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University of Iowa Iowa Research Online 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 28 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. 41 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 59 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. 60 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 61 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. 62 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 63 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. 64 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 65 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 66 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 67 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 68 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 69 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 70 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, 71 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, 72 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 73 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 74 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, 75 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 2 105 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. 107 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 109 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). 111 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 112 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 113 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°. 114 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■+ •2. Jf/' -v. I— o O 0«£»N 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. 130 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. 132 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 133 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 134 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? 135 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 137 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 138 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 151 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 162 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. 172 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 178 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 179 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. 180 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 181 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 182 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 183 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 184 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 185 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. 186 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. 188 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. 189 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. 190 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 191 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 192 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 193 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 194 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 195 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. 196 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 197 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 198 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 199 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. 200 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 201 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 202 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 203 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 204 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 205 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 206 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. 207 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 208 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 209 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 210 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 211 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 212 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. 213 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. 220 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 REFERENCES Allen, H. D. (1977). 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