European Journal of Education Studies
ISSN: 2501 - 1111
ISSN-L: 2501 - 1111
Available on-line at: www.oapub.org/edu
10.5281/zenodo.164971
Special Issue - Basic and Advanced Concepts, Theories and Methods
Applicable on Modern Mathematics Education
RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT
OF CIRCLE ON MATHEMATICS TEACHER CANDIDATES
Ayşe Yavuzi, Selin (Inag) Çenberciii
i
Department Of Mathematics Education, Elementary School Math. Teaching Program,
Necmettin Erbakan University, Turkey
ii
Department Of Mathematics Education, Secondary School Math. Teaching Program,
Necmettin Erbakan University, Turkey
Abstract:
The purpose of this study is to research the effects of visual contextual support for the
subject of circle in geometry education. The study was conducted by using one of the
qualitative research methods, special case study. The sampling of the research was
formed by 72 teacher candidates who are studying in Necmettin Erbakan University
Ahmet Keleşoğlu Faculty of Education Elementary School Math. Teaching Program. As
data collection tool, an evaluation consisting of verbally asked problems regarding the
subject of circle was used. Questions were selected according to the relevant subject
among the Olympics questions. When selecting the questions, care was taken to contain
most basic information about the relevant subject but include the attributes that are
having difficulties in transferring of concepts into shapes. As a result of the study, it
was observed that teacher candidates were having difficulties in transferring the
question into a shape while solving the questions that are directed to them without a
visual contextual support and reaching the correct answer. In conclusion, it is seen that
questions provided with visual contextual support could easily be solved by students
but prevents their spatial thinking skills.
Keywords: geometry education, circle, mathematics teacher candidates, visual
contextual support
Copyright © The Author(s). All Rights Reserved
Published by Open Access Publishing Group ©2015.
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
MATHEMATICS TEACHER CANDIDATES
1.
Introduction
One of the most basic difficulties in the process of teaching mathematics is revealed
during the process of making a connection between the representation in the mind of a
student and abstract nature of mathematics. It is not easy to find concrete examples for
each mathematics concept. Graphics, diagrams, pictures and geometrical shapes or
models are visualization tools for abstract concepts in mathematics. People can make a
connection between the physical or external world and abstract concepts through these
(Konyalıoğlu, 2003). The discipline that focuses on this connection is geometry.
Students can analyze the abstract world by using geometry knowledge; utilize visual
elements to better understand the abstract concepts (Nemirovsky & Noble, 1997).
Geometry is one of the most important fields of mathematics education
containing space and shape concepts (Fidan & Türnüklü, 2010). Tapan and Arslan
(2009) have expressed that one of the most important objectives of geometry is to
develop the visual perceptions and rational thinking skills of students. Thanks to
geometry, students start to express and understand the world around them, analyze
and solve problems (Erdoğan, Akkaya & Akkaya, 2009). Thus, they can comprehend
the shapes around them and make a connection between the daily life and geometrical
shapes. In geometry teaching, understanding the mathematical definition, making and
proving assumptions, visualization and solving problems is essential (Jones, Mooney
and Harries, 2002). According to Shield and Swinson (1997), after a concept is examined
and discussed in all aspects, providing the development of a definition for this concept
by students is considerably useful in the process of students deeply understanding this
concept.
The teacher utilizes visual representations when explaining a mathematical
concept to student. These representations are, in a sense, the tools ensuring
communication in mathematics. Mostly, representation and demonstration are utilized
in geometry teaching. Poincare divides the science of geometry into two parts as
geometrical space and representation (shape) space (Mesquita, 1998). For example,
while the definition of a triangle that is independent of the shape in the science of
geometry falls within the field of geometrical space, demonstration of that triangle by
drawing on a paper falls within the field of representation space. In our country,
representation space is used mostly in geometry teaching. Davis and Bamford (1995)
have found out that contextual support increases the correctly answering ratio of
students to mathematics questions. Also, Threlfall (1993) has shown in a study that
visual contextual support increases the answering ratio of students to geometry
questions. When the verbal expression of a shape is told, the student has to remember
European Journal of Education Studies - Special Issue
Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
MATHEMATICS TEACHER CANDIDATES
this shape; however, it is easier when the shape is given. Not giving the shape forces the
students to be productive. Transferring verbal text into shape mostly forces the student
to think and produce.
Hacısalihoğlu (1998) has pointed out that learning is a perception issue and a
stronger perception in mathematics can be ensured through three important sense
organs such as touching, hearing and seeing (writing, telling and drawing) and concept,
drawing and proofs will help the development of thinking system of students.
In geometry teaching, memorizing the attributes of shapes, not supporting with
sufficient examples is causing the students to have difficulties in learning geometrical
concepts (Fujıta and Jones, 2007). In this context, to realize the meaningful learning,
students must being able to transfer the geometrical concepts into shapes is of great
importance.
Problems containing verbal definitions transferred into shapes are very
important for studying on this subject due to no studies being available in the literature
regarding the identification of circumstances requiring distinguishing of new shapes
formed as a result of drawings made.
Based on the information above, teacher candidates were aimed to be researched
in terms of transferring problems containing verbal definitions to shapes in this study.
According to this common objective, answers were searched in the study for the
following sub problems:
1. How are the resolution processes of mathematics teacher candidates in
transferring diameter chord concepts into a shape?
2. How are the distinguishing all circumstances and drawing skills of mathematics
teacher candidates in transferring a problem containing verbal definition into
shape?
3. Are the mathematics teacher candidates having difficulties in transferring center
concept into shape for the subject of circle?
4. How are the distinguishing the shapes formed during transferring the problem
asked by verbal definitions into shape skills of mathematics teacher candidates
for the subject of circle?
1.1
Circle
For rŒ¬+ , the set of points that have the distance of r to point M on a plane is called a
circle with M center and r radius. If we show this set with C
C={ : Œ¬ , |
| = }.
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
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P
M
Figure 1.1.1
The required and sufficient condition for a variable P point to be on the circle is |
|=r
proposition is being correct. P= (p1, p2) and M= (a1, a2) and (x1, x2) on a plane. Circle
equation is shown by:
(x1 – a1)2 +(x2 - a2)2 =r2
(Sabuncuoğlu, 2000).
1.2.
Tangent of a Curve
A curve and a line intersecting means having at least one common point. In both of the
circumstances shown in the figures below L line and ß curve has a common point.
ß
L
L
ß
Figure 1.2
We can say that the intersection of L line and ß curve at point Q and at point T is
different than each other. Let's take an L line that intersects ß curve at a two different
points such as P and T. If the lines obtained when P point is approached to T point
while kept on the curve approach to a certain line passing through T point, the line is
tangent to the T point of curve (Sabuncuoğlu, 2000).
P
T
P
ß
P’
T
Figure 1.2.1
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
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1.3
Chord of Contact
Let's take a C circle with its center being M and radius being r. Q point being a point
outside the C circle, the contact points of tangents drawn from Q point to C circle are T
and K. The TK line is called chord of contact of Q point (Sabuncuoğlu, 2000).
T
Q
T’
Figure 1.3
2.
Method
The study was conducted by using one of the qualitative research methods, special case
study. As is known, the measurements can ensure us to understand how many people
acts how; however, can't answer to the question why. Researches towards
understanding of the "why?" of human and group behaviors are called qualitative
researches. Therefore, the basic objective is to provide a descriptive and realistic picture
regarding the subject of research, rather than achieving generalizable results through
numbers as in the quantitative approach based researches. Since there is no concern of
generalization, the obtained findings are limited with only the teacher candidates
participating in the research (Baştürk, 2011). In the qualitative researches, validity and
reliability of the findings are being tried to be ensured by presenting of obtained data in
as detailed and direct as possible (Yıldırım and Şimşek, 1999). Case studies are
researches where one entity is defined based on time and space and customized
(Büyüköztürk, Çakmak, Akgün, Karadeniz and Demirel, 2008).
2.1
Research Group
The sampling of the research was formed by 72 students who are studying in Necmettin
Erbakan University Ahmet Keleşoğlu Faculty of Education Elementary School Math.
Teaching Program. The questions were asked to these mathematics teacher candidates
at the end of 2015-2016 spring term which they took the Geometry lesson.
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
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2.2
Data Collection Tools
As data collection tool with in scope of research, an evaluation consisting of verbally
asked problems regarding the subject of circle was used. Questions were selected
according to the relevant subject among the Olympics questions. When selecting the
questions, care was taken to contain most basic information about the relevant subject
but include the attributes that are having difficulties in transferring of definitions into
shapes.
2.3.
Findings and Interpretation
In this section, detailed research findings, which are obtained as a result of statistical
analyses made according to the answers of mathematics teacher candidates
participating in the research and interpretations made regarding these findings were
given.
2.3.1 First Sub Problem
For the "How there are solution processes of mathematics teacher candidates in
transferring diameter chord concepts into a shape?" first sub problem, the results of
descriptive analysis are given in Figure 1.
Answered the Question
72 people
100%
Wrong Answer
38 people
52.77%
Correct Answer
33 people
45.83%
Personal
Interpretation
20 people
27.77%
Only Drawn a
Shape
18 people
25%
Figure 2.3.1
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
MATHEMATICS TEACHER CANDIDATES
As seen from the figure above, the majority of the teacher candidates
participating to research could not achieve the correct solution for the questions
directed to them. The majority of these teacher candidates have obtained the correct
solution to the questions in the required direction. However, some teacher candidates
went to obtain the solution by only drawing a shape. Also, about half of the teacher
candidates who could not provide a correct answer to these questions ended the
solution by making a personal interpretation. For example, to the question of "prove
that the longer chord is closer to center in a circle", they have given answers similar to
"length increases as the center is approached, since the longest chord is the diameter
and diameter passes through center, the long chord is closer to center" or drawn a shape
like the given below that tries to explain such definitions.
Figure 2.3.1.1
2.3.2 Second Sub Problem
For the "How are the drawing skills of mathematics teacher candidates in transferring a
problem containing verbal definition into shape?" second sub problem, themes,
percentage and frequency values regarding the answers given by teacher candidates are
given in figure 3.
Answered the Question
72 people
100%
Drawn both states
53 people
73.61%
Drawn Only First
State
19 people
26.38%
Personal
Interpretation
18 people
25%
Only Drawn the
Shape
28 people
32.88%
Completely Made
Full Solution
6 people
8.33%
Figure 2.3.2.1
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
MATHEMATICS TEACHER CANDIDATES
All of the teacher candidates (100%) who are directed the research questions have
answered the questions. But more than half of them (73.61%) have tried to answer by
thinking both of the states. As for those who only considered one state were 26.38%.
Small part of the teaching candidates who considered both states (25%) have tried to
provide a solution by interpreting. Again, part of them (32.88%) have only drawn the
shape for both states and left it at that. Only 6 people have drawn both states and
solved correctly. That is to say, 8.33% of the teacher candidates who answered all the
questions.
For example, the number of teacher candidates who considered both of the states
shown in the figure below to the question "if two circles are tangential to each other; the
line joining the centers of these circles go through the point of contact. Please show."
were less than expected.
T
T
Figure 2.3.2.2
2.3.3 Third Sub Problem
For the "Are the mathematics teacher candidates having difficulties in transferring
center concept into shape for the subject of circle?" third research sub problem, the sub
themes, percentage and frequency values regarding the answers of teacher candidates
given are shown in Figure 3.
Answered the
Questions 68
people
94.44%
Answered Correctly
to Questions
23 people
33.82%
Not Achieved Correct
Solution
20 people
29.41%
Made Same Mistakes
25 people
36.76%
Figure 2.3.3.1
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
MATHEMATICS TEACHER CANDIDATES
68 of the teacher candidates, i.e. 94.44% of them, have made an effort to achieve the
solutions regarding the questions. However, about half of the candidates which tried to
solve the research questions have achieved the correct answer by using the correct
method. Those who could not use the correct method and solve are 20 people. 36.76% of
the teacher candidates who participated in the research have achieved the same results
by making the same mistakes. The error here is created by the mistake made when
transferring the concept of center into visual shape. Teacher candidates accepting the
intersection point of lines as center on the shape without establishing the center
attributes were directed to mistake in solution. They continued the solution of question
by using center angle and chord angle attribute. But, the solutions of the questions are
solved by starting from the equality of angles seeing the same arc. In conclusion, they
have achieved the wrong result by incorrectly transferring to shape.
2.3.4. Fourth Sub Problem
For the "How are the distinguishing the shapes formed during transferring the problem
into shape skills of mathematics teacher candidates for the subject of circle?" fourth
research sub problem, the sub themes, percentage and frequency values regarding the
answers of teacher candidates given are shown in Figure 6.
Answered the
Questions
72 people
100%
Achieved Correct
Answer but
Incorrectly
Transferred into
Shape
67 people
93.05%
Not Achieved
Correct Answer
and Incorrectly
Transferred into
Shape
67 people
93.05%
Figure 2.3.4.1
According the research result, all of the teacher candidates have tried to express what
they think about the question. However, no teacher candidates were able to transfer the
question directed to researchers by verbal definition completely into shape. 5 people
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
MATHEMATICS TEACHER CANDIDATES
have achieved the correct result but made mistakes in drawing. The majority of the
remaining teacher candidates (93.05%) has used wrong demonstrations when
transferring verbal definition into shape and could not obtain the solution. For example,
they have transferred two lines that were not said to be parallels as if they were
parallels and made the operations based on alternate interior and exterior angles.
However, the inscribed quadrilateral created as a result of drawings give them hints
about the solution of the problem.
3.
Conclusion and Recommendations
In this research, the knowledge of mathematics teacher candidates, who are studying in
Primary Education Mathematics Teacher Department undergraduate program, for the
process of transferring problems given as verbal definitions into shape were tried to be
determined with the help of open ended questions. A qualitative approach was adopted
in the research and there was no generalization concern. Therefore, the obtained
findings were limited to teacher candidates participating in the research. As a result of
the study, it was observed that teacher candidates were having difficulties in
transferring the question asked into a shape and reaching the correct answer.
According to research results, it was revealed that teacher candidates need visual
contextual support in question solving. Because, they are having difficulties in
transferring the question containing verbal definition into shape. As understood from
the answers provided by teacher candidates, showing the generally known attributes of
the subject of circle in geometry lesson plays an important role in problem solving.
Majority of the teacher candidates have started solving the question after transferring
into shape. The result of drawing a shape being an important stage in solution of the
question was concluded from this. Questing transferring into shape in geometry lesson
may be referred as the first step to question solving. As for transferring the concepts
into shape, it results from the clarity of knowledge about those concepts. Therefore, to
correctly transfer the definitions given in questions to shapes in geometry lesson, there
should not be lack of knowledge about the concepts. That is to say, the reason of a
verbal geometry question not being able to transfer into shape is related to definitions
of geometrical concepts not being settled in their minds.
Transferring into shape, which is an important step in question, solving,
facilitates seeing the operation to be made for reaching the result. This helps those
students who correctly transfer into shape to distinguish the new shapes. A mistake
made in transferring into shape shows the solution of the question was completely
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Ayşe Yavuz , Selin (Inag) Çenberci RESEARCH OF VISUAL CONTEXTUAL SUPPORT FOR THE SUBJECT OF CIRCLE ON
MATHEMATICS TEACHER CANDIDATES
wrong. For example, they have transferred two lines that were not said to be parallels as
if they were parallels and made the operations based on alternate interior and exterior
angles. However, the inscribed quadrilateral created as a result of drawings give them
hints about the solution of the problem. The parallelism view in the shape has
prevented the teacher candidates from noticing inscribed quadrilateral and misdirected
them about the solution. Thus, the solution was completed wrong.
Within scope of the research, Olympic questions related to the subject of circle
were selected by taking the simplest attributes of basic concepts into consideration
rather than difficulty. Therefore, all of the teacher candidates were expected to not
make mistakes in transferring the questions into shape rather than solution. But the
success rate is below the expectation. This result also reveals a preventive role of visual
contextual support in geometry education in adequate development of spatial thinking
skills of teacher candidates. Also, it can be said that this results from the definitions of
concepts not being taught in a manner to form a picture in the minds of students.
Geometry lesson explained by without depending much on representations and shapes,
with less contextual support and emphasizing more on the definitions of concepts may
improve the spatial thinking skills of students more, which is one of the most important
objectives of geometry education.
Each concept of the subject given by shape and representation of that concept
when explaining the subject to student does not help to the objective of improving the
spatial thinking skills of students. Definitions of geometrical concepts demonstrated in
a manner as to form a picture in the minds of students has a great importance in
geometry education. Thus, it is not appropriate to readily give the shapes together with
concept definitions to students. It should be ensured for students to visualize the given
definition and concepts. Methods effective in terms of improving the spatial thinking
skills, which have important role in geometry education, should be preferred. Effective
methods cause permanence of knowledge. Deficiency in the subjects learned during
primary education years will not be useful to an individual without completing them,
in terms of education. Especially the knowledge of circle is given to students from the
third grade of primary education. In this context, the subject of circle is seen among the
important subjects in terms of teacher candidates. Teacher candidates should be more
equipped when transferring the subject of circle to students and the lesson should be
explained in a manner to improve the spatial thinking skills of students when
transferring the subject.
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