Trigonometry Learning

Abstract

Background: Trigonometry is an area of mathematics that students believe to be particularly difficult and abstract compared with the other subjects of mathematics. Trigonometry (MS5.1.2, MS5.2.3) is often introduced early in year 8 with most textbooks traditionally starting with naming sides of right-angled triangles. Students need to see and understand why their learning of trigonometry matters. Aims: In this study, particular types of errors, underlying misconceptions, and obstacles that occur in trigonometry lessons are described. Sample: 140 tenth grade high-school students participated in the study. 6 tenth grade mathematics teachers were observed. Method: A diagnostic test that consists of seven trigonometric questions was prepared and carried out. The students' responses to the test were analyzed and categorized. Observations notes were considered. Results: The most common errors that the students made in questions were selected. Several problematic areas have been identified such as improper use of equation, order of operations, and value and place of sin, cosine, misused data, misinterpreted language, logically invalid inference, distorted definition, and technical mechanical errors. This paper gives some valuable suggestion (Possible treatment of students' error obstacles, and misconceptions) in trigonometric teaching for frontline teachers. Conclusion: The study found students have errors, misconceptions, and obstacles in trigonometry lessons.

Full text

66 67
New Horizons in Education, Vol.57, No.1, May 2009
Trigonometry Learning
Hülya Gür
Balikesir University
Abstract
Background: Trigonometry is an area of mathematics that students believe to be particularly difficult and
abstract compared with the other subjects of mathematics. Trigonometry is often introduced early in year 8 with
most textbooks traditionally starting with naming sides of right-angled triangles. Students need to see and understand
why their learning of trigonometry matters.
Aims: In this study, particular types of errors, underlying misconceptions, and obstacles that occur in
trigonometry lessons are described.
Sample: 140 tenth grade high-school students participated in the study. 6 tenth grade mathematics teachers
were observed.
Method: A diagnostic test that consists of seven trigonometric questions was prepared and carried out. The
students’ responses to the test were analyzed and categorized. Observations notes were considered.
Results: The most common errors that the students made in questions were selected. Several problematic
areas have been identified such as improper use of equation, order of operations, and value and place of sin,
cosine, misused data, misinterpreted language, logically invalid inference, distorted definition, and technical
mechanical errors. This paper gives some valuable suggestion (Possible treatment of students’ error obstacles, and
misconceptions) in trigonometric teaching for frontline teachers.
Conclusion: The study found students have errors, misconceptions, and obstacles in trigonometry lessons.
Keywords: instructional misconceptions, learning trigonometry, obstacles.
學習三角數學
Hülya Gür
Balikesir University
摘要
背景
：比較其他數學的主題，三角學是學生認為特別困難和抽象的。學生通常在第八年級開始接觸三
角學，傳統上多數課本以介紹直角三角形的邊作開始。學生需要瞭解他們為什麼要學三角學。
目的
：這項研究觸及有關三角學習特殊錯誤的類型、其蘊含的誤解和在課堂上發生的障礙。
樣本
： 140名第十年級高中學生和六位有關的數學教師参加了本研究。
方法
：學生們參加了一次包括七個三角問題的診斷測試，其反應和觀察筆記被分類分析。
結果
： 列出了學生答題時的通病，像誤用方程式、運算程序、正弦和餘弦的數值和位置、誤用數據、
解錯題、錯誤邏輯推理、曲解定義、及技術運算錯誤等。本文給前線老師在教授三角學時可貴的建議 [學生
的錯誤障礙和誤解可能的治療]。
結論
： 本研究找出了學生在學習三角課時所犯的錯謬、誤解、和障礙。
關鍵詞：授課誤解、學習三角、障礙

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INTRODUCTION
Mathematics, particularly trigonometry is one
of the school subjects that very few students like and
succeed at, and which most students hate and struggle
with. Trigonometry is an area of mathematics
that students believe to be particularly difficult
and abstract compared with the other subjects of
mathematics.
Three generalizations were made because of
their relationship to Piaget’s description of formal
operations that could be drawn from the study on
misconceptions. These three generalizations are:
Many misconceptions are related to 1. a concept
that produces a mathematical object and
symbol. For example: sine is a concept and
symbol of trigonometric functions.
Many misconceptions are related to 2. process:
the ability to use operations. For example: as
representing the result of calculation of sin300
and value of sin300.
Many misconceptions are related to 3. procept
that is, the ability to think of mathematical
operations and object. Procept covers both
concept and process. For example: sinx
is both a function and a value. In addition
to this, Gray and Tall (1994) asserted that
“procedural thinking,” that is, the ability to
think of mathematical operations and object as
procept, is critical to the successful learning of
mathematics.
Many studies concerned with mathematics
education explain that students have misconceptions
and make errors, and these situations grow out of
learning complexities (Lochead & Mestre 1988;
Ryan&Williams, 2000). Of late, a few researchers
have mentioned students’ misconceptions, errors,
and related to these, learning complexities about
trigonometry (Delice 2002; Orhun 2002). Fi (2003)
states that much of the literature on trigonometry
has focused on trigonometric functions. Fi’s study
is related to the preservice secondary school
mathematics teachers’ knowledge of trigonometry:
subject matter content knowledge, pedagogical
content knowledge, and envisioned pedagogy. A
few researchers studied more specific issues in
trigonometry such as simplication of trigonometric
expressions and metaphors (Delice, 2002; Presmeg
2006, 2007). Brown (2006) studied students’
understanding of sine and cosine. She reached a
framework, called trigonometric connection. The
study indicates that many students had an incomplete
or fragmented understanding of the three major ways
to view sine and cosine: as coordinates of a point on
the unit circle, as a horizontal and vertical distances
that are graphical entailments of those coordinates,
and as ratios of sides of a reference triangle… (p.
228). Orhun (2002) studied the difculties faced by
students in using trigonometry for solving problems
in trigonometry. Orhun found that the students did
not develop the concepts of trigonometry certainly
and that they made some mistakes. The teacher-active
method and memorizing methods provide students
knowledge of trigonometry only for a brief moment
of time, but not this knowledge is not retained by
the students in the long run. Delice (2002identified
five levels for measurement of students’ knowledge
about research theme and for defining students’
skills. According to the results of the research,
students partially answered the questions at the
first and second levels and inadequately answered
the questions at the other levels. The students have
misconceptions and learning complexities, which
is attributed to the fact that before learning the

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Trigonometry Learning
trigonometry concepts, the students learn some
concepts, pre-learning concepts, incorrectly or
defectively. These concepts are fundamental for
learning the concepts of the trigonometry such as unit
circle, factorization, and so on. Delice (2002) has a
main assumption about the research that, generally
speaking, errors are not random but results from
misconceptions and that these misconceptions need to
be identied in the study of trigonometry. Therefore,
the students could not learn the procedure of solving
the verbal problems confidently. Hogbin (1998)
also reported similar findings. Additional findings
by Delice (2002) indicated that Turkish students
did much better with the algebraic, manipulative
aspects of trigonometry and that English students
did better with the application of trigonometry to
practical situations in England. This article reports
upon particular aspects of a study, the main aim of
which was to compare the performance of students
in the 16–18-age group from Turkey and England
on trigonometry and then to compare the curriculum
and assessment provision in each country to seek
possible explanations for differences in performance.
Weber (2005) investigated trigonometric functions in
a study, which involved students of two colleges. One
group was taught trigonometric functions traditionally
and second group was taught according to Gray and
Tall’s (1994) notion of procept, current process object
theories of learning. He found that second group that
was taught by Gray and Tall’s theories understood
trigonometric functions better than the other group.
Guy Brousseau claims that the errors committed by
students or the failure of students are not as simple
as we used to consider in the past. The mistake not
only results from ignorance, uncertainty, or chance as
the empirical theory of knowledge used to claim. The
mistake is the result of the previous knowledge that
used to be interesting and successful, but now it has
been proved wrong or simply uneducable. Mistake
of these kinds are not irregular and unpredictable and
these mistakes are due to obstacles. In the function
of both the teacher and student, the mistake is a
constituent part of the acquired knowledge. The
present article distinguishes among the different
mistakes committed by students, which result from
obstacles and misconceptions.
However, many errors are committed due
to the mechanical application of a rule in the
trigonometry exercises. The researchers believe that
some of the student’s errors are related to the concept
of “didactic contract”. As Guy Brousseau (1984)
says “in all the didactic situations a negotiation of
a didactic contract is taking place, which defines,
partly explicitly but mainly implicitly, what each
partner has to do and for which, in a way or another
he is held responsible towards the other. In another
part he writes “Thus, in the didactic contract three
elements are present: the pupil (the person who is
taught), the teacher (the person who teaches), who
are the partners, and the knowledge, as “material to
be taught”. The role played by the didactic contract is
that of settling the interaction between the teacher and
the pupils in connection with some knowledge. For
example, the research of Bagni (1997) “Trigonometric
Functions: Learning and Didactical Contract” gives
evidence that in many trigonometric exercises the
didactical contract forces the students to nd always
the solution to exercises that have no solution.
Description of the Research
The study focused on ve objectives: What are
the errors committed by students in trigonometry?
What is a possible categorization of these errors and
obstacles? What are the misconceptions and obstacles

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Hülya Gür
71
relating to learning trigonometric concepts? What are
the possible treatments of students’ errors, obstacles,
and misconceptions? What are the student’s answers
that help us explore the students’ thinking and
reection about learning?
Participants
In Turkey, the trigonometry is taught to
students in mathematics lessons during the 10th
grade of High school, which is also called lyceum
(i.e. age 14-17: High-school education covers the 4
years over the 14–17 age range). The students meet
comprehensive trigonometry instruction at second
semester of the10th grade. Participants were chosen at
random but proven not to be a representative sample.
The sample was taken from different high schools,
which have very different backgrounds. Teachers
conducted a trigonometry diagnostic test to all 140
students.
Instruments
An interview was carried out with six 10th
grade mathematics teachers to learn the problems of
teaching mathematics. The researcher also made a
four week observation in the 10th grade mix ability
mathematics class (Table1). The participants’ past
experiences about trigonometry: In Turkey, the
trigonometry concept is first taught to students in
mathematics lessons in the 8th grade of Elementary
schools (i.e. c. age 9-10: Elementary Education
covers the 8 years over the 6-14 age range). Brief
explanations of right angle are given in the 8th
grade (ages 12-13) and a general introduction to
trigonometry is made in the 10th grade (ages 14-
15). The formal mathematics courses, which go
on for three years, start with secondary education,
which is also called high school or Lycée. During
the observation students and teachers investigated
about trigonometry teaching and learning. The
mathematics teachers in Turkish secondary schools
usually prefer teaching with traditional techniques.
In mathematics teacher training course, complex
analysis (3credits), applied mathematics (3credits),
history of mathematics (2credits) and calculus (6
credits) course includes trigonometry subject. But
mathematics teacher trainees have a little practice
of teaching trigonometry. Mathematics teachers
tend to concentrate on solving the problems through
algorithmic approaches, rather than concept learning.
It is considered that practicing examples in this way
is the best preparation for the university entrance
examination (OSS). Additionally, the fact that the
high school mathematics curriculum prescribes a lot
of material to be covered is perceived as a real barrier
to an emphasis on conceptual learning.
Table 1:
Interview
Male Female
2 teachers 1 teachers 2 teachers 1 teachers
Age 25-35 51 24-35 52
Teaching experience 2 years
11 years
23 years 2 years
10 years
24 years
Present teaching
course
10th grade
(mix ability class)
10th grade
(mix ability class)
10th grade
(mix ability class)
10th grade
(mix ability class)
Observation four weeks observation during trigonometry teaching
(mix ability class)

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Trigonometry Learning
The interview extracts were analyzed and 20
trigonometry questions were established. The
diagnostic test was applied to thirty-two 10th grade
students for piloting. After the pilot study, three
university lecturers, 10th grade mathematics teachers,
reviewed items of the diagnostic test and researchers
tailored it. The last form of diagnostic test included
seven questions (Appendix 1).
Analysis and Scoring
The seven questions have been coded and
analyzed according to a concept-evaluation scheme
(Weis s & Yoe s, 1991). All the answers to the
questions were itemized and categorized from I to IV
using the scoring criteria (Table 2). Also, diagnostic
test paper of each student was labeled 1 to 140.
If the answer was coded as a correct answer, student
gave all components of the validated response and
correct answer (I). For example, for the right-angle
triangle (a/c)2+ (b/c)2= (a2+b2)/c2=c2/c2=1.
If the answer was coded as a partial understanding,
the student gave at least one of the components of
the validated response, but not all the components
of the correct answer or just given concept process.
This section was divided into two categories: Error
in mechanical application of a rule. Error is related
to concept of teachers’ teaching, students’ learning,
and knowledge. Forexample:x2+y2=sin2x+cos2x=1
(process). If the students veried the identity for only
a single case, it is also called partial understanding.
e.g. sin2300+cos2300= (1/2)2+(
3
/2)2=4/4=1
If the answer was coded as a Misconception or
obstacle, the student gave illogical or incorrect
information as an answer. Misconception covered
mistakes and obstacles. It is the result of previous
knowledge and obstacles.
For example: 1/(cos2x)+1/(sin2x)=1 and sin2x.
cos2x=1 (procept). Although it compromised concept
and process, it did not give the result of this question.
In the equation of sin2x+cos2 x was used and the
value of equation was not found as sin2x.cos2x=1.
Justifying why sin2x.cos2x=1 involved reasoning
about a process that could be used to produce the
value of sin2x.cos2x=1. Similarly, justifying why
“sin2x.cos2x=1” was not the range of sinx, cosx, sin2x,
cos2x involved understanding no matter what the
input for this process. If the answer was coded as an
unacceptable, the students gave Irrelevant or unclear
responses, or not answered or irrelevant answers
or repeat information in the question as if it was an
answer or blank. Thus, the coding schemes were
developed and the respondents’ ideas were coded.
The frequencies were calculated. If the students
mentioned these items, we calculated as a percentage.
Three researchers performed coding, and intercoder
reliability was found to be 89%. All represented
ndings included answers, which is given as italics.
Table 2:
Criteria for Scoring
I-Correct
answer
Included all components of the validated
response, correct answer
II-Partial
Understanding,
Misconception
or obstacle
*At least one of the components of
the validated response, but not all the
components, just concept or process or
mechanical application of a rule, did not
involve any justication
*Included illogical or incorrect information
or information different from the correct
information
III-
Unacceptable
Irrelevant or unclear responses or not
answered or irrelevant answers, repeat
information in the question as if it was an
answer or Blank
The examples presented in this article
are errors committed by students, obstacles and
misconceptions that arose from high-school lessons
in trigonometry.

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Hülya Gür
73
FINDINGS
The present study is different from the other
studies of trigonometry in terms of sampling,
methodology, data analysis techniques, and ndings.
There are many errors, obstacles, and misconceptions
in trigonometry, and these are given in this section.
Question 1: sin2x + cos2 x = 1. Why?
Please explain.
The list of students’ writing of the question
1 related to the “sin2x + cos2 x = 1” was coded and
presented below according to the various levels of
understanding. Students’ justifications are given
below:
I. Correct answer (77 responses out of 140)
(a/c)2+(b/c)2= (a2+b2)/c2=c2/c2=1 (If a, b, and c
indicate the sides of a triangle)(a2+b2)/c2=c2/c2=1
(If a, b, and c indicate the sides of a triangle)
sin2x+sin2(90-x)=a2/(b2+a2)= b2/(b2+a2)=1/1=1
II. *Partial Understanding (49 responses out of 140)
sin2x= 1-cos2x (process)
Already proved in a unit circle. (concept of unit
circle)
sin2300+cos2300= (1/2)2+(
3
/2)2=4/4=1 (process)
Thirteen students gave a mathematically valid
explanation for why this equation was true. Ten out
of thirteen students exemplified that this was true
using a right-triangle model. These results were
similar to the nding of Weber’s study. Ten students
demonstrated the identity for only a single case
such as sin2300+cos2300= (1/2)2+(
3
/2)2=4/4=1
(process).
sin2x + cos2 x=1 formula can only be obtained from
the Pythagoras equation (concept).
x2+y2= sin2x+ cos2x=1 (process)
*Misconception or obstacle (12 responses out of 140)
1/(cos2x)+1/(sin2x)=1; sin2x. cos2x=1 (procept)
In the equation of sin2x + cos2x was used and
the value of equation was not found as sin2x. cos2x=1
used a2-b2= (a-b)(a+b) formula but did not get the
result. (process)
sinx and cosx are only defined in a unit circle.
(concept)
There is an inverse relation between sinx and cosx.
(sinx+cosx)2
= sin2x+ 2cosx.sinx+ cos2x
(sinx-cosx)2
= sin2x- 2cosx.sinx+ cos2x
+--------------- +-------------------------
sin2x+cos2x=(sinx-cosx)2+(sinx+cosx)2 (concept)
In the equation of sin2x + cos2x was obtained,
but the value of equation was not 1.
III. Unacceptable (2 responses out of 140): two
students gave the answer as a blank.
According to results, 55% of the students
showed an understanding of this question: 35% of the
students had a partial understanding of the question;
nearly 8% of the students showed misconception
statements, which were identified through analysis
of the question, and five misconception statements
were identified through analysis of the question
sin2x + cos2 x = 1 and unit circle defined as sin2x.
cos2x. Therefore, the result is 1. Most students only
simply memorized the formula and they calculated
it using the unit circle. This can be attributed to the
fact that course textbooks and teachers almost always
introduced this subject using the same method as
was shown by the students. The students retained the
aspects learnt by them in secondary school when they
moved on to high school. The students knew only the
sin2x + cos2 x=1 equation, but could not explain it.
The other reason for this was the secondary-school

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Trigonometry Learning
mathematics textbooks in Turkey. The following
statements clearly identify this misconception:
“If a numeric value is given, then I could have
Question 2: tanx =
xcot
1
or tanx . cotx = 1. Please explain why?
calculated”, “if a proving question is asked, and I
could have proved it” (Student 13: S13)
“I never thought that sin2x + cos2 x is equal
one?” and “we use a2-b2= (a-b)(a+b) formula in
calculations”(Student 20:S20)
Common errors, obstacles, and misconceptions
that students made with probe “tanx.cotx=1”
equation is highlighted. The justifications given by
students are shown below:
I. Correct answer (130 responses out of 140)
tanx=(1/(cotx))=(1/(cosx/sinx))= sinx/cosx (6
responses out of 140)
tanx a/b, cotx=b/a; a/b=(1/(b/a)) then (a/b)=(a/b)=1
(100 responses out of 140)
sinx ≠0 and cosx≠0 (11 responses out of 140)
tanx.cotx=1 is always give 1. (8 responses out of 140)
It is the opposite of each other (5 responses out of
140)
Surprisingly, 130 students answered this
question correctly. But they only memorized the
denition without any understanding.
II. *Misconception or obstacle (9 responses out of
140)
It gives 1=1. (5 responses out of 140) (There was no
explanation)
tanx and cotx in relation to each other were complete
at 3600 (4 responses out of 140
III. Unacceptable (1 response out of 140)
Their answers showed that the students had
found the correct answer because most of them
knew the formula of tanx and cotx. However, they
wrote only simplified answers, which showed only
partial understanding of the question. Examples of
the answers they gave were, “because it’s opposite/
backwards/the wrong or other way round” as well as,
“is always multiplied by 1”. From this, it can be seen
that students simply memorized the formula found
in the textbooks, e.g. 7% of the students showed a
misconception for the question:
“1 = 1” (S 139, S 110, S5, S47, S72)
“tanx . cotx” always completes each other to 180o.
(S139 S8, S81, S72)
These answers were classified as
misconceptions. A statement in such a book
potentially encouraged this kind of misconception, as
stated below:
“If a radius is equal to one, this is a circle equation.
This is called unit circle” (Barış, 2000). The denition
shown in the books encourage students to develop
misconceptions. The list of students’ writing related
to the “tanx.cotx =1”was coded and presented below
according to the various levels of understanding of the
students who showed understanding of this question.

74
Hülya Gür
75
Common errors, obstacles, and misconceptions that
students make with Probe “tan 90 = undefined”
equation is highlighted. Students give justifications
below:
I. Correct answer (132 responses out of 140)
tan 90 = sin90/cos90 = 1/0, undened
Number / 0 = ∞, then it is undened
II. *Partial Understanding (4 responses out of 140)
If x=900, then the tangent curve did not intersect x=1
and x=-1 lines
*Misconception or obstacle (3 responses out of
140)
1.tan90= hypotenuse/adjacent line
III. Unacceptable (1 response out of 140), no
response
The list of students’ writing related to the
“tan90=undefined” was coded and presented below
according to the various levels of understanding.
Most of the students understood the question. Only
3 of the students showed misconception of this
question. The reason for the correct answers might
be that the students memorized the expressions they
learned from their course books. The students just
memorized everything from the textbook, regardless
of what they were taught in class.
Question 4: What is
=−
2222
)(cos)(sin xx
? Please simplify.
The list of students’ writing related to the
“
=−
2222
)(cos)(sin xx
”was coded and presented
below to the various levels of understanding.
Common errors, obstacles and misconceptions that
students make with probe “
=−
2222
)(cos)(sin xx
?”equation is highlighted. The justications given by
students are given below:
I. Correct answer (54 responses out of 140)
No result to the differences of two squares; sin2x-
cos2x
1-(cos2x- sin2x) = -cos2x
II.*Partial Understanding (11 responses out of 140)
No result to the differences of two squares
It is only considered the square of the parenthesis;
they did not consider the inside of parenthesis
*Misconception or obstacle (75 responses out of 140)
Above equation is equal to sin2x=1-cos2x (students
memorized this equation) (50 responses out of 140)
The equation is equal to 1-2cos2x. (Students were
memorizing this equation) (25 responses out of 140)
Approximately 39% of the students displayed
an understanding of the question, and 85 of the
students showed partial understanding. These answers
showed that the students had realized the difference
between two squares. Most of the students showed
misunderstanding of the question. It can be seen that
students simply memorized information and were
unable to transfer this knowledge to another situation.
Question 3: tan 90 is undened, Please explain why?

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Trigonometry Learning
Common errors, obstacles, and misconceptions that
students make with Probe “tanA=3/4 then tan2A”
equation is highlighted. Students gave justifications
as shown below:
I. Correct answer (34 responses out of 140)
Let us consider 3,4,5 right triangle.
1.tanA=3/4,then tan2A=sin2A/cos2A=2sinA.cosA/
(cos2A-sin2A)=24/7
2.tan2A=sin2A/cos2A=24/7 (16 responses out of
140)
II.*Partial Understanding (24 responses, 18%)
The formula tan (A+B) could not be converted to
tan2A.
Only a formula can be used
Error was committed during working out of the
answer
*Misconception or obstacle (72 responses out of
140)
‘I did not know which side of the triangle is called the
adjacent edge or opposite edge’. 900 (13 responses,
out of 140)
a2=9+16-8a+a2thena=25/8; tan2A=3/(4-a)= 3/
(4-25/8)= 24/7 (17 responses out of 140)
tan2A = (3/2)2=9/16 (14 responses out of 140)
sinA/cosA = 3/4 (15 responses, out of 140)
tan2A=1+tan2A= 1-tanA.cotA=1+(3/4). (4/5)=9/5
(13 responses out of 140)
III. Unacceptable (10 responses out of 140), no
responses
The list of students’ answers related to the
“tanA=3/4 than tan2A” question was coded and is
presented below according to the various levels of
understanding. Students who learnt the formula were
unable to apply the formula in different situations.
51% of the students showed misunderstanding of the
question. They were unable to remember the formula
as they only memorized it.
Question 6: cos(-
θ
) = cos
θ
. Please explain why?
The list of students’ writing related to the
“cos(-
θ
) = cos
θ
”was coded and presented below
according to the various levels of understanding.
Common errors, obstacles, and misconceptions that
students make with Probe
“cos(-
θ
)=cos
θ
”equation is highlighted. Students
give the justications below:
I. Correct answer (71 responses out of 140)
The function of cosx was positive in region I and in
region IV of the unit circle (they drew a unit circle)
It was a double function
II. *Partial Understanding (27 responses out of 140)
cos(-37)=cos37 then 0.8=0.8. and cos (-45)=cos45
cosx will be positive in region IV (10 responses out of
140)
* Misconception or obstacle (36 responses out of
140)
-θ = -180 (4 responses out of 140)
n is in the area II (5 responses out of 140)
The value of “cos (-
θ
) = cos
θ
” was same
everywhere (10 responses out of 140)
It was the same for every value (8 responses out of
140)
The symbols do not change (9 responses out of 140)
III. Unacceptable (6 responses out of 140)
Question 5: tanA=3/4 then what is tan2A=? Please explain.

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77
The students stated, “It is easier to work out the
equation if they are given values to work with, and
if the angle would be equal in the second area”. In
addition, they said that “every value would be the
same,” and that “the symbol would not change”.
The memorized formula was either forgotten, or
remembered incorrectly. Some other students wrote
that they did not have sufcient knowledge to answer
the question.
Question 7: Why are tangent and cotangent functions positive in Region III of the unit
circle? Please explain?
The list of students’ writing related to the
“why are the tan and cos functions positive?” was
coded and presented below according to the various
levels of understanding. Common errors, obstacles,
and misconceptions that students make with probe
“Why are the functions of tanx and cotx positive
in region III of the unit circle?” Students gave the
justications as shown below:
I.Correct answer (118 responses, 84%)
tanx = sinx/cosx= -/- = +
cotx= cosx/sinx=-/- = +
- + + +
- - + - (108 responses out of 140)
II. *Partial Understanding (5 responses, 4%) x and
y have negative values (5 responses out of 140)
*Misconception or obstacle (17 responses, 12%)
tanx = -1 / (-1) = 1 (7 responses out of 140)
cotx = -1 / (-1) = 1 (7 responses out of 140)
By definition is positive (3 responses out of 140).
They understood the values for tanx and cosx to be 1,
and because of its denition, they thought it would be
positive.
DISCUSSION AND CONCLUSION
Identifying and helping students overcome
obstacles and misconceptions includes 5 subsections
that give an answer for each research question.
What are the sources of errors committed by
students?
The results of this study showed that students
have some misconceptions and obstacles about
trigonometry. One of the two obstacles to effective
learning was that trigonometry and other concepts
related to it were abstract and non-intuitive. Lochead
and Mestre (1988) described an effective inductive
technique for these purposes. The technique may be
induced conflict by drawing out the contradictions
in students’ misconceptions. In the process of
resolving the conflict, a process that takes time,
students reconstruct the concept (Ubuz, 1999, 2001).
The students had problems with prior and new
knowledge about concept, process, and procept in
learning trigonometry. The reasons of errors, which
students made in trigonometry lesson, were mal-rule
teaching or teaching concepts. This may be especially
important at the introductory level. It is caused from
their habits, as well as the development of inaccurate
constructions, on the part of the learner. It may
also be useful for the teacher, when recognizing
a specific error, to point it out to the students for,
as Borasi (1994, page 166) observed, “although

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Trigonometry Learning
teachers and researchers have long recognized the
value of analyzing student errors for diagnosis and
remediation, students have not been encouraged to
take advantage of errors as learning opportunities
in mathematics instruction.” The teacher has an
important role to play in overcoming it. Teacher’s
roles are to observe the students, and if they are
making mistakes and errors; s/he could discuss and
correct them.
What is a possible categorization of these
errors?
Students make several different reasoning
errors in trigonometry. Some of these errors were
based on underlying obstacles and misconceptions,
while others, although repeatedly observed, were of
a partial understanding or misconception. Students
persist in making both types of errors.
All the answers of the students to the questions were
itemized from I to IV using the scoring criteria in
Table 1. The classification of general errors was
based on model of Movshovitz-Hadar et al. (1987).
The theoretical assumption was that “the most of
students’ error in high-school mathematics are not
accidental and are derived by a quasi-logical process
that somehow makes sense to the student” (p.3–4).
This model includes five descriptive categories of
errors and misconceptions, which were identied in
the present research, except one category, unveried
solution. Excluding general errors, the study
considered errors that resulted from misconceptions
of procedures and concept linked to trigonometry.
The categories of trigonometry misconceptions were:
misused data, misinterpreted language, logically
invalid inference, distorted definition, and technical
mechanical errors.
Misused data: e.g. tanx . cotx multiplication always
gave 1=1
Misinterpreted language: These misconceptions
were related to a concept that produced a
mathematical object and symbol (e.g. ‘I did not know
which side of the triangle is called the adjacent edge
or opposite edge’. 900; A value of “cos (-
θ
) = cos
θ
” is same everywhere),
Logically invalid inference: These misconceptions
were related to procept (e.g. sinx and cosx are only
dened in a unit circle; sin2x + cos2 x= (sinx-cosx)2
+(sinx+cosx)2; tanx and cotx in relation to each other
are complete at 3600 and it is the same for every
value),
Distorted definition: the misconceptions were
related to procept, that is, the ability to think of
mathematical operations and object. Procept covered
both concept and process. (e.g. sin2x. cos2x=1, for
every x; 1.tan90= hypotenuse/adjacent line),
Technical mechanical errors: These misconceptions
were related to process: the ability to use operations
(e.g. The equation is equal 1-2cos2x; a2=9+16-8a+a2
then a=25/8; tan2A=3/(4-a)= 3/(4-25/8)= 24/7).
To sum up, we sometimes speculate on the
underlying causes of the errors, but do not see them
as conceptual in nature. One of the following errors
such as “usage of simple formulae,” “calculation of
trigonometric values of angles,” and “to populate the
values into a formula” was determined to be correct
by one or more students. Few students made mistakes
choosing the correct ratio to use, but the manipulation
caused some problems. For example, some students
seemed not to be sure whether s/he should use the
trigonometric value of the angle. In addition to the
above-mentioned problems, few students also had
problem with notation.

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What are the obstacles and
misconceptions relating to concepts of
learning trigonometrically?
We offered our views as to the possible
underlying obstacles and misconceptions, that is,
we gave a general rule or idea, which, if believed by
a student, would result in that type of error. These
errors were considered to have a rational basis,
and we comment on how they might come about.
Another mistake made by students was when they
found the result of a division when zero divides a
number. The students gave up their errors, obstacles,
and misconceptions, which can have such a harmful
effect on learning, only with great reluctance. Not
only do students bring their experiences, obstacles,
and misconceptions to class, students suggested
that repeating a lesson or making it clearer will not
help those students who base their reasoning on
strongly held misconceptions. The other obstacle is
the exam of university entry. The university entrance
examination should be more comprehensive, and
more importance should be given to questions on
trigonometry. Students should be well motivated
while solving problems using four operations,
formula and parenthesis about trigonometry. This
set of results gave an indication that students are
prone to common errors even when teachers have
adopted different teaching strategies for teaching
trigonometry.
What are the possible treatments of students’
errors, obstacles, and misconceptions?
Students do not come to the classroom as “blank
box”. Instead, they come with their own ideas
and theories constructed from their everyday
experiences, and they use these theories. Another
suggestion of treatment is using resources, materials,
diagrams, and equipments. Explaining to each
other helps improve the students’ understanding,
and enhances confidence in their own mathematical
ability. This would also work while manipulating
the formulae to try to `solve’ a triangle combined
with the use of the algorithmic diagrams and graphs.
Presmeg (2006) emphasizes that connecting old
knowledge with new and allowing ample time and
moving into complexity slowly, connecting visual
and no-visual registers like numerical, algebraic, and
graphical signs are important for students, providing
memorable summaries in diagram form, which have
the potential of becoming prototypical images of
trigonometric objects for the students, because these
inscriptions are sign vehicles for these objects in
trigonometry teaching.
In a trigonometry example, simple material
needs to be used in order to show its spiral function.
Before introducing a new topic, teachers need to
know if students have enough knowledge. If the
students do not have enough knowledge, or have any
misconceptions, the teachers need to try and get rid
of it. For example, before teaching trigonometry, the
students need to give some examples of Pythagoras
theorem in different triangles. Meaningful learning
needs to be obtained. The meaningful learning
aids that might be used include Vee Heuristics,
concept map, mind map, etc. To sum up, the present
study concludes with a discussion of techniques
to help students overcome their errors, mistakes,
and misconceptions in trigonometry. These errors,
obstacles, or misconceptions can be got rid of
with plenty of practice. Errors, obstacles, and
misconceptions made by students working at the
board can be picked up and corrected by the other
students. When students feel comfortable in class, we
can act with them, without the error costing us very

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Trigonometry Learning
much. But this view only indicates that getting rid of
errors and misconceptions is an act of education that
integrates thinking, feeling, and acting (DES, 1981).
However, since misconceptions are within the domain
of conceptual behavior and since they are based on a
belief about a certain mathematical situation, it makes
it easier to engage them in classroom activities rather
than simply try to correct them (Ryan & Williams,
2000).
Another area of weakness was in considering
long-term action to reduce or avoid misconceptions.
Both the students and teachers recognized that
the work on misconceptions had many benefits
in addressing areas such as planning, classroom
observation, and evaluation. In doing this, it
highlighted the inter-relationships between areas and
thus avoided the fragmented and itemized approach
that the initial training regulations could be seen to
encourage. The intention is to retain and refine the
student’s diary and assignment for use in future years.
What are the student’s answers that help
us explore the students’ thinking and
reection about learning.
Inst ea d of that the re is a de scrip ti v e
presentation of student’s answers without any kind
of interpretation could be helping us to their thinking
and reflecting about their learning. Students reflect
their thinking into their writing. If we carefully
analyze their writing, it is easy to understand them.
Thus, teacher and student could understand each
other.
PROSPECTS FOR FURTHER WORK
As a res u l t of the study, t o de te r mine t he
misconceptions and obstacles about trigonometry, the
researchers must investigate the students’ cognitive
processes. The determination of the students’
misconceptions, obstacles, and errors also involves
a qualitative study, so it must be a deep study. For
an in depth understanding of the students’ problems
in trigonometry, the interviews can be done with the
students.
APPENDIX A: DIAGNOSTIC TEST
Section 1
sin1. 2x + cos2 x = 1. Why? Please explain.
tanx = 2. or tanx . cotx = 1. Please explain.
Tan 90 is undened. Please explain.3.
(sin4. 2 x)2 - (cos2 x)2? Please simplify.
TanA=3/4 then tan2A=?5.
cos(6.
-Ɵ
) = cos
Ɵ
. Please explain why?
Why are tangent and cotangents functions positive 7.
in Region III of the unit circle? Please explain.
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Author:
Hülya GÜR e-mail: hgur@balikesir.edu.tr
Secondary Science and Mathematics Education Department,
Necatibey School of Education, Balikesir University,
10100, Balikesir, Turkey
Received: 17.11.08, accepted 20.1.09, revised 22.1.09