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

AbstractThis 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.

IntroductionThis 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.

RQ1iaThe 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.

RQ1ibThe 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.

RQ1iiAlthough 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?

RQ2iThe 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?

RQ2iiThe 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)?

MethodologyMy 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 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.

1 Strictly speaking ‘questions’ as it has several sub-parts.

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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 instrumentsResearch Questions

Sample Instruments

RQ1ia Students Written tests; trigonometry test, algebra test

Interview Verbal protocol

RQ1ib Students Written tests; trigonometric word problems, trigonometric functions on right angled triangle test

Interview Verbal protocol

RQ1ii Students Written tests; trigonometry test, algebra test, trigonometric word problems test, trigonometric functions on right angled triangle test

Interview Verbal protocol

RQ2i Teachers Teachers questionnaire Observation Interview

RQ2ii Documents Curriculum Textbook questionnaire Exam papers Scheme of works Textbooks

Table 1 Linking research tools to research questions

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.

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Selected ResultsI 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 (=

), they do not know the rules of cancellation they found the result like x3-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 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

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

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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 issuesI try and give a sense of the mathematics, the tools and the classrooms in both countries.

School mathematics in England and TurkeyIn 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 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

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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. in Turkey. Both

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 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

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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 Turkeyreview of the last lessonintroductory explanationworked examples students do exercises

NB students are rarely called to the board

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 do not do exercises on their 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 issuesI 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?

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 ‘realistic-abstract’ than English students’ abstract drawings in that, say, a word problem featuring a

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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.

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Appendix 2Percentage 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

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AT-) Simplify TT-) Simplify

Appendix 3Percentage 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

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Turkish students

65

51

5

1826 25

5 6

010203040506070

AT TT

Questions

Perc

ent

C

P

I

N

TRAT7TT8CA%6551IA%2625PA

%518NAQ%56

English students

18 1615 16

4740

2027

010203040506070

AT TT

Questions

Perc

ent

C

P

I

N

UKATTTCA%1816IA%4740PA%1516NAQ%2027

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|=?

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English Students

80

51

2 011

45

7 40

102030405060708090

TORT TWP

Parallel questions

Perc

ent

C

P

I

N

UKTORTTWPCA%8051IA%1145PA

%20NAQ%74

Turkish students

83

15

2 515

77

0 30

102030405060708090

TORT TWP

Parallel questions

Perc

ent

C

P

I

N

TRTORTTWPCA%8315IA%1577PA

%25NAQ%03

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