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Aspects of English and Turkish students’ performance in trigonometry tasks
Ali Delice
School of Education, University of Leeds
Paper presented to the British Educational Research Association Annual Conference, University
of Leeds, 13-15 September 2001
Abstract
This paper is an initial report of a comparative study of English and Turkish senior high school
students’ understanding of trigonometry. Aspects of students’ performance as well as teaching
and curriculum issues are considered. This study suggests that what is meant by 'trigonometry' in
these two countries varies greatly. Performance is, not surprisingly, strongly related to what
curricula emphasise: 'context' word problems in England and 'algebra' in Turkey. Factors behind
students’ performance, such as teachers, curricula and textbooks and privileging of different
aspects of trigonometry are discussed.
Introduction
This paper reports on one aspect of my ongoing PhD. I have collected all of the data for this and I
am in the process of writing up. I have almost completed the Results chapter. I report this so that
it is clear that this is seen as an interim working paper.
My area of study is students’ performance of trigonometry (I will focus down shortly). This is an
important area of upper secondary mathematics in every country in the world. From a research
point of view it is interesting to note that barely any research studies have been conducted in this
area. I am from Turkey. It became clear to me very early in work that big differences exist in
curriculum foci of trigonometry in Turkey and in England (put simply the Turkish curriculum
appears to focus on algebraic aspects and the English curriculum is more concerned with
applications). My study is basically a comparison of the two countries from the students position
(what they do and understand). I think it important to say a few introductory words on ‘doing’
and ‘understanding’.
Student understanding is central to all cognitive based mathematics education studies. However,
the term ‘understanding’ has become a problematic term over the last decade and people seem
afraid to use it because we do not really understand exactly what understanding is. Given this I
focus on student performance. This does not mean that I have behaviorist tendencies. I remain
concerned with understanding but report on observable outcomes. Students do not do/perform in
a vacuum. They have learning histories shaped by the curriculum and the culture of education of
their countries. My central research focus is students’ understanding (performance) but because
of the importance of learning histories I have a second research focus concerned with the culture
of learning.
The students in my study are 16-17 year olds. The most important reason for choosing this age
group is that by this age their exposure to trigonometry in terms of time spent is roughly equal in
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both countries. My first research question1 concerns student performance on tasks concerned with
trigonometric identities and formulae, their manner of ‘simplifying’ trigonometric expressions
and their performance in solving trigonometry word problems. My second research question1
concerns the influence of teaching, the curriculum, examinations and resources on students
performance in this area. I now split these two research questions (RQ) up.
RQ1ia
The focus here is on students’ performance of trigonometric identities, trigonometric formulae
and their manner of ‘simplifying’ trigonometric expressions: what difficulties do they experience,
what errors do they make?; how do they use their knowledge of trigonometric identities and their
manner of ‘simplifying’ trigonometric expressions; how do these performances interact with their
knowledge and use of algebraic conventions.
RQ1ib
The focus here is on students performance of trigonometric word problems: what difficulties do
they experience, what errors do they make and what conceptions do they hold?; what ‘mental
models’ do students follow in solving trigonometric word problems; to what extent do the
context and the terminology affect the solution of trigonometric word problems; how do visual
and symbolic representations interact in the solution process.
RQ1ii
Although stated as a RQ this may be viewed as a baseline marker. The focus here is on students’
performance in manipulations with more elementary (for this age range) questions concerned
with algebra and with algebraic, verbal and diagrammatic aspects of basic trigonometry: what
difficulties do they experience, what errors do they make?
RQ2i
The focus here is on teachers in both countries: how do they teach trigonometry, what resources
do they use and not use, does the curriculum affect their teaching of trigonometry?; what
emphasis do they place on the foci of RQ1a&b , e.g. what order do they teach these?
RQ2ii
The focus here is on the curriculum in both countries: what is it (in written documents)?; how do
teachers implement this in terms of classroom activities; what aspects to textbooks ‘privilege’?
what is examined (and how important are these examinations)?
Methodology
My overall approach to this study may be called ‘naturalistic’ in the sense that I have observed, as
far as possible, ‘what is’ in both countries without manipulating the course of teaching and
learning in any manner. My student sample consists of 55 students doing A-level mathematics
from one English college and 65 similar aged students (studying mathematics) in one Turkish
school. My teacher sample (for observation and interview) were the mathematics teachers in
those schools and a wider set of similar teachers. Although there are some problems with my
1
Strictly speaking ‘questions’ as it has several sub-parts.
2
sample I do not believe they are major and it would detract from my presentation to engage in a
discussion of sampling in this paper.
My research instruments (tools) were selected to answer my research questions as best I could. I
used a wide variety of tools (details in Table 1 below). Opportunity and time (to analyse) were
constraints but, I believe, I have explored the issues as far as is practically feasible in the time I
had available. For example, to obtain data on students’ performance of trigonometric identities,
trigonometric formulae and their manner of ‘simplifying’ trigonometric expressions I initially
used a questionnaire/test. This enabled a large number of students to be sampled over a wide
range of items. I followed up these questionnaires/tests with interviews with a subset of the
students in order to understand reasons for students’ responses. I also, on a smaller subset of
students, used concurrent verbal protocols as students solved similar problems to gain further
insight into the thinking behind their performance. Appendix 1 presents a diagram (which I hope
makes sense) that links research questions to instruments and my intentions.
Main instruments
Research
Sample
Questions
Students
RQ1ia
RQ1ib
RQ1ii
RQ2i
RQ2ii
Table 1
Instruments

Written tests; trigonometry test,
algebra test
 Interview
 Verbal protocol
Students
 Written tests; trigonometric word
problems, trigonometric functions
on right angled triangle test
 Interview
 Verbal protocol
Students
 Written tests; trigonometry test,
algebra test, trigonometric word
problems
test,
trigonometric
functions on right angled triangle
test
 Interview
 Verbal protocol
Teachers
 Teachers questionnaire
 Observation
 Interview
Documents
 Curriculum
 Textbook questionnaire
 Exam papers
 Scheme of works
 Textbooks
Linking research tools to research questions
3
In order to collect data from students in their natural teaching and learning environment, English
students allowed to use calculators and formula sheets whilst Turkish students were merely
allowed to use trigonometric tables.
Selected Results
I have huge amount of data2 which I could report on. To keep a sense of balance I report in detail
on some aspects of RQ1 and link them with some general observations concerning RQ2. I
confess at the outset that I present details where differences between the two countries were
significant. I comment on selected items from the algebra test, trigonometry test, trigonometric
word problems test and trigonometric functions on right angled triangles test.
The overall performance of English and Turkish students on the four tests reveals that English
students were better in all trigonometric word problems and Turkish students were considerably
better in algebra and trigonometry questions. The proportion of the correct answers of English
students was merely slightly higher than Turkish students’ in one question in the algebra test and
in the trigonometry test. In the trigonometric functions on right angled triangle test there was no
clear leading of English or Turkish students.
Non contextual Algebra and Trigonometry questions
Appendix 2 shows the percentage of correct (C), partial (P), incorrect (I) and not attempted (N)
responses in parallel Algebra (AT) and Trigonometry test (TT) questions, which are specific
examples, by English and Turkish students. The only difference between two expressions of the
each question was replacing sinx by x and cosx by y. Turkish students clearly do much better in
both questions. Both English and Turkish students did slightly better in the algebra question than
in the trigonometry question. Almost one half of the English students solved the algebra and
trigonometry questions incorrectly whereas about one quarter of the Turkish students produced
incorrect solutions. At the other extreme more than one half of the Turkish students gave correct
answers to both algebra and trigonometry questions whereas less then one fifth of the English
students solved both questions correctly. The number of the English students who did not attempt
the algebra and trigonometry questions were four times higher than the Turkish students. Turkish
students gave more partial answer to the trigonometry question than the algebra question.
Students have two kind of difficulties in the trigonometry question: one is not being be able to
use algebraic operations with trigonometric expressions and the other is not being able to
recognise the Pythagorean identity3 or to know which trigonometric identity to use. Such students
could not factorise the difference of two squares in the form of the x4-y4 (= (x 2  y 2 )(x 2  y 2 ) ),
they do not know the rules of cancellation they found the result like x 3-y3 after cancellation. A
majority of the students did not convert trigonometric expressions into algebraic expression to
simplify the trigonometric expressions.
Trigonometry word problems and trigonometric functions on right angled triangles
2
When I have finished my PhD I plan, as an amusement for and a warning to starting PhD students, to send an e-mail
to the Nottingham mathematics education network, on how many pieces of paper I processed in the course of my data
analysis. My current estimate is 19000.
3
sin 2 x  cos 2 x  1
4
The second question of the trigonometric functions on right angled triangle test was context free
form of the second trigonometry word problem. As it can be seen in the Appendix 3, there is no
much difference between the performance of the English and Turkish students, the only
difference is all of the Turkish students attempted the all questions whereas a few of English
student did not.
Students in both countries were successful with context free question, however they did less well
with the corresponding trigonometry word problems. Turkish students results were really bad
compared to English students’ result. Though almost same percent of English and Turkish
student did not attempted the question, English students correct answers were more than incorrect
answers whereas Turkish students incorrect answers were 5 times of their correct answers. None
of the English students answered the trigonometry word problem question partially, but few of
the Turkish students did.
All students have mainly three types of the difficulties with trigonometry word problems; first
thing was transforming their mental representation, (they could not draw the correct diagram), the
second thing was using the terminology in the problem and the last thing was mislabelling the
diagram (which mostly occurred because of the misuse of the terminology…they did not know
the diagram they drew was wrong or right). The students could not use the terminology correctly
e.g. the angle of elevation and angle of depression, especially the English students said they know
what is angle of elevation because they solved many questions about it, but they did very few
questions with angle of depression. They were not good at using terminology and the misuse of it
led students to incorrect answer.
DISCUSSION
I split this section into two sub-sections: broader curriculum/teaching and learning issues in the
two countries; and cognitive (roughly speaking) issues that have emerged from my research that
are relevant to a discussion of the results presented above.
Broader curriculum issues
I try and give a sense of the mathematics, the tools and the classrooms in both countries.
School mathematics in England and Turkey
In England, class sizes were, relative to Turkey, quite small, 15-25 in a class. Classrooms were
dedicated mathematics classrooms and the walls were full of posters on basic trigonometric
functions, trigonometric ratios and geometric figures. Overhead projectors, whiteboards and
computers were used in the lessons. Every student had her/his own calculator. Teachers had their
own (departmental) room which was full with textbooks, worksheets, calculators and a computer.
When students forgot to bring something like a textbook or calculator the teachers provided it.
Teachers closely followed the syllabus and paid heed to the exam board throughout the teaching
year. They mainly use textbooks, handouts, their own notes and worksheets during teaching.
Teachers present and students solve a number of ‘application’ problems.
In Turkey, in contrast to England, classes are quite crowded, 40-45 students in a class. The
classrooms are not dedicated mathematics classrooms and there is nothing on the walls relevant
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to mathematics. Calculators are not allowed to be used in lessons. Overhead projectors and
computers are not used in the lessons. Teachers mainly use chalkboards and students sit on
benches. There is no departmental room, just a common staffroom which did not contain
mathematics resources. Teachers are required to follow the national curriculum, but they said
they can change the order for their schemes of works. They mainly use textbooks, tests, their own
notes and worksheets during teaching. Trigonometry lessons are mostly on abstract mathematics,
e.g. simplification, solving equation and inequalities, trigonometry on right angled triangles and
other geometric figures. Teachers complained about the shortage of application problems in the
curriculum and also said trigonometry word problems are not part of the university entrance
examination, so they almost never do application problems.
Teaching and learning tools in England and Turkey
 Formula sheets are used in classrooms and in UK but not in Turkey. Question: to what extent
does this influence the better performance of Turkish students in the algebra and trigonometry
tests?
 Teachers in England use a wider variety of resources: overhead projectors, computers,
calculators and whiteboards as well as the textbooks, worksheets and blackboard used in
Turkish classes. Does using a wide variety of resources enrich lessons? If so, then how so?
 As mentioned above calculators were common place in English classrooms but were not
allowed in Turkish classrooms. This appears to partially explain at least two noted
phenomena: little emphasis on secants, cosecants and cotangents in England; an emphasis on
special angles, e.g. 300, 450, 600 and 900, and surd forms, e.g. sin60 0  3 in Turkey. Both
2
Turkish students and teachers complained about the range of angles considered. Because of
this they also rarely used trigonometric tables. English students use all kinds of angles.
Vygotsky spoke of tools altering the structure of mental functioning. I accept this and merely add
that they can also alter the structure of the curriculum. I have to consider the import of tool use
(and non-use) in much more detail in my thesis. I apologise for not doing so here but I want to
wait until I have clearer thoughts on the matter.
Mathematics classrooms in England and Turkey
 Trigonometry lessons in Turkey are more ‘abstract’ than in England.
 Teachers present and students in England solve more application problems than in Turkey.
 English teachers appear happy with the textbook they use but they also prepare their own
handouts and worksheets. Turkish teachers do not appear to be happy with the textbooks, and
use a variety of textbooks.
 Teachers in both country follow the given curriculum (and syllabi and exam board in
England). They prepare schemes of work but rarely write a lesson plan. Lessons in both
countries could be said to be ‘driven’ by examination foci (modular exams in England and
questions banks and tests from private institutes which prepares students for university
entrance examinations in Turkey).
 Teachers in both countries give homework. In England some homework is assessed but, and
this, I feel, is important, all homework is checked by teachers and returned to students.
Turkish teachers do not give homework on regular bases and they usually do not check
6
homework (they sometimes check HW during the lesson, but they do prefer not to do this
because this wastes time).
 Teachers are the important part of the teaching system they are involved with curriculum,
syllabii, exam boards, textbooks. They teach to children, they know the advantages or
disadvantages of the instructional tools and the most important thing is they know the
students, the difficulties they meet and they are the best spotlight point to observe teaching
system, so if the teachers could be a part of the community working on all resources
(curriculum…textbook), productive and effective resources, plans, instructional tools could
be prepared
 The pattern of lessons
England
review of the last lesson
introductory explanation
worked examples
students do exercises
Turkey
review of the last lesson
a) introductory explanation
b) worked examples (sometimes by
students at the board)
(a) and (b) repeated several times
NB students are rarely called to the NB Students do not do exercises on their
board
own
Clearly there are similarities and differences. I do not yet have a real sense of how classroom
differences (apart from ‘abstract’ and ‘application’ foci) may contribute, if at all, to the overall
differences in students’ performance.
Emergent (socio-) cognitive issues
I note areas for further exploration. They are ‘emergent’ in the sense that they arose during data
collection and analysis, i.e. they were not, with the exception of simplification, explicitly
considered in my research questions. In every case I have a great deal more work to do in
clarifying the issues and their import.
 Simplification appears to be a tool whose use is determined in practice  there is no definition
of ‘simplification’. Students’ ‘apprenticeship’ into simplification practices clearly varies in
the two countries. English students, more than Turkish students, appear not to know where to
stop or where to go with many expressions. The meaning of ‘simplification’ with regard to
trigonometric expressions does not appear as clear to students as it does with regard to
algebraic expressions.
 The apparent reasons behind the partial and incorrect answers to trigonometry questions were
mostly the use of inappropriate trigonometric identities or algebraic manipulation. Teachers
and students (especially Turkish ones) emphasised the importance of the solving a lot of
problems. To what extent is ‘solving a lot of problems’ a factor in students’ performance?
 Both English and Turkish students were better at algebra than trigonometry. Students often
did not treat trigonometric expressions as algebraic expressions. They did not, in general,
convert trigonometric expressions into algebraic expressions. But if they did, then would this
be enough to simplify a trigonometric expression?
7
 Protocol analysis suggests a recognition/doing ‘dialectic’, i.e. students see something and
write down something on the basis of what they see but what they write down also affects
what they see. I need to do further work even to clarify my statement of the dialectic but I feel
there is an important issue here.
 With regard to diagrammatic representations of word problems many Turkish students in
particular clearly had difficulty transforming their mental representation onto paper and
drawing appropriate diagrams. Turkish students’ drawings were more what I call ‘realisticabstract’ than English students’ abstract drawings in that, say, a word problem featuring a
statue would have a drawing of a statue and not simply a line representing the statue. Does
this realistic-abstract representation impede their word problem solving or is the better
performance of English students simply, again, a matter of them having more practice in such
problems?
CONCLUSION
What is meant by ‘trigonometry’ in England and Turkey varies greatly. Student performance of
is, not surprisingly, strongly related to what curricula emphasise: ‘context’ word problems in
England and ‘algebra’ in Turkey.
8
Appendix 2
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Percentage of correct (C), partial (P), incorrect (I) and not attempted (N) responses in
parallel Algebra (AT) and Trigonometry test (TT) questions by English and Turkish
students
sin 4 x  cos 4 x
x 4  y4
AT-) Simplify
TT-) Simplify
xy
sinx  cosx
Percent
English students
70
60
50
40
30
20
10
0
47
C
40
P
27
18
20
15
16
I
16
AT
N
TT
UK
CA%
IA%
PA%
NAQ%
Questions
AT
18
47
15
20
TT
16
40
16
27
Percent
Turkish students
70
60
50
40
30
20
10
0
65
51
C
P
26
25
I
18
5
6
5
AT
N
TT
Questions
Appendix 3
10
TR
AT7 TT8
65
51
CA%
26
25
IA%
5
18
PA%
5
6
NAQ%
Percentage of correct (C), partial (P), incorrect (I) and not attempted (N) responses in
parallel Trigonometry on right angled triangles (TORT) and Trigonometry word problem
(TWP) questions by English and Turkish students
TWP) A rocket flies 10 km. vertically, then 20 km. at an angle of 150 to the vertical and finally
60 km. at an angle of 640 to the horizontal. Calculate the vertical height of the rocket at the end of
the third stage.
TORT) Find the length |BD|=?
Percent
English Students
90
80
70
60
50
40
30
20
10
0
80
C
P
51
I
45
N
11
2
7
4
0
TORT
TWP
Parallel questions
11
UK
TORT TWP
80
51
CA%
11
45
IA%
2
0
PA%
7
4
NAQ%
Percent
Turkish students
90
80
70
60
50
40
30
20
10
0
83
77
C
P
I
N
15
2
15
5
0
TORT
3
TWP
Parallel questions
12
TR
TORT TWP
83
15
CA%
15
77
IA%
2
5
PA%
0
3
NAQ%