, and tangent in a
cluster; and secant, cosecant, and cotangent in another cluster, and then related the two
clusters with the connective cofunctions. Another participant used both cofunctions and
inverses for the same two clusters. Five participants linked the following pairs: sin
esc, cos
sec, and tan ■*->cot with the connective inverse. Another participant formed
inverse pairs as follows sin
sec, cos
esc, and tan <-> cot. Two participants formed
the following linkages sin ** esc, cos <-> sec, and tan ^ cot but did not specify any
relations. Yet another participant claimed that the arc function o f / i s the reciprocal o f the
function f. For example, ArcCosine was claimed to be the reciprocal o f cosine. In
summary eleven of the fourteen participants displayed errors that suggest a weak
knowledge of inverse trigonometric functions.
Card sort 1
The study participants received fifteen propositions (see appendix C) and three
labels each on a separate 4-inch by 6-inch card. The purpose o f this card sort was for the
participants to correctly place each o f these fifteen propositions into one o f three piles:
Always True, True Sometimes, and Never True. To better focus the study and gain
meaningful insight into the participants’ understanding o f fundamental high school level
trigonometry, the propositions were restricted to the six basic trigonometric functions:
sine, tangent, secant, cosine, cotangent, and cosecant. Table 8 shows the correct
classification for each proposition, the responses o f the participants, and the number of
correct responses given by the participants. For example, AX incorrectly classified
propositions 4, 5, 8, 11, 12 as always true. She also incorrectly classified propositions 9
and 13 as never true. A X ’s 6 points in card sort 1 came from her accurate classification of
six of the nine true-sometimes propositions. The number o f correct responses ranged
from 5 to 10 out o f 15 possible.
92
Table 8. Correct classification for each o f the fifteen propositions o f card sort 1,
participant responses, and the number o f correct responses by participant
Classification
A T (Always True)
TS (True Sometimes)
N T (Never
True)
1,3,1 3
2, 5, 6, 7, 9 , 1 0 , 1 1 , 1 4 , 1 5
4, 8 , 1 2
AX*
NM*
4, 5, 8, 11, 12
1 , 2 ,3 , 11
9, 13
8
6
10
ES*
LN*
AB*
IB
8,
9,
8,
3,
9
9
6
5
ZN
AD
Al
El
1 ,2, 5 ,7 , 13, 15
1, 3, 8, 11, 15
2, 4 , 5 , 7 , 10, 11
1 ,2 , 4 , 5 , 6 , 7, 8, 10, 11,
13, 15
2 ,8 , 11, 13, 15
1 ,2 , 3, 4, 9, 10,11
1 , 2 , 3 , 5 , 10, 11, 12, 15
5, 11, 13, 15
1,2 , 3, 6, 7, 10, 14, 15
4 , 5 , 6 , 7, 9, 10, 12, 13, 14,
15
3 ,4 , 6, 9, 10, 11, 14
2, 4 , 5 , 6 , 7, 10, 13, 14
1 , 3 ,6 , 9, 13, 14, 15
9, 14
5 ,9
8, 13, 14, 15
8
5
6
7
7
PM
CT
1 ,4 , 5, 10, 13, 14, 15
1 ,5 ,1 1
3, 8
8
8
IA
SY
1 , 2 , 3 , 7 , 8 , 10, 11, 13, 15
1 , 5 ,8 , 11, 13, 15
1 ,3 ,4 , 6, 7, 10, 12, 14
5 ,6 , 7, 12
4, 6, 7, 9, 13, 14
1 ,2 , 3, 4, 6, 7 , 8 , 9 , 10, 12,
14
2, 6, 7, 9 ,1 1 ,1 2
2 , 3 , 4 , 6, 7 , 8 , 9 , 10, 12, 13,
14, 15
5 ,6 , 9, 14
2, 4, 6, 7, 9, 10, 14
4, 12
3, 12
9
9
Ke y
N um ber Correct
Name
12
12
12
12
Note: The names o f the participants have been replaced by pseudonyms
* Represents participants that were interviewed in the second phase o f the study. The
bolded numbers represent the key.
The next discussion o f results centers on emergent themes and profiles from
participants’ responses in card sort one. The discussion focuses on the propositions that
the majority o f the participants classified incorrectly.
Emergent themes for card sort 1
Table 9 shows that majority o f the fourteen participants classified propositions 2,
3, & 4 incorrectly. These three propositions involve notions o f periodicity o f the six basic
trigonometric functions. Thus it appears that the participants lack a thorough
understanding o f periodicity and how that notion is used in the study o f trigonometry.
93
Four o f the fourteen participants correctly classified proposition 5 as true
sometimes. The fifth proposition (given triangles o f sides a, b, and c the trigonometric
functions are ratios o f the lengths o f two o f the sides) was intended to assess the
participants’ understanding o f the implied conditions in the definitions o f the six basic
trigonometric functions. This result indicates a lack o f understanding that the
trigonometric ratios are ratios of the lengths of two sides o f a right triangle. Thus, perhaps
the recall o f SOFICAHTOA (the mnemonic for sine is the opposite over the hypotenuse,
cosine is the adjacent over the hypotenuse, and tangent is the opposite over the adjacent)
may not adequately explain how the participants understand the conditions under which
SOHCAHTOA can be applied.
Table 9. The placements of the propositions into the three piles (AT, TS, NT) by the 14
participants
N um ber o f participants placing propositions in the piles
Propositions
1
2
3
4
5
6
7
8
9
10
11
12
13
14
■
AT
TS
NT
10*
4
6*
6
8
0
0
3
1*
1
0
0
8
5*
5
9
1
4
6
1
6
12
1
4*
13*
10*
i
10*
8*
2*
1
5
7*
1
12*
9
4*
6*
3
0
0
6*
2
1
1
* Represent the number o f participants out of the pool o f 14 participants that placed that
particular proposition in the correct pile.
Note: Highlighted propositions indicate those questions for which there are more
incorrect placements than correct placements.
94
Proposition 8 highlights the difference between radian and degree measure. Only
six of the fourteen participants were able to correctly classify the proposition (one radian
is equal to 180 degrees) as never true. This lack of understanding is troubling since
radian measure is a versatile and useful way o f measuring angles because o f its non­
dimensionality (Radian measure is a ratio o f two distances: the length o f the arc with
central angle
0
o f a circle, and the radius o f the circle. Thus the units o f distance cancel
out and the radian measure is rendered dimension free).
The results for proposition 11 (the inverses o f trigonometric functions are
functions) are very much o f concern. As table 9 shows only 2 participants classified this
item correctly. This result agrees with earlier studies (Even, 1989, Bolte, 1993, Howald,
1998) that preservice and inservice teachers have difficulties with inverse function ideas.
The lack of understanding o f inverse trigonometric functions is most troubling because o f
the intricacies that are involved in the pedagogy o f this concept. This result may also be
symptomatic of the participants’ limited understanding o f the following foundational
concepts and ideas needed for meaningfully teaching inverse trigonometric functions: 1-1
ness, domain restrictions that yield 1-1 functions, principal values that arise due to the
restrictions, and thus the need to resort to the periodicity o f the trigonometric functions in
order to generate other values. An equally disappointing finding is that only 6 participants
correctly classified proposition 12 (for a trigonometric function there are situations when
a particular domain value has two range values) as never true. Proposition 12 was used
to assess the participants’ knowledge of what a functions is. Again, this result supports
the findings o f Even (1989), Bolte (1993), and Howald (1998).
The intent o f proposition 15 (inverse trigonometric functions yield angle
measures) was to gauge the participants’ breadth o f knowledge about the utility of
trigonometric functions. The applications o f trigonometry go beyond just angular notions.
Even at the high school level, the level of focus for this study, applications of
trigonometry involve situations where the argument is a time component; albeit one can
95
relate the units of time to angular measurements, say in the case o f modeling circular
motions. So if one were to accept angular notions as sufficiently adequate for the delivery
of trigonometric concepts and ideas in the classroom, then one would combine the
Always True and True Sometimes categories for this proposition. In that case, one could
conclude that the participants possess a working understanding of the inverse
trigonometric functions. As table 8 shows, thirteen of 14 participants placed proposition
15 in either o f Always True or True Sometimes categories. In general though, the inverse
trigonometric functions yield real numbers, if we restrict the study to only real number
arguments, which is the case at the high school level.
So what do preservice secondary mathematics understand about trigonometry
based on the results from card sort one? They seem to understand that the six basic
trigonometric functions are periodic (10 o f 14 participants correctly classified proposition
1), that the general theorem o f Pythagoras a 2 + b2 = c 1 (with a, and b representing the
sides of a right triangle and c the hypotenuse) applies to right triangles (13 o f 14
participants correctly classified proposition 6), that graphs o f trigonometric functions are
sometimes sinusoidal (10 o f 14 participants correctly classified proposition 7), although
most of them were not sure what sinusoidal means, that sin2(0) + cos2(50) = 1 is
sometimes true (10 of 14 participants correctly classified proposition 9), and that if a
phenomenon is periodic, that does not imply that the graph o f that phenomenon is going
to be one o f the graphs o f the six basic trigonometric functions (12 o f 14 participants
correctly classified proposition 14 as true sometimes). The aforementioned claims of
understanding shown by the participants assume that correct reasons were used to support
the classifications. However an analysis o f the reasons the participants provided for their
classifications shows that the attribution of understanding might be overly generous (see
appendix L).
Appendix L contains the reasons provided by the fourteen participants for the
classifications of the eight propositions that majority o f the participants classified
96
incorrectly. Participants tended to give surface reasons that did not reflect deep structural
understanding of high school level trigonometric topics. There were also times when the
reason given included mathematically incorrect claims. There were at least 22 Null
Reasons given, such as 1) No stated reasons, “guess” as a strategy, or “not sure” in the
AT category, 25 Null Reasons in the TS category, and 5 Null Reasons in the NT category
(see appendix L). This suggests perhaps that the participants often resorted to guessing to
complete many o f their propositions. Ideally, if all participants in phase one had correctly
placed the fifteen propositions in the piles in which they belong, there would have been
fourteen correct classifications and fourteen correct reasons for each o f the fifteen
propositions. There were however no correct reasons given for the four correct
classifications for proposition fifteen (see table 10). Four o f the six correct classifications
for proposition twelve were supported with correct reasons. There were two correct
classifications for proposition eleven, but only one was correctly supported. Four o f the
six correct classifications for proposition eight had supporting reasons. The sole correct
classification o f proposition four was not supported with a reason. There were no
compelling conceptual reasons provided for the five correct classifications of proposition
three (see appendix L). The implications o f the participants’ lack of supporting reasons
are discussed later.
Table 10. Number of participants that provided valid reasons for selected propositions
Pronosition
15
12
11
8
5
4
3
2
No Reason
6
8
5
6
5
10
7
5
Placed Correctly
4
6
2
6
4
1
5
6
Valid Reason
0
4
1
4
3
0
0
2
97
Card sort 2
Card sort two involved ordering thirty-four cards into an instructional sequence
that the participants thought would make pedagogical sense for students in high school to
experience. The concomitant objective was for the participants to consider the role of
prerequisite knowledge. The notion o f prerequisite integrity is fundamental to
understanding why the study de-emphasized the use o f calculators and focused instead on
conceptual notions that underlie even the simplest task o f determining the measures of
missing angles in right triangles. The study was not oriented towards determining what
calculator can do, but rather what the participants understood about high school level
trigonometry. If we assume that the teacher is the most important variable in the
classroom, then what they know and how they can connect what they know to help
unpack the mathematics students are to learn is fundamental. This is not to argue,
however, that teachers cannot use, say the inverse trigonometric functions keys on the
calculator before formally teaching inverse trigonometric functions.
The argument is that teachers ought to understand that knowledge of inverse
trigonometric functions is prerequisite knowledge to using inverse trigonometric
functions. Figure 8 shows the demarcation of trigonometric terms into early,
intermediate, and advanced topics. Hirsch and Schoen (1990), and Senk et al. (1998)
were used as guides for deciding which topics should be considered early, intermediate or
advanced topics. The researcher also drew from his high school mathematics teaching
experiences, having taught courses such as algebra, algebra two, pre-calculus, and
advanced placement calculus, to guide the demarcation o f topics into early, intermediate
and advanced categories. The researcher’s experiences also influenced the order of the
sequence shown as well.
On average, four of the nine Early Concepts and Ideas cards were accurately
identified by the participants as such. That represents a 45% agreement with Hirsch and
Schoen (1990) and Senk, et al. (1998). For the Intermediate Concepts and Ideas, there
98
was 50% agreement. That is, participants on average correctly identified six out of the
twelve cards that were classified as Intermediate Concepts and Ideas. And finally,
participants correctly identified four of the thirteen Advanced Concepts and Ideas cards.
Sequencing of Topics of Trigonom etry (Card Sort 2): See Hirsch & Schoen ( 1 9 9 0 ) , Trig onometry an d its
a p p lic a tio n s ; Senk a t al. ( 1 9 9 8 ) , Functions, Statistiscs, an d T rig ono m e try (UCSMP In te g r a t e d M athem atics)
AX*
Participants
Early Concepts
& Ideas
Cards 1 — 9
In term e d iate
Concepts &
Ideas
Cards 11 13. 15 - 23
Advanced
Concepts &
Ideas
IB
ZN
A D
A I
El
PM
CT
IA
SY
Degree
Sim ilar Riqht Trianqles
Coordinate
P la n e /A n g le s /R o ta tio n s /Q u
adrantal A ngles/Coterm inal
A n a le s
5, 1,
2, 3,
Pythaqorean Theorem
8, 4,
1
2
Six Basic Trigonom etric
R atios
Special Angles/Reference
Angles/Trig Ratios of these
an ales
C o m p le m e n ta ry /S upplem en
ta r v /C o - fu n c tio n s
Radian
M e asure /A rcle n g th /A n g u la r
- Linear Velocity
T rigono m etric Ide ntities
15,
(Reciprocal, Quotient,
17,
P v th a o o re a n l
18,
18,
C ircular Functions
14,
23,
13,
Unit Circle
16,
20
14,
P e rio d ic ity
20
S in u s o id s
G ra p h s /D o m a in /R a n g e /A s y
m p to te s /F re q u e n c y /1 1/O dd-E ven Functions
2,
2,
2, 1,
3,
3, 1 3, 8
4, 8
16,
15,
21,
22,
11,
12,
18,
17
12,
13,
19,
14,
15,
16,
18,
17,
23
28,
24,
34,
25,
26,
32
24,
31,
25,
26,
30,
29,
32
5,
3,
1,
3,
3,
2,
1,
5,
2,
5,
8,
2,
2,
2, 1, 3 1, 4
3, 4, 3
3,
4, 7
1, 4
5, 8
4, 1 1, 4
20,
13,
14,
15,
18,
16,
19,
17
19,
22,
18,
13,
12,
11,
14
14,
15,
12,
11,
13
30,
28
24,
30,
31,
32,
28
29,
30,
28,
24
12,
14,
19,
13
12,
16,
20,
14,
19,
11,
13
13,
14,
15,
17,
18,
11,
12,
22
14,
15,
16,
17,
18
29,
30,
24
24,
28,
30,
32
29,
24,
30,
28
29,
30,
28,
24
3,
1,
8,
7,
4, 2
14,
15,
16,
19,
20
13,
20,
12,
11,
21,
15,
17,
16,
18,
22
14,
11,
19,
20,
15,
16
29
34,
28,
29,
25,
26,
30,
24
28,
30,
29,
24
Inverse Trig F u n c tio n s // =
x line/P rincipal Values
T riq o n o m e tric Equations
Triangles &
Trigonom etry/Law of
Cosine/Law of
Sines/S olvino Trianales
S u m /D iffe re n c e /D o u b le /H a lf
/Product from
Sum/Difference/Sum from
Half fDifference or Suml
Cards 10, 14,
24 - 34
N M * ES* L N* A B *
Polar Equations
30,
34,
33,
28,
32
24,
28,
32
Vectors & T riqono m etry
Complex Numbers and
T ria o n o m e trv
Circular Functions and
S eries
Figure 8. Sequence agreement with Hirsch & Schoen (1990) and Senk et al. (1998)
99
That represents 31 % agreement with Hirsch and Schoen, and Senk, et al. The results
show that participants had the most difficulty with the advanced topics. Overall, there
was a minimum agreement o f 9 out of 34, and a maximum of 21 out o f 34. The results of
the analysis o f prerequisite integrity are discussed later with the phase two results of the 5
interviewees. Those results will show that these preservice teachers possessed limited
knowledge o f prerequisite knowledge for trigonometric concepts. Furthermore, their
pedagogical content knowledge was lacking.
Test of trigonometric knowledge
This section first presents the overall performances o f the 14 participants on the
test of trigonometric knowledge. That is followed by item performance results for each of
the 17 items that made up the test o f trigonometric knowledge. After that, some salient
features of the performances o f the participants are presented, and the section ends with a
summary o f the types o f insights that were gleaned from the test results. Recall from
chapter III that the test o f trigonometric knowledge was analyzed both quantitatively and
qualitatively. The quantitative aspects o f the analysis involved scoring the 17 items for
correctness. Each item had a possible score, ranging between 3 and 9 points (see table 11)
with partial credit given using a holistic scoring rubric (see figure 9) to score the
problem-solving tasks, definitional tasks, and proof-writing tasks on the test. Detailed
information about how each item was scored is provided with the presentation of results
for each item.
Descriptive statistics of mean, mode, maximum, and minimum were used to
analyze the scores obtained by the participants (see table 12). Further qualitative analysis
was conducted to elucidate discemable patterns of response and error patterns. These
patterns were categorized into emergent themes o f responses o f the preservice teachers.
100
3
2
1
0
All procedures and solutions are accurate, com plete and appropriate. Theorem s, definitions, and all
conventions are spelled out correctly and used appropriately
M inor errors in at m ost one part o f the solution process (could be definitional, related to a theorem,
conventions: In a triangle if one o f the angles is labeled A, then the side opposite that angle is
labeled a; algebra, or the final answer
Serious and m ajor errors in process, solution and/or m ultiple algebra m istakes, or understanding
and use o f definitions, theorem s, or conventions, but shows an understanding o f the question
Did not understand the problem , solution process is irrelevant to the question asked, or
inappropriate process or solution w as provided. Lack o f understanding in use o f theorem s,
definitions, or conventions______________________________________________________________________
Figure 9. A holistic model for scoring free-response items on the test of trigonometric
knowledge
Table 11. Participants’ scores on the 17 items of the test of trigonometric knowledge
Item N um bers
Item
V alue
1 2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
3
3
3
3
3
3
3
4
3
3
6
3
3
6
3
9
3
Participants’ Item Scores
N ame
Total
Score
Percent
Correct
AX*
3
3
1
0
3
3
1
2
0
1
1
0
1
0
2
2
0
23
36
NM*
3
2
3
1
1
3
1
3
4
2
2
6
3
1
6
2
3
46
72
ES*
3
3
3
0
3
3
1
2
3
3
2
6
3
3
1
0
9
48
75
LN*
3
2
0
0
3
1
1
2
4
3
1
6
3
3
5
0
0
37
58
AB*
0
1
0
0
1
1
1
0
2
1
1
3
1
1
1
1
9
24
38
IB
3
2
0
0
3
3
1
3
2
1
0
6
1
1
4
1
0
31
48
ZN
3
2
0
0
1
0
1
0
1
1
1
3
1
0
0
0
6
20
31
AD
3
0
3
2
3
3
2
3
2
1
0
6
0
3
2
1
3
37
58
Al
3
1
0
0
0
1
2
2
1
1
1
3
1
3
5
2
3
29
45
El
3
2
0
0
3
1
0
1
1
1
1
0
1
0
1
3
0
18
28
PM
3
3
0
1
3
3
1
1
3
1
0
6
3
1
2
3
0
34
53
CT
3
1
0
0
3
0
1
0
1
1
1
6
1
0
2
0
0
20
31
IA
3
3
0
0
3
3
2
0
4
3
3
0
1
0
0
0
0
25
39
SY
3
2
0
0
3
1
0
1
3
1
0
3
1
3
2
2
0
25
39
* Interview participants
101
Table 12. Test o f trigonometric knowledge item statistics
TTK Item L,ow Item Score H ish Item Score M ode Item Score Total Score on Ite:m Item M ean 1tern
3
39
2.79
1
0
3
2
27
1.93
3
2
0
0
10
.71
0
3
3
4
.29
4
0
2
0
3
33
2.36
5
0
3
1.86
0
3
3
26
6
1
15
1.07
7
0
2
2
20
1.43
8
0
3
4
1
31
2.21
9
0
1
21
1.50
10
1
3
14
1.00
0
3
1
11
6
54
3.86
12
0
6
1
21
1.50
0
3
13
0
19
1.36
14
0
3
6
2
33
2.36
15
0
0
17
1.21
16
0
3
0
9
0
33
2.36
17
Value
3
3
3
3
3
3
3
3
4
3
3
6
3
3
6
3
9
The overall mean score across participants was approximately 30, which gives an
approximate mean correct percentage o f 47. The lowest score was 18 and the highest
score was 48 (see table 11). On fifteen o f the items, at least one participant achieved a
perfect score. However, most participants’ scores were lower, with item scores of zeros
and ones (also see table 12).
The following results o f the item performance analyses include descriptions of the
17 items for the purposes of clarifying the discussion of the results. Description of each
item includes difficulty rating, types of representational modalities, and anticipated
success rates are presented in figures 10, 12, 15, 18, 21, 24, 27, 30, 33, 34, 37, 38, 39, 40,
43, 46, and 48. A mathematician assisted with developing the item descriptions during
the pilot phase of the study. Types o f representational modalities are indications of
whether the item required graphical, tabular, symbolic (equation), diagram, or geometric
representations. Anticipated success rates were predictions o f how well the study
participants would perform on the test items. The descriptions are accompanied by results
of qualitative analyses and some sample participants’ responses in figures 10 through 30.
102
Figure 10 contains the description of test item 1 which had a possible score value
of 3. The quadrant system is one o f the foundational concepts that undergird the study of
trigonometry. It is part o f the conventions, definitions, and reference points for sharing
ideas and concepts about trigonometric functions. So knowledge o f the basic idea of
quadrants is a necessary knowledge that teachers need in their repertoire to be effective in
teaching trigonometry.
The only incorrect response for item 1 on the test o f trigonometric knowledge
involved a circle with the following marks 0,
••• ’
(see figure
11). This incorrect response was scored zero points using the holistic rubric (see figure
9). All together the marking on the circle created 16 sectors in the circle. This incorrect
response was focused on demarcating angular rotations in radians, but the demarcations
also included some purported radical values interspersed within the angular measures.
There must have been confusion, on the part of the participant, as to what exactly the
intent of the questions was.
1. W h a t is the co n v e n tio n a l n u m b e r in g o f th e q u a d r a n ts o f a c o o r d in a te p la n e ? D r a w a
p ictu re w ith lab els fo r th e q u a d ra n ts.
C h a r a cteristics o f Q u estio n 1: Low level o f difficulty, G raphical/D iagram m atic,
Convention/Definition, and Factual
F u n d a m e n ta l to the study o f trigonometry: O rganizes the plane into quadrants in reference to
the origin (a point o f reference) and encapsulates the standard position for angles o f rotation,
and conventions for positively and negatively directed angles o f rotation. Furtherm ore, it
encapsulates East, North, W est, and South directions, w hich are very useful in navigation.
P ro jected S u c c e ss R ate: All
Possible Score: 3 points
Figure 10. Description o f item 1
103
Figure 11. The only incorrect response to item 1
Item 2 on the test of trigonometry focused on the knowledge o f the two special
triangles used in trigonometry to find exact range values o f trigonometric functions (see
figure 12). Knowledge of the two special triangles is fundamental to resolving
trigonometric equations and solving triangles without the aid o f electronic computational
or graphical devices. Furthermore, the two special triangles when used in conjunction
with the notions o f reference angles and the behavior of trigonometric functions in the
four quadrants, provide a powerful and effective means o f resolving trigonometric
problems.
104
Four o f 14 participants scored 3 points, another six scored 2 points, three scored 1
point, and one participant got a zero on item 2 o f the test of trigonometric knowledge.
Figures 13 and 14 show samples of participants work on item 2. Seven o f 14 participants
omitted length measures for the sides of the special triangles. Three participants ascribed
3-4-5 length measures to the 30°-60°-90° special triangle. One participant provided no
response to the question. And one participant drew a triangle with 45°-45°-45° angle
measures.
Figure 13 shows a participant’s response that received a score o f one point. The
response had serious deficiencies. For example, the participant did not give measures of
the lengths o f the sides of the special triangles. The measures o f the lengths of the special
triangles are important because without them it will be extremely difficult to compute the
trigonometric ratios that are vital to determining exact values to trigonometric problems.
Moreover, the angles measures are incorrect. It was inferred that this participant intended
the angle measures to be in degrees. However, the unit of degree measures was missing.
By default then the angle measures should be interpreted as radians. Figure 14, on the
other hand shows the response o f a participant who was cognizant o f the unit of degree
measure, and this participant also provided the measures o f the lengths o f the sides o f the
special triangles.
2. D r a w tw o sp ecial trian g les co m m o n ly u sed in tr ig o n o m e tr y to ca lcu la te e x a c t solutions.
L ab el all a n g les w ith th eir m easu res.
C h a r a cteristics o f Q u e s tio n 2: Low level o f difficulty, Pictorial/D iagram m atic,
C onvention/D efinition, Factual, Geometric
F u n d a m e n ta l to the study o f trigonometry: Instrumental and essential for finding exact values
o f w ithout the aide o f electronic com putational or graphical technologies. U seful for resolving
special obtuse angle (m ultiples o f the special angles.)
P ro jected S u ccess R ate: All
P o ssib le Score: 3 points
Figure 12. Description o f item 2
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Figure 13. Sample response to item 2
Figure 14. Second sample response to item 2
106
Item 3 on the item o f trigonometric knowledge assessed the knowledge o f the law
of sines (see figure 15). Knowledge of the law o f sines was considered fundamental to the
study of high school level trigonometry because it is a useful tool for resolving triangles,
especially, non-right triangles. It is particularly useful in the SSA (side-side-angle) case
where no triangle, one triangle, or two triangles can possibly satisfy the given
information. A related knowledge that is needed for effective application o f the law o f
sines to a triangle, is the convention that ascribes a to the side across from the angle A.
3. S tate th e la w o f sines
C h a r a cteristics o f Q u e s tio n 3: A dvanced level o f difficulty, Pictorial/D iagram m atic,
Theorem , Factual, Geom etric, Symbolic
F u n d a m e n ta l to the study o f trigonometry: Useful for resolution o f triangles
(especially non-right triangles, for w hich the Pythagorean theorem does not hold).
N eed to use the convention that the side across from angle A is labeled a.
P ro jected S u c c e ss R ate: Few
Possible score: 3 points
Figure 15. Description o f item 3
Three o f 14 participants scored 3 points, one scored 1 point, and 10 participants
scored zero on item 3 o f the test of trigonometry. Figures 16 and 17 show two sample
participants’ responses to item 3. Five of 14 participants did not respond to this question.
Another 4 participants gave the definition o f sine o f x as their response. O f those four,
one incorrectly defined the sine o f x as the ratio of the adjacent side to the hypotenuse
(see figure 16). And finally, one participant gave the following response:
fflZ_Ci w / n m / r
= -------= --------(see figure 17). The incorrect response in figure 17 shows that the
a
b
c
participant is aware that the law of sines involves proportions. However, the recalled
facts were not congruent with the intent of the question. So it was scored one point. The
incorrect response in figure 16, on the other hand does not reflect an understanding of the
law of sines. Moreover, it includes a wrong definition of the sine function.
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Figure 16. Sample response to item 3
Figure 17. Second sample response to item 3
108
In item 4 o f the test o f trigonometric knowledge, the study participants grappled
with the definition o f the law o f cosines (see figure 18). This law is the sister law to the
law of sines and taken together, both laws represent powerful tools for the resolution of
triangles. The law o f cosines is also useful in generating proofs o f addition formulas. This
law relates the squared lengths o f the sides o f triangles to the cosine o f one o f the angles
o f the triangle. Again, it assumes the conventional usage o f “a ” for the length of the side
across from the angle A.
4. S tate th e la w o f cosines
C h a r a c te r is tic s o f Q u estio n 4: A dvanced level o f difficulty, Theorem , Factual,
Geometric, Sym bolic
F u n d a m e n ta l to the study o f trigonometry: Useful for resolution o f triangles (especially
non right triangles, for w hich the Pythagorean theorem does not hold).
P ro jected S u c c e ss R ate: Few
Possible score: 3 points
Figure 18. Description o f item 4
One o f 14 participants scored 2 points, two scored 1 point each, and eleven
participants each scored zero points on item 4. Sample participants’ responses are
presented in figures 19 and 20. Five participants had no response to the question. Four
gave the definition o f the cosine o f x with one participant stating incorrectly that the
cosine o f x is the ratio o f the opposite side to the hypotenuse. Three other participants
provided incorrect responses that showed that the participants had some understanding of
the law o f cosines, albeit the facts they recalled were wrong.
The participant whose response is shown in figure 19 displayed some rudimentary
understanding of the law of cosines. This participant attempted to connect the cosine of
angle C to sides a and b. So, this response was scored 1 point for the minimal
understanding shown. Figure 20 shows a much broader understanding o f the law of
cosines because the response shown in this figure has almost all the necessary ingredients
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for an accurate statement o f the law of cosines. The participant’s response depicted in
figure 20 left out the factor of 2 in the product abcosO. Moreover, this participant did not
seem to have followed the convention o f ascribing the label “a ” to the length of the side
across from angle A. As a result, angle 0 is incorrectly placed on the triangle. Theta
should have been enclosed by side a and side b. The score o f 2 points for this response
was in accordance with the holistic rubric (see figure 9).
Figure 19. Sample response to item 4
110
Figure 20. Second sample response to item 4
One o f the fundamental concepts in trigonometry is the concept of angular
rotations. Knowledge o f the conventions of clockwise and counterclockwise rotations
form part o f the prerequisite knowledge base needed for meaningful understanding o f the
behavior of trigonometric functions in the four quadrants o f the plane. Knowledge of
these rotations is also useful in understanding the behaviors o f even and odd functions.
Additionally, knowledge o f the standard position of angles o f rotation is necessary so that
direction and bearing are not confused. The convention of the standard position of the
direction o f an angle o f rotation uses East (the positive horizontal axis) as its initial side,
while conventionally, bearing uses the North Pole (the positive vertical axis) as its initial
side. Knowledge o f these basic concepts allows students of trigonometry to be better
problem solvers. The knowledge of clockwise rotation was assessed in item 5 o f the test
of trigonometric knowledge (see figure 21).
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5. W h a t d oes a n eg a tiv e a n gle m ea su re rep resen t? A s s u m e th a t th e a n gle is in
sta n d a rd p ositio n .
C h a r a c te r is tic s o f Q u estio n 5: Low level o f difficulty, Pictorial/D iagram m atic,
D efinitional, Factual
F u n d a m e n ta l to the study o f trigonometry: U nderstanding the counterclockw ise and
clockw ise conventions, and standard position are essential to understanding
trigonometry. The conventions provide a com m on reference base for angles o f
rotation.
P r o jected S u c c e ss R ate: All
Possible score: 3 points_________________________________________________________
Figure 21. Description o f item 5
Figure 22. Sample response to item 5
Ten o f the 14 participants scored 3 points, three scored 1 point, and one
participant scored zero points on item 5 o f the test of trigonometric knowledge. Sample
participants’ responses are presented in figures 22 and 23. A participant gave
“counterclockwise direction,” two other participants argued that the negative angle
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measures are angles subtracted from 360° and they gave responses similar to the
following: -25° = (360° - 25°) = 335° (see figure 22). The arguments by these two
participants show that they have some understanding o f the conventions o f angular
rotations. However, they confused equal and coterminal angles as having the same
meaning. Thus these responses indicate limited knowledge o f what negative angle
measures represent and the responses were score one point apiece. Yet another
participant argued that negative measures represent angles “in the 2nd or 3rd quadrant”.
Figure 23, on the other hand, represents a correct response to item 3.
Figure 23. Second sample response to item 5
Item 6 of the test o f trigonometric knowledge assessed the study participants’
knowledge of the Pythagorean theorem for trigonometric functions and the ability to
provide justifications for a trigonometric identity (see figure 24). This identity is arguably
the most fundamental o f all trigonometric identities. It is used to derive the other
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Pythagorean identities, it is useful in simplification o f trigonometric expressions, and it
helps in writing proofs for other identities.
2
2
6. P ro v e th a t sin x + cos x = 1 .
C h a r a cteristics o f Q u e s tio n 6: M edium level o f difficulty, Pictorial/D iagram m atic,
Theorem /Identity, Factual, Symbolic
F u n d a m e n ta l to the study o f trigonometry: A fundam ental identity in trigonom etry. This
is the T heorem o f Pythagoras for sine and cosine
P ro jected S u c c e ss R ate: M ost
Possible score: 3 points
Figure 24. Description o f item 6
Seven o f the 14 participants scored 3 points, five scored 1 point, and two
participants scored zero points on item 6 of the test o f trigonometric knowledge. Sample
participants’ responses are presented in figures 25 and 26. The participants that provided
correct proofs used trigonometric ratios o f the sides of right triangles and the theorem of
Pythagoras. Some embedded the right triangle in a unit circle (see figure 26). Two
participants did not respond to this item. Another five attempted but failed. One of the
five that failed presented a “justification” based on the specific case when the argument
equals 45° (see figure 25).
The response presented in figure 25 supports the findings o f proof studies
(Chazan, 1989; Simon & Blume, 1996; Sowder & Harel, 1998) that students tend to use
and accept examples or empirical evidence as proofs. Moreover, the sine and cosine
ratios, used in this question, were incorrectly defined. The participant defined the sine
functions as adjacent over the hypotenuse, and the cosine functions as opposite over the
hypotenuse. Thus the example itself was based on faulty definitions. However, this
participant was using an argument of 45°, so the result was not affected by the mismatch
of the sine and cosine functions because the sine and cosine functions yield the same
value at 45°.
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Figure 25. Sample response to item 6
Figure 26. Second sample response to item 6
Item 7 o f the test of trigonometric knowledge assessed the study participants’
knowledge of radian measure (see figure 27). The radian measure is a powerful and
115
versatile measure o f angles that is not encumbered by unit o f measurement, because it
represents a ratio o f two magnitudes that have the same unit o f measurement. It is widely
used in advanced mathematics. For example, is a more convenient measure to use in
computing lengths o f arcs.
7. a. D efin e th e r a d ia n m e a s u r e o f an angle.
b. S tate th e r e la tio n sh ip b etw een the rad ian m ea su re a n d th e d e g r e e m e a s u r e o f an angle.
[N ote th at p a rt (a) a n d p a rt (b) req u ire d ifferen t an sw ers]
C h a r a c te r is tic s o f Q u estio n 7: M edium level o f difficulty, Pictorial/D iagram m atic,
D efinitional, Factual, Geometric
F u n d a m e n ta l to the study o f trigonometry: A pow erful w ay o f m easuring angles that is not
encum bered by units o f measure that could be difficult to control for in com putations. The
radian m easure, w hich is a ratio, is dim ensionless and as such is greatly facile and useful in
resolving com putational situations.
P r o jected S u c c e ss R ate: Few
Possible score: 3 points
Figure 27. Description o f item 7
Three o f the 14 participants scored 2 points, nine scored 1 point, and two
participants each scored zero points on item 7 of the test of trigonometric knowledge.
Samples o f participants’ responses on item 7 are presented in figures 28 and 29. Eleven
of the fourteen participants gave the correct conversion between radians and degrees.
However, no participant was able to accurately define the radian measure. The definition
that came closest to a true definition was “one radian is the arclength of that angle” (see
figure 29). Participants didn’t express the idea that the radian measure is a ratio of two
lengths: the length o f the arc o f a central angle o of a circle and the radius o f the circle.
But the fact that eleven participants were able to recall the correct conversion between the
radian measure and the degree measure speaks to the instrumental understanding (Skemp,
1978) that might inadvertently be promoted over conceptual understanding in the study of
trigonometry at the high school level.
116
Figure 28. Sample response to item 7
Figure 29. Second sample response to item 7
117
One o f the major organizing tools in trigonometry is the unit circle. It can be used
to encapsulate the trigonometric functions, behaviors o f the trigonometric functions in the
four quadrants (in both clockwise and counterclockwise rotational systems), generate the
trigonometric values o f quadrantal arguments, generate identities, prove identities and
addition formulas, and other uses. Item 8 sought to assess the study participants’
knowledge of the utility o f the unit circle in trigonometry (see figure 30). This item was
scored using the holistic rubric. A correct response to part (a) o f this item received one
point while a reasonable response to part (b) received two points. Part (a) is mostly a
recall question and as such requires lesser cognitive load than part (b), which assessed the
scope of utility and versatility of the unit circle in the study of trigonometry.
8. a. W h a t is a u n it circle?
b. H o w c o u ld y o u u se th e un it circle in tr ig o n o m etry ?
C h a r a cteristics o f Q u e s tio n 8: M edium level o f difficulty, Pictorial/D iagram m atic, Factual,
Geom etric
F u n d a m e n ta l to the study o f trigonometry: Encapsulates the fundam ental ideas o f trigonometry:
A ngle rotation and quadrants, definitions o f the six basic trigonom etric functions, reference
triangles and angles, and the Theorem o f Pythagoras (x 2 + y 2 = 1) w hich is also the equation o f the
circle o f radius 1 centered at the origin. A fundam ental notion that accom panies the unit circle is
that it be centered at the origin to retain its utility in trigonometry.
S u ccess R ate: Few
Possible Score: 3 points
Figure 30. Description o f item 8
Three o f the 14 participants scored 3 points, four scored 2 points, three scored 1
point a piece, and four participants each scored zero points on item 8 o f the test of
trigonometric knowledge. Two sample responses are presented in figures 31 and 32. A
sample response that received a perfect is thus: “A circle with radius = 1; the unit circle,
combined with the Pythagorean theorem is the basis for trig. That is, the fundamental
relationship o f cos2 d + sin20 = 1” (see figure 32) It should be clear from the excerpted
response that the grading on the test o f trigonometry was not the most stringent. Overall,
10 participants gave the correct definition of the unit circle. Flowever, they have weak
118
understanding of the functionality of the unit circle in the study o f trigonometry. They did
not express the encapsulating power o f the unit circle: relative to trigonometric functions
of angles greater than 90°, quadrantal angles, clockwise rotations, periodicity, coterminal
angles, proof o f identities and formulas, and the notions of even and odd functions, to
name just a few. The response depicted in figure 31 was scored one point for correctly
defining the unit circle in part (a). The participant mentioned the trigonometric functions
and identities in this response, accepting that response as correct would have required an
enormous assumption o f understanding that the researcher was not able to make. The
participant whose response is shown in figure 31 ought to have delineated how the unit
circle could be used “to show trigonometric functions and trigonometric identities.”
Figure 31. Sample response to item 8
119
Figure 32. Second sample response to item 8
Item 9 of the test of trigonometric knowledge assessed four trigonometric
concepts: part (a) involved addition formula and algebra o f functions; part (b) assessed
knowledge of identities; part (c) tested the study participants’ knowledge of even and odd
trigonometric functions; and part (d) assessed knowledge o f inverse trigonometric
functions and the relationships between the domains and ranges o f trigonometric
functions and the related inverse functions (see figure 33). The assessed concepts are all
major ideas in trigonometry and are useful ideas to have in one’s repertoire in order to
navigate effectively in trigonometry. Item 9 was not scored using the holistic rubric of
figure 9. Rather, each part was scored zero or one point for a correct response.
120
9. T r u e or F alse: G iv e rea son s
a) sin (a + /3) = sin a + s in p
_________
b) sec"^ x + 1 = t a n ^ x
___________
c) c o s ( - jc ) = COSX
________
d) Sin ^(2) exists
_________
C h a r a cteristics o f Q u estio n 9: A dvanced level o f difficulty, Pictorial/D iagram m atic,
Functional U nderstanding, Factual, G eometric, Symbolic
C o n cep tu al: Requires an understanding o f the domain, range, and behavior o f the six
basic trigonom etric functions and their inverses. Furtherm ore, an understanding o f the
dom ain and range o f com posite functions is also required. N ote that part (d) exists if we
consider com plex, non-real arguments. However, for high school level trigonom etry, nonreal argum ents do no obtain. As such the answ er for part (d) should be false.
P ro jected S u c c e ss R ate: Few
Possible score: 4 points (1 point per part)
Figure 33. Description o f item 9
Three of the 14 participants scored 4 points, three scored 3 points, three scored 2
points, four each scored 1 point, and one participant scored zero points on item 9 of the
test o f trigonometric knowledge. Ten participants gave the correct choice offalse on part
(a), two participants did not respond, and another two participants gave the incorrect
response o f true. On part (b), seven participants gave the correct response offalse, three
participants did not respond, and four gave the incorrect response o f true. Nine
participants answered part (c) correctly, two participants did not respond, and three gave
the incorrect response offalse. There were four non-responders for part (d), five
participants gave the correct response o ffalse and the remaining five gave the incorrect
response o f true. Four o f the five incorrect responses argued that it is true because it
“gives the angle measure whose sine is 2”, and the fifth incorrect response argued that “2
can be an angle measurement in degrees.”
Item 10 was one of three questions that assessed the study participants’ ability to
solve triangles (see figure 34). These triangle resolution problems assessed knowledge of
fundamental triangle trigonometric concepts. Part of the assessment involved whether the
121
participants could introduce auxiliary items that would help simplify the question. For
example, dropping a perpendicular (altitude) to side b would divide triangle ABC into
two triangles (30°-60°-90° & 45°-45°-90°) that are basic to understanding trigonometry.
Thus the altitude simplifies this problem into finding the dimensions o f the two special
triangles. Due to this possible simplification o f the problem, the researcher predicted that
all the study participants ought to be able to resolve this problem.
10. F in d all m issin g sid es an d angles: S h o w all y o u r w ork .
C h a r a cteristics o f Q u estio n 10: M edium level o f difficulty, Pictorial/D iagram m atic,
Geom etric, Factual, Symbolic, Reasoning
A p p lic a tio n o f T rig o n o m e try : Resolution o f triangles.
P ro jected S u c c e ss R ate: All
Possible score: 3 points
Figure 34. Description o f item 10
Three o f the 14 participants each scored 3 points, one scored 2 points, and ten
each scored 1 point on item 10 of the test o f trigonometric test. Sample participants’
responses are presented in figures 35 and 36. Three participants applied the trigonometric
ratios on the non-right triangle that was provided. Two participants were defeated
because they could not come up with sine and cosine o f special triangles (see figure 36).
[J
IT
And another incorrectly argued thus “ J — + J — = 1” (see figure 35). The copy (in figure
35) o f this participant’s work includes the researcher’s marks that crossed out the “equal
to” sign to indicate that the mathematical sentence was not true. That is, the “not equal
122
to” sign in figure 35 came to be when the researcher crossed out the equality to indicate
that the response was incorrect during the grading process.
Figure 35. Sample response to item 10
Figure 36. Second sample response to item 10
123
Item 11 is another question that asked the study participants to solve a triangle
(see figure 37). However, in this item the purported triangle was an illusion because the
dimensions provided do not satisfy any triangle. Dropping a perpendicular (altitude) from
vertex B onto side b creates a 30°-60°-90° triangle with a side-length o f 5 units. This sidelength violates the length o f hypotenuse (4 units) in the second right triangle that resulted
from the introduction o f the altitude. This item was scored using the holistic rubric of
figure 9. Responses that highlighted the non-existence o f the purported triangle and also
gave reasons for the fallacy received three points.
One o f the 14 participants scored 3 points, two scored 2 points, seven scored 1
point, and four participants each scored zero points on item 11 o f the test o f trigonometric
knowledge. Four of the participants did not respond. Another two argued that the
measures of angle B and angle C equal 75°, making the triangle isosceles. Two other
participants applied the trigonometric ratios on the non-right triangle provided in item 11.
11. F in d all m issin g sid es an d angles: S h o w all y o u r w o rk .
C h a r a c te r is tic s o f Q u estio n 11: M edium level o f difficulty, Pictorial/D iagram m atic,
Geom etric, Factual, Reasoning
P r o b le m Solving: Resolution o f triangles (N o solutio n situ a tion )
P r o jected S u ccess R ate: Few
Possible score: 3 points
Figure 37. Description of item 11
Item 12 assessed the fourteen study participants’ knowledge o f sinusoids (see
figure 38). The intent o f the item was for the participants to match two graphs of
sinusoids with their symbolic representations. Knowledge o f the effect o f transformations
(vertical stretch/shrink, horizontal stretch/shrink, vertical translation, horizontal
124
translation (also known as phase shift)) were needed for success in this item. The holistic
rubric of figure 9 was not applied to the scoring of item 12. Each correct match of a graph
with its symbolic representation was scored three points. So correct responses to the two
matches received six points. Incorrect responses were scored zero points.
Figure 38. Description of item 12
Seven o f the 14 participants scored 6 points, four scored 3 points, and three scored
zero points on item 12 o f the test o f trigonometric knowledge. There were seven perfect
scores. Four other participants correctly matched one out o f the two graphs. Two
125
participants did not respond. Four chose option E for the second graph, perhaps due to
confusion about when a sinusoid is stretched horizontally and when it is shrunk
horizontally.
Item 13 was a re-couched 30°-60°-90° special triangle. The dimensions provided
have been scaled by a factor o f 1/ V3 (see figure 39). The intent o f the item was for the
study participants to identify the scaling factor and use that in conjunction with the
dimensions o f the 30°-60°-90° triangle to resolve the item. Alternatively, the study
participants could have employed the law o f sines and the sines o f the special angles (30°
& 60°) to resolve this item. The holistic rubric o f figure 9 was applied to the scoring of
item 13. Four o f the 14 participants scored 3 points, nine scored 1 point, and one scored
zero points on item 13 o f the test of trigonometric knowledge. There were no discemable
patterns o f response.
Figure 39. Description o f item 13
Item 14 assessed the study participants’ knowledge o f fundamental concepts of
domain and range of functions (see figure 40). The researcher was particularly interested
in the participants’ knowledge o f the effects transformations on sinusoids have on domain
126
and range of the sinusoid. Item 14 was scored using the holistic rubric o f figure 9. Correct
responses received three points.
Figure 40. Description o f item 14
Five of the 14 participants scored 3 points, four scored 1 point, and five scored
zero points on item 14 o f the test o f trigonometric knowledge. Samples o f participants’
responses are presented in figures 41 and 42. Five participants did not respond to the
question. Three other participants gave a range of -3 s y <, 3 (see figure 41), perhaps
applying the vertical stretch but failing to apply the vertical shift of 2 units. Nine out of
the 14 participants gave the correct domain o f all real numbers.
P re se rv ic e S e c o n d a ry S c h o o l MrUttenuKfcx T e a c h e r* K n o w le d g e O f T rig o n o m e try
Fall 2 0 0 2
T u e sd a y . N ovem ber 1 2 , 2 0 0 2
Name;
322 I SC
/
| ^ W r T im l Ihc (liimain and raiiRC « f 3sinjz.r
"• t
Figure 41. Sample response to item 14
^ 1 X.
~[| ♦ 2.
127
Prescrvicc ^^ot:>inwJairy
r 11 2 0 0 2
TV
M
M
&
taers Km>wiedg.e: of r«jcMww»eir>
'I"*iKSSMd4*y,. N o v e m b e r 12.. 2IKJ2
4
3 3 2 I SMC
p amc::
3?
|
FI***! the cl«****a*trii m«*1 rwiiRe of 3a»iw
|2tJr ■*-™
J■+
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Jf/'
-v.
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Figure 42. Second sample response to item 14
Item 15 assessed the study participants’ ability to solve trigonometric equations
(see figure 43). Correct resolution of this item required the recall and application of the
following trigonometric concepts: behavior o f the six basic trigonometric function in the
quadrants, periodicity o f the six basic trigonometric functions, coterminal angles, and six
basic trigonometric functions are one-to-infinitely many. That is, a range value has
infinitely many pre-image values. Each equation was scored out o f possible three points.
Thus correct resolution o f item 15 would have received six points. The rubric o f figure 9
was applied twice in this item, once to each part of the item.
15. S olve the fo llo w in g tr ig o n o m e tr ic eq u ation s: F in d all solu tio n s.
a .
t a n x = t a n ax/ 5
b.
s i n jc = c o s x
C h a r a cteristics o f Q u e s tio n 15:
A dvanced level o f difficulty
Symbolic
Reasoning
Functional U nderstanding
P ro b lem Solving: Requires know ledge o f the behavior o f the basic six trigonom etric
functions in the 4 quadrants, and the use o f the periodicity o f the six basic
trigonom etric functions to solve trigonometric equations.
P ro jected S u ccess R ate: Few
Possible score: 6 points (3 points for each part)____________________________________
Figure 43. Description o f item 15
128
One o f the 14 participants scored 6 points, two scored 5 points, one scored 4
points, five scored 2 points, three scored 1 point, and two scored zero points on item 15
of the test o f trigonometric knowledge. Sample participants’ responses are presented in
figures 44 and 45. Two participants did not respond to the question. Again, there were no
discemable patterns of response on item 15. The participant whose response to item 15 is
shown in figure 44 received 2 points for part (b) of the item. This participant did not
consider coterminal angles or the periodicity o f the sine and cosine functions, so the
participant gave the principal solutions in the first and third quadrants. The participant
whose response to item 15 is shown in figure 45 received the full credit for part (a) but
missed one point in part (b) because the participant used the wrong cycle (it/2) instead of
K.
Figure 44. Sample response to item 15
129
Figure 45. Second sample response to item 15
Item 16 was a reasoning question about when a triangle with sides o f known
lengths will have the largest area (see figure 46). This item required versatility in finding
the area o f triangles. For example, it was not enough that the participants knew that the
area o f a triangle is half of the base times the height. The correct resolution o f this
question rested on the trigonometric formula for the area o f the triangle:
(l/2)afrsinC = (l/2 )acsinfi = (l/2)&csin A . It also required the knowledge that the sine of
an angle ranges from negative one to positive one, and that the maximum area will occur
when the sine assumes a value o f one. This item was scored using the holistic rubric of
figure 9. The maximum possible correct points were 3 points.
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16. W h e n d oes a tria n g le w ith sid es o f k n o w n le n gth s a, an d b h a v e th e larg est area?
J u stify y o u r con jectu re.
C h a r a c te r is tic s o f Q u estio n 16: M edium level o f difficulty, Graphical/Diagram m atic,
U nderstanding o f Function, Reasoning, Problem Solving
R ea son in g: Requires know ledge o f the area o f a triangle
1
1
1
2
2
2
— ab sin C = — ac sin B = — be sin A , and that sine o f an angle is bounded betw een [-1, 1]
P r o jected S u c c e ss R ate: Few
Possible score: 3 points
Figure 46. Description o f item 16
Figure 47. Sample response to item 16
Two o f the 14 participants scored 3 points, four scored 2 points, three scored 1
point, and five scored zero points on item 16 on the test o f trigonometry. Five o f the
131
participants did not respond to this item. The sample participant’s response presented in
figure 47 did not receive the full 3 points because the participant argued about the known
sides as opposed to the included angle. The two known sides do not necessarily have to
have equal dimensions. Thus the argument presented in figure 47 is a particular case
when the known sides have the same dimensions.
Item 17, the last item on the test of trigonometric knowledge, was intended to
assess the study participants’ knowledge of inverse trigonometric functions (see figure
48). The item required knowledge of the graphs o f the six basic trigonometric functions,
the restrictions on their domains that yield one-to-one portions o f the graph, knowledge
of reflecting graphs over th e y = x line to get the inverse functions, and knowledge o f co­
functions. The resolution could have been approached graphically, or via a combination
of geometric/graphical and symbolic approaches. This item was rated difficult and the
predicted success rate was few to none. The scoring did not follow the rubric in figure 9
because this item involved matching graphs with their symbolic representations. So a
correct match received three points and an incorrect match received zero points.
Two o f the 14 participants scored 9 points, one participant scored 6 points, three
participants scored 3 points, and eight participants scored zero points on item 17 o f the
test o f trigonometric knowledge. Five of the participants did not provide an answer to
item 17. There were two perfect scores. One participant correctly matched 2 out o f the 3
graphs. Three other participants correctly matched 1 out of the 3 graphs. Five participants
correctly matched the inverse tangent function, three correctly matched the inverse cosine
function, and three correctly matched the inverse sine function. Overall, six out o f the
fourteen participants were able to correctly match 1, 2, or 3 o f the inverse functions with
their graphs.
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Figure 48. Description of item 17
Phase two results: Interviews and case studies
The format o f interview 1 (see appendix F) and interview 2 (see appendix G) was
clinical, semi-structured, and content-oriented (Merriam, 2001). The interviewees were
chosen after a preliminary analysis o f phase one data. Seven possible interviewees were
identified based on their results on Card Sort 1 (CS1) and the Test o f Trigonometric
Knowledge (TTK). The Concept Maps (CM 1 & CM2) and Card Sort 2 (CS2) were open
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to multiple interpretations and as such were not used in the identification process.
However, they provided additional sites that the researcher used to analyze the
participants’ understanding of trigonometry.
Five interviewees (NM, ES, LN, AX, AB) participated in phase two o f the study.
Three of the interviewees (NM, ES, & LN) had the top three scores on the test of
trigonometric knowledge and top two scores in card sort one (see table 13). These three
were classified as the high knowledge group. The remaining two (AX & AB) scored
lower. Their scores on the test of trigonometric knowledge were tenth and eleventh o f the
fourteen scores, and they (AX & AB) received the second lowest scores on card sort one.
AX and AB formed the low knowledge group for the interview. The interviews served to
disaggregate the data and focus on the plethora of information from phase one o f the
study and to create profiles of the kinds of understanding that the preservice teachers had
about trigonometry.
Table 13. Rating o f participants’ knowledge o f trigonometry
Low
M ed iu m
H ig h
Knowledge Rating
Name
ES**
NM**
LN**
AD
PM
IB
Al
IA
SY
AB**
AX**
CT
ZN
El
Percent Correct on
TTK
75 (1)*
72 (2)
58(3)
58(3)
53(5)
48 (6)
45 (7)
39(8)
39 (8)
38(10)
36(11)
31 (12)
31 (12)
28(14)
Number Correct on
CS1
9(2)
10(1)
9(2)
6(10)
8(6)
5(13)
7(8)
9(2)
9(2)
6(10)
6(10)
8(6)
5(13)
7(8)
*() represents the rank o f the participants’ scores relative to the group o f fourteen. For
instance, ES had the highest score (ranked 1) in the test o f trigonometric knowledge and
she had the second highest score (ranked 2) in card sort one
** Interview Participants
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This section presents overall results for the 5 interviewees. A conjecture that was
posited and assessed in phase two of the study was “given enough time, the participants
would be able to correct most o f the mistakes in their responses to phase one activities.”
Thus the researcher asked the interviewees to review their concept maps, and to change,
refine, or elaborate on anything they wanted. To the researcher’s surprise only three
changes were made and as the report will show those three changes represented a small
part of the set o f misconceptions that were discovered in phase one o f the study. Thus it
became apparent quite early that the results o f phase one were not caused by a lack of
time to complete tasks and that perhaps two other related conjectures needed to be
explored. That (1) the weak conceptual understanding o f trigonometry suggested by the
results o f phase one data is a result o f forgetting information over time or (2) these
preservice secondary mathematics teachers did not adequately learn or understand
trigonometry when they were exposed to the ideas in high school or in their subsequent
use o f trigonometry in their college mathematics courses. Further discussions of these
conjectures are presented in chapter V.
Interview 1
Concept map 1
The interviewees reported that they followed the direction provided for this activity as
their guide to developing their concept maps. That is, they listed terms/ideas, grouped
them into clusters, and then went about the task of creating relations both within clusters
and between clusters. One member o f the high-knowledge group and the low-knowledge
group members highlighted triangles as the focal concept for their maps as illustrated in
the following part o f the interview with ES.
I:
That was my next question, about time. Um, if
you’re to choose a focal concept, what would you
choose, o f all the terms on this concept map?
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ES: I think, in my experience with using
trigonometry, 1 think I would focus on the right
triangle trig, ‘cuz that’s really the basis of most
o f the trig that is used throughout mathematics.
‘Cuz even when you get to the unit circle and trig
and calculus it kinda goes back to that basic right
triangle trig. Um, I also have to say I had no
experience with the hyperbolic trig functions, so
I haven’t studied those, and, um, I mean, you use
the sine, cosine, and tangent functions, um, but to
really kind o f - and I don’t know that my
understanding o f those is much larger - but to
really understand the right triangle trig in relation
to the unit circle I think is kind of the foundation.
So I would probably focus on that.
The remaining members o f the high-knowledge group chose functions
(trigonometric functions) at the focal concept for their concept maps.
I:
OK. Looking at this, what would you chose as
your focal concept? The major concept?
LN: I think the trigonometric functions.
I:
OK. Any other thing you want to say about your
diagram? Or your map?
LN: I don’t think so.
I:
OK.
Concept map 2
As in concept map 1, the interviewees reported following the directions provided
as their guide to developing their concept maps. Since the interviewees were provided
with terms and ideas for use in concept map two, the researcher wanted to know if there
were terms/ideas provided that they had not encountered before or did not recognize. The
responses to that question included terms and ideas such as convention, coterminal,
ArcCosine, ArcSine, ArcTangent, Sinusoidal, Derived Identities, Fundamental Period,
Even/Odd functions. The focal concepts and ideas used for concept map 2 were similar to
those used in concept map 1. The high-knowledge group highlighted functions, and the
low-knowledge group highlighted numbers and angle as their focal concepts and ideas.
One of the three changes that were made was in concept map two. A member o f the high-
136
knowledge group changed her initial phase one use o f reciprocity as a connector among
the Arc functions and their respective functions to inverse as a connector for the Arc
functions and their respective functions.
I:
And then you have trig functions and then you
have graphs. It’s like your sub-concepts. Um, can
you talk a little bit about, um, you know where
you have trig functions including, um, cosecant,
right? esc?
LN: Mm-hmm.
I:
Is that what you meant? OK. And then
arccosecant, cosine, arccosine, cotangent,
arccotangent.
LN: Yeah. That shouldn’t be reciprocal, that should
be inverse.
I:
OK.
LN: So the reciprocal o f like, cosecant should be sine,
I think. Is that right? one over sine?
I:
OK, I don’t know.
LN: That’s what I, when inverse here, and then
reciprocal over there.
I:
OK. So this part’s where you have it is the
reciprocal of. Y ou’re looking at it right now,
you’re saying you meant to write
LN: The inverse
I:
The INVERSE of, I see. OK. What about in the
form of a quotient that it is right below that
cluster? What are you trying to do there?
LN: OK. That. Yeah. OK, so, what I was thinking is,
like the first one where it’s arccosecant and
cosecant?
I:
Mmm-hmmm.
LN: It should be sine and cosecant and this would
stay it’s the reciprocal of
I:
Mmmm
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LN: And then, in the form o f a quotient 1just meant
like 1 over sine is cosecant. And, like 1 over
secant would be cosine. And that should just be
tangent, I think. Yeah.
I:
I see. So all the arcs you are talking about now
are, you’re saying, the reciprocals of the stuff on
the right side o f that cluster?
LN: Yeah. Those should be replaced with the actual
reciprocal.
Comparison o f concept maps one and two
The interviewees used similar approaches to construct their concept maps.
Furthermore, the focal concepts used in concept map 1 were also used as focal concepts
in concept map 2. Participants commented on the fact that their second concept maps
contained more items than their first concept maps. However, one of the high-knowledge
group members felt that her first concept map had bigger ideas than her second concept
map because, as she put it, she was caught up in trying to fit terms into places in making
concept map 2. Other interviewees expressed similar sentiments about the difficulty of
concept map 2. They felt that it was more difficult to construct the second concept map
because they were working with someone else’s ideas and terms. From the researcher’s
perspective, observations o f the differences in the difficulty levels o f the two concept
maps were positive outcomes and ones that were not unexpected. Working in an expert
paradigm is always more difficult because it forces participants to situate their knowledge
within structured and abstracted forms o f knowledge.
Student task number one
In this task interviewees were asked to either support or refute a student’s
explanation of inverse trigonometric functions as follows: Just like real numbers, i f you
want to fin d the inverse o f the function, say sin(x), you should multiply sin(x) by the
multiplicative inverse (or reciprocal) 1/sin x = cscx. That way you get the identity 1
because sinx • 1/sinx = 1. Follow the same procedure fo r the other trigonometric
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functions. They completed the task first without the aid o f electronic devices and then
were given the option to redo the task using the TI-83+ graphing calculator. Four out of
five interviewees decided to reconsider the task using the graphing calculator.
Without the graphing calculator
Four out the five interviewees supported the student’s claim that the inverse
function is essentially a reciprocal action just as in taking the multiplicative inverse of
non-zero real numbers. This misunderstanding of the fundamental operation underlying
inverting functions is quite severe. The four interviewees that supported the student’s
work claimed that
'(a:) = 1/sinx . Thus in effect, arguing that the underlying operation
is multiplication. The only interviewee that refuted the student’s work was classified as a
high-knowledge student. But when pressed to either explain why the student is wrong or
how to correct the student’s misconceptions, she exhibited some weakness in her grasp of
the conceptual underpinning o f inverse trigonometric functions. With further discussion it
became obvious that she was refuting the use of the word multiplication and may not
have had a profound understanding of composition as a functional operation, albeit she
displayed some fundamental and instrumental yet rudimentary understanding of
composition. The following excerpt sheds light on the discussion.
I:
Hmmm. OK. OK. Alright. Um, so let’s go to
Student Work Number 1. And, in this case, what
I would like you to do is to read what the student
has done carefully, and then choose to support or
refute, and whatever you choose, if you could
give me some reasoning why, in your supporting
or refuting.
LN: OK. You could probably stop the tape for a
minute.
(turns off tape)
LN: Um, I guess I would refute the work because um
the student has the right idea about the
multiplicative inverse, but, the definition o f like
the multiplicative inverse of, the multiplicative
inverse and inverse o f a function aren’t the same.
139
But I made the same mistake on my concept map
when I did the arccosine is the reciprocal of
cosine. So, I can understand where they’re
coming from I guess. But, um, I ’m not, do you
want like what I would tell the student or why?
I:
Yeah, what would you, what would you do to
help the student understand? If you’re refuting, to
show them that, you know, what they’re doing is
not quite right.
LN: I think I would, I would try to come up with an
example where that didn’t work. And I would try
to convey the idea of inverse of a function as,
um, like if you have the function sine x equals y
I:
You know you can write on this.
(laughter)
LN: OK. If the function sine x equals y then you plug
in the value x and you, like the function splits off
the value y, sort of. So, for the inverse of that
function you want something where you can plug
in y and it will spit out the value x so I might use,
I’m not sure. I would, you’d maybe use the
example they had, one over sine x equals
cosecant x, I don’t know what I ’m thinking here.
I am confusing myself!
(pause)
LN: Maybe it was pi. I can think of the sine, the sine
of pi, o f pi over 2 is 1, is that right? It’s been a
long time. OK, so that would be 1.. .the cosecant
pi over 2.. .so th en .. .1 think I ’m thinking circles.
But I would also, I guess if I were talking to the
student this is probably counterproductive. But I
was, um, also point out the fact that, like where
we used radians for the value of x we used an
angle, I guess.
I:
Let’s try to recount what you’ve said so far. You
said you will refute this, right?
LN: Yes.
1:
Alright. And you refute it because?
LN: Um, I think the student is mixing up
multiplicative inverse with inverse o f the
function and saying that they are the same thing.
I:
How are they different?
140
LN: Um.
I:
It doesn’t have to be sine, of course in general,
LN: I see a multiplicative inverse as a value, like a
like x instead o f sine x where x is just a value,
instead of a function of x.
I:
S o / o f x is equal to x?
LN: I guess it’s .. .well for the inverse of a function
you’re not necessarily multiplying the two
functions together to get an identity. Whereas the
multiplicative inverse, the definition is if you
multiply a value with its inverse you get 1. But
with the inverse function you’re sort of using one
function as the value for the other and then you
get x, like/ of (g o f x), if it’s x then your inverse
is, I think. I don’t know.
I:
You can write it down, so that when I, you know,
kind of look, listen to the tape, I will know what
you have written down.
(long pause)
I:
Shall we move on and come back to it in a bit?
LN: Yeah! Let’s do that.
The following conversation took place with an interviewee who supported the
student’s incorrect claim in this task. It is similar to the other three interviewees rationale
for supporting the claim.
I:
So now, let’s look at, w e’ve done kind of 3 as
well, so let’s go to Page 4 [referring to question
4: Student Work 1 of interview 1], And now
Page 4 , 1 may have to stop the tape, so you can
think about it, or if you want to read through it
and talk at the same time, that’s fine too. Student
Work 1.
AX: Um, I would - do you want me to write on this?
I:
You can write on this, yeah.
AX: OK, I would support.
I:
You would support Student Work 1?
AX: Yes.
141
I:
And why would you do that?
AX: Um, because, the, um, your multiplicative inverse
you should be able to get 1. You take a number
times its inverse, you should get 1. So this part,
the second part sine of x times 1 of sine of x
equals 1. Because you should be, well you should
get the identity, which is 1 when you’re
multiplying. So I would agree with that part and
the beginning.
I:
Which part? Wanna read that part, the beginning
part?
AX: OK. If you want to find the inverse of a function,
say sine o f x, you would simply multiply sine of
x by the multiplicative inverse. Oh. Wait a
minute. You want to find the inverse of a
function. OK, NO, I don’t support it. OK, stop,
let me read this for a minute. OK, if you want to
find the inverse o f a function. Say sine of x of the
inverse, that way you get the identity. Well, the
way she’s wording it, (pause), define what it is.
OK. I would agree with her math, the way she
words it is kind o f weird. She’s saying that to
find the inverse of a function, so you’re actually
trying to find the, she said a sine of x, you’re
actually trying to find the inverse of sine of x,
OK. (pause) OK, in order to find, to know that
it’s the correct inverse, then the number or like
sine o f x times its inverse should equal 1. So you
could manipulate it in a different way to find
what your 1 over, your reciprocal. But yes, the
inverse o f a function times the function should
equal 1 ,1 think that’s what she’s trying to get at
here.
I:
So you want to go with that?
AX: Yes, I agree with that.
With calculator
The graphing calculator did not change any minds. The interviewee that refuted
the student’s work without the calculator refused to use the calculator because she was
sure that the underlying operation is not multiplication. The other four interviewees used
the calculator in a similar way as AB who used the TI-83+ to check two operations such
142
as: sin(l) •—- = 1, and sin(Sm_1(l)) = 1 to support his incorrect conclusion that the
inverse sine function is the same as the cosecant function.
Summary o f student’s task number one
The responses to this task strongly suggest that these preservice teachers possess
limited understanding of inverse trigonometric functions. This finding was corroborated
by the results on item 17 o f the test of trigonometric knowledge as well and their use of
reciprocal for inverses in their concept maps. By claiming that the reciprocal functions
were in fact the inverse functions, the preservice teachers incorrectly argued that the
underlying functional operation was multiplication.
Student task number two
In this task, interviewees examined a constructed case o f a common student
misconception of sinusoids: [1] I f 0 < b < 1, then there is a horizontal shrink because bx
is smaller than x; [2] I fb > 1, then there is a horizontal stretch because bx is bigger than
x, [3] I fb is negative and -1 < b < 0, then there is a horizontal stretch opposite that
obtained in [1], and [4] I fb is negative and b < -1, then there is a horizontal shrink
opposite that obtained in [2], The focus of the task was on a student’s misconception of
the effects o f horizontal shrink and stretch transformations of y = sinx into y = s'mbx.
The mistake that students tend to make in situations similar to this one is to follow the
magnitude of b and incorrectly argue that bigger b values stretches the graph o f y = sin x
horizontally. And smaller b values cause the graph o f y = sinx to shrink. The same
procedures were applied in delving into interviewees’ rationales and understanding as in
the first student work task.
Without the graphing calculator
Four o f the five interviewees could not resolve this task without using a graphing
tool (TI 83+). LN, who correctly refuted student’s work 1 also refuted parts [1] and [2] of
143
student’s work 2, and quarreled with the use o f the word opposite in number [3] and
number [4] o f student’s work 2. Again, she refused to use the graphing calculator because
she argued that she was quite confident o f her response without the calculator. The
following extended excerpt highlights LN’s reasoning about this activity.
I:
We are back to Student Work #2.
LN: OK. I would refute the student’s work because,
um, I would probably draw these [drew sin (x),
sin (2x) on her sheet without the aid o f the TI53+] to help myself think about it and to sort of
show the student what happens, er, maybe even
use my graphing calculator and let the student
experiment with different values for b. But, I
guess I sort o f think about it like, if b is greater
than 1, then the, I guess that’s not the x value. If
the x equals y, er not y, w e’ll just call it z. So it’s
sine z. Then z is increasing faster than x was, so
the graph is sort o f smooshed, since it’s sort of
covering all these values twice as fast, I guess.
That’s how I think about it, and if it’s between 0
and 1 then the graph is stretched because it’s not
covering the values as quickly. And I hope that
would make sense to the student. And, oh boy, I
have these backwards [realized that her spoken
word and her written work did not agree]
(pause). Are you allowed to be the student and
tell me what the student was thinking?
I:
(laughs).
LN: No? That’s alright.
I:
OK. It’s, well, I did number [1] and I realized
that when b is between 0 and 1, that I have a
shrink. Because bx is smaller
LN: Mmm-hmm.
I:
than x, so I felt it was gonna shrink, where your
smaller x values, um, and I was thinking that I ’m
going to have the same y values, um, so that’s
what I was thinking. Well, yeah, and I ’m gonna
have the same y values, same smaller x values
but the same y values, so, ‘cuz aren’t we like
closer, so we will have a shrink? And then, um,
when b is greater than 1, of course the x ’s are
gonna be bigger. But I still have the same y so
I’m gonna have to travel farther out, which is
same y values as I ’m stretching it. So then I said,
144
well 1 heard something about when something is
negative there’s some kind o f switch and stuff, so
I was thinking, OK, b is negative means that
there has to be something opposite the first one,
‘cuz if you look at number [1] and number [3],
it’s like I ’ve done some kind o f opposite and I
don’t know exactly, um, ‘cuz I ’m not strong in
trig, so I decided, I said there’s horizontal stretch
opposite that obtained in number [1], I mean I
don’t, I mean, I think I ’m right, but I don’t know.
You’re to tell me whether I am wrong.
LN: OK. So, because number [1] shrinks, then the
opposite would be to stretch it?
I:
Is it a shrink? The opposite of a shrink is a
stretch?
LN: OK.
I:
Is that right, though?
LN: Um, I would refute that also [seems to refute
number [3] o f the acivity].
I:
OK. Why?
LN: Um,
I:
So you say number [3] is shrink, then?
LN: No. OK, but the reasoning, I refute the reasoning.
I:
OK.
LN: But, I would agree with the final answer, sort of
[agreed with the conclusion o f number [3] o f the
activity].
I:
What do you mean you, um, disagree with the
reasoning? What part of the statement do you
disagree with?
LN: OK. It would, I think, the graph would be, there
would be a horizontal stretch,
I:
Mmm-hmm.
LN: But, instead o f the stretch being the opposite of
the shrink, both o f them stretch and the opposite
is the graph is sort o f flipped, I guess, o r.. .so ...
I:
What you graphed [LN graphed sin(-x)\, is this,
what is the relationship between this one and that
145
one? [referring to the graphs o f sin(x) and sin(-x)
that LN drew on her sheet]
LN: OK. This is the opposite, I ’m thinking, rather
than shrinking and stretching.
I:
Oh, ‘cuz 3 [number [3]] is the opposite o f 1
[number [If], according to your graph? That’s,
according to what you’ve graphed, is that right?
LN: Yes.
I:
OK.
LN: So, maybe, I will use that then in talking to the
student and ask if, um, what if b is equal to -1
then w hat’s the opposite of sine x if there’s no
shrink or stretch on the original?
I:
Oh!
LN: OK, here’s my original graph o f sine x.
I:
OK.
LN: And the student is saying that a negative value
for b would stretch it because it’s the opposite of
shrinking, but in this first graph we haven’t
stretched it or shrank it
I:
OK. OK.
LN: So, I would ask the student what the opposite of
that is, then.
I:
Hmmmm. Since there is no stretch or shrink then,
the opposite would be just the same? But how
would [3] be opposite [1]? OK. So you’re, you’re
saying, for number [3], you’re OK with the
horizontal stretch part, but that reason about
opposite number [ 1] you disagree with?
LN: Right.
I:
OK. What about [4]? [referring to number [4] o f
student task 2]
LN: Um, same idea. I agree that there will be a
horizontal shrink, but if they use the same
reasoning as being the opposite of
I:
Mmm-hmm
146
LN: The answer for number 2 is that 1 would disagree
with it.
I:
OK. Alright. Did you want to say anything more
about Student Work 2, or?
LN: Mmmmm, no, 1 don’t think so.
I:
So what does b do, in general?
LN: Ummm
I:
What is the effect o f 6? What do we call b after
we looked at the graph, beyond just stretch and
shrink? Can you say a little bit more? In terms o f
functions?
LN: Um, what do you mean? If you want me to use
technical terms or do you want me to sort of,
I:
No, no, no, no, you decide. You decide. If you
are, if you are, let’s say describe what the effect
of b would do. And, there is something that you
said that I will kind o f pick up on later on, about
you have the student explore different b 's. OK?
‘Cuz that plays into your use of technology in the
classroom, um, OK? so I ’ll pick up on that later
on. But, how would you, kind of, you know,
describe the effect of b if you are to, if you want
to teach this? W hat’s the effect o f b l How would
you go about doing that?
LN: Hmmm.
I:
Do you recall what b is called? Does it have a
name?
LN: I don’t remember.
I:
Don’t remember, OK. That’s alright.
LN: Is it on my sheet, from th e ...
(laughing)
I:
I don’t know. I don’t want to answer that, I don’t
want to give too much, I don’t want to give it
away, I don’t know, um, OK.
LN: It changes the, um, period of the graph.
1:
How so?
147
LN: Um, if b is greater than 1, it, then the period
decreases. OK, so, the change in period is the
multiplicative inverse o f b. Something like that.
I:
So what is the period of the first graph you have
on that sheet? [referring to sin(x)\
LN: 2 pi
I:
And what is the period of the second one?
[referring to sin (2x)\
LN: Pi.
I:
OK.
With the graphing calculator
LN talked about using the calculator to help her students visualize the effect of
parameter b (the frequency), but she did not use the calculator in the resolution of this
question. The other four interviewees refuted parts [1] and [2] when they used the
graphing calculator (TI-83+) to graph different sine functions with different b values. The
interviewees wanted to support parts [3] and [4] but they quarreled with the use o f the
word opposite in the student’s argument. AB called parameter b the slope. So he was
asked to comment on the effect of 2 in the following functions: y = 2sin(x)and
y = sin(2x) in terms o f the slope idea. AB responded, “you caught me.”
I:
Let me ask you a question. Why does 3 and 4
make sense? That 1 and 2 do not make sense?
According to what you’re saying. [/, 2, 3, & 4
refer to the fo u r parts o f the activity\
AB: Um, because the multiplier, um, the multiplier
tightens the graph. If it’s larger than 1 it has an
absolute value larger and it shrinks it if its, or
wait! Yeah, it tightens it, and it spreads it out, if
it’s less than 1. [AB used the word multiplier as a
referent fo r 6]
I:
Why does it do that?
AB: Good question. That’s why I’m probably looking
at it wrong but I just can’t think
I:
(laughs) So you?
148
AB: I would’ve, I would just have been thinking a
normal graph, (pause) I guess it tightens. I guess
a multiple, with a larger slope. So like a slope o f
2 or 3 would bring a normal graph tighter
towards the y-axis, and so that makes sense, in
this respect, um. I think it pulls it away from the
x-axis sort of.
I:
OK. Alright. That’s good.
AB: Yeah, I guess that makes sense. Multiplier that is.
It’s absolute value o f less than 1 would flatten the
graph out, like, and the slope would be lower, so
it, it flattens the graph out. So that’s kinda what
this is doing. I think, with the graph, it seems
like.
I:
What do you mean by this slope?
AB: Um, along with the slope o f 1 it’s, if I give it a
slope of 2 it’s, it shifts it obviously [talking about
when b = 1 or 2]
I:
Can you draw the same stuff on the same axis?
AB: Yeah.
I:
What would it look like?
I:
OK. and how does that relate into the sine of 2x
as opposed to the sine o f x?
AB: With respect to the, taking either o f these with
respect to the y axis, it, it brings the graph tighter
with respect to the y-axis if the slope is higher,
and it brings it, it spreads it, um, and brings it
more towards the x-axis if the slope is lower. [AB
is discussing the graphs o f sin(x) and sin(2x) that
produced on the TI-83+]
I:
OK. Alright.
AB: But, um,
I:
Alright. I ’m going to behave as if I don’t
understand it for the sake o f the study, and I will
ask you this question. Um, what about y is equal
to 2 sine o f x?
AB: Y is equal to 2 sine o f x?
I:
Without graph. D on’t graph i t .
AB: OK. Um,
149
I:
Using your argument, I ’m trying to figure out
what you mean by slope, actually. That’s what
I’m trying to figure out.
AB: Oh. Um,
I:
So how is y equal to 2 sine of x, in terms o f
behavior you have described different or the
same as y is equal to sine o f 2x? [an attempt to
delve deeper to the slope idea that AB brought up
w ith y = 2sin(x) a n d y = sin(2x)]
AB: Uh-huh. I ’m thinking that it [referring to the
numeral 2] stretches the entire graph and
stretches the limitations of the box, I mean the,
um, the graph o f the sine o f x just goes between 1
and -1 on the y-axis, and multiplying the entire
thing by 2 would, um, stretch that out, so that
would go between 2 and -2. [referring to vertical
stretch - amplitude]. Um, you caught me on the
slope. I don’t, I’m not sure. I’m thinking of, the
way I ’m thinking o f this multiplies is in terms of
slope. I don’t know. Not sure if that term is right.
I:
And I don’t know either. I’m just
AB: OK. Yeah.
I:
Alright. OK. So is that OK for Student Work 2,
then?
AB: Yeah. I ’m pretty good with that.
I:
Great. So I ’m going to turn this off and then
have you take a look at Card Sort 2.
Card sort 2
In the interview, card sort 2 activities were completed before card sort 1 activities
and the discussion of the results o f the card sorts follow that order. The interviewees
described how they went about sequencing the cards and how they determined
prerequisite knowledge to aid their sequencing. To guide their sequencing, the
interviewees focused on different goals. AX focused on the unit circle and used that to
guide her sequencing o f the cards. NM and LN used the order they remembered
encountering trigonometric topics in high school. ES put cards that were unfamiliar to her
towards the end. And AB said he first organized related concepts into clusters and created
150
his sequence from the clusters. AB also mentioned starting with simplest groups o f
concepts. However, AB considered Addition Formulas early in his sequence because he
interpreted Addition Formulas as formulas for adding angles. When asked to elaborate on
the need for formulas for adding angle measures and the difference between adding
angles and regular addition, AB answered that he was not sure. Two interviewees, AX
and LN wanted to introduce ideas o f trigonometry via study o f sine, cosine, Theorem of
Pythagoras, and other basic ideas.
The interviewees gave varied responses to the question: What is trigonometry?
AX said that trigonometry is numbers. NM said that trigonometry is the study of sine,
cosine, tangent, unit circle, radian, and function. ES defined trigonometry as the study of
triangles, angles and side relations and how to find missing data. AB described
trigonometry as a tool for figuring out different angles and sides o f triangles.
An analysis o f the interviewees’ sequences revealed areas o f weak prerequisite
knowledge on their part. The discussion of the interviewees’ sequences is presented
casewise in the following order: AB, LN, ES, NM, and AX. The discussion focus on
prerequisite misplacements that were gleaned from the interviewees’ sequences. For
instance, for AB a low-knowledge interviewee, the addition form ulas card was the first
card in the sequence. Trigonometric expressions, and equations came before
ideas/concepts such as identities, inverse functions, definitions o f the trigonometric ratios
and functions, periodicity, domain, and range. The laws o f sines and cosines came before
definitions o f trigonometric ratios and functions. Moreover, inverse trigonometric
functions came before definitions o f trigonometric ratios and functions, domain, range,
even and odd functions, and one-to-one. Finally, AB placed reference angles before
quadrants, and angular rotations. These decisions suggest that AB, like the other
interviewees, does not seem to have a sufficiently deep understanding o f trigonometric
concepts to facilitate construction o f a coherent sequence o f trigonometric concepts and
topics that embody pedagogical and prerequisite integrity. The following excerpt
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highlights A B’s method for ordering, some of his misinterpretations, and some his
misplacements.
AB: Um, I just kinda went through and organized
these as well as I can, I guess to just skipped to
one o f your next few questions [referring to the
interview questions on the interview instrument],
probably time was a factor in this question, once
I got down towards the end I remembered a little
bit. But um, and um, actually I got down right
about two times, so, um, But I just kinda tried to
find some o f the simplest concepts, kinda moved
those towards the front, um organized concepts
that I thought kinda went together. Organized
those in a group, and then also like, um, I ’m
doing that, take some o f the more complex
concepts and kinda set those into, 1 don’t know a
pile, or another further down the line anyways.
Um, so I started, tried to start with the simplest
group o f concepts and then progressed from
there, and then. So within that, some o f these are,
are grouped, some of them go before each other
but other ones I w asn’t, didn’t necessarily have a
preference in terms o f order, um, so I just kinda,
like, took like, I don’t know say for example this
one, this one could’ve been switched, [the
referent will become obvious in the next
exchanges]
I:
When you say that one? What do you mean?
AB: Just as a generic example.
I:
Which is, a graph o f trig functions, and
periodicity, could have been changed?
AB: Right. Mmm-hmm. I ’m not sure if those [still
referring to the referents above] per se could’ve
been changed, but in general there may have
been two or three cases where a didn’t have to
come before b before c, it could’ve been a, then
c, then b. But, um, it’s just, addition formulas, I
may have been thinking, um, of that incorrectly
but I was thinking these are just basic addition or
addition of angles, um, then with the lesson
intended, was supposed to mean. But that’s just
how I read it and so I was thinking just addition
of angles, 30° angles, 25°, and 55°, I thought that
was extremely basic, um, I guess looking at it
now, if I was thinking o f it as angle measure, er,
adding angles the, um, maybe degree measures
should have been before that, because if you’re
152
going to add two angles measures, you need to
know how to add them. So I guess some of those
first three I just, I probably may have rushed at
the end and just kinda thrown the simple
concepts up there towards the front, so probably
could’ve rearranged those a little bit. Um, similar
to right triangles, you know I thought that was,
um, that m ay’ve even been able to go ahead of
both of those. It was just a, simple concept that,
um, yeah, just a very simple concept. Um, a2 plus
b2 equals c2, um I had that towards the front, it
was just something, um, I thought it kinda led
into some o f the trig stuff. Um, (pause), um trig
expressions, um, what was meant by trig
expressions, um, kinda just thought o f that as
kinda basic introduction to trig.
I:
Like what?
AB: I was thinking, um, just basically, um, sine and
cosine, tangent, um, it’s just those kinda things.
And maybe a brief introduction o f what they
mean or do. The bottom o f the trig equation is
kinda how you can meet those, maybe. Um,
reference angles. I was just thinking, um, I ’m not
sure where that one should’a gone. I was just
thinking o f it as, um, just the angles that you use
as a reference for, to base off of t, you use a trig
function to find something else. Um, application
of trig, um, it’s like only used in those equation.
Um, law of sines and law o f cosines, I don’t
remember what those were, I can’t remember
exactly law o f sines and law of cosines for this,
that probably is something I could at the
beginning of, the starting of trig. Um, but 1
thought they were a little bit more complex but, I
don’t remember what they were, but I was
thinking that maybe there’s something more to it
Pythagorean theorem for trig functions, um, that
would probably be once you introduce trig,
something kinda relating back to the Pythagorean
theorm [referring to a 2 + b2 = c 2]
I:
Do you know what it is, the Pythagorean theorem
or trig functions?
AB: Um, I don’t think I do
I:
OK. You talk about application o f trig. Can you
name an application o f trig, what would be an
example o f an application?
153
AB: (long pause) Um, just take the sine of, I can’t
even remember angles, sides. I get, I’m, I really
don’t remember much lecture. Um, take the sine
o f an angle to find, um the other corresponding
side or angle that you’re trying to find, I guess
sine o f something
I:
And let me just pick on some o f this here since
we don’t have a lot of time. You said the radian
measure came after, um, something like, um, let
me see, um, solving triangles in general. Um, you
also said the radian measure came after the oneto-one function idea. And that a radian function
also came after solving right triangles. And it
also came after, let’s see, after application o f trig
came after inverse trig functions, and it came
after, um, the addition formulas, it came after,
let’s see, trig, expressions, trigonometric
equations, and law o f sines, and law of cosines,
um, so you will have done all of that before
talking about the radian measure, um, think w e’re
out of time.
[side one o f tape one ended]
I:
So what is the radian measure? If you are to
describe it, what would you say?
AB: Distance around the unit circle.
I:
OK
AB: So, I guess I was just looking through these, I
was just trying to relate them back to what I
remembered I had done, and I don’t know, part
of the reason I was thinking that radian measure
did something that was kinda done later on, so,
these are all pretty general and I was trying to
come up with what they were and what order
they would go in, um, and how they would relate
to each other but, I w asn’t coming up with a
whole lot of how some o f these relate to each
other. And I was kinda thinking that [referring to
radian measure] would come later on but I was
probably remembering incorrectly.
LN, a high-knowledge interviewee, started her sequence with degrees. Her second
card was solving triangles in general. The second card was placed before any mention of
concepts or ideas o f trigonometry. The Pythagorean identity sin2 6 + cos2 d = 1 came
before definition o f trigonometric ratios and functions. LN also placed reference angles
154
before clockwise/counter clockwise angular rotations, and quadrants. Moreover,
trigonometric expressions came before using cosines and sines to define the other
trigonometric functions. LN also placed derived identities before fundam ental
identities/properties. Furthermore, LN placed trigonometric equations before coterminal
angles and cofunctionality. These misplacements represent a serious imbalance in her
pedagogical and prerequisite integrity (knowledge) o f trigonometry.
ES, a high-knowledge interviewee, started her sequence with similar right
triangles. The second card is solving right triangles. The second card was placed early in
the sequence before any mention of degrees, radians, and trigonometric functions. ES
placed reference angles before quadrants and angular rotations, both clockwise and
counterclockwise. ES also placed trigonometric expressions and equations before using
sine and cosine to define other trigonometric functions, domain, range, odd and even
functions, cofunctionality, degree, radian, periodicity, sin2 0 + cos2 0 = 1, inverse
trigonometric functions, and coterminal angles. Furthermore, ES placed the addition and
half-angles before the law o f sines and cosines. Finally, ES placed derived identities
before the fundamental Pythagorean identity sin2 0 + cos20 = 1. Thus ES’s sequencing
also suggest that she did not fully understand the pedagogical and prerequisite
implications of the 34 trigonometric topics provided in card sort 2.
The only participant that left out cards in their sequence is NM, a high-knowledge
interviewee. He left out coterminal angles, cofunctionality, even and odd functions,
transformations, addition formulas, and half-angle form ulas because according to him,
he did not know either what they meant or how they fit into the overall sequence o f the
thirty-four cards. NM started with similar right triangles. In the sequence he placed law
o f sines and cosines, and solving triangles in general before degree measure, radian
measure, and definition o f trigonometric ratios. Furthermore, NM placed trigonometric
expressions and equations before domain, range, inverse functions, using sine and cosine
to define other trigonometric functions, sin2 0 + cos2 0 = 1, and fundam ental identities and
155
properties. The nature o f the interview with NM about sequencing is illustrated in the
following excerpt.
I:
OK, w e’re back. So how did you go about
arranging Card Sort 2?
NM: Um, I did them pretty much the same way we did
the concept maps. I just kinda spread them out
and tried to sort them into ideas first. And then I
tried to group them, in the order in which, I think
I generally did what I remembered, how I
remembered learning things. I don’t know if
that’s the best way to sort them, but that’s the
way I put them in because that was, how I
remembered, I was familiar the order in which I
learned them.
I:
When you say learned them, do you mean how
you were presented the material in class?
NM: Yeah. When they were, the order in which I
remember understanding them, I guess, I
might’ve learned those then, earlier or something
and not grasped it but I remember actually
putting them together, this kind o f went with this
[was not referring to anything specific], I don’t
know.
I:
Was it the way the teacher presented it, or just
the way you organized it? I ’m just...
NM: It could be either. I ’m assuming it’s pretty much
the order in which it was presented to me, in
school, but I mean some things, there’s some
things especially in math that, where I ’ll see it,
and I ’ll kinda grasp it but not really and then later
on when I went in to a different concept or
something they just intertwined, all make a lot
more sense. I didn’t really do any prerequisites
for this. I didn’t like actually sit down and
determine, well you need to do this first, then
this, that’s what I kinda remember trying to do it
in the order I remembered learning that it. I think.
I think that the same thing happens when this
runs into this. But I didn’t actually intentionally
sit down and say what do you need to know first,
and w e’re gonna understand that. I think there is
a lot of, so many ways, especially in
trigonometry that things can be connected that
the order in which they learned is.
I:
OK. Any other thing about Card Sort 2?
156
NM: Um, these cards [NM talks about these cards
later on in the excerpt] that I didn’t include I
either couldn’t remember exactly what they
meant or figure out how to incorporate them into
any of the other groups I had. There were more in
here, that I had when I sorted the cards and I
finished and I kept going through these over and
over again, and then I ’d kinda think o f something
as it related to, and I ’d go back through the card
sort and try to slip it in somewhere, kinda. I can’t
figure any specific ones or anything.
I:
But this one, out o f six of them, there is no place
for them, basically? Or you, you haven’t found a
place for them?
NM: There probably is, I just didn’t, couldn’t figure
out where to put it in there.
I:
For the record, which ones are those, that you are
holding now?
NM: Even and odd functions, transformations,
coterminal angles, addition formulas, co­
functionality, and half-angle formulas.
I:
In terms of sequence, now that, you know,
you’ve had some time to reflect on how your
sorted, if you are to teach how students are
supposed to solve triangles, do you teach inverse
functions first? Are inverse functions related to
solving triangles, you know what I mean by
solving triangles? That you are supposed to find
all the missing angles and sides.
NM: Mmm-hmm so it can be any triangle?
I:
Yeah. So would you teach inverse functions first?
Does it even relate to any solving triangles?
NM: I ’m sure it does but I can’t think how. I don’t
remember using inverse functions, I don’t think
solving triangles, um, yeah, unless I don’t
really...
AX, a low-knowledge interviewee, started her sequence with quadrants. For her,
fundamental identities and properties came before domain, range, similar right triangles,
angular rotations (clockwise and counterclockwise), periodicity, coterminal angles, even
and odd functions, and cofunctionality. She placed trigonometric equations before using
cosine and cosine to define the other trigonometric functions, and inverse functions.
157
Moreover, AX placed inverse trigonometric functions before theorem o f Pythagoras
( a 2 + b2 = c 2), one-to-one functions, and graphs o f trigonometric functions. And finally
she placed law o f sines and cosines before sin2 6 + cos2 0 = 1.
As the preceding discussion o f misplacements show, the interviewees displayed
minimal prerequisite integrity in their sequences o f trigonometric concepts and ideas.
These misplacements highlight flaws in the interviewees’ pedagogical content knowledge
and curricular knowledge. For instance, how does a teacher approach trigonometric
identities before defining the trigonometric ratios or functions? These are coherence
issues and are not easily overcome. There were no discemable differences between the
high-knowledge group and the low-knowledge group. Both groups committed similar
errors.
Card sort 1
The interview discussions that ensued regarding how the interviewees placed
propositions into the three different piles revealed that these preservice secondary
mathematics teachers have a weak understanding of trigonometric concepts and ideas.
The interview did not shed much additional light on the participants’ depth of
understanding of trigonometry. Rather it corroborated the results o f the first card sort of
phase one. The two additional changes of phase one responses were made during
discussions related to card sort one. Recall that the first change had occurred in
discussion of the concept maps when LN changed her use o f the connecter reciprocal,
among the six basic trigonometric functions and their Arc functions, to the connector
inverses in her second concept map. In card sort 1, LN moved sin2 9 + cos2(50) = 1 to the
true sometimes pile from her initial placement of the card in the never true pile. She
argued that for integer multiples of 90°, sin26 + cos2(5d) = 1 would be true. In addition,
ES moved graphs o f trigonometric functions are sinusoidal, and 2n radians represent the
fundamental period fo r trigonometric functions to the true sometimes pile. ES had
158
originally put those two cards in the always true pile in phase one o f the study. What
follows is the entire interview transcript for ES speaking about her first card sort.
I:
Now let’s see. Card Sort 1.
ES:
Oh it just keeps going!
1:
(laughs) ... OK w e’re ready. So in this you want
to describe how you grouped this statement into
the categories and, you know, would you like to
change any o f the cards?
ES: (laughs) probably.
I:
So, so to help you out, one thing I suggest is if
you can like put the category level in front and
then put the cards underneath them so that we
can kind o f take a look at them.
ES: Um, I think I tried to actually write my reasoning
on the back of these cards, which maybe a good
plan at this time, to see what in the world I was
thinking when I was doing this. ‘Cuz obviously
I ’ve changed my mind several times tonight, so.
I:
OK. OK.
ES: Who knows, ‘tis the season. OK. W e’ll start with
my “always true” pile.
I:
Alright.
ES: Trig functions are periodic. I couldn’t think of
one that w asn’t.
I:
OK. and you actually only considered only these
six ones, right? The sine, cosine, tangent,
and...those ?
ES: Yeah, yeah. And on the back I wrote yes, all trig
functions repeat themselves. Sine, cosine,
tangent, cosecant, secant, cotangent. I could think
of functions that had vertical asymptotes but they
still repeated themselves, so I decided that
needed to be always true. Um, inverse trig
functions yield angle measures, um, yes and I
give the example inverse tangent o f adjacent over
the hypotenuse equals theta. W e’ve been there
before tonight.
I:
(laughs)
159
ES:
So that one’s true. When you transform the six
basic trigonometric functions in a plane you get
other functions that are also trigonometric
functions, not necessarily one of the six basic
trigonometric functions. And I said yes, not sure
why, seems to be true, always so um, I don’t
know if I can give you a better answer than that.
Um, I guess I was thinking transforming which
would be a horizontal shift, a vertical shift, um,
and it was hard for me to think of an example
that’s counter to that. So it, it seemed good to me.
I’ll just stick with that and w e’ll go with gut
feeling for that one. Graphs o f trig functions are
sinu...
I:
Sinusoidal.
ES:
Sinusoidal. Yeah, sure. I don’t know what that
means. I ’m not familiar with sinusoidal, so um,
sure and I said not sure why, sounded so, I’m just
going with that too. OK. Given triangle o f sides
a, b, and c the trig functions are ratios o f the
lengths of two o f the sides and I said yes,
definition o f trig functions. Um, and I suppose I
was thinking specifically o f right triangle trig. If I
think about it now, um, other triangles, OK, so I
guess if you look at law o f sines, I mean, you
kind of do look at the ratios o f two o f the sides,
kind of, but not really. So maybe that should go
to the sometimes true pile.
I:
Why would you change it, why do you want to
change it?
ES: Oh. Well I ’m just, it’s always true if you look at
a right triangle. Um, so if I could add that in there
I’d keep it in the always true pile. And again this
is me questioning my reasoning so, um, just
thinking of triangles that aren’t right triangles,
um, well, are ratios o f lengths of two of the
sides? Which yeah, ‘cuz if you think of law of
sine, sine o f a over a equals sine b over b, and
you could play with that ratio ‘til you get, what?
b over a equals sine - I ’m just thinking - so a
over b equals sine a over sine b.
I:
You can write on this sheet.
ES: Oh sure. So, I mean it would work for law of
sines and I honestly couldn’t remember law of
cosines, so if sine a over a equals sine b over b.
If you manipulate that we get b over a equals
sine b over sine a, so, yeah that’s a ratio of sides.
160
I don’t know. Um, I don’t know. It can always be
true. I ’m comfortable with that.
I:
OK.
ES: And the last card in that, yeah, the last card in
that pile is 2Jt radians represent the fundamental
period for trig functions, and um, yeah, ‘cuz you
do a full period in 2lt radians, that’s generally
what we consider, so I said yes on the basic
principle of trig functions was my reasoning for
that.
I:
OK.
ES:
Um, so then moving into the true sometimes pile.
Um, I had sine squared theta plus cosine squared
5 theta equals 1. Um, my reasoning was true
when theta equals 5 theta because the
Pythagorean theorem for trig functions says sine,
er sine squared theta equals cosine squared theta
equals 1, um and theta wouldn’t necessarily
always give the same, um trig identity as 5 theta.
So, that’s why I said sometimes true.
I:
OK.
ES: Um, iff and g are two trigonometric functions,
then the period off over g is period o f f over
period o f g and I said not sure but it didn’t seem
like it would always be true.
I:
OK.
ES: Um, if a phenomenon is periodic, then the graph
of the phenomenon is the graph o f one o f the six
basic trig functions. And I said not always true
because other functions can be periodic. I’m not
sure which ones, right off the top o f my head but
I recall that others can be periodic, so, that’s only
sometimes true. Um, the inverse of a
trigonometric function, or the inverses of
trigonometric functions are functions, and I said
um, that problem’s true isn’t it? Um, my counter
example was looking at a different function such
as the square root o f x if you consider both the
positive and negative solution it doesn’t make
that a function, and so um, but if you looked at
trig functions as, or the inverses as secant,
cosecant, cotangent, um, then they are functions
in which case that would always be true. So, but
I’m trying to think if we are talking about other
trig functions, which would be. If you consider
those hyperbolic ones that I have no idea about,
161
then it might fit for those, if we are just talking
about sine, cosine, tangent, then it would always
be true.
I:
Are you changing that?
ES: Um, I would change it with the understanding
that w e’re just looking at sine, cosine, and
tangent.
I:
OK.
ES: Are we just looking at sine, cosine, and tangent?
I:
Um, no w e’re looking at six basic ones: sine,
cosine, tangent, and cosecant...
ES:
...secant...
I:
yeah.
ES: OK. Then it’s always true. ‘Cuz the inverses of
those are just sine, cosine, tangent, OK. The
domain of a trig functions is the set o f real
numbers, um but I said trig functions can also
apply to imaginary numbers, so that was true
sometimes, ‘cuz for, as long as you’re not
including imaginary numbers but trig functions
do apply to imaginary numbers.
I:
So if you don’t need to use imaginary numbers as
input, then you’d be OK with that?
ES: Yes.
I:
OK. So then why don’t we do that?
ES: Oh. OK.
I:
So we don’t want to include imaginary numbers,
what do you think?
ES: If we don’t include imaginary numbers then it
would always be true. Because you have real
numbers and you have imaginary numbers, so,
but if w e’re just, if w e’re not considering
imaginary numbers than the domain would be
real numbers. So I ’ll move that to always true,
‘cuz w e’re not considering imaginary numbers.
OK. The general theorem o f Pythagoras applies
to triangles o f sides a, b, c, and I said that’s
sometimes true as long as you have the right
triangle, but the statement does not, um, say that
you have to have a right triangle, so that’s
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sometimes true. And 2jt radians can be the
period o f any trig function, um and it is the entire
period for some functions but not all the
functions. Um, ‘cuz sometimes you get two
periods within 2 ji radians, so. I ’m not sure of an
example, but.
I:
And um you have one for never true?
ES:
I have a couple for never true. For a trig function
there are situations when a particular domain
value has two range functions and I said no
because then it wouldn’t be a function.
I:
OK.
ES: OK. And one radian is equal to 180°, and I said
no because one, well 180° equal n radians
I:
OK. Before we move on, let me ask you, well let
me make a clarification on two issues. One is
sinusoidal and the fundamental period, you say
you’re not quite sure about what these are. The
graphs o f sinusoidal functions look like the sine
wave.
ES: OK.
I:
I wanted to make that clarification and then ask
you to re-sort that card. And, a fundamental
period is a period of the shortest length.
ES: OK.
I:
Fundamental period is a period of the shortest
length.
ES: OK.
I:
So let me, I mean, let me give you time to resort
those cards.
ES: OK. I can do that. OK. Graphs of trig functions
are sinusoidal would only be true if you have a
sine function, I would say, because a trig
function cannot be, as you defined them to be,
what’d you say a sine wave?
I:
Sine wave, yeah.
ES: Um, hmmm
I:
What about the cosine function?
163
ES:
It has, it is similar to a sine wave, but I ’m
thinking o f the tangent function, which is entirely
different.
I:
So would the cosine be a sine wave or would that
be sinusoidal?
ES: Um, they have the same shape, the difference is
where they start. So, um, but I’m still going to
say that my counterexample is the tangent
function, which is a trig function. And that has
some definite different properties to its wave.
I:
OK. Alright. So let’s go to the fundamental
period.
ES: Um, 2jt radians represents the fundamental
period for trig functions, and I would say
sometimes, um, in the case of the, you know it
completed one period within 2 tc radians, um and
there are functions that meet half a period in 2:t
radians or a period in 2n, so that would be true
sometimes.
I:
So you’d move the card with the sinusoidal
function and the fundamental period to
sometimes true or true sometimes, from always
true?
ES: Yes.
I:
OK. Any other thing you want to say about this
card sort?
ES: I don’t think so. No.
The excerpt above showed that ES had a strong subject matter content knowledge
of trigonometry, relative to the other 13 study participants. Her responses indicated that
she had a fertile foundation upon which in-depth knowledge o f trigonometry can be
rebuilt.
Technology
In interview one, the interviewees discussed their envisioned use of technology in
the mathematics classroom with special emphasis on the use o f graphing calculators in
teaching and learning trigonometry. All five interviewees indicated that they would use
the graphing calculator for parameter exploration and to facilitate visual representations
164
of concepts and ideas in trigonometry. Moreover they stated that they would use the
technology to facilitate conceptual understanding, but not to supplant students learning o f
the basics and computations. Therefore, the interviewees said that they will use the
calculator after the students have been introduced to concepts and have had time to
practice computations, presenting visual representations of trigonometric functions and
problem situations, and justifications in trigonometry. The interviewees were then asked
to complete a related task: How would you use the graphing calculator to graph the
ArcCotangent function (same as Cot~l(x))? State all your steps
(\\
AX argued that Cot '(x) is equal to Tan 1 — and proceeded to produce the
\x)
graph o f the inverse tangent function of the reciprocal o f x. NM presented the following
work:
Cot'U x) = —-— = tan x , and proceeded to graph the tangent function as the inverse
cotx
cotangent function. ES made a similar misinterpretation o f the inverse function as a
reciprocal o f some sort and argued that C o r'(x ) = ___ -____So she graphed the
v ’ Tan\x)
reciprocal o f the inverse tangent function in place o f the inverse cotangent function. AB
provided the same argument as ES, arguing that C o f \ x ) = ----- . . LN could not
Tan" (x)
reach a conclusion. She tried the following approach but would not commit to any
interpretation: C o f'( x ) = y
COS V
Cot(y) = x <->
—= x. Then she wanted to rewrite the
siny
new expression forx in terms o fy in order to produce a ratio o f Arccosine to Arcsine. LN
Arccosine
reasoned that Arccotangent = -------------- . However she could not come up with a
Arcsine
procedure that will allow her to move f r o m
COS V
siny
= x to produce the ratio that she
desired, as illustrated in the following excerpt from her interview.
165
I:
Pi? OK. Now, so let’s do this one. How would
you use the graphing calculator to graph the
inverse cotangent o f x?
LN: OK. You just want me to sort of talk through
what I’m doing?
I:
Mmm-hmmm.
LN: OK. I ’m going to the y equals [referring to th e y
= key on the TI-83+].. .and then I ’m going to just
put in the inverse cosine o f x and push graph.
Maybe.
I:
OK. Now let’s read the question again. It is the
graph o f the inverse cotangent.
LN: Oh! OK. Alright. So, oh shoot. OK. So then. OK.
[works on calculator]. So I ’m not sure if this will
work. But I would do 1 over tangent o f x. And
then that to th e .. .probably won’t come o u t...
I:
Are you trying to do inverse tangent of x or 1
over tangent to the -1 ? What are you trying to
do?
LN: I ’m trying to do this, the inverse of that [referring
to 1/tan x to the negative 1 power]
I:
Oh. OK. OK.
LN: So I’m not sure if 1 over the inverse of tangent is
the inverse cotangent.
(pause)
I:
What do you think you graphed?
LN: The tangent.
I:
And why do you think that will be, looking at
what you have?
LN: Um, because the tangent to the negative first
power would be 1 over tangent. So it’s the
multiplicative inverse, the inverse function, I
think.
I:
OK. And let me try this. If you have this, right? x
to the negative 1? What is that?
LN: 1 overx.
166
I:
Now, if you have a function/ o f x, and then you
decide to do/ to the negative 1. Is this equal to/
inverse ofx? No, I don’t know. I ’m just asking.
And the other question is, is this the same thing
as, right, what is, I mean how is this, all o f this
related? That’s what I ’m trying to get at.
(pause)
I:
Is this, which i s / o f x that quantity to the
negative 1. Is that the same thing as 1 over/ of x?
I mean is that the same thing as the/ to the
negative 1 ofx?
LN: Um, I think I would see these two as being the
same [referring to ( / (x)) and l / / ( x j ]
I:
OK.
LN: But then, but this is the inverse o f the function.
So
I:
OK. And let’s see. Alright. Why don’t we start
this, why don’t we start this way. Can you graph
the inverse tangent o f x?
LN: I hope so! W e’ll find out.
(pause)
I:
Ah. Are you satisfied with that?
LN: I think so.
I:
Why?
LN: Um, my, um, the domain and range sort of work
- well no not really. I was just looking a t.. .this
one.. .and that one piece o f the tangent function. I
would see the domain as negative pi over 2 to pi
over 2. And the range as all the real numbers. So
for the inverse function I would expect that to
b e .. .1 don’t know how to explain it. It makes
sense to me and I think it’s right. But, I don’t
know how to explain it.
I:
OK. But you’re certain that that is the inverse
tangent, right?
LN: Yeah.
I:
So now can we try the inverse cotangent? What
would we do to graph the inverse cotangent?
167
LN: Can I play with this? [working on the calculator]
I:
Mm-hmm. Just tell me what you are doing, so
that we can hear.
LN: Um, I ’m just gonna graph 1 over the tangent o fx
just to look at the cotangent function, (pause) Do
I need parenthesis on this calculator for...
I:
For the x you mean?
LN: For, um, like if I ’m just graphing 1 over tangent,
do I need these parenthesis here? Not for the x
but for the whole thing? [wanted to know i f she
needed parenthesis around tan(x) in 1 over
tan(x)]
I:
No.
LN: OK.
(long pause)
I:
Is there any other way you can think of inverses?
That would help you with the problem.
LN: Um, on the graph or just inverses in general?
1:
Inverses in general, that might give you an idea.
LN: Oh. As far as, like what the inverse does, or? I’m
not sure...
(long pause)
(long pause)
I:
So what you have is inverse cotangent of x is
equal to y and you’re saying cotangent o fy is
equal to x, and then you writing the cotangent
ratio? Cosine o fy to sine o fy is equal to x, OK?
LN: Um,
(pause)
LN: I was going to try to p u ty in terms of, um,
arcsine and arccosine since I have both of those
on my calculator.
I:
I see. I see. Nice strategy, though. That’s
interesting.
(long pause)
168
LN: This is a lot harder than I thought it was going to
be. I really don’t know.
Interview 1 Summary
The interview results show that the interviewees view triangles and functions as
central to the study of trigonometry. The interviewees found working within expert
conceptions to be more difficult than working within their emic perspectives. Only one of
the four interviews correctly identified composition as the underlying operation for
inverting functions. The other four interviewees, two low-knowledge and two highknowledge interviewees, incorrectly accepted a student’s argument that the underlying
operation was multiplication. The same four interviewees could not adequately explain
the effect o f the parameter b in y = sin (bx) without the help o f the graphing calculator
(TI-83+). This seeming lack of understanding o f b highlighted the interviewees’ limited
knowledge o f the frequencies o f sinusoidal functions. All five interviewees correctly
identified the effect o f b in the sinusoid with help from the TI - 83+ graphing calculator.
The interviewees’ ordering of trigonometric topics into a pedagogical sequence
revealed that the interviewees have gaps in their knowledge o f the prerequisite sequence
of these topics. This weakness suggests that the preservice teachers would not be able to
develop a coherent learning trajectory for students. For instance, teaching students about
law of sines or law of cosines before introducing the definitions o f the trigonometric
functions makes little pedagogical sense. The envisioned use o f technology to explore
effects o f parameters and for visual reasoning, professed by the interviewees, are
congruent with reform recommendations for mathematics education (NCTM, 1989,
2001). However, the results also suggest that these preservice teachers are lacking in their
abilities to effectively conceptualize and articulate effective pedagogies that connect and
integrate trigonometric topics.
169
Interview 2
The presentation o f results of the second interview follows the sequence of
activities completed by the interviewees: First, results of the conversations about problem
solving, proof and justification in trigonometry are presented. That is followed by a
discussion of the results o f the interviewee’s responses to the task o f defining radian
measure. A discussion o f the results from a problem situation (model a Ferris wheel ride)
follow the discussion o f the results o f the interviewees’ proofs o f the claim: There are
360° in one revolution. Then a discussion o f the results o f a second proof task {prove that
a + (3 = y; see appendix G, interview 2, question 4) is presented. The presentation of the
results from interview 2 concludes with a discussion o f the results o f a task that asked the
interviewees to generate domain values for specified range values for a sinusoidal
function.
Problem solving, proof and justification
The interviewees displayed knowledge of Polya’s problem-solving methodologies
of understanding the question (or problem), identify givens, identify the familiar, and
relate problem to a fam iliar process. They all stated the aforementioned methodologies
as their starting (entering) approach to problem solving. The interviewees also revealed
uncanny similarities in what keeps them resilient in problem solving. AX enjoys “doing
mathematics problems”. NM is motivated by the challenge mathematics problems present
and he is driven to resolve mathematics problems. Curiosity and the desire to find
solutions to mathematics problems propel ES to persevere in problem solving. LN
claimed that her motivation comes from working on fun mathematics problems and if the
mathematics problem is not fun, then she measures her resilience by “how much it [the
problem] is worth” relative to a test grade or homework grade. AB stated that he is driven
by the desire “to find solutions” to mathematics problems.
In discussions about the roles of justification and proof in trigonometry, the
interviewees revealed that they value justification and proof ideas, because as AX put it,
170
“it helps one understand why things work.” NM, LN, and AB echoed A X ’s views. ES, on
the other hand, argued, “mathematics is based on sets o f rules and definitions.” E S ’s view
on proof reveals a different understanding of the nature o f mathematics relative to her
peers. ES seemed to have made the argument for viewing proof and justification as the
foundation for mathematics. The other interviewees did not profess similar conceptions
and they did not hint that they considered the foundations o f mathematics relative to
proof or justification.
The remainder o f interview two involved tasks which required defining radian
measure, proving that there are 360° in one revolution, solving a Ferris wheel problem,
showing an equality, and determining the domain values that yield a specified set of
range values.
Interview question one: What is the radian measure?
For NM, the conversion between radians and degrees: 2n radians are equal to
360°. When asked to define the quantity that he converted to degrees, he replied, “I will
get a textbook and see how they define it.” ES described the radian measure as another
measure for angles. She also stated that there are 2n radians in a circle. However, she
could not give a definition for radian measure. LN defined radian measure as the
“measure of an angle in terms of n.” For AB, radian measure is the “distance traveled
around a unit circle.” However, he could not describe a unit circle. He was further asked
if the size o f the circle matters. To that AB responded, “radian measure is the angle
traveled through a circle.” The intent o f the question was to gauge whether he understood
that radian measure does not depend on the size o f the circle.
According to AX, radian measure is in terms o f n. She also considered it as the
“angles on the unit circle,” and as “a fraction of n.” Further discussions with AX revealed
that she considered 180° to be an equivalent angle measure as 1 radian. She concluded by
171
stating, “you include n because it is related to a circle.” What follows is a brief account
of AX discussing what she thought the radian measure meant.
I:
No? OK. Alright. So now what w e’ll do is w e’ll
move to the next page and then look at um,
number 1. And the question is, what is the radian
measure? Tell me everything you know about
radian measure.
AX: A radian measure. Um, a radian is in terms o f pi.
And you use it to measure angles of a circle.
Well, angles in general but more specifically on
the unit, you’d use the unit circle to do that.
I:
Alright.
AX: That’s it.
I:
So what is it, then? What is a radian?
AX: Well, it’s, it’s a fraction o f pi.
I:
How so?
AX: Well, um, on your unit circle there are, 180° is
one radian. And 360° is two radians. But you
write it in terms o f pi because it’s on a circle.
1:
OK. Now I have to ask you some more questions,
OK? So now you say 180° is equal to one radian,
360’s gonna be two radians.
AX: Mmm-hmmm.
I:
And then you included pi because it’s related to a
circle? Is that what you’re saying?
AX: Yes.
The results presented here mirror the results on item 7 o f the test o f trigonometric
knowledge in which eleven o f the fourteen study participants accurately provided the
conversions between radian measure and degree measure o f angle. In item 7 o f the test o f
trigonometric knowledge as in question 1 o f the second interview, preservice teachers
could not define radian measure. The results strongly suggest that the preservice teachers
possess a limited understanding o f radian measure, including its versatility, utility in
advanced study o f mathematics, and its non-dimensionality.
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Interview question two: Prove that there are 360° in one
revolution
The purpose o f this activity was to assess interviewees’ knowledge o f definitions
and the fact that definitions need not be proved. AX argued that there are four 90° in one
rotation. She drew four circles and started with 90° clockwise rotation in the first circle
and completed a full cycle with her fourth circle. NM argued that a straight line has 180°
and furthermore four right angles make up one revolution. ES correctly pointed out that
the proof is by definition. LN argued that “2k = 360°, so there are 360° in one
revolution.” LN ’s alternative argument was that there are four 90° angles in one
revolution. AB drew a coordinate system and posited that there are four 90° angles in one
revolution.
The results show that the interviewees understand that there are 360° in one
revolution. However, only one o f them was keenly aware o f the fact that the veracity of
such a claim is by definition. The fact that there are four 90° in one revolution is because
we have defined perpendicular angles as having measures o f 90°. At an even more
profound level, the whole idea is based on the base system o f the Babylonians, which
gave rise to defining a complete revolution as going through 360°. That does not have to
be the case. For example, one can as easily use the grad measure that measures 400° in
one revolution. Or one can use 2n radians in one revolution.
Interview question three: Ferris wheel problem
The interviewees grappled with the following situation in question three:
A person is seated on a Ferris wheel of radius 100ft that makes one
rotation every 30 seconds. The center o f the wheel is 150ft above the
ground at any time t o f a 2-minute ride. Assume uniform speed from the
beginning to the end o f the ride and that the person is at the level o f the
center o f the wheel and headed up when the ride begins. Find a function
that models the height of the person, with time t as the independent
variable. What if the person starts at the lowest point when the ride
begins?
AX, a low-knowledge interviewee, gave the following response to this question:
h = 100 sin t +150. She used a diagram of a circle to represent the problem situation. She
173
used the TI-83+ to check if sinO = 0. She struggled with changing time into an angular
measure. And she did not know how to use the frequency o f the ride to generate the
expression for the argument.
NM, a high-knowledge interviewee gave f { t ) = 50sin(t) + 150. He also used a
diagram of a circle to represent the wheel, but he also drew a sine function to model the
motion of the rider. He confused the diameter with the radius, hence the amplitude o f 50
in his solution. And NM did not know how to apply the frequency o f the ride to generate
an expression for the argument.
ES, a high-knowledge interviewee, gave the same function as AX. Again, ES did
not know how to use the frequency of the ride to generate an expression for the argument.
ES also used a circle to represent the wheel and a sine graph to represent the motion of
the rider.
LN, a high-knowledge interviewee, and AB, a low-knowledge interviewee,
correctly modeled the motion o f the rider in the first part with height =
100sin(jtf/15) +150. LN also generated the correct model for part (b) of the problem that
involved a phase shift with the function height = 100sin(jrr/15 —tt/ 2) +150. LN used a
circle to represent the wheel, she constructed a T-table, used the period o f the ride, and
she translated time in seconds into radians. AB completed all his work on the TI-83+. He
did not show much work.
The conclusion is that only two (one low-knowledge, and one high-knowledge) of
the five interviewees were able to resolve part (a) of the problem. And only one (the same
high-knowledge that resolved part (a)) was able to answer part (b) correctly. Thus, the
results seem to indicate the interviewees’ knowledge o f periodic phenomena that can be
solved using high school level trigonometry is limited. Only one o f the five interviewees
was able to correctly complete the task. The inability o f the interviewees to correctly
model the frequency o f the ride again highlights their limited understanding o f sinusoids.
Moreover, the lack o f appropriate resolution o f the second part o f question 3, by four of
174
the interviewees, was further indication that phase-shift, a transformational idea, was not
adequately understood. Thus the preservice teachers seemed to struggle with articulating
the effects of transformations on sinusoidal functions as was revealed by the results of
this interview question and the study participants’ poor to average performances on items
12 and 14 o f the test o f trigonometric knowledge, and proposition 13 o f card sort 1.
Interview question four: A proof question
Question 4 o f interview 2 asked interviewees to prove that a + (3 = y in a 1-by-3
rectangular figure (see appendix G, interview two, question four for the exact wording of
and diagram for the problem). None of the interviewees were able to resolve this question
without the aid of the calculator. The difficulties that the interviewees experienced,
without the use o f the calculator, were due to the fact that use o f addition formulas was
required, if you approached the problem without the calculator. With the calculator, two
(one low-knowledge and one high-knowledge) interviewees resolved the problem and
proved the desired result. AX used sine functions of the angles and their inverses to show
that a + {3 = y = 45°. ES, on the other hand, resorted to tangent functions o f the angles
and their inverse values to show that a + /3 = y = 45°.
NM, LN, and AB could not resolve the problem. NM could not move outside the
geometric format o f the problem into a trigonometric format. He was able to find the
sines o f the angles but he could not bring this idea to fruition because he was working
within a geometric interpretation that did not avail him the flexibility to take the inverse
functions and compare the measures of the angles. NM also tried law o f sines, and
attempted to resolve the problem by using the areas o f the squares and related rectangles.
AB argued that y = 45°, ft = 30°, and a = 15° by proportional reasoning. He incorrectly
argued that since tan (3 = 1/2 and ta n a = 1/3, the measures of /3 and a are in a 2:1 ratio.
Hence, AB concluded that 13 = 30°, and a = 15°. He was not able to move beyond this
175
viewpoint and consider alternatives. His commitment to his initial intuition of some
proportionality interfered with his ability to resolve question 4 o f the second interview.
LN should have been able to resolve this problem because she seemed to have all
the necessary knowledge and skills. She was only a step away from the solution but could
not escape an apparent cognitive block. LN had the following work: Arctangent(l/3) = a ,
Arctangent(l/2) = /3, and Arctangent(l) = y. She commented, “ideally I will like to show
that Arctangent(l/3) + Arctangent(l/2) = Arctangent(l).” But she could not bring her
desire to fruition. Repeated probes such as what are you trying to show? proved
ineffective as illustrated in the following excerpt.
I:
Alright. Now w e’re on number 4. And you have
a diagram in front o f you. The objective is to
prove that alpha plus theta is equal to gamma.
And, um, I can stop the tape and have you work a
little bit.
LN: That would probably be a good idea.
I:
OK
(turns off tape)
I:
OK, w e’re on number 4. And you have a diagram
and it says you’re supposed to show that alpha
plus theta is equal to gamma. What have you
found so far?
LN: Um, I ’ve found the tangents o f alpha, theta, and
gamma which were, the diagram looks to me like
it’s 3, um, squares whose sides all have equal
lengths. Um, yeah, the sides all have lengths, say
x. So I have the tangent of alpha as 1/3, the
tangent o f beta as 1/2, and the tangent of gamma
as 1. But now I ’m not even, well I guess,
I:
What are you looking for?
LN: OK. Never mind. It is. So, OK. Sorry.
I:
You are doubting whether they are squares?
LN: Yeah. But then I read the problem.
I:
OK.
(lines 1153 - 1236)
Interview 2, #4. The diagram
below shows three equal
squares, with anglesa,/3,y
as marked. Prove thal a + fi = y.
176
LN: So (laughs). The tangent o f alpha is 1/3, beta is
1/2, and gamma is one. So then translated that to
the arctangent o f 1/3 equals alpha, arctangent of
1/2 equals beta, and arctangent o f 1 equals
gamma. And now I ’m not sure where I ’m going
with that.
I:
What do you want to show?
LN: I want to show that alpha plus beta equals
gamma.
I:
OK.
LN: So, ideally I would like to show that the
arctangent of 1/3 plus the arctangent o f 1/2
equals the arctangent o f 1.
I:
Mmm-hmm.
LN: Nothing’s really jumping out at me.
(long pause)
LN: I just labeled the side o f each square as being one
unit. So the bottom o f the figure would be 3 units
and I’m finding the lengths o f the hypotenuses of
each of the, um, right triangles formed by the left
side o f the bottom and th e ...
(long pause)
LN: I’m probably just going in circles, but I’m just
looking at the sine of each o f the angles.
(long pause)
I:
What are you thinking? What are you finding
with the sine?
LN: Um, found the sines and then I was looking at the
cosines and I was sort o f hoping that the
cosines.. .now I ’m not even sure.
(long pause)
LN: I want to use some kind o f equation like sine
squared plus cosine squared equals 1 ‘cuz that’s
the only one I remember. And I don’t know if
there’s any that would even help me if I could
remember them, so.
I:
I see. Sine squared plus cosine squared is equal to
1.
177
LN: I was trying, I don’t . ..
I:
How would we use that here?
LN: How would I?
I:
Just the sine squared plus cosine squared is equal
to 1.
LN: Oh. I w as.. .trying.. .1 don’t know. I don’t think it
would, it’s right anyway.
(pause)
LN: I don’t remember what I was doing. I was trying
to find some relationship betw een.. .alpha and
beta so that I could.. .oh, I don’t know. So that I
would have the same angle so that I could use the
formula.
I:
Mmmm-hmmmm.
I:
OK. At this point, um, why don’t I ask, what if
we used the graphing calculator, would you be
able to solve it, in any way? Would that help, if
it’s a computation problem, that’s what I ’m
thinking.
LN: No.
I:
Not at all?
LN: Oh, I think if I knew what I was doing it might
help, but I don’t.
I:
OK.
(long pause)
LN: thought I had something but then.. .don’t think
it’s it.
I:
I thought you had something too. (laughs)
LN: No?
I:
OK.
One possible conclusion that could be made from the above excerpt is that LN did
not have deep knowledge o f inverse trigonometric functions. If she did, she would have
been able to compute the inverse trigonometric functions, especially considering the fact
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that she had the TI-83+ at her disposal. The fact that none o f the interviewees were able
to answer this question without the aid of electronic computing devices indicated that
these interviewees had limited knowledge o f addition formulas and inverse trigonometric
functions and their uses in trigonometric computations.
Interview question five: Find domain values for a given set
of range values
In this question, interviewees were asked to resolve the following question:
Given / ( x ) = 3sin(2x —jc/3), when is f(x) > 0 for 0 < x < 42
Four out o f the five interviewees could not resolve the question without the aid of
the graphing calculator (TI-83+). Two more interviewees (AX and LN) were able to
resolve the question with the help o f the graphing calculator. ES and AB were not
successful at this problem, with or without the graphing calculator. NM was the lone
interviewee who used algebra with the period of n to resolve the question. The following
excerpt reflects N M ’s thinking strategies.
I:
Alright, let’s look at #5. In this one you have the
function and we want to know where the function
is positive.
NM: If the function is positive?
I:
Mmm-hmmm. Greater than zero.
NM: OK. [working problem on paper, N M describes
his work later on in the excerpt].
I:
Do you want some time to take a look at it before
we talk? Alright.
I:
W e’re back to number 5, and can you explain
what you were thinking and how you got your
solutions?
NM: I had, we wanted to find where the function of 3
sine of (2x minus (pi over 3)) is greater than 0
between an interval from 0 to 4, so first I took the
3 in front, this is indifferent because it is a
positive number, so it’s not going to affect
whether or not the answer is positive or negative.
So whenever sine o f (2x minus (pi over 3)) is
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negative that x be negative, otherwise it will be
positive. So I plugged 0 in for x and I got sine of
negative pi over 3, which is negative because it’s
in the fourth quadrant, no, let me think, no it’s in
the third quadrant, yeah but no, fourth. Anyway
that doesn’t matter. I knew that it was negative.
So when I set (2x minus (pi over 3)) equal to 0
and solved for x because once that was equal to 0
I knew that the function would be equal to 0.
I:
Why?
NM: Because once 2x minus pi over 3 is equal to 0,
then sine o f 0 is 0, so the function is going to 0 . 1
just noticed this now that the problem asked for/
o f x greater than, strictly greater than 0, and I
solved it for, I guess I can put it in the other ones
that are not inclusive. So when I solved that, I
came up with pi over 6 for x. And then I solved it
again for (2.x minus (pi over 3)) for pi because I
knew once it hit pi inside, once (2x minus (pi
over 3)) hit pi that after that point the sign would
be negative again. Then I got 2 pi over 3, then I
subtracted (2 pi over 3) minus (pi over 6) to
come up with the period and I got (pi over 2) for
the period, so I added (pi over 3) to (2 pi over 3),
got (7 pi over 6), and then there was less than (pi
over 2) remaining in the stated interval so I had
my intervals off o f x greater than 0 from (pi over
6) to (2 pi over 3) and the second one is (7 pi
over 6) to 4.
I:
OK. Do you want to review or revise any work
you did? That was a very good explanation.
NM: I don’t think so.
I:
OK. Do you mind putting the, your solutions in a
box for me? Just put a box around it, just so that
when I go over that, I can identify your answer.
The fact that only one interviewee was able to resolve question five, using
algebraic methods, was a further illustration o f the interviewees’ limited understanding of
the effects transformations have on sinusoidal functions. Coupled with earlier results that
have shown the same pattern, it should be obvious that these preservice teachers lack
deep knowledge o f sine waves, and the effects o f vertical stretch and shrink, phase shifts,
horizontal stretch and shrink on the domain and range o f sinusoids.
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Five profiles o f understanding
The presentation o f the five cases follows a case-by-case analysis along the
categories o f subject matter content knowledge, pedagogical content knowledge, and
envisioned practice. The delineation of these categories developed in Shulman (1986,
1987) was employed as guide for further refinement o f the discussion. The interview
results were used to support the various tasks (concept maps, card sorts, and test of
trigonometric knowledge). In addition, the interview data were used to support the
discussion o f each case’s envisioned use o f technology, problem-solving orientation, and
envisioned use o f proof and justification in the teaching and learning o f trigonometry.
The presentation o f the cases does not follow in any special order. The presentation of
each case begins with a brief description o f the knowledge level o f that case, and then the
case’s depth o f conceptual understanding is presented. A case summary concludes each
presentation. At the end o f the fifth presentation, a summary o f summaries is presented to
close the presentation o f the results o f the study.
AX
AX was classified as a low trigonometry knowledge interviewee due to her poor
showing on the test o f trigonometric knowledge (23/64) and the first card sort task (6/15).
As has been alluded to, these two tasks were considered more objective than the concept
maps and card sort 2 for selecting participants for the interviews.
A X ’s subject matter content knowledge of trigonometry was fraught with
conceptual holes. It lacked deep conceptual connections and integration, and it failed to
satisfy what Ma (1999) called Profound Understanding o f Fundamental Mathematics
(PUFM); here referred to as Profound Understanding o f Fundamental Trigonometric
Ideas and Concepts (PUFTIC). A X ’s knowledge o f functions lacked depth. For example,
she accepted as always-true proposition 12 (for a trigonometric function there are
situations when a particular domain value has two range values) o f card sort 1. Related
to this was her limited knowledge o f trigonometric functions. She claimed that some of
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the six basic trigonometric functions might not be periodic. In trigonometry, periodicity
and the characteristic behaviors o f the six basic trigonometric functions are considered to
be conceptual underpinning for learning and teaching trigonometry because o f their
fundamental nature. AX also gave incorrect responses to other basic notions such as what
radian measure means and how to convert between radian and degree measures. She
claimed that one radian is equal to 180°, because n comes with radians; sort o f a unit of
measurement for radian measure
A X ’s knowledge o f other fundamental trigonometric concepts was mixed. She
gave correct definitions o f the six basic trigonometric ratios, but she incorrectly applied
those ratios to a non-right triangle in item 10 of the test o f trigonometric knowledge
indicating that her knowledge o f the ratios was not profound. AX gave correct
conventional demarcation o f the plane into four quadrants, and she also stated the correct
conventions o f angular rotations that are used in trigonometry. However, her knowledge
of the behaviors o f the six basic trigonometric functions in the four quadrants was weak,
because she assumed that coterminal angles have equal angle measures and she did not
demonstrate the fact that trigonometric functions of coterminal angles yield the same
range values. For example, cos (n/4) = cos (97t/4) = cos (-7tt/4), and so on. However, n/4
J 9rc/4 f -lnl4 .
A X ’s knowledge o f the unit circle, a truly encapsulating tool in trigonometry, was
not deep. She correctly defined the unit circle as a circle with radius o f 1 unit, but she
could not explain the utility o f the unit circle in learning and teaching trigonometry at the
high school level. The two special right triangles (30°-60°-90°, and 45-45°-90°) are
important tools for computing exact values of trigonometric functions o f arguments that
are multiples or fractional parts of 30°, 45°, and 60°. Used in conjunction with angular
rotation, the behavior of the six basic trigonometric functions in the four quadrants, and
the unit circle, many computations can be greatly simplified and resolved without use of
electronic computing devices. So knowledge o f the two special right triangles is
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fundamental to learning and teaching trigonometry at the high school level. AX was able
to generate the two special right triangle with their standard measurements, but she was
unable to apply them in problem solving situation where the side opposite the 60° angle
had been reduced by a factor o f v/3 . Her inability to transfer knowledge o f the standard
special right triangles to the problem situation was an indication o f A X ’s fragmented
understanding of the special right triangles.
A X ’s knowledge o f inverse trigonometric functions was limited. She equated
reciprocals with inverse functions. For example, she classified the cosecant function as
the inverse o f sine. And when asked why, she stated that “it is like you flip it, sine is
equal to O/H, and cosecant is H/O ... on the graphing calculator, you use Sin~'{ ) key.”
It is quite possible that the confusion experienced by AX and other participants regarding
reciprocal and inverse functions might be a case o f over-generalizing the notation
x"1= 1/x for the multiplicative inverse o f any nonzero real number x to functional
situations. However, such notational interference suggest that A X ’s pedagogical content
knowledge was not robust and will need redevelopment if she intends to be successful at
unpacking trigonometric content knowledge for students.
AX described trigonometry as numbers. She stated that numbers “go to triangles,
circles and graphs” referring to the measurements o f angles and lengths o f sides of
triangles. Further discussion with AX revealed that she conceived o f the trigonometric
quantities sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x) as computational tools for
measuring the dimensions o f the sides and angles of triangles. This coupled with A X ’s
errant definition o f function necessitated the conclusion that AX did not have a fully
developed understanding o f trigonometric functions or a robust pedagogical content
knowledge about trigonometric functions. Another area where A X ’s limited pedagogical
content knowledge was observed was in card sort 2. She had an overall agreement o f
approximately 41% (14/34) with Hirsch and Schoen (1990) and Senk et al. (1998). Her
pedagogical sequence o f trigonometric topics contained several missteps such as placing
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identities and fundam ental properties ahead o f domain, range, similar right triangles,
angular rotations, periodicity, and coterminal angles. She also placed trigonometric
equations ahead of using cosine and sine to define the other trigonometric functions and
inverse functions. Her pedagogical ordering o f trigonometric topics also included the
misstep of placing law o f sines and law o f cosines ahead o f sin2(0) + cos2(0) = 1. Put
together, these errors indicated that AX lacked pedagogical and prerequisite integrities.
AX envisioned using technology to facilitate parameter exploration, such as
investigating the effects of a, b, c, and d in asin[bx - c ) + d. She also intimated that she
would use electronic graphing calculators to help student visualize graphs o f
trigonometric functions. The results of the interview also revealed that A X ’s envisioned
practice involved 5 stages. At the first stage, she envisioned teaching about graphing,
algebra, shapes, and geometry. Then she would move on to sine, cosine and the unit
circle in her second stage. Her third stage comprised o f relating stages 1 and 2 to
triangles. Solving triangles and graphing sine and cosine formed the fourth stage o f her
envisioned practice. The final stage of her envisioned practice involved applying stages 1,
2, 3 and 4 to real life problems. She also envisioned using proof, justification and
reasoning processes to help students understand where “it came from”, help them
understand “why something is true, how it is connected to other things you learned”, in
reference to trigonometric concepts and ideas.
In summary, AX was able to accomplish some things and she failed at others. The
nature o f her knowledge o f trigonometry was complex and variegated. She successfully
proved the Pythagorean theorem (identity) for trigonometric functions, but she could not
adequately define the radian measure or convert between degrees and radians. She was
able to recall the special triangles in item 2 o f the test o f trigonometric knowledge, but
she could not use the special triangles to resolve item 13 of the same test. She could not
adequately reason with transformational ideas but she used the graphing calculator to
resolve a constructed student’s task rife with misconceptions about horizontal shrink and
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stretch. Her envisioned practice was progressive and reflected ideals supported by
NCTM, but her subject matter content knowledge and her pedagogical content
knowledge performances did not support her envisioned practice. The results support her
classification as a participant with low knowledge o f trigonometry.
NM
NM was considered a participant with high knowledge o f trigonometry, relative to
the 14 study participants, for the purposes o f the interviews. He had the second highest
score (46/64) on the test o f trigonometric knowledge and the highest score (10/15) on
card sort 1. However, his concepts maps were two o f the more poorly developed ones.
They were disconnected and there were no clear focal concepts. Moreover, his concept
maps contained few explicit relationships between nodes (items).
The results of the interviews showed that N M ’s subject matter content knowledge
of trigonometry was mixed. He was aware of the conventional numbering o f the
quadrants of the plane. He correctly recalled the law of sines. He correctly proved the
Pythagorean theorem (identity) for trigonometric functions. He was aware o f the
conversion formula between degrees and radians, but he could not define radian measure.
His knowledge of the unit circle and its utility in trigonometry was weak. NM provided
correct definitions o f the unit circle, but he did not provide uses o f the unit circle as was
requested in item 8 o f the test of trigonometric knowledge. N M ’s knowledge of
trigonometric functions was also weak. He confused periodicity with continuity. He
argued that sine and cosine functions were periodic, but he was not sure whether tangent
and the other three reciprocal functions were periodic since they have asymptotic
behaviors. He further stated, “since the functions approach infinity at the asymptotes,
tangent, for example, would not be periodic.” He did not understand what a fundamental
period was. He also stated that the proposition fo r a trigonometric function there are
situations when a particular domain value has two range values was true sometimes. He
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also incorrectly concluded that the period of the ratio o f two o f the six basic
trigonometric functions/ and g was equal to the ratio o f the period off to the period o f g.
These misconceptions indicated that NM lacked depth and connectedness in his
knowledge of trigonometric functions.
N M ’s organization of trigonometric topics was disconnected and fragmented. His
concept maps (see figures K-7 and K-8 in appendix K) contained three disconnected
clusters with few stated relationships between nodes (items). However, he attempted to
use functions and the unit circle as his focal organizing concepts, albeit the effort was
neither integrated nor coherent.
The results o f N M ’s ordering o f trigonometric topics into a pedagogical sequence
indicated that he lacked pedagogical and prerequisite integrities. Hence his pedagogical
content knowledge was deemed poor. His overall agreement with Hirsch and Schoen
(1990) and Senk et al. (1998) was approximately 38% (13/34). He did not consider
prerequisites when he put the 34 topics into an order that students ought to be exposed to
trigonometry. In fact, he had emulated the order he remembered learning trigonometric
concepts in high school. His pedagogical sequence contained missteps such as placing
law o f sines, law o f cosines, solving triangles ahead of degree measure, radian measure
and definition o f trigonometric ratios o f the sides o f right triangles. He also misplaced
trigonometric expressions and trigonometric equations before topics such as domain,
range, inverse functions, using cosine and sine to define other trigonometric functions,
sin2(0) + cos2(0) = I, and the fundam ental identities and properties. Further evidence of
his weak pedagogical content knowledge was gleaned from the content o f the cards he
excluded from his pedagogical sequence. He stated in the interview that he did not know
what to do with them, where to put them, or how to connect them to other topics,
referring to the following six cards: even and odd functions, transformations, coterminal
angles, addition form ulas, cofunctionality, and half-angle form ulas.
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N M ’s envisioned practice was also progressive, as A X ’s, perhaps highlighting
their currency with the ideals o f NCTM. NM professed the goal o f using electronic
technologies, including graphing calculators, to help students develop visual
comprehension o f trigonometric concepts. However, he also stated that he would first
have students develop conceptual understanding o f trigonometric concepts before
employing electronic technologies. His professed problem solving strategies involved the
processes delineated by Polya in his 1945 book: How to solve it. He also stated, as did
AX, that he would employ proof, justification, and reasoning processes to help students
get at the “why” questions, and help students understand “why” and not just “how.”
ES
ES was classified as having high knowledge o f trigonometry relative to the 14
study participants. Her score (48/64) was the highest score on the test o f trigonometric
knowledge. Her score (9/15) on card sort 1 was the second highest, second only to N M ’s
score o f (10/15). However, E S’s knowledge o f trigonometry was mixed. She correctly
stated the conventional demarcation o f the plane into four quadrants. She produced the
two special right triangles complete with angular measures and side-lengths. She
provided the correct law of sines, but she did not respond to item 4 of the test of
trigonometric knowledge regarding the law o f cosines. She was aware o f the convention
of clockwise and counterclockwise angular rotations. She correctly proved that
sin2(x) + cos2(x) = 1. She provided the correct conversion between radian and degree
measures, but she did not attempt to define radian measure as was requested in item 7 of
the test of trigonometric knowledge. She provided correct definition for the unit circle,
but she was weak in her knowledge o f the utility o f the unit circle in trigonometry. For
ES, the unit circle was used “to create right triangles that always have a hypotenuse o f 1
unit length.” She gave incorrect response to the reasoning question about maximizing the
area o f a triangle given lengths o f two of the sides in item 16 o f the test of trigonometric
187
knowledge. She correctly identified the graphs o f Tan~l(x ), C o G '(x ), and S in ~ \x). But
she could not produce the graph of Cot~l(x ) with the TI-83+ in the first interview.
ES argued incorrectly that the period of the ratio of two o f the six basic
trigonometric functions/ and g was equal to the ratio o f the period o f f to the period of g.
She also incorrectly classified as always true the proposition given triangle o f sides a, b,
and c the trigonometric functions are ratios o f the lengths o f two o f the sides. She also
claimed that graphs o f trigonometric functions are sinusoidal. But when the word
sinusoidal was explained to ES, she re-classified that proposition as true sometimes. She
incorrectly stated that cofunctions are reciprocal functions. She could not resolve item 15
of the test of trigonometric knowledge or question 5 o f interview 2, which dealt with a
trigonometric equation and a trigonometric inequality, respectively. She also had
difficulties with the Ferris wheel problem of interview 2. In general, ES displayed a much
more robust knowledge of trigonometry than her peers, but still, her knowledge was
mixed.
ES’s organization o f trigonometry was one o f the better ones. E S’s concept maps
(see figures K-5 and K-6 in appendix K) were considered mixed. Her concept maps
contained mostly early or intermediate level concepts, and there were misconceptions in
them. ES’s notion o f “related functions” in her concept maps was not clearly articulated,
so it was not obvious whether she wanted to specify cofunctions, inverse functions or
reciprocal functions (see figure K-5 in appendix K). In concept map 2, she labeled
cotangent, cosecant and secant as cofunction/reciprocal functions; and she connected the
six basic trigonometric functions to an inverse node without explicitly stating what she
meant by that. However, in the interview, she provided more details about those
relationships. She stated incorrectly that tangent and cotangent have cofunction/reciprocal
relationship, cosine and secant have cofunction/reciprocal relationship, and sine and
cosecant have cofunction/reciprocal relationship.
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ES displayed a weak pedagogical content knowledge. Her pedagogical sequence
of trigonometric topics had an overall agreement o f approximately 53% (18/34), the third
highest score, with Hirsch and Schoen (1990) and Senk et al. (1998). She had the most
(8/12) agreement with the intermediate concepts and lower agreements with early (4/9)
and advanced (6/13) concepts. ES’s pedagogical sequence lacked pedagogical and
prerequisite integrities. There were several misplaced concepts in the learning trajectory
embodied in ES’s pedagogical sequence. For example, she had incorrectly placed solving
right triangles before any mention of degrees, radians, or trigonometric functions. In
fact, solving right triangles was ES’s second card in the sequence, after similar right
triangles. She misplaced reference angles ahead o f quadrants and angular rotations. She
also misplaced trigonometric expressions and equations ahead o f using sine and cosine to
define other trigonometric functions, domain, range, even and odd functions,
cofunctionality, degrees, radians, periodicity, sin2(0) + cos2(0) = 1, inverse trigonometric
functions, and coterminal angles. Furthermore, she misplaced addition and half-angle
form ulas ahead o f law o f sines and cosines. Finally, if we implement ES’s pedagogical
sequence, students would first learn about derived identities before they are exposed to
sin2(0) + cos2(0) = 1, a truly fundamental identity.
ES’s envisioned practice was progressive, forward looking, and it reflected the
nature of mathematics. She envisioned using electronic technologies to aid students’
visualizations o f trigonometric functions. She mentioned that she would use the
Geometer’s Sketchpad (Key Curriculum Press, 2001) as a tool to help students tie unit
circle to the study o f trigonometric functions. ES’s stated problem solving process
meshed with Polya’s conceptualizations. She intimated that she would endeavor to help
students understand mathematics as a system based on rules and definitions. For her, as
well as the other four interviewees, reasoning, justification, and proof were avenues for
students to reconcile misconceptions and develop deep mathematical understanding.
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LN
For the purposes o f the interview, LN was another participant that was classified
as having high knowledge o f trigonometry relative to the 14 study participants. Her score
(37/64) was the third highest score on the test o f trigonometric knowledge. And her score
(9/15) on the first card sort was the second highest score. However, her subject matter
content knowledge was mixed. She provided the correct conventional demarcation o f the
plane into four quadrants. She gave the correct angular measures o f the two special right
triangles, but did not provide the length measures for the sides. She gave correct
conversion formula between radian and degree measures, but she could not define radian
measure. Her knowledge of inverse trigonometric functions was mixed, as well. She
argued incorrectly that the period o f the ratio of two o f the six basic trigonometric
functions/ and g is equal to the ratio o f the period o f f to the period o f g. She did not
attempt item 17 o f the test of trigonometric knowledge, but in discussing her concept
maps (see figures K -1 and K-2 in appendix K) in the interview, it was apparent that LN
had studied up on some o f the ideas presented in phase one o f the study and she
confirmed that she had looked up some o f the ideas prior to the interview. Hence she
displayed a much more robust understanding o f inverse functions in her interviews. For
example, she changed the connective reciprocal between the group o f six basic
trigonometric functions and their arc functions to inverse. And she also correctly argued
that the student thinking in task 1 was incorrect since the student had used multiplication
as the underlying operation for functions. However, she could not explain the use of the
identity 1 in student task 1. Moreover, she could not adequately resolve question 4 of
interview 2 on her own, because she was not able to apply the inverse trigonometric
functions to resolve the question. She was only able to resolve that question after heavy
prodding and hints, after we had concluded the interview (officially). She was also not
able to produce a graph o f the inverse cotangent function on the TI-83+ graphing
calculator. Hence her knowledge of inverse trigonometric functions remained mixed.
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LN’s organization o f trigonometry was not robust. She used trigonometric
functions, their uses in right triangles to measure lengths o f sides, and their graphical
representation as her focal concepts. Her concept maps were connected, but the concepts
included in her maps were mostly o f the early or intermediate types as shown in figure 8.
LN displayed weak pedagogical and weak prerequisite integrities in her
pedagogical sequence o f trigonometric topics. Her pedagogical sequence of trigonometric
topics had an overall agreement of approximately 56% (19/34), the second highest
agreement o f the 14 study participants, with Hirsch and Schoen (1990) and Senk et al.
(1998). Her pedagogical sequence agreement was strongest (75%) at the intermediate
concepts level and weakest (33%) at the early concepts level. Thus, it seems that LN
would have started students off with developmentally inappropriate content. Her
pedagogical sequence also contained several misplaced concepts and ideas. For example,
she misplaced solving triangles ahead of any mention o f trigonometry, just after degree
measure. She also misplaced sin2(0) + cos2(0) = 1 ahead o f definition o f trigonometric
ratios and functions. Students would have encountered reference angles well before they
were introduced to angular rotation, and quadrants in LN ’s pedagogical sequence. She
misplaced trigonometric expressions ahead of using cosine and sine to define other
trigonometric functions. She also misplaced derived identities ahead offundam ental
identities and properties. Finally, if we implement LN’s pedagogical sequence, students
will first solve trigonometric equations before they have the opportunity to learn about
coterminal angles and cofunction ideas.
LN ’s envisioned practice was similar in progressiveness and forward looking as
the other interviewees. She envisioned using graphing calculators for parameter
exploration to investigate effects of amplitude, shifts and shrinks on graphs of
trigonometric functions. She also envisioned approaching problem solving ala Polya. She
stated that she would use proof, justification and reasoning processes to help students
understand “why certain ideas work, instead of taking it at face-value.” Finally, she stated
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that she would also use reasoning processes to help students connect current ideas to
previous learning.
AB
AB was classified as having low knowledge o f trigonometry due to his score
(24/64), the fourth lowest score of fourteen scores, on the test of trigonometric
knowledge and his score (6/15) on card sort 1. A B’s knowledge o f trigonometry was
weak across the board. He was not aware o f the conventional demarcation o f the plane
into four quadrants. He gave the definition o f the sine ratio in place o f the definition of
the law o f sines. His response to item 5 of the test of trigonometry also indicated that he
was not aware of the conventions o f angular rotation used in the study o f trigonometry.
He could not define radian measure or the unit circle. And his knowledge o f the utility of
the unit circle was in its infancy. He stated that the unit circle provides a “way of
measuring angular movement or distance, ... , and helps students learn common results.”
A B’s knowledge o f trigonometric functions was weak. His knowledge of even
and odd functions was non-existent. His knowledge o f inverse functions was mixed. For
example, he argued that S in ~ \2) exists in the real number system, but he correctly
identified the graphs o f 7an~‘(x), C os~ \x), and Sin~l(x). But, again he incorrectly
argued in his concept maps and in the first interview that inverse functions and reciprocal
functions held the same meaning. He was aware that the six basic trigonometric functions
were all periodic. He argued incorrectly that the period of the ratio o f two o f the six basic
trigonometric functions/ and g is equal to the ratio o f the period off to the period o f g.
He incorrectly associated SOHCAHTOA with any triangle, indicating a lack of
understanding o f the necessary conditions for the trigonometric ratios. He also incorrectly
argued that the six basic trigonometric functions were sinusoidal.
A B ’s organization o f trigonometry was weak. He centered his concept maps (see
figures K-4 and K-5 in appendix K) around measuring angles. Hence his characterization
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of the addition formulas as formulas for adding angles in his pedagogical sequence of
trigonometric topics. His concept maps revealed that his knowledge o f trigonometric
functions and properties was not reified. He was still working at the elementary
geometrical level o f angles.
A B ’s pedagogical content knowledge was weak. His pedagogical sequence of
trigonometric topics had an overall agreement of approximately 40% (14/34) with Hirsch
and Schoen (1990) and Senk et al. (1998). His agreement at the advanced concepts level
was extremely low (2/13). He did not consider prerequisites in determining the order in
which to place the cards. AB had several misplaced cards in his pedagogical sequence,
indicating lack of pedagogical and prerequisite integrities. The first card in his
pedagogical sequence was addition form ulas because as he indicated in the interview, it
reminded him of formulas for adding angles. He misplaced trigonometric expressions
and equations ahead o f identities, inverse functions, definitions o f trigonometric ratios
and functions, periodicity, domain, and range. The law o f sines and law o f cosines were
considered prior to definition o f trigonometric ratios and functions. He misplaced inverse
trigonometric functions before definitions o f trigonometric ratios and functions, domain,
range, even and odd functions, and one-to-one functions. He also misplaced reference
angles ahead o f quadrants and angular rotations (clockwise and counterclockwise
rotations).
The interview results provided further evidence o f A B ’s weak pedagogical
content knowledge. He stated in interview 1 that he had not previously encountered ideas
such as coterminal angles, derived identities, or fundam ental period. And he was not sure
how even and odd functions, range, rate o f change, and radians were implicated in
trigonometry. Moreover, he was not sure where and how to fit continuity and identities
into his organizational scheme of trigonometry.
A B ’s envisioned practice was as optimistic, progressive and forward looking as
the other interviewees. He stated that he would use electronic technologies, such as
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graphing technologies, to help students visualize trigonometric ideas. However, he
intimated that he would first teach concepts without the aid o f electronic technologies.
But if students do not understand the concepts presented without the aid of electronic
technologies, then he would marshal the resources o f the electronic technologies to help
the students. In this sense he was like NM. His problem solving processes were also
related to Polya’s conceptualizations. He also stated that he planned to use proof,
justification, and reasoning processes to help students understand “what is behind the
concept”, and to help students understand “why answers and concepts are correct.”
Summary o f cases
The case profiles show that these preservice teachers, who by all accounts have
successfully completed advanced mathematics courses with advanced mathematics
content, struggled with fundamental ideas of trigonometry that students are exposed to at
the high school level. The preservice teachers had clear deficiencies in their knowledge o f
high school level trigonometry. They were weak in subject matter content knowledge and
pedagogical content knowledge o f trigonometry. Their envisioned practices reflected the
ideals o f NCTM, but one must wonder if talk is not always cheaper than actual action.
Moreover, their knowledge o f trigonometry does not support the loftiness o f their
envisioned practices. All the cases intimated in the interviews that they would definitely
look up these concepts before they actually taught them, no doubt an effort to reassure the
researcher that they were once knowledgeable in high school level trigonometry. They
also stated that their knowledge o f trigonometry was weak because their last encounter
with the underpinning and conceptual framework of trigonometry was in high school,
which for them was, at the minimum, four years prior to the time o f the study. Depending
on their student teaching assignments, some o f these preservice teachers might encounter
serious trigonometric content as student teachers and they will then need to redevelop
some o f their lost competencies. But for some, they may not have that opportunity and
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might end up teaching trigonometry without ever being re-introduced to it from “an
advanced perspective.” For those preservice teachers, their grasp o f trigonometric
concepts will not be far removed from the students that they will be called upon to teach.
Therefore, the preservice teachers’ envisioned practices would have to be re-envisioned
at a later time, after they have become better acquainted with students’ misconceptions,
and common cognitive difficulties associated with trigonometry. Hopefully, at that later
time the preservice teachers will be able to develop better learning trajectories for their
students.
Summary of results
This section discusses study results using M a’s (1999) categorization o f teachers’
knowledge along the dimensions o f connectedness, multiple perspectives, basic ideas,
and longitudinal coherence, and Even’s (1990) framework for analyzing teachers’ subject
matter knowledge along the dimensions o f essential features, different representations,
alternative ways o f approaching, strength o f concept, basic repertoire, knowledge and
understanding o f a concept, and knowledge about mathematics. Brief descriptions of
these categories are presented next, which are then followed by an integrated
summarization o f the findings.
For Ma, teachers that possess connectedness o f knowledge are able to integrate
subject matter content topics. If teachers have multiple perspectives o f mathematics, then
they should be able to model flexibility and versatility in approaching mathematical
problem situations for their students. Additionally, teachers with multiple perspectives
should also be able to highlight advantages and disadvantages o f different ways of
approaching problems. Knowledge of basic ideas is the knowledge o f the “simple but
powerful basic concepts and principles of mathematics” (p. 122). The teacher with
knowledge o f the basic ideas in mathematics ought to be able to use such knowledge to
facilitate students’ mathematical inquiry. Finally, Ma argued that, the knowledge and
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articulation o f the pre-, present, and post-ramifications o f mathematical concepts and
ideas in the school curriculum allow teachers to better scaffold students’ learning and
inquiry. Furthermore, teachers with the knowledge o f where mathematical concepts and
ideas are situated in the continuum o f school mathematics, again, are better suited to help
students attain a coherent understanding o f the mathematics that they encounter.
For Even, teachers understand the essential features o f school mathematics if they
have knowledge o f the critical attributes and prototypes o f any given school mathematics
concept. An understanding o f the different ways that mathematical concepts can be
manifested and the ability to navigate amongst the varied representational systems
encapsulate the different representations category. Alternative ways o f approaching
involves the teacher’s ability to apply mathematics both to mathematical and nonmathematical situations. If teachers grasp the scope o f utility and limitations o f given
mathematical concepts, and are able to apply such knowledge to render mathematics
useful and applicable, then they would have manifested the knowledge o f the strength o f
the concept. Furthermore, Even argued that teachers ought to have a basic repertoire of
routinized essential and fundamental mathematics. An indication o f knowledge and
understanding o f a concept is the teacher’s ability to integrate both conceptual and
procedural knowledge and use both readily in problem solving situations. Finally,
teachers who understand school mathematics have knowledge about mathematics: They
have an understanding o f the nature of mathematics, its truth structures, and understand
mathematics’ progression, accretion, and development.
To further organize study results into a coherent picture, this section is subdivided
into the following four subsections that unify both M a’s and Even’s perspectives: (1)
Basic ideas, essential features, and basic repertoire; (2) Multiple perspectives, different
representations, and alternative ways of approaching; (3) Connectedness, strength o f the
concept, and knowledge o f a concept; and finally (4) Longitudinal coherence and
knowledge about mathematics.
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Knowledge o f basic ideas, essential features, and basic
repertoire of trigonometry
The majority o f the fourteen participants incorrectly classified propositions
involving periodicity ideas. Ten o f the fourteen participants displayed limited
understanding that the six trigonometric ratios encapsulated in the mnemonic
SOHCAHTOA applies only to right triangles. Eight o f the fourteen participants could not
adequately define a function. Ten o f the fourteen participants did understand that the six
basic trigonometric functions are periodic. Thirteen o f the fourteen participants also
understood that the general theorem o f Pythagoras ( a 2 + b 2 = c 2) applies only to right
triangles. Thirteen o f the fourteen participants correctly drew the conventional
demarcation of the plane into quadrants, but only four o f the fourteen participants gave
accurate accounts o f the two special triangles. Five participants accurately resolved a 30°30°-90° triangle presented in a problem situation. Ten o f the fourteen participants gave
accurate accounts o f the conventions o f clockwise and counterclockwise angular
rotations. None o f the participants gave an accurate definition o f the radian measure.
However, eleven of the fourteen participants gave accurate conversions between radians
and degrees. Ten of the fourteen participants gave a correct definition o f the unit circle.
In summary, the preservice teachers seemed to possess basic and essential
trigonometric concepts and ideas that they can build on. There were areas o f serious
deficiencies, such as knowledge o f radian measure, but the preservice teachers should be
able to easily read about those things and update their understanding o f the essentials of
high school level trigonometry.
Multiple perspectives, different representations, and
alternative ways o f approaching trigonometry (including
problem solving ideas)
The 14 study participants could have capitalized on the multiple entry points
afforded by the some o f the tasks. Items 6, 10, 11, 13, 14, 15, and 16 o f the test of
trigonometric knowledge (appendix B) could have been approached and resolved from at
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least two perspectives. For example, item 14 (domain and range o f 3sin[2x + jt/ 3] + 2) of
the test of trigonometric knowledge could have been approached from a transformation
perspective and tabular perspective that could then have been connected to a graphical
perspective. Seven o f the 14 participants correctly resolved item 6, three o f the 14
participants correctly resolved item 10, one of the 14 participants correctly resolved item
11, four o f the 14 participants correctly resolved item 13, five o f the study participants
correctly resolved item 14, and two o f the 14 study participants resolved item 16. Perhaps
if the participants have been more flexible in their approaches it is conceivable that they
would have generated many more correct responses. The lack of flexibility was also
observed in the interviewees’ responses to the problem-solving tasks in the interviews.
The performance o f the interviewees on student tasks one and two, the Ferris wheel
problem, and the “three squares” problem showed that they were lacking in their ability
to approach questions from different perspectives and to use different representations.
The interviewees’ seeming lack o f flexibility, in the non-calculator environment, may
have been a result of their limited subject matter content knowledge of trigonometry.
Such flexibility is predicated on having strong subject matter content knowledge and the
ability to draw upon it.
Connectedness, strength o f trigonometric concepts,
knowledge and understanding o f trigonometric concepts
Eleven o f fourteen participants displayed a weak knowledge o f inverse
trigonometric functions. Twelve o f the fourteen participants agreed incorrectly that
inverses o f functions are always functions. Ten of fourteen participants understood that
graphs of the six basic trigonometric functions are sometimes sinusoidal. However, their
knowledge of sinusoids was not adequately reflected in their responses to a part of item
14 of the test of trigonometry that involved determining the domain and range of a
complex sinusoidal function. Nine of 14 participants gave accurate domain values for the
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given trigonometric function. However, only five were able to generate the range values
for the same trigonometric function. Twelve o f the 14 participants understood that
periodic phenomena are not always represented by the graphs o f the six basic
trigonometric functions. Only three o f the 14 participants gave accurate accounts of the
law of sines. The law o f cosines met an even worse fate: none o f the participants
remembered what it was. H alf o f the 14 participants proved the fundamental Pythagorean
identity: sin2 x + cos2 x = 1. Only three of 14 participants presented acceptable ideas on
the utility of a unit circle in the study o f trigonometry. The preservice teachers’ limited
knowledge o f the unit circle is significant because the unit circle is an apparatus for
connecting and tying together many fundamental and foundational trigonometric ideas.
Ten of the 14 participants recognized the falsity of the claim that
sin(a + /3) = sin a + sin/3. Hence arguing that the addition formulas are not distributive as
multiplication over addition is. The aforestated discussions reveal that these preservice
teachers operated in a fragmented framework o f trigonometry. They were not quite able
to connect their knowledge bits into a coherent whole. Moreover, as all the results have
shown, the preservice teachers displayed limited strength and knowledge o f trigonometric
concepts.
Longitudinal coherence and knowledge o f trigonometry
Only one o f five interviewees realized that definitions are accepted as true and
need no proof to establish their veracity. The participants’ pedagogical and prerequisite
sequencing of trigonometric topics achieved a maximal agreement o f approximately 62%
(21/34) with expert conceptions. There were several misplaced cards in the sequence that
caused the researcher to conclude that the participants had not sufficiently considered
prerequisite ideas. The results of the interviews confirmed that indeed the participants
had not considered the prerequisite integrity of their sequences. Therefore, one can
conclude that these preservice teachers’ knowledge o f trigonometric concepts and ideas
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was not adequately robust that they could contemplate satisfactorily the connections
between concepts and ideas.
Summary
This chapter presented the results from the analysis o f the data from the two
phases o f the study. The general picture painted is that o f a mathematics content area that
needs to be re-emphasized in our high schools and in preservice education courses for the
future cadre o f high school mathematics teachers. What the preservice teachers were able
to recall and use in problem solving involving trigonometry was limited. They showed
limited understanding o f basic ideas, weakness in their knowledge and understanding of
trigonometric concepts, and an inability to operate flexibly among different perspectives
and representations. And finally, they showed that their knowledge o f trigonometry was
not sufficiently robust that they could envision a meaningful learning trajectory for
students to follow in their study o f high school level trigonometry. In chapter V, how the
data relates to each o f the study’s research questions is discussed and related to the
flexibility, adaptability, and robustness of the preservice teachers knowledge of
trigonometry.
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CHAPTER V
CONCLUSIONS, DISCUSSION, LIMITATIONS AND
IMPLICATIONS
In general, the available literature on preservice teachers’ and inservice teachers’
knowledge o f school mathematics suggests that their knowledge o f school mathematics is
not as robust and as connected as the mathematics education profession would prefer
(Even, 1989; Bolte, 1993; Howald, 1998; Ma, 1999; Ball, Lubienski, &Mewbom, 2001).
The purpose o f this study was to extend the literature base in this area and contribute to
an area (trigonometry) that has received minimal attention for a long time, starting in the
New Math Era. The specific aims o f the study were to (1) characterize the depth of
preservice secondary school mathematics teachers’ subject matter content knowledge and
pedagogical content knowledge o f trigonometry in the school mathematics curriculum;
(2) explore how preservice secondary school mathematics teachers envision applying
their content knowledge in teaching trigonometry; and (3) provide a description of the
relationships among preservice secondary school mathematics teachers’ subject matter
content knowledge, pedagogical content knowledge, and their envisioned practice.
This chapter presents an overview and discussion o f results and their relation to
the research questions. This is followed by a discussion o f the limitations o f the study.
The chapter concludes with suggestions for further research and ways o f enhancing
studies similar to the present study.
Overview and discussion o f the results
This section presents a discussion of how the reported results addressed the
research questions and concludes with a discussion of two conjectures: (1) the weak
conceptual understanding o f trigonometry shown in the results o f phase one data is a
result o f a loss o f information over time or (2) these preservice secondary mathematics
teachers did not adequately learn or understand trigonometry when they were exposed to
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the ideas in high school or in their subsequent use o f trigonometry in their college
mathematics courses.
What content knowledge of trigonometry do preservice
secondary school mathematics teachers possess?
Definitions and terminology
The participants displayed some understanding o f the definitions of sine, cosine,
and tangents as SOHCAHTOA. That is, in a right triangle, the sine o f either acute angle
is found by computing the ratio o f the length o f the opposite side to the length o f the
hypotenuse. For cosine, we compute the ratio o f the lengths o f the adjacent side to the
hypotenuse. And for tangent, we compute the ratio of the lengths o f the opposite side to
the adjacent side. However, not all o f the participants understood or recalled that these
definitions of trigonometric ratios apply only to right triangles. There was also
overwhelming confusion with the inverse symbol in Cos~l{x), S in ~ \x ), T an~ \x),
Sec~l{x), Csc"1(x ), and Cot"'(x). The participants knew that these were the inverse
functions but regularly and incorrectly interpreted the inverse functions as reciprocal
functions. Part o f the confusion may have arisen due to the fact that for a non-zero real
number x, the multiplicative inverse is written x ' 1 and it is equal to —, the reciprocal of
x
x. However, for functions, the underlying operation is not multiplication and the
symbolism f ~ \ x ) represents the inverse of / ( x ) relative to composition and not
multiplication. This notational problem indicated that these preservice teachers did not
have a well-developed pedagogical content knowledge o f trigonometry.
Degree and radian measures
The participants showed an understanding o f the methods for converting between
degree and radian measures, but they could not define radian measure. Thus they showed
an instrumental understanding of radians, although they could use it in problem solving
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because o f its relation to degree measure. There was also some misunderstanding o f the
presence o f n in radian measures. There was an instance in the interview when n was
misunderstood as the unit for the radian measure and the interviewee argued that 1 radian
equaled 180°. To summarize, these preservice teachers were much more comfortable with
degree measure than radian measure. And they can move easily between degrees and
radians, but they do not have a deep understanding o f what radian measure means.
Co-functions
The study did not find any participants with deep conceptual knowledge of co­
functions. The lack o f understanding of the relations between the complementary pairs
(sine - cosine, tangent - cotangent, and secant - coseceant) was quite apparent. Neither
the concept maps nor the interviews showed any sign that the participants even thought
about the connections and the use of the prefix co for the complementary nature of the
aforementioned function pairs. Knowledge of this idea allows one to meaningfully
engage the relationships among these functions and provides versatility and adaptability
in problem solving. It is also helpful in simplifying expressions to yield equivalent yet
much simpler expressions and thus facilitate proofs and problem resolution. The co­
function idea also shows up in discussions about inverse functions and their rates of
change, where the rates of change o f the inverse co-functions are opposites of one
another. More specific to this study, the task of graphing the inverse cotangent function in
interview 1 required knowledge o f cofunctions and since this idea was not properly
understood, the preservice teachers were not able to complete that task.
Angles o f rotation, coterminal angles, and reference angles
The participants showed a thorough understanding o f the conventional
counterclockwise and clockwise rotations and the signing o f the size of the angles of
rotation. There was a considerably weak understanding of coterminal angles, however,
and how to use the idea in solving trigonometric equations. Weakness in this area will
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invariably affect one’s ability to work with inverse trigonometric functions. In the case of
finding the angle measures for which a particular function attains a certain value, the
inverse function only gives its principal values. To generate more solutions, we resort to
finding co-terminating angles by employing the periodicity o f the functions and also the
behavior o f the function in the four quadrants.
Special angles (30°. 45°. 60°). their triangles, and their use
to simplify computation
There was weak understanding o f the two special right triangles (30°-60°-90° &
45°-45°-90°) that are employed to resolve and simplify trigonometric computations. There
was also confusion about special angles, reference angles, quadrantal angles and the use
of these ideas to simplify expressions and computations. As a starting point, knowledge
of special angles and special triangles is crucial to finding values o f trigonometric
functions without scientific or graphing calculators. Coupled with knowledge of addition
formulas, and by extension formulas for half-angles, one can begin to generate values of
trigonometric functions of small or large values of arguments without the aid of
electronic devices.
Trigonometric functions and their graphs
The participants showed considerable weakness in their knowledge of sinusoids.
Their knowledge o f inverse trigonometric functions was also weak. Moreover, their
graphical understanding o f inverse functions as reflections o f the original functions about
the y = x line was also lacking. Knowledge of inverse functions allows us to be versatile
and adaptive in problem-solving situations. An important action in mathematics is the
ability to undo or reverse actions in order to reclaim the beginning. Inverses allow one to
do that. Inverse functions, to be particular, guarantee our return into a unique domain.
This difficulty with inverse functions has also been shown to be prevalent with other
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groups o f preservice teachers and some in-service teachers (Even, 1989; Bolte, 1993;
Howald, 1998).
Domain and range
The participants had a strong knowledge o f the domain o f the basic trigonometric
functions. However, they showed a lack o f understanding o f how to generate the range of
trigonometric functions. The weak knowledge o f the range o f functions may be
attributable to shallow knowledge o f the effects of transformations on sinusoids as noted
in the next section.
Transformation o f trigonometric functions
The participants’ knowledge o f the effect o f transformations on trigonometric
functions was at best average. The interviews revealed that the preservice teachers have
deficiencies in their understanding of the effects of parameters on sinusoids. Moreover,
the preservice teachers could not recall how to transform trigonometric function into their
inverses. That is, they showed weak understanding o f the effect o f reflecting a graph of a
function about the line y = x.
Even and odd functions
The participants seemed to associate even and odd with numbers and did not
necessarily associate oddness or evenness with functions. This supports a finding that the
preservice teachers have weak understanding of the prerequisite knowledge needed to
understand inverse functions. Another area o f concern with the preservice teachers’
knowledge o f trigonometry was restriction o f domains o f trigonometric functions so that
they yield one-to-one functions, whose inverses are functions.
Laws of cosines and sines
Most preservice teachers could not recall the laws o f sines and cosines. This lack
o f recall may not be as serious as the inability to use the laws if provided with them. A
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future study could investigate such problem-solving episodes. Furthermore, future studies
could also investigate preservice teachers’ knowledge o f the range o f applicability o f the
laws of sines and cosines.
Trigonometric identities
The participants showed weak understanding of trigonometric identities. The
concept maps revealed that the preservice teachers have some misconceptions and
misunderstanding of what identities are. However, they showed that they could prove the
fundamental identity, the theorem of Pythagoras involving sine and cosine functions.
Algebra and calculus o f trigonometry
The participants did not know the addition formulas for trigonometry. The
interviewees could not resolve a question about proving that the sum o f two angles was
equal to a third, without the aid of graphing calculators. A resolution o f that task, without
the calculator, would have involved the use o f addition formulas and the interviewees
could not come up with any addition formula. The study did not investigate proof ideas in
relation to addition formulas, difference formulas, or half-angle formulas.
The use of trigonometry in solving and modeling
mathematical and real-world situations
The preservive teachers showed some capacity to resolve triangles. But even here,
the participants’ skills were not uniformly strong. For example, only one participant
correctly identified and adequately explained the reasons why an illusory triangle in item
11 of the test of trigonometry was not actually a triangle. Two other participants’
responses indicated that they knew there was a problem with the triangle, but they could
not interpret their computational fallacies to argue for the non-existence o f the purported
triangle. The participants tended to approach the resolution o f triangles by introducing
altitudes so that the triangles were always divided into two smaller right triangles. A
participant applied the law o f sines to item 10 o f the test o f trigonometric knowledge but
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then could not bring it to fruition because the participant could not generate the exact
values for sine o f 30° or 45°. The interviews revealed that the participants have some
knowledge o f how sinusoids could be employed to model real-world situations. The
interviewees correctly identified the Ferris wheel problem situation as being modeled by
a sinusoidal function. All five interviewees generated sinusoidal functions o f the form
asin(bx - c ) + d. However, only two were successful at using the frequency of the ride to
resolve the first part o f that problem. So these preservice teachers’ knowledge of
sinusoids was not as robust as one would have expected. In modeling periodic
phenomena, as in the Ferris wheel problem, a related and necessary knowledge is that of
coterminal angles. An adequate knowledge of coterminal angle should entail knowledge
that any trigonometric function o f coterminal angles yields equal range values. This
knowledge is also very useful in generating values other than the principal values that
inverse trigonometric function provide. None o f the 14 participants demonstrated an
adequate knowledge o f coterminal angles, and hence the idea o f periodicity was a
problematic area for them.
What pedagogical content knowledge o f trigonometry do
preservice secondary school mathematics teachers possess?
What prerequisite knowledge is necessary for the learning
of trigonometry?
The results o f the second card sorting activity showed that the participants have
some understanding o f the sequential coherence o f trigonometric topics. They did best
with basic and intermediate topics. Not unexpected, their agreement with expert
sequencing was smallest with the advanced topics. The interview revealed that the
process used by the participants to sequence the topics relied almost exclusively on the
preservice teachers’ recollection o f how they were exposed to the trigonometric topics in
previous courses. They did not necessarily consider prerequisites.
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Weak prerequisite integrity is indicative o f the preservice teachers’ lack o f deep
structural articulation o f trigonometry. The lack o f prerequisite integrity hindered the
preservice teachers’ abilities to develop viable learning trajectories for students in the
area of trigonometry. The presence of weak prerequisite integrity may also indicate a lack
of experience with the subject matter from the vantage point o f a teacher. Thus weak
integrity may highlight weak pedagogical content knowledge. That is, a weak
prerequisite integrity may signal that the preservice teacher is not cognizant o f the feeder
knowledge structures required for understanding trigonometric concepts. The lack of
understanding o f feeder knowledge may also imply that the participant is not aware of the
cognitive developmental trajectory inherent in learning trigonometry. Moreover, a
teacher with weak prerequisite integrity may not be able to anticipate cognitive
difficulties students might encounter. A further implication is that teachers with weak
prerequisite integrity may possess a diminished grasp o f the structural connections among
trigonometric topics at the high school level. Thus the teachers’ delivery of trigonometric
content would be impaired and students would not be meaningfully exposed to a coherent
and meaningful body of trigonometric concepts.
How do the preservice secondary mathematics teachers
understand multiple representations that will prove useful
to unpacking the content of trigonometry for students?
The preservice teachers were quite comfortable discussing trigonometric
situations that involved geometric interpretations: Right triangle trigonometry. They did
less well with functional representations and graphical representations. Their dependence
on the geometric representational system limited their flexibility, adaptability, and
responsiveness in problem-solving situations. The results o f the interview items on
constructed students’ tasks, the exercise on graphing the inverse cotangent o f x, and the
Ferris wheel problem, were indications of the impairments imposed on the preservice
teachers by their limited ability to work in multiple perspectives and representations. For
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instance, the question about graphing the inverse cotangent functions could have been
resolved by graphing the cotangent function, restricting the domain to 0 < x < n, and
reflecting that portion over the line y = x. Or, the interviewees could have generated table
o f values (or ordered pairs (x, y)) for y = cot x , switched the ordered pairs into (y, x) and
then graphed the new ordered pairs to generate the inverse cotangent function, with
restriction of course.
How do preservice secondary mathematics teachers
sequence and organize trigonometric concepts for teaching?
The previous discussions have argued that the preservice teachers have
considerable weakness in their pedagogical integrity. They relied on their high school
experiences to guide their ordering of trigonometric topics into a pedagogical sequence.
There were several cases o f misplaced trigonometric topics in the sequences produced;
misplacements so severe that they would have caused the collapse o f the pedagogy o f the
trigonometric concepts along the lines suggested by the sequences.
Do the sequence and organization of the concepts anticipate
both students’ preconceptions and misconceptions, and
possible approaches to help students overcome such
misunderstanding?
The sequence and organization o f the topics did not indicate that the preservice
teachers considered students’ preconceptions or misconceptions. In fact, the preservice
teachers’ sequencing o f the provided trigonometric topics revealed their own
misconceptions.
If preservice secondary mathematics teachers were
presented with difficulties that students might encounter,
how would they help students get better conceptualizations
of trigonometry?
The interview results were mixed. The preservice teachers showed weak
understanding o f possible misconceptions when they did not use graphing calculators.
More were able to provide better explanations about students’ misconceptions when they
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were allowed to use graphing calculators as part o f their explanations. However, it is not
certain that they clearly understood the conceptual underpinnings o f the changes they saw
when they used the graphing calculator.
How are preservice secondary school mathematics
teachers’ content and pedagogical content knowledge of
trigonometry organized?
The concept maps in appendix K are representative of the concept maps produced
by the fourteen preservice teachers in phase one o f the study. The samples show that
preservice teachers have some fundamental, yet weak understanding o f trigonometry.
They understood right triangle trigonometry reasonably well. They displayed diminished
knowledge o f the functional approach to trigonometry. Some o f the relationships shown
on their concept maps were incorrect and some functional ideas such as evenness and
oddness were misconstrued. Their organization o f trigonometry into pedagogical
sequences did not fare better. There was a serious disregard for the prerequisites that are
necessary for a meaningful learning trajectory.
How do preservice secondary school mathematics teachers
envision teaching trigonometry?
How will they develop the six basic trigonometric ratios?
The results indicate that the preservice teachers will accurately define the six
basic trigonometric ratios via right triangles. However, most preservice teachers did not
make the appropriate connections to similar triangles, which supports understanding of
similar right triangles, and which in turn yields the notions o f trigonometric ratios. There
was little discussion o f functional definitions or the use o f the unit circle.
What pedagogical approaches (didactic or heuristic) will
the preservice secondary mathematics teachers employ?
The data was weak on this aspect of the preservice teachers’ knowledge of
trigonometry. However, the interview data indicate that the preservice teachers anticipate
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using a problem-solving based approach. They showed good understanding of Polya’s
model of problem solving. They also argued that proof is important in trigonometry and
mathematics in general and they posited that proof processes reveal the veracity of
claims. They also indicated that they would use electronic technologies to explore the
conceptual underpinnings o f trigonometry. They emphasized parameter exploration as an
example of such conceptual explorations.
How are preservice secondary school mathematics
teachers’ content and pedagogical content knowledge of
trigonometry related to their envisioned application o f their
content and pedagogical content knowledge in mathematics
classrooms?
The results show that the preservice teachers have forgotten a lot about
trigonometry since they last encountered the topics either in high school or in their
college classes. They showed deficiencies in definitions, terminology, basic concepts,
intermediate concepts and advanced concepts. They showed strengths in conventional use
of quadrants, angle o f rotations, SOHCAHTOA, and right triangle trigonometry. Their
envisioned practices involve use o f technologies to explore conceptual underpinnings of
trigonometry, use o f problem solving approaches, and the use o f proof and justification
ideas to concretize students’ understanding.
The interviewees recognized that they had deficiencies in their knowledge of
trigonometry. They intimated that they would consult textbooks before they teach the
kinds of topics that were explored in this study. Some o f them reported that they may not
have studied trigonometry conceptually and would like to change that for their students.
The researcher concluded that the weak conceptual understanding o f trigonometry
suggested by the results o f phase one data was partially a result o f forgetting information
over time. The preservice teachers had not seriously revisited the basic notions or the
conceptual foundations o f trigonometry, for most o f them, since high school. Moreover,
there was evidence that these preservice secondary mathematics teachers did not
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adequately learn or understand the conceptual foundations o f trigonometry when they
were exposed to the ideas in high school or in their subsequent use o f trigonometry in
their college mathematics courses. The preservice teachers’ inability to define radian
measure, apply special right triangles to solve problems, distinguish between inverse
trigonometric functions and reciprocal functions, their lack o f knowledge o f coterminal,
reference angles, cofunctions, and their limited knowledge o f periodicity, and range ideas
suggested that these preservice teachers may have only developed an inadequate
conceptual understanding o f trigonometry.
Limitations
The assessment o f the preservice secondary mathematics teachers’ pedagogical
knowledge was not situated in actual practice and thus may not have reflected the
participants’ full understanding o f the pedagogy of trigonometry. The researcher
attempted to use the preservice teachers’ envisioned practices as a proxy for the
preservice teachers’ classroom pedagogical practices. However, the envisioned practice
component o f the study did not yield a robust picture as was intended. This was due to
the limitations imposed by the weaknesses of the participants’ knowledge of
trigonometry. The intended discussion of implementation was relegated once it became
obvious from phase one o f the study that the preservice teachers’ knowledge of
trigonometry was weak. Nonetheless, the results on the envisioned use o f technology, the
role o f proof and justification in trigonometry, and problem solving practices paint an
optimistic picture o f possible progressive pedagogies that the preservice teachers intend
to employ.
Time allotted for the activities in phase one o f the study was very short in the
scheme of things. That limitation was accounted for in phase two o f the study, where the
interviewees directed the pace and time required to reevaluate and synthesize their work
from phase one of the study. Nonetheless, more time on specific tasks may be needed to
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meaningfully explore the multi-dimensions of the preservice teachers’ knowledge of
trigonometry.
Implications
High school trigonometry (teaching and learning)
The preservice teachers voiced concerns about the way they were taught in high
school. They argued that they had not learned “relationally” and that the difficulties they
displayed may have resulted from their lack of conceptual understanding. The discussion
of trigonometric topics in Algebra II and Precalculus courses at the high school level may
not provide students with the in-depth knowledge o f trigonometric concepts they need for
teaching. Moreover, college courses are not filling in the gaps that these students bring
with them to postsecondary education, because it is often assumed that students have the
essential understanding o f those topics.
As college and high school teachers, we may not be sufficiently stressing
definitions, say the radian measure and its non-dimensionality, or the importance of right
triangles in the study of trigonometry. We may not be focusing on such fundamental
ideas as cofunctions, coterminal angles, periodicity, and the algebra o f trigonometric
functions including prerequisite knowledge for understanding inverse trigonometric
functions. Other areas that might need additional attention include addition formulas,
difference formulas, half angle formulas, sinusoids and transformations on trigonometric
functions.
Further emphases need be placed on solving problems from multiple perspectives.
There should be a refocus o f effort to emphasize the importance of the special angles and
the related special right triangles. There needs to be further discussion o f composition as
the underlying operation on functions. Related to that, there should be discussions of
conventional restrictions o f the domains o f the trigonometric functions that yield inverses
that are functions.
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Preservice teacher education
There was a considerable lack o f trigonometric knowledge on the part of the
preservice teachers. The argument put forth by the preservice teachers that they had not
been exposed to the foundational ideas in their college mathematics courses is a criticism
that the mathematics education profession needs to consider. And it would seem that the
profession is moving in that direction with the promotion o f activities surrounding
providing capstone courses that reexamine high school mathematics content from an
advanced perspective (CBMS, 2001; Usiskin, et al., 2003). It is the researcher’s hope that
the mathematics education profession will continue and sustain the push to re-acquaint
preservice teachers with the fundamental mathematics ideas that they will be teaching.
Furthermore, mathematics educators need to help preservice teachers
reconceptualize school mathematics. The preservice teachers ought to have the
opportunity to develop integrated curricula knowledge of high school mathematics. They
should explore the conceptual sequencing and organization o f topics that are studied at
the school level. The researcher applauds the use of professional development schools
and wishes that they were adequately supported, utilized and expanded at the high school
level. Professional development schools have the potential to provide preservice teachers
with opportunities to develop pedagogical content knowledge, curricular knowledge,
knowledge o f students’ cognition, and awareness of the types o f preconceptions and
misconceptions that students bring to mathematics learning.
Concluding remarks and suggestions for further research
The results o f this study provide insight into the status and organization of
preservice teachers’ knowledge o f school mathematics. It was not intended and should
not be read as contributing to the negative findings o f previous studies that have
investigated the same population. Rather, the results o f this study should be viewed as an
additional pane on a window through which we can explore what preservice teachers
know. The study encompassed a large content domain and it has delivered on its goals.
214
What comes next will be even more important than this study. This study was bold in the
sense that it examined an area o f school mathematics that has seen its share of the school
curriculum diminish over the years. This researcher argues, as did Markel (1982) and
Hirsch, Weinhold, and Nichols (1991), that knowledge o f this content area is vital if
students are to mathematicize the world around them, problem solve, and develop an
appreciation for the relevance and utility of mathematics.
There are many possible extensions of this study. One can investigate preservice
secondary mathematics teachers’ knowledge of trigonometry more deeply, but restructure
the investigation to account for growth. For instance, the researcher envisions a
longitudinal study that takes a snap shot of incoming preservice teachers’ knowledge of
trigonometry, provide intervention in methods courses to increase their knowledge,
posttest them at the end o f the semester, and continue with differing degrees of
interventions and posttests throughout the preservice years. And this can be done in other
mathematics content areas as well.
Alternatively, one can focus on experienced teachers. Related studies can
investigate the differences in conceptions o f trigonometry by expert teachers, novice
teachers, and preservice teachers. One can also subdivide the study and focus on specific
aspects such as the subject matter content knowledge or pedagogical content knowledge.
One could carry out descriptive studies o f teachers’ in-class practices when engaged in
teaching trigonometric topics. A possible extension o f a descriptive observational study
could entail an inservicing after initial observations, conducting further observations and
delineating the growth that was achieved. A researcher could decide instead to explore
finer grain size understanding o f smaller components o f trigonometry such as right
triangle trigonometry, functional approaches, applications o f trigonometry, or advanced
topics (vectors, complex numbers, calculus).
It is the researcher’s hope and dream that sooner rather than later, studies on
preservice teachers’ and inservice teachers’ understanding o f school mathematics will
215
begin to reflect a positive and strong knowledge base. To get there, we must begin to
effect changes in preservice education and continue to expand partnerships with schools
to provide valuable learning experiences for teachers and educators.
216
APPENDIX A
CONSENT FORM
217
Project Title: Preservice Secondary School Mathematics Teachers’
Understanding o f School Mathematics
Investigator(s): Cos Fi
Purpose
This is a research study. The purpose of this research study is to assess the
content knowledge, pedagogical content knowledge, and envisioned practice of
preservice secondary school mathematics teachers in school mathematics.
We are inviting you to participate in this research study because you are a
preservice secondary mathematics teacher.
Procedures
If you agree to participate, your involvement will last for no more than three
hours at three separate meetings. However, if you are selected to participate in the
interview portion o f the study, then you will be interviewed for an additional hour.
The following procedures are involved in this study. If you decide to participate in
the study, here are some o f the things you may be asked to complete:
1) A subject-area test that will last about one hour to one and a half hour. I will
take some notes as you take the test. I will only answer clarification questions
and will not answer questions about procedures or definitions.
2) Two card-sort activities will last about 30 minutes. In the first card sort, you
will be asked to sequence mathematical concepts on index cards into an order
you think students should be exposed to those concepts. In the second card
sort, you will be asked to sort mathematical propositions on index cards into
three piles: Always true, True Sometimes, and Never True.
3) Two concept-mapping activities will last about one hour. In the first concept
mapping activity, you will be asked to generate mathematical terms associated
with a specific mathematical strand, and then use the terms to construct a
218
hierarchical concept map of that mathematical strand. In the second concept
mapping activity, you will be provided with the researcher generated terms
and asked to use the list of terms to construct a second map o f the same
mathematical strand. However, you are permitted to supplement and add to
the list o f terms.
4) Based on the initial analysis of the subject-matter test, the concept maps, and
the card sorts, participants will be selected for a one-on-one interview. If you
are selected for the interview, then you will be interviewed for about an hour.
The interview will be audio taped, transcribed and interpreted. I will also take
note o f your responses and actions during the interview.
Risks
The possible risks associated with participating in this research project are as
follows: There are no foreseeable risks associated with participation in this research
project.
Benefits
There may be no personal benefit for participating in this study. However it is
hoped that, in the future, society and the mathematics education field could benefit from
this study by gaining better understanding of preservice teachers’ knowledge o f school
mathematics.
Costs and compensation
You will not have any costs for participating in this research project.
You will not be compensated for participating in this research project.
Confidentiality
Records of participation in this research project will be kept confidential to the
extent permitted by law. However, federal government regulatory agencies and the
219
University Institutional Review Board (a committee that reviews and approves research
studies) may inspect and copy records pertaining to this research. It is possible that these
records could contain information that personally identifies you. In the event of any
report or publication from this study, your identity will not be disclosed. Results will be
reported in a summarized manner in such a way that you cannot be identified.
If you are selected for the interview
Audio or visual recording
By initialing in the space provided, you verify that you have been told that audio
recordings will be generated during the course of this study. The audiotapes will serve as
reference and verification tools for the researcher’s notes and your responses to interview
questions and activities. The audiotapes will be retained until after the final acceptance
and publication of the dissertation; at which time they will be destroyed.
_________________Subject’s initials
Voluntary participation
Taking part in this research study is voluntary. You may choose not to take part
at all. If you agree to participate in this study, you may stop participating at any time. If
you decide not to take part, or if you stop participating at any time, your decision will not
result in any penalty or loss o f benefits to which you may otherwise be entitled.
Questions
Questions are encouraged. If you have any questions about this research project,
please contact: Cos Fi, XXX-XXX-XXXX; or email: XXXXX. You can also contact my
academic advisor Professor Douglas Grouws at XXX-XXX-XXXX, or email: XXXXX.
If you have questions about the rights of research subjects or research related injury,
please contact the Human Subjects Office, The University, City, State, Zip Code,
Telephone, or e-mail.
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Your signature indicates that this research study has been explained to you, that
your questions have been answered, and that you agree to take part in this study. You
will receive a copy o f this form.
Subject's Name (printed):
(Date)
(Signature o f Subject)
Investigator statement
I have discussed the above points with the subject or, where appropriate, with the
subject’s legally authorized representative. It is my opinion that the subject understands
the risks, benefits, and procedures involved with participation in this research study.
(Signature o f Person who Obtained Consent)
(Date)
221
APPENDIX B
TEST OF TRIGONOMETRIC KNOWLEDGE4
4 The test of trigonometric knowledge and other instruments presented in the appendices
have been collapsed to save paper. They were formatted differently in the actual instruments that
were employed in the study, with one question per page in most instances
222
Main test o f trigonometric knowledge
Directions
No calculator is allowed.
You should have received a test booklet, and some blank sheets o f paper for your
work. All items in the envelope that you received should indicate your identity. If you
need extra sheets of paper, extra paper will be provided.
This is a test of trigonometric knowledge. As such it contains items that are basic,
intermediate, and advanced. Answer all questions to the best o f your ability. If an
explanation or justification is required, give an algebraic, or geometric justification.
Graphs can be used to help you crystallize your reasoning and explanations. However,
graphs do not represent explanations or justification. State all properties and laws that
you call upon to reach the desired goal. Use standard mathematical conventions. Check
your work before you turn it in (use the envelope provided).
Place all items (including all scratch work) into the envelope before you turn in
your envelope.
223
1. What is the conventional numbering of the quadrants o f a coordinate plane? Draw
a picture with labels for the quadrants.
2. Draw two special triangles commonly used in trigonometry to calculate exact
solutions. Label all angles with their measures.
3. State the law o f sines.
4. State the law o f cosines.
5. What does a negative angle measure represent? Assume that the angle is in
standard position.
9
9
6. Prove that sin x + cos x = l.
7.
a. D efine the radian m easure o f an angle.
b. State the relationship between the radian measure and the degree measure o f an
angle.
[Note that part (a) and part (b) require different answers]
8. a. What is a unit circle?
b. How could you use the unit circle in trigonometry?
9. True or False: Give reasons
a) sin(ct + /3) = sin a + sin/3
________
b) sec2 jc + 1 = tan2 x
________
c) cos(-x) = COSX
________
d) Sm_1(2) exists
________
224
10. Find all missing sides and angles: Show all your work.
11. Find all missing sides and angles: Show all your work.
225
12. Match the graphs shown below with the correct symbolic representation listed at
the right. Assume that the x-scale is one radian and the y-scale is one unit. Place
the letter indicating your choice in the space provided. Some choices will NOT be
used
13. Find all missing sides and angles: Show all your work
226
15. Solve the following trigonometric equations: Find all solutions.
a.
t>.
tan x
—tan j t / 5
s in x = co s x
16. When does a triangle with sides o f known lengths a, and b have the largest area?
Justify your conjecture.
17. Use the following six graphs to answer this question. Choose the graph that
represents the inverse function given below. Write the inverse function in the
label box next to the graph.
a. T a n ~ \x )
b. Cos~l (x)
c. Sin~l (x)
227
228
229
The pre-version o f the test o f trigonometric knowledge
scrutinized by a mathematics professor
1.
Conventionally how do we organize the plane into quadrants? Give a picture with
labels for the quadrants.
2.
Draw two special triangles used in trigonometry to calculate exact solutions.
Label all angles with their magnitudes.
3.
Prove that —— = — = —-— in any triangle with sides a, b, and c.
sin A sinB sinC
4.
Prove that a 2 = b2 +c2 - (2be) cos A in any triangles with sides a, b, and c.
5.
What does a negative angle represent? Assume that the angle is in standard
position.
9
9
6.
Prove that sin x +cos x = 1.
7.
a. D efine the radian m easure o f an angle.
b. State the relationship between the radian measure and the degree measure o f an
angle.
[Note that part (a) and part (b) require different answers]
8.
a. What is a unit circle?
b. How could you use the unit circle in trigonometry?
c. Could you use any circle for the purposes you state in part (b)?
9.
If 0 < a < 90° and 0 < b < 90°, then sin(a) = sin(b) <^> a = b, and cos(a) = cos(b) <=>
a = b, but one o f these is not true if 0 < a < 180° and 0 < b < 180°. Which one is it?
Why?
10.
A person is seated on a Ferris wheel o f radius 100ft that makes one rotation every
30 seconds. The center o f the wheel is 150ft above the ground at any time t o f a 2min ride. Assume uniform speed from the beginning to the end o f the ride and that
the person is at the level of the center of the wheel and headed up when the ride
begins.
230
Find a function that models the height o f the person, with time t as the
independent variable.
11.
12.
True or False: Give reasons
a) sin(a + /J) = sina + sin/3
________
b) sec2 x + 1 = tan2 x
________
c) co s(-x) = COSX
________
d) Sin~l(2) exists
________
e) Tan-1 (200) exists
________
f) C o s m o s '1(2)) = 2
_________
Find all m issin g sides and angles: Show all y o u r w ork.
231
13.
Write YES next to a graph o f a function that you think is periodic, and NO next to
a graph o f a function that you think is not periodic. Assume the graphs continue
indefinitely in the same pattern.
14.
In any acute triangle ABC, show that c = a cos B + b cos A
15.
Find all missing sides and angles: Show all your work.
232
16.
Match the graphs shown below with the correct symbolic representation listed at
the right. Assume that the x-scale and y-scale is one unit. Place the letter indicating your
choice in the space provided. Some choices will NOT be used
233
17.
If sina = a , where 0 < a < 1, express in terms o f a the value o f the other five
trigonometric functions o f a.
18.
Find all missing sides and angles: Show all your work
_ I ^25 - x 2 )
19.Rewrite y = esc Tan 1 ------------ as an algebraic equation that has only x as the
x
/.
V
variable and has no trigonometric function in it.
20.
Graph the following functions: Label and scale the axes.
h (x ) = Sin~l x
21.
The diagram below shows three equal squares, with angles a ,( f y as marked.
Prove that a + /? = y.
JC
22.
Find the domain and range o f 3sin 2x +— +2.
23.
Given /(x ) = 3sin 2 x - ^ , when is f(x) > 0 for 0 < x < 4?
234
24.
25.
Solve the following trigonometric equations: Find all solutions.
cl.
t a n
x
—
t a n x r
b .
s i n
x
=
c o s
2
/5
x
When does a triangle with sides of known lengths a, b and c have the largest area?
Justify your conjecture.
235
APPENDIX C
CARD SORT TASK 1
236
Direction
Trigonometric functions refer to the six basic trigonometric functions: sin
(x), cos (x), tan (x), esc (x), sec (x), and cot (x). Do not combine, or transform the
functions.
The purpose o f this activity is to give you the opportunity to assess the veracity of
trigonometric propositions. You should have 18 index cards in total. Fifteen of the cards
have propositions on them. The remaining three cards are the labels for the three piles,
Always True, True Sometimes, and Never True that you will sort the cards into. Write the
reason why you placed the card in a specific pile on the back of the card. When you
are done, place the label cards on each pile and clamp the piles with the binder-clips
(provided). Place the cards back into the envelope and leave it on your desk for pick-up.
Open the second envelope labeled Card Sort 2. You will have the remainder o f the time
for the card sort.
Always True_______________ True Sometimes____________ Never True
1. Trigonometric functions are periodic
2. 2n radians represent the fundamental period for trigonometric functions
3. 2n radians can be the period o f any trigonometric function
4. If f and g are two trigonometric functions, then the period o f f/g is Per*Q(* ° ^ f
period of g
5. Given triangle of sides a, b, and c the trigonometric functions are ratios o f the lengths
of two o f the sides
6. The general theorem o f Pythagoras ( a 2 + b2 = c 2) applies to triangles of sides a, b, c
7. Graphs of trigonometric functions are sinusoidal
8. One Radian is equal to 180 degrees
9. sin2(0) + cos2(50) = 1
10. The domain of trigonometric functions is the set o f real numbers
11. The inverses of trigonometric functions are functions
237
12. For trigonometric function there are situations when a particular domain value has
two range values
13. When you transform the six basic trigonometric functions in the plane you get other
functions that are also trigonometric functions (not necessarily one o f the basic
trigonometric functions)
14. If a phenomenon is periodic, then the graph o f the phenomenon is the graph o f one of
the six basic trigonometric functions
15. Inverse trigonometric functions yield angle measures.
238
APPENDIX D
CARD SORT TASK 2
239
Description
The main purpose for this activity is to provide you the opportunity to organize
and sequence concepts of trigonometric knowledge into an instructional sequence. The
concepts and ideas are written on the index cards you have received. You may want to
group the cards into clusters o f related concepts and ideas. Then think about which
clusters or concepts and ideas in the clusters should be prerequisite to other concepts and
ideas. To further guide your classification, think about what prerequisite knowledge and
skills students would need to learn and understand the concepts that you have been
provided. Moreover, how do you think they should be exposed to those concepts?
When you finish your sequencing, go back and check your work. Use the binderclip (provided) to secure the sequence that you generate. Put the finished work in the
envelope (provided) with your name on it. Leave the envelopes on your desk for pick-up.
The Ideas and Concepts for Card Sort 2
Addition formulas
Application o f trigonometry
Clockwise and counterclockwise angular rotations
Co-functionality
Coterminal angles
Definition o f trigonometric functions using the unit circle and right triangles
Definition o f trigonometric ratios of the sides o f right triangles
Degree measure
Derived Identities
Domain and range o f functions
Even and Odd functions
Fundamental Identities/Properties
Fundamental period
240
Graphs of trigonometric functions
Half-Angle formulas
Inverse trigonometric functions
Law o f cosines
Law o f sines
One-to-one functions
Periodicity
Pythagorean theorem ( a 1 + b2 = c 2)
Pythagorean theorem for trigonometric functions
Quadrants
Radian measure
Reference angles
Similar right triangles
Solving right triangles
Solving triangles in general
Transformations
Trigonometric equations
Trigonometric expressions
Trigonometric functions o f acute angles
Trigonometric functions o f obtuse angles
Using cosine and sine to define the other trigonometric functions
241
APPENDIX E
CONCEPT MAPS
242
Directions for first concept map
1. First write down all you know about trigonometry on sheets o f paper (sheets are
provided).
2. After you have done that, sort the ideas/terms into clusters according to the extent
of relatedness among the terms.
3. Create a map o f the terms in each cluster. Note that any one term could possible
reside in more than one cluster. Use connecting phrases or words to connect the
terms to create a knowledge web. Use directed arrows or arcs to connect the terms
in the clusters.
4. Connect your clusters; you can use dotted directed arrows or arcs for this purpose.
Again identify/explain your connections with phrases or words.
5. Check your work.
6. Place your map into the envelope (your first concept map must be handed in
before you start the second concept map).
7. At the conclusion of the second map, you will have a five-minute break before
you start the next activity.
8. Make sure that you have done a final check before you hand in your concept map.
243
Directions for second concept map (including terms)
1. You have been provided with a list o f terms
2. Use as many of the terms as you wish in this concept map.
3. Sort the ideas/terms into clusters according to the extent o f relatedness among the
terms. To avoid confusion, circle terms on the list that you have included in a
cluster. Note that some terms could possibly reside in more than one cluster.
4. Create a map o f the terms in each cluster. Use directed arrows or arcs to connect
the terms in the clusters. Label the arcs or connecting arrows with phrases or
words that explain/identify the relationship between connected the terms.
5. Connect your clusters; you can use dotted directed arrows or arcs for this purpose.
Again identify/explain your connections with phrases or words.
6. Check your work.
7. Place your map into the envelope
8. At the conclusion of the second map, you will have a five-minute break before the
start of the next activity.
9. Make sure that you have done a final check before you hand in your concept map
10. The next activity will be an assessment of trigonometric knowledge
244
Acute
Addition Formula
Adjacent Side
Amplitude
Angle
Angle of Depression
Angle o f Elevation
Arccosecant
Arccosine
Arccotangent
Arclength
Arcsecant
Arcsine
Arctangent
Argument
Asymptote
Bearing
Circle
Clockwise rotation
Cofunctions
Complement (ary)
Complex numbers
Composition
Continuous
Convention
Cosecant
Cosine
Cotangent
Coterminal
Counterclockwise-rotation
Degree
Derived Identities
Direction
Domain
Even and Odd
Formula
Frequency
Function
Fundamental Period
Graph
Horizontal Shrink
Horizontal Stretch
Hypotenuse
Identity
Image
Initial side
Inverse
Law
Minute
Obtuse
One-to-One
Opposite Side
Period
Periodic
Phase-shift
Principal values
Pythagorean
Quadrantal Angles
Quadrants
Quotient
Radian
Range
Rate o f change
Real numbers
Reciprocal
Reference
Reflection
Relation
Representation
Right
Secant
Second
Sine
Sinusoidal
Special angles
Special triangles
Standard position
Supplement (ary)
Symbolic
Table
Tangent
Terminal
Transformation
Triangle
Unit
Vertical Shrink
Vertical stretch
Vertical translation
y = x line
245
APPENDIX F
INTERVIEW 1
246
Descriptions
The main purpose o f this interview is to revisit your work with the concept maps,
card sorts, and check your pedagogical content knowledge with respect to trigonometry.
You were selected for this interview based on your responses to the concept maps, card
sorts, test of trigonometric knowledge, and your course taking history.
In 45 minutes today, I will ask you questions about the concept maps and card
sorts that you constructed on November 12, 2002. You are free to modify and synthesize
your previous work. I will also ask you to respond to students’ work in trigonometry and
discuss ways to further students’ understanding of the subject domain. The interview will
be recorded on audiotapes to serve as a reference to your work and my notes. We will
take a five-minute break at the conclusion of this interview before we begin the second
interview on problem solving in trigonometry.
247
1. Concept Map 1
a. Describe the steps you used to construct the first concept map.
b. What were you trying to convey?
c. Why did you decide to organize the map as you did?
2. Concept Map 2
a. The same questions as in concept Map 1. Describe the steps you used to construct the
first concept map.
b. Were there terms on the list provided that you had not encountered or did not
recognize?
c. What were you trying to convey?
3. Comparison o f Concept Map 1 and 2.
a. What about the first and second concept maps are the same?
b. What about the first and second concept maps are different?
4. Student Work 1
Choose to support or refute the following work by a student. Provide reasons for your
decision.
Asked to describe inverse trigonometric functions, a student gave the following
explanation
Just like real numbers if you want to find the inverse o f the function say
sinx you should multiply sinx by the multiplicative inverse (or
reciprocal) —-— = esc x . That way you get the identity 1 because
sinx
sinx • -------= 1. Follow the same procedure for the other trigonometric
sinx
functions.
248
5. Student Work 2
Choose to support or refute the following work by a student. Provide reasons for your
decision.
Asked to explain the effect o f the parameter b 'm y = sin(bx) on the parent function y =
sin(x), a student argued as follows:
[1] I f 0 < b < 1
then there is a horizontal shrink because bx is smaller than x
[2] If b > 1
then there is a horizontal stretch because bx is bigger than x
[3] If b is negative and -1 < b < 0
then there is a horizontal stretch opposite that obtained in [1]
[4] If b is negative and b < -1
then there is a horizontal shrink opposite that obtained in [2]
6. Card sort task 2
Describe how you went about arranging these cards in an order to represent how you
think students should be introduced to these aspects o f trigonometry.
How did you determine the prerequisites?
7. Card sort 1
Describe how you grouped the statements into the categories o f “always true”, “true
sometimes”, and “never true.”
Would you like to make any changes to the card sort?
8. Technology
How do you envision using technology (graphing calculator or computers software) to
facilitate the teaching and learning o f trigonometry?
How would you use the graphing calculator to graph the Arccotangent (same as C o f'x )
function? State all your steps.
249
Data Collection Design: Interview Notes
Time
Activity/Action/Questions
(Complimentary to the Audio Record)
Comments vis-a-vis status of
trigonometric knowledge,
pedagogical content
knowledge, representational
systems, thoroughness and
flexibility, facilitating
(FAC), or blocking (BLC)
Start
End
Closure: Is there anything else that you would want to share?
Thank you for contributing to the advancement o f mathematics education.
250
APPENDIX G
INTERVIEW 2
251
Directions
The problems used for this survey were selected from a Test of Trigonometric
Knowledge Item Bank that was created for this study. Particular consideration was given
to items that yielded interesting conceptual misunderstandings or presented good problem
solving situation in the pilot study.
Procedure:
In the next 45 minutes you will solve trigonometric problems. I will
present the problems one problem at a time. The focus will be on problem
solving. I will focus on your depth of explanation, justification, and flexibility via
the approaches you use. Complete the tasks to the best o f your abilities. The
interview will be recorded on audiotapes to serve as a reference to your work and
my notes.
To start the interview, tell me what you do first when you start solving a
mathematics problem.
How do you stay focused and resilient in problem solving?
What is the role o f justification and proof ideas in trigonometry?
252
1. What is the radian measure?
2. Prove that there are 360° in one revolution.
3. A person is seated on a Ferris wheel of radius 100ft that makes one rotation every
30 seconds. The center o f the wheel is 150ft above the ground at any time t o f a 2min ride. Assume uniform speed from the beginning to the end o f the ride and that
the person is at the level o f the center of the wheel and headed up when the ride
begins.
Find a function that models the height o f the person, with time t as the independent
variable.
What if the person starts at the lowest point when the ride begins?
4. The diagram below shows three equal squares, with angles a,/3,y as marked.
Prove that a + !3 = y.
5. Given f ( x ) = 3sin 2 x ~ — , when is f(x) > 0 for 0 < x < 42
253
Data Collection Design: Interview Notes
Time
Activity/Action/Questions
(Complimentary to the Audio Record)
Comments vis-a-vis status of
trigonometric knowledge,
pedagogical content
knowledge, representational
systems, thoroughness and
flexibility, facilitating
(FAC), or blocking (BLC)
Start
End
Closure: Thank you for participating in this research project. I have enjoyed working with
you. I wish you the best in your student teaching, in-service work and other things you
get involved with. Again your cooperation has been invaluable and profoundly
appreciated. Thank you.
254
APPENDIX H
INFORMATION AND INVITATION TO PARTICIPATE
255
I am a student at the University working on my doctoral degree in mathematics
education. I am particularly interested in teachers’ knowledge o f school mathematics.
There is a large body o f research that describes teachers’ levels of understanding of
elementary school mathematics. There are also studies on teachers’ knowledge of
secondary school mathematics, but not quite as extensive as those on teachers’
understanding o f elementary school mathematics. The consensus in the profession is that
teachers’ knowledge impacts their actions in the classroom. This includes preservice
teachers. This study is an effort to contribute to the literature on preservice teachers’
knowledge o f school mathematics and the ways they intend to teach such knowledge to
students. Your participation will help further the mathematics education field’s
understanding o f preservice teachers’ knowledge o f school mathematics.
If you decide to participate in the study, here are some o f the things you may be
asked to complete:
1) A subject-area test that will last about one hour to one and a half hour. I will
take some notes as you take the test. I will only answer clarification questions and will
not answer questions about procedures or definitions.
2) Two card-sort activities.
3) Two concept-mapping activities.
4) If you are selected for the interview, then I will interview you for about an hour. The
interview will be taped, transcribed and interpreted. I will also take note o f your
responses and actions during the interview.
Participation in this study is voluntary. Your decision whether or not to participate
will not adversely affect you in any way, shape or form. The use o f pseudonyms will
protect you from ever being identified and thus ensure confidentiality. Your instructors at
256
the University will not receive any data or identification information pertaining to this
study. To further safeguard your anonymity, 1 will advise you not to share your
participation or refusal to participate in this study with your peers or your superiors. The
data and interpretations will be available in the form o f a dissertation and anyone will
have access to that document, but your identity will not be apparent or disclosed.
257
Sign-Up Sheet
□ Yes, I wish to participate (Provide Name and Contact Information Below)
□ And if selected I will participate in the interviews
□ But I do not want to participate in the interviews
□ No, I do not wish to participate (You should not provide name or contact
information)
Print Name:
Signature:
Date:
(Please Sign)
Contact Information (1) Tel (Indicate best time to reach you):
(2) Email:
258
APPENDIX I
INTRODUCTION TO CONCEPT MAPPING
259
Resources:
1. Concept Mapping by Douglas McCabe of the Hong Kong Polytechnic University
(assigned reading for this activity)
2. Novak and Gowin (1984)
3. Bolte (1999)
A concept map is a visual representation o f how information (facts, concepts and
ideas in a knowledge domain are connected. Concepts, ideas and facts are represented as
nodes and the relationships (propositions) represented by connecting lines (or arcs) that
have descriptive phrases attached to them. There are four major categories o f concept
maps: Spider Concept Map (also called web-like); Hierarchy Concept Map; Flowchart
Concept Map; and Systems Concept Map. In this presentation, we will focus on the
hierarchy concept-mapping model (see figure 1). According the Novak and Gowin (1984)
the hierarchical model allows for an organization of knowledge domains such that the
more inclusive and general concepts are put at the top and the more concrete and less
inclusive ones included at lower and lower levels in the hierarchy (see figure 1). The
coherence of the hierarchy is supremely fundamental to the integrity o f the model.
Therefore, the validity of the hierarchy receives utmost importance. The validity o f the
propositions is only secondary to the hierarchy.
260
Figure I - 1. Hierarchical Concept Map Schematic CM
CM Concept Map Schematic was adapted from Novak and Gowin (1984, p. 37).
261
Steps in Constructing a Concept Map
1. First write down all you know about the subject domain on sheets o f paper.
2. After you have done that, sort the ideas/terms into clusters according to the extent of
relatedness among the terms.
3. Create a hierarchy of the terms in each cluster, with the more general or abstract
ideas/terms put above the less general ones. Note that any one term could possible
reside in more than one cluster.
4. Create an inter-hierarchy among the clusters. Specify relationships among the
terms in each hierarchy. Represent a relationship with a line segment or a curve
between two ideas/terms and a descriptive phrase connecting a subordinate term
to a super-ordinate one.
5. If there are ideas and concepts that appear in two or more clusters, then include
those ideas as members o f one cluster and use cross-links (arcs) to connect the prolific
concepts to the other clusters.
6. Check your work.
For the exercise that we will do in class the next week, we will try to map a
concept map of the domain o f functions. The following list is intended to facilitate the
“brainstorming” process. Study the terms carefully, and generate some additional terms
that you may want to include in a concept map of the function domain. When we recongregate as a class, I will like you to share and discuss with a partner and come up with
a unified list. We should have at least 8 addenda lists (since we have about 17 people
in this class). We will have the original list o f 54 terms on the overhead. I will ask each
dyad to add a term they came up with to the original list. We will go around a second
time if there are more terms to add. We will use the terms so generated to construct a
concept map o f the function domain as a class. If we run out o f time, part of your
homework will be to finish the map for the following week.
262
FUN CTION
R eal num bers
Range
C o o r d in a t e s y s t e m
X - a x is
Y - a x is
D o m a in
R e l a t io n
O n e -t o - o n e
In d e p e n d e n c e
D ependence
D e f in e d
A sym pto tes
E n d -b e h a v io r s
V e r t ic a l l in e t e s t
Z eros
R oots
R ate of change
P o l y n o m ia l s
R oot
L in e a r
Q u a d r a t ic
P ow er
E x p o n e n t ia l
A l g e b r a ic
T ranscendental
T r a n s f o r m a t io n s
Sets
G raph
Table
A rgum ent
In p u t
O utput
M a p p in g
E q u a t io n
In v e r s e s
In v e r s e f u n c t io n s
C u b ic
R a t io n a l
Sym m etry
M a x im u m / m in im u m
S o l u t io n s
In t e r s e c t io n s
S h r in k
Stretch
S h if t s
V a r ia b l e s
O r d e r e d p a ir s
E ven
O dd
C o n t in u it y
X - y LINE
R ule
M a c h in e
263
Figure I - 2. A concept map o f how to construct a hierarchical concept map ACES
ACES
Retrieved from the University o f Illinois at Urbana-Champaign’s
Agriculture, Consumer, & Environmental Sciences (ACES) web site’s Mind
Module: http: ://classes.aces.uiuc.edu/ACES 100/Mind/c-m2.html
264
APPENDIX J
PILOT VERSIONS OF THE TEST OF
TRIGONOMETRIC KNOWLEDGE
265
P il o t F o r m A
1.
of
T est
of
T r ig o n o m e t r ic K
no w ledg e
Conventionally how do we organize the plane into quadrants? Give a picture with
labels for the quadrants.
2.
3.
4.
5.
Give two special triangles used in trigonometry.
Prove that
a - = —— = —— in any triangle with sides a, b, and c.
sin A sinfi sinC
Prove that a 2 = b2 +c2 - ( 2 be) cos A in any triangles with sides a, b, and c.
What is the relationship among the sides of the triangle inscribed in a semicircle
with diameter EG?
6.
What does a negative angle measure mean?
Assume that the angles are in standard position.
7.
Find the amplitude, period, frequency and phase-shifts o f the following graph.
Write an equation for the graph o f the function.
266
8.
Give at least two proofs o f the Pythagorean theorem sin x + cos
9.
Define the radian measure of an angle?
10.
The diagram below shows a circle with diameter AC = 1.
x=1
Find the line segment in the diagram equal in length to sin a and to sin/1. Draw in the
line segment equal in length to sin(a + /3). State the formula for sin(a + /J).
11.
What makes the unit circle particularly useful in the study o f trigonometry?
Could we use any circle instead of the unit circle?
12.
If 0 < a < 90° and 0 < b < 90°, then sin(a) = sin(b) <£> a = b, and cos(a) = cos(b) <^>
a = b, but one of these is not true if 0 < a < 180° and 0 < b < 180°. Which one is it? Why?
13.
A person is seated on a Ferris wheel o f radius 100ft that makes one rotation every
30 seconds. The center of the wheel is 150ft above the ground at any time t of a 2-min
ride. Assume uniform speed from the beginning to the end o f the ride and that the person
is at the level o f the center of the wheel and headed up when the ride begins.
Find a function that models the height o f the rider, with time t as the independent
variable.
267
14.
True or False:
a) sin(a + /3) = sin a + sin/3
b) sin(—x )
=
cos x
c) cos(-x) = sinx
d) sin(180° + x) = -sinx
e) sin(180°-x) = sinx
f) sin(90° + x) = cosx
g) cos(90°-x) = sinx
h) sin(270°-x) = -sinx
i) sin(360°-x) = -sin x
15.
Describe at least two relations among the interior angles in a right triangle?
16.
Solve the following triangle:
17.
What is the angle of elevation of the sun when a 30-foot flagpole casts a 40-foot
shadow?
18.
Given /(x ) = 3sin 2 x - ^ , find
a. Amplitude of the function;
b. Phase shift o f the function
268
19.
W rite Y E S n ex t to a g rap h o f a fu nction th at y ou thin k is p erio d ic, and N O next
to a graph o f a function that you think is not periodic. Assume the graphs continue
indefinitely.
20.
True or false: Give reasons for either choice.
sin2a + cosa
21.
2
In any acute triangle ABC, show that c = a cos B + b cos A
269
22.
A central angle in a circle o f radius 2 units intercepts an arc o f 5 units. What is the
radian measure o f this angle?
23.
For an acute angle a , which is larger, sina or the radian measure o f a ? Explain.
24.
c - , c ,i
• sin(a + B) - cos a sin 6
Simplify the expression —
—--------------1cos(a +f$) +sina sin/3
25.
Factor sin(a + ji) + sin(a - yS).
26.
(
kn\
Show that for all values o f x, cos x +— =
I
2
- sin x, if k = 4n +1 for some integer n,
27.
Graph the following function: f ( x ) = S in '1 x
28.
Given the following function, state and explain the effect o f the parameters.
h(x) = a sin(bx - c) +d
29.
How are the graphs of a function and its inverse related graphically, symbolically
(equation), via mapping, and either via ordered pairs or tables?
30.
If tan /3 = a , express in terms of a the value of sin2/3, cos2/3, and tan2/E
31.
Show that AB = 2r sina, and
CD = 2r tana in the circle with center O
270
P il o t F o r m B
1.
of
T est
of
T r ig o n o m e t r ic K n o w l e d g e
Conventionally how do we organize the plane into quadrants? Give a picture with
labels for the quadrants.
2.
Give two special triangles used in trigonometry.
3.
Prove that
4.
Prove that a 2 = b2 +c2 ~(2bc) cos A in any triangles with sides a, b, and c.
5.
What is the relationship among the sides of the triangle inscribed in a semicircle
- ■- = —— =
in any triangle with sides a, b, and c.
sin A sinfi sinC
with diameter EG?
6.
What does a negative angle measure mean? Assume that the angles are in
standard position.
7.
Find the amplitude, period, frequency and phase-shifts o f the following graph.
Write an equation for the graph of the function.
271
8.
Give a proof o f the Pythagorean theorem sin x + cos x = 1
9.
Define the radian measure of an angle?
10.
The diagram below shows a circle with diameter AC = 1.
Find the line segment in the diagram equal in length to sina and to sin/3.
Draw in the line segment equal in length to sin(a + /3). State the formula for sin(a + ft).
11.
What makes the unit circle particularly useful in the study o f trigonometry?
Could we use any circle instead o f the unit circle?
12.
If 0 < a < 90° and 0 < b < 90°, then sin(a) = sin(b) <=> a = b, and
cos(a) = cos(b) <=> a = b, but one o f these is not true if 0 < a < 180° and 0 < b < 180°.
Which one is it? Why?
13.
A person is seated on a Ferris wheel o f radius 100ft that makes one rotation every
30 seconds. The center o f the wheel is 150ft above the ground at any time t o f a 2-min
ride. Assume uniform speed from the beginning to the end of the ride and that the person
is at the level o f the center of the wheel and headed up when the ride begins. Find a
function that models the height o f the person, with time t as the independent variable.
272
14.
Solve the following triangle:
15.
Given that tan0 = -3 , find sin26
16.
Given f( x ) = 3sin 2x - - j , find
a) The fundamental period o f the function;
b) Sketch two fundamental periods o f the graph o f the function;
17.
Write YES next to a drawing o f a chain that you think is periodic, and NO next
to a drawing that you think is not periodic. Assume the graphs continue indefinitely.
273
18.
In a circle o f diameter 10 units, how long is a chord intercepted by
an inscribed angle of 60°?
25
19.Show that for any triangle, sina = — ; where S is the area o f the
be
triangle, and b and c are the sides that include a .
20.
Define odd functions and even functions.
21.
Match the graphs shown below with the correct symbolic representation
listed at the right. Place the letter indicating your choice in the space provided.
Some choices will NOT be used
A.
B.
C.
D.
f ( x ) = cosx
f( x ) = sinx
/(x ) = 2cosx
/(x ) = 2sin2x
E. / ( x ) = 2 c o s ^ x j
F. /(x ) = tanx
G. f ( x ) = -tan
H ./(x ) = ic o s
jt(x - 4 )
+2
I. / ( x ) = cos jt( x + 4) + 2
J. f( x ) = i - s i n ; t ( x - 4 ) + 2
274
22.
sm a
Give a geometric justification for lim
= 0. [Hint: Use a unit circle]
a-*o a
23.
Simplify cos(a + /3)cos/l + sin(G! + /J)sin/f.
24.
What is the maximum value of sin x cos x?
25.
{
kn\
Show that for all values o f x, cos x + — =
I
2 j
-cos x, if k = 4n +2 for some integer n
26.
Graph the following function: g(x) = Tan~l x
27.
How would you write a linear combination of y = sin&x and
y = cos &x as
y = asink(x +q?)?
28.
Find the length o f QR in the following diagram.
Line PR is tangent to the graph o f the sine function at P.
(3/25— 2 y
30. Rewrite y = esc Tan~x --------v
V
31.
as an algebraic equation.
Find an expression for e,nx; « G 3 . Use two different procedures to generate the
formula for cos 36.
275
P il o t F o r m C
of
T est
of
T r ig o n o m e t r ic K
n o w led g e
1.
Prove that
2.
Give two special triangles used in trigonometry.
3.
Conventionally how do we organize the plane into quadrants?
sin A
= —— = c in any triangle with sides a, b, and c.
sin 5 sinC
Give a picture with labels for the quadrants.
4.
Prove that a 2 = b2 +c2 - ( 2 be) cos A in any triangles with sides a, b, and c.
5.
What is the relationship among the sides o f the triangle inscribed in a
semicircle with diameter EG?
6.
What does a negative angle measure mean? Assume that the angles are in
standard position.
7.
Find the amplitude, period, frequency and phase-shifts o f the following graph.
Write an equation for the graph of the function.
8.
9
9
Give a proof of the Pythagorean theorem sin x +cos x = 1
276
9.
Define the radian measure o f an angle?
10.
The diagram below shows a circle with diameter AC = 1.
Find the line segment in the diagram equal in length to sin a and to sin/3. Draw in the
line segment equal in length to sin(a + /3). State the formula for sin(a + /3).
11.
What makes the unit circle particularly useful in the study o f trigonometry? Could
we use any circle instead o f the unit circle?
12.
If 0 < a < 90° and 0 < b < 90°, then sin(a) = sin(b) o a = b, and cos(a) = cos(b) <^>
a = b, but one of these is not true if 0 < a < 180° and 0 < b < 180°. Which one is it? Why?
13.
A person is seated on a Ferris wheel of radius 100ft that makes one rotation every
30 seconds. The center of the wheel is 150ft above the ground at any time t of a 2-min
ride. Assume uniform speed from the beginning to the end of the ride and that the person
is at the level o f the center of the wheel and headed up when the ride begins.
Find a function that models the height o f the person, with time t as the independent
variable.
277
14.
Solve the following triangle:
15.
State the domain and range o f the inverse to the six trigonometric functions.
16.Given f( x ) = 3sin 2 x - ^ , solve f(x) = 0 for 0 < x < 4
17.
Below are graphical representations o f periodic functions, in some certain
domains. The graphs continue indefinitely. In each drawing a part o f the graph is bolded.
Write YES next to the drawings in which you think the bolded part is a period of the
function and NO next to a drawing in which you think the bolded part is not a period of
the function.
278
18.
What is the degree measure of a semicircle? A quarter o f a circle?
19.
Three riders on horseback start from a point X and travel along three different
roads. The roads form three 120° angles at point X. The first rider travels ata speed o f 60
MPH, the second at a speed o f 40 MPH, and the third at a speed o f 20 MPH. How far
apart is each pair of riders after 1 hour?
20.
Which of the following functions are odd, even, or neither.
279
21.
A wheel whose radius is 1-meter rolls along a straight path. The path is marked
out in 3-meter lengths, with red dots three meters apart. The wheel has a wet spot of blue
paint on one point. When it starts rolling, that point is touching the ground at a red dot.
As the wheel rolls, it leaves a blue mark every time the initial point touches the ground
again.
a. How far apart are the blue marks?
b. Through what angle has the wheel rolled between the time it makes a blue
mark and the time it makes the next blue mark?
c. Will a blue mark ever coincide with a red mark again after the wheel is set
in motion? If so, after how many times around? If not, why not?
22.
For any acute angle a and /3 for which cosa cos
cos (a + /3)
-----------— = l - t a n a t a n p .
cos a cos f3
23.
* 0, show that
Show that tana tan2a tan 3a = tan 3a - tan2a - tana whenever all these expressions
are defined. For what values o f a are some o f these expressions not defined?
24.
What is the maximum value of sin x + cos x ?
25.
Show that for all values o f x, cos x + — =
I
2 j
sin x, if k = 4n +3 for some integer n
26.
Graph the following function: h(x) = Sec~x x
27.
The diagram below shows three equal squares, with angles a,(3,y as marked.
(
Prove that a + /3 = y.
k n \
280
28.
Describe the rate o f change of a phenomenon that is modeled by a cosine
function.
29.
The displacement of the pendulum on a clock is 10 in. It makes one complete
cycle in 4 s. Determine a sinusoidal function that represents its motion as a function of
time.
30.
A plane approaching an airport is being tracked by two devices on the ground that
are 746 ft apart. The angle o f elevation from the first device is 45°, and from the second it
is 30°. Determine the height o f the plane.
31.
Describe the domain and range o f y = cos(sinx). Provide a sketch. Is y a function?
Give the equation for the inverse o f y. Is the inverse o f y a function?
281
P il o t F o r m D
1.
of
T est
of
T r ig o n o m e t r ic K n o w l e d g e
Conventionally how do we organize the plane into quadrants? Give a picture with
labels for the quadrants.
2.
Give two special triangles used in trigonometry.
3.
Prove that
4.
Prove that a 2 = b2 +c 2 ~{2bc) cos A in any triangles with sides a, b, and c.
5.
What is the relationship among the sides o f the triangle inscribed in asemicircle
q - = —— = c in any triangle with sides a, b, and c.
sin A sin 5 sinC
with diameter EG?
6.
What does a negative angle measure mean? Assume that the angles are in
standard position.
7.
Find the amplitude, period, frequency and phase-shifts o f the following graph.
Write an equation for the graph o f the function.
9
9
8.
Give a proof o f the Pythagorean theorem sin x + cos x = 1
9.
Define the radian measure o f an angle?
282
10.
The diagram below shows a circle with diameter AC = 1.
Find the line segment in the diagram equal in length to sina and to sin/3. Draw in the
line segment equal in length to sin(a + /3). State the formula for sin(a + /3).
11.
What makes the unit circle particularly useful in the study of trigonometry? Could
we use any circle instead o f the unit circle?
12.
If 0 < a < 90° and 0 < b < 90°, then sin(a) = sin(b) <=> a = b, and cos(a) = cos(b) <*=>
a = b, but one o f these is not true if 0 < a < 180° and 0 < b < 180°. Which one is it? Why?
13.
A person is seated on a Ferris wheel o f radius 100ft that makes one rotation every
30 seconds. The center o f the wheel is 150ft above the ground at any time t of a 2-min
ride. Assume uniform speed from the beginning to the end o f the ride and that the person
is at the level o f the center of the wheel and headed up when the ride begins. Find a
function that models the height o f the person, with time t as the independent variable.
283
14.
Solve the following triangle. Find the area o f the triangle using two different
procedures.
15.
Find the domain and range o f 3 sin 2x + -j +2.
16.
Given f{ x ) = 3sin 2x - - j , when is f(x) > 0 for 0 < x < 4
17.
If sin a = a , where 0 < a < 1, express in terms o f a the value o f the other five
trigonometric functions o f a.
18.
1- t 2
21
Suppose that t is a number between 0 and 1. If a = ------ — and b =----- -,s h o w
1+ r
1+ r
that there is an angle 6 such that a = cos 6 and
b = smd.
19.The Circumradius o f a triangle: Circumscribe a circle about a given triangle ABC as
shown.
Show that — = 2R, where R is the radius of the triangle’s circumscribed circle. Length
sina
a is the length o f the side opposite angle A. Use the formula to show that the area o f the
inscribed triangle has area A =
4R
284
20.
Using the law o f cosines, show that in any triangle ABC o f area S,
c 2 = a 2 + b2 - 4ScotC.
21.
Use the law o f cosines and the following figure to give a direct derivation of the
formula for cos(a + /3). Assume that segment AD is perpendicular to segment BC.
22.
What is a sinusoidal function?
(
k n \
23.Show that for all values of x, cos x +— =
I
2 )
cos x, if k = 4n for some integer n
24.
What is the difference between Arc tan x
and
25.
If n is a positive integer, write a function o f the form y = sinfcc with period n.
26.
Solve the following trigonometric equations:
s i n x = —1/2
ta n x — t a n ^ r / 5
c o s2 x = 3 /4
s in x = c o s 2 x
t a n 2 x = c o t x, t a n x ^ O
t a n 2 x + 1 = (l h—\f3 ) ta n x
tan-1 x?
285
27.
What is the procedure for computing reference angles in the quadrants? Give
procedures for both clockwise and counter-clockwise rotations.
28.
When a Ferris wheel is in motion, do all riders move at the same linear velocity?
29.
Determine the area o f the shaded region. PRST is a square. Points P and S are the
centers for the circular arcs with the same radii.
30.
When does a triangle with sides o f known lengths a, b and c have the largest area?
Justify your conjecture.
31.
If you know that a phenomenon’s rate o f change can be modeled with a sine
function, then what type o f function should be used to model the phenomenon?
286
APPENDIX K
REPRESENTATIVE SAMPLE CONCEPT MAPS
Figure K - 1. Concept map 1 produced by LN in phase one of the study
287
Figure K - 2. Concept map 2 produced by LN in phase one of the study
288
289
Figure K - 3. Concept map 1 produced by AB in phase one of the study
290
Figure K - 4. Concept map 2 produced by AB in phase one o f the study
Figure K -5 . Concept map 1 produced by ES in phase one of the study
291
Figure K - 6. Concept map 2 produced by ES in phase one of the study
292
293
Figure K - 7. Concept map 1 produced by NM in phase one o f the study
294
Figure K - 8. Concept map 2 produced by NM in phase one o f the study
Figure K - 9. Concept map 1 produced by AX in phase one of the study
295
Figure K - 10. Concept map 2 produced by AX in phase one of the study
296
297
APPENDIX L
COMPARISON OF CM 1 & CM2, AND REASONS
GIVEN IN CARD SORT 1
298
Table L - 1. Comparison o f CM1 and CM2 including the terms/ideas/relations used in
both activities and some salient features o f both concept maps.
CM1
Sin *(x), Cos *(x), Tan ^ (x ),
cos2( b )
A
2
+B
2
+ sin2 ($ ) = 1,
2
/
\
- C , sin(/t± B ],
cos(<4 ± /?),
1 1
cos2x = —+ —sinxcosx,
2 2
2
2
1 + tan 9 = sec 6 ,
2
2
1+ cot d - esc 9
2
2
2
sin 6 + cos 6 = tan Q ,
2
2
1 - sec 9 = tan 9 ,
2n radians = 360° , Adjacent,
,
T erm s/Ideas/Concepts
Amplitude, A stronomy,
A sym ptote, Axes, Calculus,
Continuous, Cosecant, Cosh,
Cosine, Cotangent, Degree,
Derivatives, D iscontinuous,
Domain, Finding angle
m easure, Finding side length,
Frequency, Function, Graphs,
Hypotenuse, Inverses, Law o f
Cosine, Law o f sines,
M athem atics, Opposite, Period,
Phase-shift, Physics, Polar
representations, Quadrants,
Radians, Range, Repeating
sections, Right angle, Right
triangles, Roots, Secant, Shift,
Shrink, Sine, Sinh,
SOH CA H TO A , Stretch,
Tangent, Tanh, Triangle, Trig
Identities, Trigonom etry, U nit
Circle, Vertical shift,
W avelength
N = 65*
CM 2
A cute, A ddition Formula,
A djacent Side, A m plitude
Angle, A ngle o f Depression,
A ngle o f Elevation,
A rccosecant, Arccosine,
A rccotangent, A rclength,
A rcsecant, Arcsine, Arctangent,
A rgum ent, A sym ptote, Bearing,
Circle, Clockw ise rotation,
Cofunctions, Com plem ent(ary),
Com plex numbers,
Com position, Continuous,
Convention, Cosecant, Cosine,
C otangent, Coterminal,
Counterclockw ise-rotation,
D egree, D erived Identities,
D irection, D om ain, Even and
Odd, Form ula, Frequency,
Function, Fundam ental Period
G raph, Horizontal Shrink,
Horizontal Stretch, Hypotenuse,
Identity, Image, Initial side,
Inverse, Law, Minute, Obtuse,
O ne-to-O ne, O pposite Side,
Period, Periodic, Phase-shift,
Principal values, Pythagorean,
Q uadrantal, A ngles, Quadrants,
Quotient, Radian, Range, Rate
o f change, Real numbers,
Reciprocal, Reference,
Reflection, Relation,
R epresentation, Right, Secant,
Second, Sine, Sinusoidal,
Special angles, Special
triangles, Standard position,
Supplem ent(ary), Symbolic,
Table, Tangent, Terminal,
Transform ation, Triangle, Unit,
V ertical Shrink, V ertical stretch,
V ertical translation, y = x line
N = 89
299
Table L - 1 Continued
CM1
CM 2
Explicitly Stated Trigonom etric
Ratios
N = 7 out o f 14
M ention o f Triangles
N = 11
M ention o f A pplication
(Solving Triangle)
N =6
M ention o f Radian M easure
N = 11
N =9
W rong Interpretation o f
Inverses
N = 6; another 5 were
unspecified
N = 7; another 4 were
unspecified
Mention o f Sinusoidal Ideas**
N =2
r~
II
Z
N = 11
* Some o f the terms/ideas/relationships are not correct.
** Sinusoidal Ideas include Amplitude, Phase-Shift, Stretch, Shrink, Vertical Shift,
Frequency, Period
300
Table L - 2. Reasons provided for placing the propositions for which the participants had
the most difficulties in card sort one.
Propositions*
is
AT
NT
Reasons for placem ent in one o f the three piles: AT. TS. & N T
Proposition 2: 271 radians
represent the fundam ental
period for trigonom etric
functions
(B) Trigonom etric
functions are based on
2tt as the period unless
the function is
m anipulated in some
way: 2k is a circle
(F) Basic functions
repeat after 2n period
(G) The basic function
y = sin 0 goes through
its com plete cycle in 2k
radians
(H) {}
(J)
{}
(K)
TT
^
©
(A) Stretching and
shrinking changes
periods
(C) It is true for sine
and cosine, but not
tangent
(D) O nly for those that
hav e n ’t been
transform ed
(£){}
(L) Y es for sine, cos;
not for co / tan
f 2
TT
(P)
{}
N = 6
(M ) Yes, basic
principle
(N ) {}
N = 8
(B) 2 k is always a
possible period for a
trigonom etric function
Proposition 3: 2 ji radians
can be the period o f any
trigonometric function
(F) {}
(G) It can be, but it
doesn’t have to be
(L) M ultiply by some
constant to
stretch/shrink the
period
(N) {}
(9 0
(E) {}
(H) {}
(K) {}
(M) 2 ti is period for
som e but not all
(P)
{}
N = 6
N =5
(A) Some have
sm aller period
because o f shrinking
and stretching
(D) The period o f
cos2x
4 2k
(J) N o, there are
functions w here the
period is 3 tc, so 271
couldn’t be the
period
N =3
301
Table L - 2 Continued
Propositions*
TS
AT
NT
Reasons for placem ent in one o f the three piles: AT. TS. & NT
(A) sin x , cos x have
(D) It is logical
(G) I guessed
(J) {}
(K) Guess
Proposition 4: I f f and g
are two trigonometric
functions, then the period
(P)
{}
c c . . period o f f
o f f/g is
period o f g
N = 5
period o f 2 n;
sin x
tan x cos X
2 jt
= — = 1 * 2it
2n
(C) {}
(E) {}
(F) N ot necessarily, it
m ay be less
(H ){ }
(L) {}
(M ) N ot sure
(N) True for period o f f
= 1 and period o f g = 1
N =8
(B) {}
N = 1
/ A\
■
0PP
(A) sin x = ----- ,
hyp
COS X
adj
= ----- ,
hyp
opp
tan x = ----adj
opp
(C) — = tan,
adj
opp
adj
----- = sine, ------ = cos,
hyp
hyp
Proposition 5: G iven
triangle o f sides a, b, and
c the trigonom etric
functions are ratios o f the
lengths o f two o f the sides
inverses also
opp
(D) sin = ----- ,
hyp
(B) This is true for
right triangles
(G) {}
(L) I f A B C is right
triangle
(N) O nly i f said
triangle is a right
triangle
(H) {}
adj
opp
cos = ----- , tan = -----hyp
adj
(E) {}
(F) Definition
N = 1
(J) (}
(K) Functions are
S
C
T
o
a
h
h
o1
f sides
al
(M ) Yes, definition o f
trig functions
(P) {}
N = 9
N =4
302
Table L - 2 Continued
Propositions*
Proposition 8: O ne radian
is equal to 180°
AT
is
NT
Reasons for placem ent in one o f the three piles: AT. TS. & NT
(A) 2 k = 360°, k =
(D) 2n rad = 360°, it
180°; Look at unit
rad = 180°,
circle
180°
1ra d =---(B) This is part o f the
ji
(C )0
conversion from
(E) {}
5* 1
degrees to radians
(F) Jt= 180° (not
(H) 1 8 0 ° = Jt = 1 (I
radian)
think 1 radian)
(G) {}
(J) {}
(K)
{}
(L) Full rotation = 2n
radians
= 360 degrees
(P) {}
360°
1ra d =---(M ) N o,
N
=2
180°
2lt
ji
(N) k radians is
equal to 180°
N =6
N =6
(A) {}
(B) Inverses are
functions with their
ow n properties - they
are one-to-one
(C) Pass vertical line
test
(E) {}
(F) {}
(G) T hey are functions
w ith real num bers as
inputs and radians or
degrees as outputs
Proposition 11: The
inverses o f trigonom etric
functions are functions
(D) I d o n ’t know about
hyperbolic
(M ) N o t alw ays true.
Sim ilar to ±~\[x
(H) ----- 1----- is a
fu n c t io n
function as well
(J) {}
(K) Trig functions
involve 2 o f the sides,
inverse simply flips
these and gives another
trig function
(L) {}
(N) The inverse o f any
function is a function
(P) Trig function
inverses - you get other
trig functions so they
are all functions
N = 12
N =2
303
Table L - 2 Continued
Propositions*
Proposition 12: For a
trigonometric function
there are situations w hen
a particular dom ain value
has two range values
AT
IS
NT
Reasons for placem ent in one o f the three piles: AT. TS. & NT
(A) I f so, not a
(F) {}
function
(C) {}
(P) Yes, there are some
(B)
It
w ou ldn ’t be a
(D) {}
situations w hen it
function
(E) {}
m ight be two values
(J) N o, it is the other
(G) {}
(true)
w ay around
(H) {}
(K) {}
(N) {}
(L) sin/cos, co/tan,
co/sec all pass the
vertical line test
(M ) N o, w ould not
N =6
be a function
N =2
N =6
(A) D efinition o f
inverse
0
(B) sin Q = — , so
H
Sin
-\(
— =0
\H )
(D) Sin
-1 ( A
-
= 45°
\2 )
Proposition 15: Inverse
trigonometric functions
yield angle measures
(E) {}
(F) D efinition o f
inverse equation
(H) Trig function yields
angle m easurem ents, so
their inverses do too
(J)
{}
(L) Domain becomes
range and vice versa
(M ) Yes, for example,
I
-1 a“J
tan
- 6
(C) {}
(K) IFF both sides have
m easures in equation
(N) {}
(P)
(G) {}
{}
N
=1
N =4
\hypj
N= 9
* Participants were asked to consider only the six basic trigonometric functions (sinx,
tanx, secx, cosx, c o tx , esc x ) when making decisions on the veracity o f the
propositions. Attempts were taken to report the exact reasons provided with minimal
clerical (non-mathematical) editing.
(A), (B), (C), ... mark the reasons given by various participants and they keep track of
the reasons given by each participant
{} imply that the participant did not provide a reason for the placement of that
proposition in the chosen pile. For example, (C) {} means that participant C did not
provide any supporting reason for placing the proposition in the chosen pile.
304
APPENDIX M
VITAE
305
Cos Dabiri Fi
T h e U n iv e rs ity o f Io w a , C o lle g e o f E d u c a tio n
D iv is io n o f C u r r ic u lu m a n d In s tru c tio n
2 5 9 L in d q u is t C e n te r N o rth , I o w a C ity , IA 5 2 2 4 2
P h o n e (H o m e ):
P h o n e (O ffic e ):
E -m ail:
E d u c a t io n ________________________________________________________________________
2003
1999
1996
1996
1994
Ph.D., M athem atics Education, Curriculum and Instruction. The U niversity o f Iowa, Iowa
City, Iow a (A nticipated com pletion date o f M ay, 2003)
D issertation: P reservice Secondary M athem atics T ea ch ers’ K now ledge o f Trigonom etry:
Subject M atter Content Knowledge, P edagogical C ontent K now ledge a n d E nvisioned
Practice
Supervisor: D ouglas A. G rouws
Illinois Teaching Certificate, G rades 6 — 12 M athem atics, V alid through 7/30/2005
M .A., M athem atics Education, Curriculum and Instruction. The U niversity o f Iowa
Com pleted D ecem ber 1996. Areas o f Concentration: Secondary M athem atics Education,
M athem atics, and Educational Psychology
Licensure, Iow a Standard Teaching, K — 12 M athem atics, Valid through 1/312007
B.S., M athem atics. T he U niversity o f Iowa, Iow a City, 5/13/1994
U n iv e r s it y E x p e r ie n c e __________________________________________________________
2002 - 2003
2001 — 2002
Research A ssistant/Teaching Assistant, The University o f Iow a (Iow a City, IA)
Supervised the elem entary student teaching com ponent o f the Secondary M athem atics
E ducation program: O bserved and assessed student teachers’ pedagogy, content knowledge,
and professionalism in area elem entary schools
Course T aught
Elem entary Student T eaching Com ponent o f the Secondary M athem atics Teachers
Education Program
Research A ssistant/Teaching Assistant, The U niversity o f Iow a (Iow a City, IA)
Supervised 4 high school m athem atics student teachers and taught student teaching
seminar. T aught the introductory mathem atics education practicum course and supervised 8
practicum students in 2 m iddle schools and 2 high schools. W orked on Core Plus
M athem atics Project (CPM P) evaluation under Dr. H arold Schoen: C onducted library
research, conducted phone interviews o f C PM P graduates, scored C PM P assessm ents using
rubrics, coded Post HS surveys using N U D *IST qualitative analysis software, worked on
C PM P longitudinal data, worked on the revision o f C P M P course m aterials (C ontem porary
M athem atics in Context)
Courses Taught
(7S 95) Introduction and Practicum: M athem atics Education, 3 sem ester hours
D esigned and taught lessons that had varying instructional intent and that used multiple
instructional strategies; students read the N ational Council o f T eachers o f M athem atics
2000 Standards (PSSM ), they learned how to plan for instruction, spent 30 - 40 hours
observing and helping teachers in high schools and ju nior high schools, and the students
taught a lesson in the jun ior high schools
(7S 187) Seminar: Curriculum and Student Teaching, 1-3 sem ester hours
Provided opportunity for student teachers to discuss, role-play, present group and individual
reports, analyze critical incidents and classroom m anagem ent, and tape and view videotapes
o f their (student teachers’) classroom perform ances. Student T eachers wrote short papers on
classroom managem ent, m eeting the needs o f all students, and lesson planning
(7S 191) O bservation and Laboratory Experience in the Secondary School, 6 sem ester
hours
306
1997 - 1999
1997
Preservice teachers perform ed the duties o f a regular classroom teacher as part o f their
student teaching experience
(7S 192) O bservation and Laboratory Experience in the Secondary School, 6 semester
hours
Preservice teachers perform ed the duties o f a regular classroom teacher as part o f their
student teaching experience
Research Assistant, C PM P evaluation under Dr. H arold Schoen
Conducted library research, coded observations using N U D *IST qualitative research
software, ran statistical analysis on SPSS, scored tests
T eaching Assistant, The U niversity o f Iow a (Iow a City, IA)
W orked with a G oals 2000 grant and supervised 12 practicum students in W est Liberty (IA)
Schools with a high proportion o f Hispanic students. Provided a sem ester of in-service
training (assessm ent a nd cooperative learning) to W est Liberty cooperating teachers
K - 1 2 E X P E R I E N C E ___________________________________________________________________________________________
2001 - 2003
1999 - 2001
1996 - 1997
M athem atics T eacher, City H igh School, Iow a City, IA
T aught A lgebra 2, C PM P Course 4, A P Calculus
M athem atics T eacher, W heaton N orth H igh School, W heaton, IL
T aught Honors Pre-Calculus, A lgebra 2, and A lgebra
Substitute T eacher, Iow a Com m unity School District, Iow a City, IA
T aught grades 4 - 1 2
P u b l ic a t io n s
2003
2003
2002
2001
1999
1998
and
P r e s e n t a t io n s _____________________________________________
Schoen, H. L., Cebulla, K. J., Finn, K. F., Fi, C. (2003). T eacher variables that relate to
student achievem ent w hen using a standards-based curriculum . Jo u rn a l f o r research in
m athem atics education.
Fi, C. (2003, February). Preservice Secondary M athem atics Teachers Know ledge o f
Trigonom etry. Iow a Council o f Teachers o f M athem atics (ICTM ) Annual Meeting, Ames,
Iowa
Fi, C. (2002, June). PR and politics o f reform m athem atics. C P M P LEA D ERS W orkshop,
Iow a City, IA.
Schoen, H. L., Finn, K. F., Field, S., Fi, C. (2001). Teacher variables that relate to student
achievem ent in a standards-oriented curriculum . Paper presented at the Annual M eeting of
the A m erican Educational Research A ssociation (Seattle, W A , A pril 10 - 14, 2001). (ERIC
D ocum ent R eproduction Service No. ED 453 265)
Fi, C. (1999, February). Problem solving/investigations. Presentation at the N CTM Central
Regional Conference, Des M oines, IA.
Fi, C. and Finken, T. (1998, February). Goals 2 0 0 0 : P rofessional D evelopm ent
C ollaboration W est Liberty School D istrict and UI C ollege o f Education: Pre-Service
Teacher E ducation Project. Presentation at the Annual C onference o f the Iow a Council of
Teachers o f M athem atics, Des M oines, IA.
R e l a t e d E x p e r ie n c e ____________________________________________________________
2002 - 2003
1995 - 2000
1998
M athem atics Item W riter for T eacher Certification Exam for A CT, Iow a City, IA.
M athem atics Tutor: The University o f Iow a (New D im ensions in Learning & Tutor
Referral Service), Iow a City, IA.
Tutored L inear Algebra, Calculus, and College A lgebra
Grader, University o f Iow a M athem atics D epartm ent, Iow a City, IA.
G raded Linear A lgebra assignments
H o n o r s __________________________________________________________________________
2003
C om m encem ent speaker, class o f 2003, City H igh School, Iow a City Com m unity School
District, Iow a City. H ancher A uditorium, M ay 31, 2003.
Title: Evolution o f Peace
307
Se r v ic e ___________________________________________________________________________
2003
2002
2002 - 2003
2000 - 2001
2001
1999 - 2001
1995
Hosted and m entored visiting South Korean Teachers fo r a day. The teachers w ere here for
tw o w eeks to learn about gifted education. Belin-Blank Center, U niversity o f Iowa, Iowa
City, IA
(October) Redeveloped the course description for A P Calculus for City H igh School, Iowa
City C om m unity School District, Iow a City, IA
Co-sponsored the Kung-Fu Club at City High School, Iow a City C om m unity School
District, Iow a City, IA
Co-sponsored W heaton North High School M athem atics Club, W heaton, IL
(M arch) M oderated a working session on A lgebra 2 at the 20th D upage Valley
M athem atics Conference, Glenbard North High School, Carol Stream , IL
K eynote Speaker: Lee Stiff, President o f N CTM , The Value o f R eform : A H igh-Q uality
M athem atics Education fo r Every Child
Judge/Facilitator, A CTSO - Dupage County, Illinois
Judged m athem atical presentations by African A m erican students in the 2000 and 2001
A frican A m erican Cultural, Technological, and Scientific O lym piad at the College o f
D upage under the directorship o f Mrs. Sadie Flucas.
V olunteer m athem atics teacher for a S um m er Program for A frican A m erican
Elem entary School Children at the University o f Iowa, Iow a City, IA
P r o f e s s io n a l
a f f il ia t io n s _______________________________________________________
National Council o f Teachers o f M athem atics (NCTM )
Association for Supervision and Curriculum D evelopm ent (A SCD )
M athem atical Association o f A m erica (M AA)
Iow a Sate Education Association/N ational E ducation A ssociation (NEA)
C oursew ork
to w ards th e
P h .D .
Seminars in M athem atics Education (including Learning T heories, Philosophies o f M athem atics
Education, Problem Solving, Professional D evelopm ent o f Teachers, G eom etry, M athem atics
Education Research, Reform Curricula)
Philosophy o f M athem atics
Classical A nalysis
Theories o f G raphs
A bstract A lgebra I and II
Discrete M athem atics M odels
A nalysis I
M athem atical Logic
Cognitive Theories o f Learning
Educational Research M ethodology
Design o f E xperim ents
C oursew ork
tow ards th e
M .A ._________________________________________________
Problem Solving
Teaching o f A lgebra
Teaching Elem entary M athem atics
Teaching o f G eom etry
Foundation o f M athem atics Education
Current Issues in M athem atics Education
T echnology and M athem atics Teaching
History o f M athem atics
N um ber Theory
General T opology
Situated Cognition
Interm ediate M ath Statistics
Interm ediate Statistical M ethods
308
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