European Journal of Education Studies
ISSN: 2501 - 1111
ISSN-L: 2501 - 1111
Available on-line at:
www.oapub.org/edu
Copyright © The Author(s). All Rights Reserved 90
Published by Open Access Publishing Group ©2015
10.5281/zenodo.154550
Volume 2│Issue 5│2016
A CONCEPTUAL FRAMEWORK FOR EXAMINING MATHEMATICS
TE“CHERS’ PED“GOGIC“L CONTENT KNOWLEDGE IN THE
CONTEXT OF SUPPORTING MATHEMATICAL THINKING
”erna Tataroğlu Taşdan
1i
, Adem Çelik
2
1
PhD, Department of Mathematics Education,
Dokuz Eylul University, Izmir, Turkey
2
PhD, Professor, Department of Mathematics Education,
Dokuz Eylul University, Izmir, Turkey
Abstract:
This study has been aimed to propose a conceptual framework that helps researchers
examine mathematics teachers PCK in the context of supporting students
mathematical thinking. “dvancing Children’s Thinking Framework which is a
pedagogical model developed by Fraivillig, Murphy and Fuson (1999) that supports the
development of students conceptual understanding of mathematics has been adopted
as the theoretical foundation. Pedagogical content knowledge knowledge of students
thinking and knowledge of instructional strategies and representations) has been
examined in the context of supporting mathematical thinking and has been
interconnected to “dvancing Childrens Thinking Framework. Then, a new framework
has been obtained. Instructional examples included within the framework suggested as
a result of the interconnection have become the indicators regarding PCK of
mathematics teachers in the context of supporting mathematical thinking. Some
examples from a performed research where this framework has been used as an
analytical framework have been presented. As a conclusion, it can be said that the
suggested framework may be a useful tool for the researchers and teacher educators
who are dealing with teachers knowledge focusing on students mathematical thinking
and a guide for the teachers.
Keywords: conceptual framework, pedagogical content knowledge, mathematics
teachers, mathematical thinking
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CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
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Introduction
Effective teaching requires teachers to have specific knowledge and skills. The starting
point of many research aimed at determining the knowledge that a teacher needs to
have is Shulmans studies. Shulman  suggested that a person who knows
something does not mean that this person can teach this issue. According to Gearhart &
Saxe (2004), teachers who have knowledge about the subject and have flexible
pedagogical knowledge are called perfect teachers. This calls the concept of pedagogical
content knowledge (PCK) to our mind.
Shulman (1987) has defined PCK as a special amalgam of subject matter
knowledge and pedagogical knowledge. PCK is described as the most beneficial
representations, the most powerful metaphors/analogies as well as best examples and
explanations used to make a subject of a special field understandable to others
Shulman, . Subsequent to Shulmans definition, PCK has been discussed and
examined by many researchers (Grossman, 1990; Fennema & Franke, 1992; Magnusson,
Krajcik & Borko, 1999; An, Kulm & Wu, 2004; Ball, Thames & Phelps, 2008). The agreed
components for PCK that has been modeled through different components by several
researchers can be listed as knowledge of students thinking, knowledge of instructional
strategies, knowledge of curriculum, content knowledge and knowledge of assessment.
It is known that many studies have been carried out where the PCK of mathematics
teachers has been analyzed within the context of one or a more of these components.
An, Kulm & Wu (2004) emphasized that deep and broad PCK is important and
necessary for effective teaching. We can say that many researchers such as An, Kulm &
Wu have agreed upon the fact that a more effective and quality teaching depends on
teachers PCK. In other words, it is possible to assume that teachers with improved PCK
could be more successful in achieving the goals of teaching mathematics and providing
a meaningful mathematics education for students. So, teachers PCK can also be
considered as an unavoidable component when providing students with mathematical
thinking skill and supporting this way of thinking which is one of the objectives of
mathematics teaching in our country (Ministry of National Education [MNE], 2011). In
fact, it wont be easy to develop students mathematical thinking if a teacher is unable to
understand how his/her students comprehend a particular issue and is unable to
estimate what type of misconceptions he/she will have, or which strategies he/she has to
refer to in particular cases. In National Council of Teachers of Mathematics (NCTM)
(2000) Standards, it has been emphasized that effective teaching involves observing the
students, paying close attention to the thoughts and explanations of students, having
mathematical objectives and using knowledge when taking instructional decisions.
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Teachers using these practices motivate their students to mathematical thinking and
reasoning and provide learning opportunities for students at every level of
understanding that will challenge them (NCTM, 2000: 19). Therefore, the PCK of a
teacher is one of the concepts that have to be considered as first priority when it comes
to supporting/developing students mathematical thinking.
Teachers who are knowledgeable of the behaviour of their students have more
flexibility, capacity and creativity in constructing lessons and tasks that meet student
learning needs (Lee, 2006: p. 1-2). Professional development programs that focus on
students mathematical thinking have produced results that consistently indicate the
value of the approach for both students and teachers (Norton, McCloskey & Hudson,
2011). Research projects such as Cognitively Guided Instruction (ter et al., 1989), the
Purdue Problem-Centered Mathematics Project (Cobb, Wood and Yackel, 1990; Cobb et
al. 1991), SummerMath (Simon and Schifter, 1991), the Kenilworth Project (Maher,
Davis and Alston, 1991, 1992; Maher and Martino, 1992), the Mathematics Case
Methods Project (Barnett, 1998), the work of Gordon and MacInnis (1993) and the work
of Putnam and Reineke (1993) have found the following to be of potential benefit for
both teachers and students when teachers tend to their students mathematical thinking:
The ability on the part of teacher to construct or select appropriate, worthwhile
mathematical tasks;
A shift from teacher-centered didactical instruction to student-centered problem-
solving instruction;
Higher levels of conceptual understandings by students without compromising
their computational performances;
More positive beliefs of teachers and students toward mathematics (cited in
Chamberlin, 2002: p. 1-2)
Although there are numerous research that examine the pre/in-service teachers
knowledge based on the knowledge of students or researches that examine the
knowledge of students thinking in relation to mathematical thinking (An, Kulm & Wu,
 Jenkins,  Kılıç, ,  Lee,  Norton, McCloskey & Hudson, 
Sleep & Boerst,  Yeşildere-İmre & “kkoç, , studies that examine a model
regarding mathematical thinking within the scope of PCK are limited. An, Kulm & Wu
 classified the knowledge of students thinking into four categories in their studies
which they aimed to compare to the PCK of the middle school mathematics teachers in
“merica and China “ddressing students misconceptions, engaging students into math
learning, promoting students thoughts regarding mathematics, and building on
students math ideas. On the other hand, Lee (2006) built a conceptual framework in
order to analyze teachers knowledge of middle school students mathematical thinking
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of algebraic word problem solving therefore benefiting from the study of An, Kulm &
Wu (2004). In another study, Cengiz, Kline & Grant (2011) built a framework
Extending Student Thinking Framework by gathering “dvancing Childrens Thinking
model built by Fraivillig, Murphy and Fuson (1999) and other studies together. They
examined this framework within the scope of Mathematical Knowledge for teaching
developed by Ball, Thames and Phelps (2008) and focused on whole-group discussions
based on students existing mathematical thinking. “s it is seen, by examining PCK and
mathematical thinking all together it can help with the search for an answer to the
question What type of knowledge should a teacher have who wants to support/develop
students’ mathematical thinking and what should he/she do for this?. Although current
studies help to answer this question, there is still a need for deeper studies regarding
this issue. ”ased upon this idea, mathematics teachers PCK has been examined within
the context of supporting mathematical thinking in this study. Therefore, the purpose of
this study is to propose a conceptual framework that helps researchers examine
mathematics teachers PCK in the context of supporting students mathematical
thinking.
In this study, examination of mathematics teachers PCK is attempted within the
context of supporting mathematical thinking. Supporting/developing students
mathematical thinking has been considered as the most significant idea that forms the
theoretical foundation of the study. “dvancing Children’s Thinking Frameworkwhich is
a pedagogical model developed by Fraivillig, Murphy and Fuson (1999) that supports
the development of conceptual understanding of mathematics by students has been
adopted as the theoretical base.
This model has been preferred, because it does not only suggest that students
mathematical thinking should be supported and developed, but also shows a concrete
way as to how teachers can do this. Another theoretical idea that has been adopted as a
base in the study is Shulmans, ,  the idea of PCK. The focal points of this
study are the knowledge of students thinking as well as knowledge of instructional
strategies and representations components of PCK. In the next part, first of all, each
adopted theoretical framework will be introduced and explanations regarding the
conceptual framework that has been built by associating to these will be presented.
Afterwards, examples from performed research where this framework has been used as
an analytical framework will be presented.
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Mathematical Thinking
One of the skills that are aimed to make the students gain in mathematics teaching is
mathematical thinking (MNE, 2011). Stacey (2008) has specified the importance of
mathematical thinking in three ways:
1) Mathematical thinking is an important goal of schooling,
2) Mathematical thinking is important as a way to learn mathematics,
3) Mathematical thinking is important for teaching mathematics.
When the learning and mathematical thinking are examined together, it can be
said that many components come to the forefront. Schoenfeld (1992: 5) has listed the
fundamental aspects of mathematical thinking as core knowledge, problem-solving
strategies, and effective use of resources, having a mathematical perspective and
engaging in mathematical practices. Mathematics teaching should present practices that
develop a students knowledge in each of these fields Swan & Ridgway, .
Many people think/may think that mathematical thinking is a way of only
thinking related to mathematics. According to Burton (1984), this type of thinking is
mathematical not only because it is about mathematics, but because the operations it is
based on are mathematical operations and its field of application is general. Therefore,
regardless of a person being a mathematician or not, all individuals use mathematical
thinking in their lives, in the events or facts they are confronted with or in solving
problems. In other words, mathematical thinking is not a way of thinking peculiar to
only mathematicians. On the contrary, its a way of thinking that each person having a
profession should use it at the present time (Alkan & Bukova-Güzel, 2005).
Consequently, individuals use mathematical thinking in every phase of their lives or to
solve their problems wittingly or unwittingly “rslan & Yıldız, .
On the other hand, NCTM (2000) points out the increase in the mathematical
level that is necessary for individuals in the workplace, in professional areas ranging
from health care to graphic design and also the increase in mathematical thinking and
problem-solving levels. According to Umay (2003), one of the fields (possibly the first
one) where thinking skills and logic is used intensively is mathematics. One of the
objectives of learning mathematics should not only be learning mathematical terms,
concepts and language of mathematics, but learning to think by using them (Umay,
2007). To put it more clearly, practicing mathematics is a way of thinking beyond using
ample formulas, keeping technical data in mind and re-proofing an already proven
theorem Yıldırım, . For this reason, the desired mathematical education is the one
that prioritizes the students to gain thinking, reasoning, problem-solving skills and the
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ability to relate these to daily life while obtaining mathematical knowledge (Umay,
2003).
Mathematical thinking is also emphasized in the standards and programs
developed for the learning and teaching processes of mathematics. One of the standards
suggested by NCTM   regarding mathematics teaching is To provide students
with mathematical thinking skills. This objective has also been included within the
general objectives of mathematics education within the scope of the renewed
mathematics lesson curriculum in our country and mathematical thinking skills have
been determined as one of the skills that the curriculum aims to develop (MNE, 2011).
The expression of ”esides gaining basic concepts and skills, learning mathematics also
involves thinking mathematically, developing general problem-solving strategies, maintaining a
positive attitude towards mathematics and understanding that mathematics is an important tool
used in real life is one part of the MNE (2011) mathematics lesson curriculum that brings
emphasis to the curriculum places on mathematical thinking. In addition, it has been
indicated that the activities brought to the class by the teachers (within the scope of
mathematics lesson curriculum) should be aimed at providing the students with high-
level mathematical thinking skills such as analysing, synthesising, assessment,
connection, classification, generalization and deduction (MNE, 2005).
NCTM objectives have shown a change from its traditional practice that was
summarizing the required mathematical outputs such as skills, concepts and practices
knowledge through wider trends, attitudes and beliefs regarding the nature of
mathematical knowledge and own mathematical thinking of the individual (Romberg,
1994). Expectations and objectives aimed at developing the mathematical thinking skills
of students arise accordingly. These expectations and objectives can only be put into
practice within teaching environments composed of teachers carrying out effective
teaching. For this reason, it is in evidence that teachers play a significant role in
supporting/developing students mathematical thinking.
“dvancing Children’s Thinking Framework
Fraivillig, Murphy and Fuson (1999) have emphasized that teachers should consider the
components of eliciting students solutions, supporting students conceptual
understanding and extending their mathematical thinking in an instruction where
students mathematical thinking are supported and developed. In this direction, they
have developed the “dvancing Children’s Thinking Frameworkwhich is a pedagogical
model that supports the development of students conceptual understanding of
mathematics. This model is composed of three components eliciting students
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solutions, supporting their conceptual understanding and extending their mathematical
thinking.
The instruction of eighteen mathematics teachers has been observed in the study.
Lessons of six teachers (that are characterized as qualified) have been monitored
through extra observations and one teacher has been examined as special case.
Instructional strategies that the teachers refer in advancing students mathematical
thinking have been listed within the scope of data. Things that a teacher can do in order
to develop students mathematical thinking in a questioning class environment where
the thoughts and solutions of students are found are presented in Table 1.
Eliciting
Supporting
Facilitates students’ responding
Elicits many solution methods for
one problem from the entire class
Wait for and listen to students’
descriptions of solution methods
Encourages elaboration of
students’ responses
Conveys accepting attitude toward
students’ errors and problem
solving efforts
Promotes collaborative problem
solving
Orchestrates classroom discussions
Uses students’ explanation for
lesson’s content
Monitors students’ levels of
engagement
Decides which students need
opportunities to speak publicly or
which methods should be discussed
Supports describers’ thinking
Reminds students of conceptually
similar problem situations
Provides background knowledge
Directs group help for an individual
student
Assists individual students in
clarifying their own solution
methods.
Supports listeners’ thinking
Provides teacher-led instant replays.
Demonstrates teacher-selected
solution methods without endorsing
the adoption of a particular method
Supports describers’ and listeners’
thinking
Records symbolic representation of
each solution method on the
chalkboard
Asks a different student to explain a
peer’s method
Supports individuals in private help
sessions
Encourages the students to request
assistance (Only when needed)
Table 1: Examples of Instructional Strategies of ACT Framework
(Adapted from Fraivillig, Murphy & Fuson, 1999, p. 155)
“s it is seen, the model aimed at advancing students mathematical thinking is
composed of three components as eliciting, supporting and extending. Eliciting is
considered to enable the students to explain their thoughts. Knowing what students
think and finding out their answers is considered significant in supporting students
thinking. Yackel (1995) argued the reason for this as the teacher can provide learning
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opportunities for all the students by these means (as cited in Fraivillig, Murphy &
Fuson, 1999, p. 149).
The supporting component of the model involves encouraging the students to
explain their own solutions that they bring out through their current cognitive abilities
and the teacher to take pedagogical decisions in this direction. Instructional
components of eliciting and supporting involve instructional strategies aimed at
students to reach their thoughts regarding solutions that they are familiar with and
easing this process. However, these components do not involve the methods that
teachers refer to in order to challenge and extend students thinking. Extending, which
is the last component addresses the strategies that could be used to advance the
students progress through their zones of proximal development Fraivillig, Murphy &
Fuson, 1999)
Pedagogical Content Knowledge (PCK)
Shulman (1987) has defined PCK as knowledge of teaching where subject matter
knowledge intersects with pedagogical knowledge and where practice knowledge
integrates with theoretical knowledge. According to Shulman (1987), PCK, is the
knowledge that differs a specialist in a particular field (for instance, a mathematician)
from an educationist (mathematics teacher). Fennema and Franke (1992) have
emphasized the important aspects of PCK in their definitions for teacher knowledge:
Knowledge of mathematics teaching includes knowledge of pedagogy, as well as
understanding the underlying processes of the mathematical concepts, knowing the
relationship between different aspects of mathematical knowledge, being able to interpret
that knowledge for teaching, knowing and understanding students’ thinking, and being
able to assess student knowledge to make instructional decisions. (p. 161)
Shulman (1986) defined PCK as the most useful forms of representation of a
subject, the most powerful analogies, illustrations, examples, explanations and
demonstrations. In other words, the knowledge is used to represent the subject to make
it comprehensible to others. In addition, he included what makes it easy or difficult to
learn of specific concepts, especially knowledge regarding the conceptions and
preconceptions that students of different ages and backgrounds bring with them to the
learning within the scope of PCK. ”ased on Shulmans definition, Kovarik 08) has
divided PCK into two categories and sub-categories as ways of knowledge of
representations and approaches and knowledge of student thinking. When the models
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examining PCK are analyzed, it is seen that these two components which are prominent
in Shulmans definition and which Kovarik has emphasized are examined within the
scope of PCK with different names by many different researchers. In this study, it has
been decided to examine PCK within the scope of knowledge of students thinking and
knowledge of instructional strategies and representation components.
Knowledge of students thinking involves knowing what makes it easy or
difficult to learn specific concepts (Ball et al., 2008; Shulman, 1986), to know how
students perceive a concept and how they think (Ball et al., 2008; Fennema & Franke,
1992), to determine the misconceptions and learning disabilities of students (An, Kulm
& Wu, 2004; Fennema & Franke, 1992; Kovarik, 2008; Magnusson, Krajcik & Borko,
1999; Park & Oliver, 2008; Schoenfeld, 1998; Shulman, 1986), and to be aware of the
prior knowledge of students (Kovarik, 2008; Magnusson et al., 1999; Schoenfeld, 1998;
Shulman, 1986). Knowledge of instructional strategies and representation involves the
demonstrations, activities and examples that the teacher will use and the strategies
peculiar to the topic and subject (Ball et al., 2008; Kovarik, 2008; Magnusson et al., 1999;
Park & Oliver, 2008; Shulman, 1986). How the knowledge of students thinking has
been defined and under which names they have been examined within the frame of the
analyzed models have been summarized in Table 2.
Researcher
Component
Content of the Component
Shulman (1987)
Knowledge of Learners and
Their Characteristics
An understanding of what makes the learning of
spesific topics easy or difficult
The conceptions and preconceptions that students of
different ages and backgrounds bring with them to
the learning of those most frequently taught topics
and lessons
Preconceptions and misconceptions (Shulman, 1986).
Grossman (1990)
Knowledge of Students’
Understanding
Fennema & Franke
(1992)
Knowledge of Learners
Cognitions in Mathematics
Knowledge of how students think and learn
Understanding the processes the students will use and
the difficulties and successes that are likely to occur
Knowledge of how students acquire the mathematics
content
Magnusson, Krajcik
& Borko (1999)
Knowledge of Students’
Understanding of Science
Knowledge of requirements for learning (prequisiste
knowledge, abilities and skills tahta students might
need, ability levels or different learning styles)
Areas of student difficulty
An, Kulm &Wu
(2004)
Knowing Students’ Thinking
Building on students’ mathematical ideas
Addressing students’ misconceptions
Engaging students’ in mathematics learning
Promoting students’ thinking mathematically
Ball, Thames &
Phelps (2008)
Knowledge of Content and
Students
Anticipating what students are likely to think and
what they will find confusing
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Predicting what students will find interesting and
motivating when choosing an example
Anticipating what students are likely to do with it and
whether they will find it easy or hard when assigning
a task
Knowledge of common student conceptions and
misconceptions about particular mathematical
content
Park & Oliver (2008)
Knowledge of Students’
Understanding
Misconceptions
Learning difficulties
Motivation and interest
Need
Kovarik (2008)
Knowledge of Student Thinking
Student Prior Knowledge
Mathematical Background
Student Misconceptions
Conneceting Prior Knowledge to New Knowledge
Anticipating Students Questions
Assessing Understanding
Table 2 Knowledge of Students Thinking in Different PCK Frameworks
It is seen that knowledge of students thinking takes place in teacher knowledge models
with different terms such as knowledge of learners and their characteristics, knowledge
of students understanding, knowledge of learners cognitions in mathematics, knowing
students thinking and knowledge of student thinking. When the definitions are
examined, although there are some varying points, it is predominantly seen that a
similar scope is pointed out. The sub-components of knowledge of students thinking
could be listed as follows based on the relevant literature determining students current
knowledge, connecting prior knowledge to new knowledge, knowing students
misconceptions, valuing students questions and thoughts, foreseeing students
thoughts and considering students individual differences.
Determining students’ current knowledge: One of the components required for
the teacher to support students mathematical thinking is to firstly determine the
students current knowledge. Knowing the current situation of the students helps the
teacher to take instructional decisions and to plan his/her instruction (Fennema &
Franke, 1992). Shulman (1986) has indicated that students at different ages with
different knowledge bring some previous knowledge with them and that these should
be known by the teacher as there is a high possibility of this prior knowledge
transforming into misconception later. For this reason, determining previous
knowledge and precognitions of students by the teacher is considered extremely
important.
Connecting prior knowledge to new knowledge: According to Bransford et al.
(2000) students use their prior knowledge to understand and configure the new ones
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and sometimes this knowledge may cause the new knowledge to be misinterpreted (as
cited in Kovarik, 2008: p. 32). In other words, students build new knowledge upon the
previous knowledge. For this reason, it is quite important that the prior knowledge of
students is configured correctly. It would be beneficial for the students having teachers
who relate the prior knowledge of students with new knowledge, so that the students
can configure their mathematical thinking correctly.
Knowing students’ misconceptions: Determining the misconceptions of
students, knowing the source of misconceptions and referring to ways that remove
these is another most important component which teachers should consider in
supporting students mathematical thinking. Examining misconceptions has caught the
attention of many researchers and they have shown great effort in finding the sources of
misconceptions (Even & Tirosh, 2008). In their research, An, Kulm and Wu (2004) have
found that teachers are using various activities, graphics, manipulatives and processes
in order to correct misconceptions and are focusing on use of concrete models for
configuring abstract thoughts. Knowing the misconceptions of students is a necessary
component for teachers in supporting the students to configure their mathematical
thinking correctly.
Valuing students’ questions and thoughts: Considering the questions of
students is also an important component that will direct teachers instructions. Park and
Oliver (2008) suggested that the questions asked by the students are one of the factors
which affect the development of teachers PCK. “ccording to the researchers,
challenging questions asked by students deepen and expand the subject matter
knowledge of the teacher. According to NCTM (2000), paying close attention to the
thoughts and explanations of students is one of the necessities for effective teaching.
The teacher should listen to students answers and should try to understand the
students thinking when he/she asks a question to the students or wants an explanation
from the students. This component is also included within the Fravillig, Murphy and
Fusons model as one of the strategies that a teacher can use to elicit the students
thoughts.
Foreseeing students’ thoughts: According to Ball et al. (2008), teachers should
predict what students are thinking and what they see as confusing. They should also
foresee what would be interesting and motivating for students when they choose an
example. They should predict using tools where students can participate when they
carry out an activity and they should predict if that activity would be easy or difficult
for the students (Ball et al., 2008). It can be said that a teacher who acts by foreseeing the
thoughts of students can make the students the focal point and plan his/her instruction
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 101
in this manner. Thus, this would be an important step in supporting students
mathematical thinking.
Considering students’ individual differences: Park and Oliver (2008) have
emphasized the importance of the abilities, learning styles, development levels and
different needs of students within the scope of knowledge regarding the learning of
students in their PCK model. Magnusson et al. (1999) have indicated the required
abilities, skills, students needs and learning styles as the requirements of learning. This
component has been named as considering students individual differences in the
study presented.
Knowledge of instructional strategies and representations which have been
emphasized by Shulman as another component of PCK has been defined as one of the
knowledge components which a teacher should have by many researchers. Table 3
shows under which name and scope this component has been examined in the
literature.
Researcher
Component
Content of the Component
Shulman (1986)
Representations
The most powerful analogies, illustrations, examples,
explanations and demonstrations
Grossman (1990)
Knowledge of Instructional
Strategies
Magnusson, Krajcik
& Borko (1999)
Knowledge of Instructional
Strategies
Subject-specific strategies
Topic-specific strategies (Representations, activities)
Ball, Thames &
Phelps (2008)
Knowledge of Content and
Teaching
Choosing which examples to start with and which
examples to use to take students deeper into the content
Knowing the instructional advantages and disadvantages
of representations used to teach a specific idea
Identifying what different methods and procedures
afford instructionally
Park & Oliver (2008)
Knowledge of Instructional
Strategies
Subject-specific strategies
Topic-specific strategies
Kovarik (2008)
Knowledge of Representaions
and Approaches
Demonstrations (Graphs, Tables, Formulas)
Examples (Real World Examples, Problems)
Analogies
Table 3: Knowledge of Instructional Strategies and Representations in Different
PCK Frameworks
While some researchers have examined representations within the scope of the
component named instructional strategies, some of them have given place to the
representations by the name of the said component. Although the component names
defined by the researchers differentiate, it can be said that they are considerably similar.
Magnusson et al. (1999) have examined knowledge of instructional strategies in two
categories; as subject-specific strategies and topic-specific strategies. Subject-specific
Berna Tataroğlu Taşdan, “dem Çelik -
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CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 102
strategies represent the general approaches used to teach a particular area (science,
mathematics etc.. Teachers knowledge of subject-specific strategies involves the ability
to define and realize a strategy and its phases. According to Magnusson et al. (1999),
topic-specific strategies are strategies that may be used to help the students understand
certain (science) concepts. This strategy has been divided into two; as representations
and activities. Knowledge of representations is the knowledge regarding the ways of
representing certain concepts or principles used to ease the learning process of students.
Examples of representations are illustrations, examples, models and analogies.
Activities peculiar to the subject are activities that may be used to make the students
comprehend certain concepts or relationships. Examples of such activities are problems,
demonstrations, simulations, researches and experiments (Magnusson, Krajcik & Borko,
1999).
Kovarik (2008) has indicated knowledge of representations and approaches as a
component of PCK in the model she has developed based on Shulmans PCK definition.
Kovarik has divided knowledge of representations and approaches as demonstrations,
examples and analogies. Demonstrations include graphics, tables and formulas.
Examples are real-world examples and problems. Analogies have also been included
within the knowledge of representations and approaches.
As it is seen, these researchers have also examined knowledge of instructional
strategies and representations in a similar way in terms of scope. Kovariks knowledge
of representations and approaches classification corresponds to topic-specific strategies
definition of Magnusson et al. (1999).
“s a result, PCK has been examined within the scope of knowledge of students
thinking and knowledge of instructional strategies and representations components in
this study. Six sub-components (that have been summarized as a result of literature
review have been defined for knowledge students thinking. The classification
suggested by Kovarik (2008) has been used for knowledge of instructional strategies
and representations. In this way, the PCK framework adopted in the study and
presented in Table 4 has been obtained.
PEDAGOGICAL CONTENT KNOWLEDGE
Knowledge of Students’ Thinking
Knowledge of Instructional Strategies and Representations
Determining students’ current knowledge
Representations
Connecting prior knowledge to new knowledge
Examples (Real life examples-Problems)
Knowing students’ misconceptions
Analogies
Valuing students’ questions and thoughts
Foreseeing students’ thoughts
Considering students’ individual differences
Table 4: PCK Framework Adopted in this Study
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 103
The theoretical framework chosen for PCK in the study is presented in Table 4.
But it is known that teacher knowledge is not monolithic, it is a large, integrated,
functioning system with each part difficult to isolate (Fennema & Franke, 1992). Park
and Oliver (2008) indicated that an improvement in one of the components of PCK will
affect the other components and that only in this way an improvement can be provided
in the whole PCK. In addition to this, they have suggested that PCK can be seen as a
combination of other components of teacher knowledge. It is considered that
improvement in only one component of PCK may not provide much benefit to the
teacher and lack of compliance between the components would cause trouble in terms
of PCK development of the teacher (Harel & Lim, 2004; Park & Oliver, 2008). For this
reason, the interaction between the components of PCK should always be taken into
consideration and other fields of PCK should also be considered when examining
knowledge of students thinking, knowledge of instructional strategies and
representations components. However, the theoretical aspect of this study has been
limited by the two components of PCK mentioned. The reason for this is the idea that
the teacher prioritizes these two types of knowledge during instruction that supports
mathematical thinking and also the difficulty of studying on all the components of PCK.
Our Conceptual Framework: PCK in the context of Supporting Students’
Mathematical Thinking Framework
Finding a concrete answer to the question What type of knowledge should a teacher have
who wants to support/develop students’ mathematical thinking and what should he/she do for
this? is hard for mathematics teachers and mathematics educators. The main purpose
of this study is to make a contribution to the field in terms of finding an answer to this
question. For this purpose, mathematics teachers PCK has been examined theoretically
in the context of supporting students mathematical thinking and the theoretical
frameworks chosen for PCK has been interconnected to mathematical thinking. The
interconnected framework has been started to derive from ACT Framework (1999) of
Fraivillig, Murphy & Fuson composed of three components including examples of
instructional practices. Then the PCK model modified reflecting onto Shulmans model
has been integrated on this model. Each instructional example included within the ACT
has been re-examined within the context of two components of PCK (knowledge of
students thinking and knowledge of instructional strategies and representations) and
the sub-components of these components. While making this interconnection, first of all
PCK components were placed horizontally and three components of ACT were placed
vertically on a table. Afterwards, instructional practices within ACT were placed to the
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 104
appropriate cell in this table (considering within the scope of sub-components of PCK).
While the same component has been placed in more than one cell sometimes, no
component has been assigned for some cells. In this way, the theoretical frameworks
shown in Table and Table have been obtained. For instance, determining students
current knowledge component of PCK has been interconnected to eliciting and
supporting steps of ACT. No relationship has been established for extending step.
“ccording to the interconnection made a teacher who wants to determine students
current knowledge should ask the students to explain their own solutions and listen to
them, encourage the students to explain their answers in detail, decide which students
should be provided with answering opportunities in front of the class and ask these
students to explain their thoughts, listen to them and share the students
thought/solution with the whole class within the scope of Eliciting. In terms of
Supporting, it is considered that a teacher who is trying to determine students current
knowledge can assist them when explaining their solutions or thoughts on an
individual basis.
Since interconnection has been started with ACT, one component of ACT might
be placed under more than one component of PCK. For instance, asking students to
explain their solutions and listen to them which are some of the eliciting components of
“CT have been examined within the scope of both determining students current
knowledge and knowing the misconceptions component of PCK.
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG
M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 105
PCK
Component
Components
of ACT
Knowledge of Students’ Thinking
Determining students’ current
knowledge
Connecting prior
knowledge to new
knowledge
Knowing students’
misconceptions
Valuing students’ questions
and thoughts
Foreseeing students’
thoughts
Considering students’
individual differences
ELICITING
Wait for and listen to students’
descriptions of their solution
methods
Encourages students to explain
their responses in detail
Decides which students need
opportunities to speak in front of
the class and requesting these
students to explain their thoughts
Shares a student’s thought/solution
with all class
Wait for and listen to students’
descriptions of their solution
methods
Encourages students to explain
their responses in detail
Uses students’ misconceptions (that
are determined through their
explanations) for lesson’s content
Uses students’
thoughts/questions for
lesson’s content
Conveys accepting attitude
toward students’ errors and
problem solving efforts
Directing lesson’s content by
predicting what students will
find easy or confusing
through their explanations
Elicits many solution
methods for one problem
from the entire class
Conveys accepting
attitude toward students’
errors and problem
solving efforts
Decides which students
need opportunities to
speak in front of the
class
SUPPORTING
Assists individual students in
clarifying their own thoughts or
solution methods
Reminds students of
conceptually similar
problems/ situations
Provides background
knowledge
Provides teacher-led instant
replays
Reminds students of conceptually
similar problems/ situations
Assists individual students in
clarifying their own thoughts or
solution methods
Asks a different student to
explain a peer’s solution
method
Encourages the students
to request assistance
(when needed)
EXTENDING
Encourages students to
consider and discuss
interrelationships among
concepts
Encourages students to analyse,
compare, and generalize
mathematical concepts in terms of
removing the misconceptions
Uses students' creative and
different responses, questions,
and problems as core lesson
Table 5: Conceptual Framework of The Research (Connecting ACT Components to Knowledge of Students’ Thinking)
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG
M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 106
Table 6: Conceptual Framework of The Research (Connecting ACT Components to Knowledge of Instructional Strategies and Representations
PCK
Components
Components of
ACT
Knowledge of Instructional Strategies and Representations
Representations
Examples
Analogies
Real Life Examples
Problems
ELICITING
Wait for and listen to students’ detailed descriptions of the
representations in their solution methods
Encourages students to explain the representations that they used
in their solution methods in detail
Shares students’ representations that they used in their solution
methods with all class
Uses students’ representations and explanations for lesson’s
content
Wait for and listen to students’ real life examples
Shares students’ real life examples
with all class
Encourages students to explain their real life examples in
detail
Uses students’ real life examples for lesson’s content
Elicits many solution methods for one problem from the
entire class
Wait for and listen to students’ descriptions of their solution
methods
Shares students’ solutions with all class
Encourages students to explain their responses in detail
Promotes collaborative problem solving
Wait for and listen to students’ analogies
Encourages students to explain their analogies in detail
Shares students’ analogies with all class
SUPPORTING
Reminds students of similar representations
Provides prior representations
Provides teacher-led instant replays about using representations
Assists individual students in clarifying their own representations
Demonstrates teacher-selected representations without endorsing
the adoption of a particular representation)
Asks a different student to explain a peer’s representation in her/his
solution method
Reminds students of conceptually similar real life examples
in problems/ situations
Provides prior real life examples
Provides teacher-led instant replays about real life examples
Assists individual students in clarifying their own real life
examples
Demonstrates teacher-selected real life examples without
endorsing the adoption of a particular example)
Asks a different student to explain a peer’s real life example
Reminds students of conceptually similar problems/
situations
Assists individual students in clarifying their own thoughts or
solution methods
Demonstrates teacher-selected solution methods without
endorsing the adoption of a particular method
Asks a different student to explain a peer’s method
Demonstrates an alternative solution method for one problem
Reminds students of conceptually similar analogies
Provides prior analogies
Provides teacher-led instant replays about analogies
Assists individual students in clarifying their own analogies
Demonstrates teacher-selected analogies without endorsing
the adoption of a particular analogie)
Asks a different student to explain a peer’s analogie
EXTENDING
Asks all students to attempt to solve difficult problems and to try
using various representations
Promotes use of more efficient representations in the solution
methods for all students
Encourages students using representations to analyze, compare,
and generalize mathematical concepts
Encourages students using different representations to consider and
discuss interrelationships among concepts
Lists all representation in students’ solution methods on the
chalkboard to promote reflection
Uses students' creative and different responses, questions, and
problems as core lesson
Encourages students using real life examples to analyse,
compare, and generalize mathematical concepts
Promotes use of more efficient real life examples in the
solution methods for all students
Lists all real life examples in students’ solution methods on
the chalkboard to promote reflection
Uses students' creative and different real life examples as
core lesson
Asks all students to attempt to solve difficult problems and to
try various solution methods
Promotes use of more efficient solution methods for all
students
Encourages students to analyze, compare, and generalize
mathematical concepts for encountered problem
Encourages students to consider and discuss
interrelationships among concepts
Lists all solution methods on the chalkboard to promote
reflection
Uses students' responses, questions, and problems as core
lesson
Cultivates love of challenge
Encourages students using analogies to analyze, compare,
and generalize mathematical concepts
Lists all analogies on the chalkboard to promote reflection
Encourages students using different analogies to consider
and discuss interrelationships among concepts
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 107
Using PCK in the context of Supporting Students’ Mathematical Thinking
Framework
Examples from the results of a research Tataroğlu Taşdan,  where the above-
mentioned interconnected conceptual framework has been used as an analytical
framework will be given in this section. Mathematics teachers PCK development in
the context of supporting mathematical thinking) has been examined in the research
mentioned.
The study was a qualitative research carried out by Tataroğlu Taşdan  as a
PhD Thesis. The aim of the research was to improve mathematics teachers PCK in the
context of supporting students mathematical thinking. In the research that has been
carried out with six volunteer teachers, mathematics teachers teaching of function
concept have been observed for two years consecutively prior to and after the
implementation (a workshop, meetings, interviews). Observations have been recorded
on video. All the video recordings were watched before starting the analyses. The
actions done by the teacher in the in-class practices have been taken into consideration
in order to decide which category within the framework these will be included. At this
stage, some of the components have been divided into two and some of them have been
renamed when considered necessary. For instance, reminding prior knowledge
component differs depending on who has reminded it and since this is important
within the scope of the research, it has been deemed more suitable to examine this sub-
component as two sub-components as reminding prior knowledge by the teacher and
teacher asking the student to remember the prior knowledge. In the analysis, the teachers
approach has been included in the suitable sub-component. However, it has been
determined that there are negative approaches concerning this sub-component. With
the thought that indicating these cases is necessary for reflecting the PCK of teachers, it
has been decided to arrange these findings by classifying them as positive and negative.
Negative findings show that a negative approach has been observed towards the
teacher regarding that component or the teacher cannot use the opportunity positively
although there is a very convenient classroom environment to establish a positive
approach.
Transcribed lesson sections and some examples regarding the analysis of these
scripts by the help of framework are shown below. Source of expression (teacher,
student, blackboard, smartboard), expression, basic components of the framework, sub-
components of the framework and some descriptions/notes have been included in the
tables. The Stud. Abbreviation is used for student in the tables. The situations where
students talk as a crowded group have been indicated as Stud. (together). In order to
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 108
distinguish situations where the conversation belongs to the same student, indications
such as Stud.1, Stud.2 has been used. Screen quotations have also been used where
necessary. Explanations have been included (within the expressions) in square brackets
and in italics in order to describe the current situation. Since some rows could not be
included in any component within the scope of the framework, abbreviations as …
have been used.
Source
of
expression
Expression
Basic
Components of
the framework
Sub-
components of
the framework
Descriptions/
Notes
Ersin
... Here are three relation diagrams. Let’s
look and try to see common and different
properties of them. Ziya?
ACT -
Knowledge of
Students’
Thinking
Determining
students’ current
knowledge -
Eliciting
Wait for and
listen to
students’
descriptions of
their solution
methods
Teacher
illustrated the
Venn
diagrams of
three relations
and asked the
students to
examine these
relations.
ACT -
Knowledge of
Instructional
Strategies and
Representations
Representations
- Eliciting
Demonstrates
teacher-selected
representations
(without
endorsing the
adoption of a
particular
representation)
Smart
board
Stud.
Teacher, the elements of the sets are the
same.
Ersin
The elements of the sets are the same. We
can say this by looking at their common
properties.
ACT -
Knowledge of
Students’
Thinking
Determining
students’ current
knowledge-
Eliciting
Shares a
student’s
thought/solution
with all class
A student
expressed his
thought, then
the teacher
shared the
student’s
thought with
other students.
Stud.
a goes to 1 in each one.
Ersin
a goes to 1 in each one, okay.
ACT -
Knowledge of
Students’
Thinking
Determining
students’ current
knowledge-
Shares a
student’s
thought/solution
with all class
Teacher
repeated the
student’s
answer and
shared with
others.
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 109
Eliciting
Stud.
The elements of the relation diagrams are
the same.
Ersin
Are the elements of the relation diagrams
the same?
ACT -
Knowledge of
Students’
Thinking
Determining
students’ current
knowledge-
Eliciting
Encourages
students to
explain their
responses in
detail
The student’s
answer was
wrong and the
teacher
provided
whole students
to think about
this answer by
asking a
question.
Stud.
No.
Ersin
Let’s write some of them. For example the
elements of the relation β
1
.
ACT -
Knowledge of
Students’
Thinking
Determining
students’ current
knowledge-
Eliciting
Wait for and
listen to
students’
descriptions of
their solution
methods
The teacher
suggested to
write the
elements of
the relation to
help the
students
examine.
Stud.
(a,1),(a,2),(b,3),(c,4)…
Ersin
The elements of relation β
1
are [the students
are saying, the teacher is writing] (a,1)
,(a,2), (b,3), (c,4). Okay. Now let’s try to
look them as a whole not only to the
elements.
ACT -
Knowledge of
Students’
Thinking
Connecting prior
knowledge to
new knowledge-
Extending
Encourages
students to
consider and
discuss
interrelationships
among concepts
Teacher
encouraged
the students to
make
generalization
based on a
spesific
example.
Smartboard
Stud.
Teacher, if we calculate the number of
subsets, all will be equal, 2
12
.
Ersin
Yes, 2
12
and we can say all are the subsets
of the same set. What did we say when we
defined the relation? We described relation
as each subset of the Cartesian Product.
Now we are focusing on some special ones.
We will pass through to the function
concept. Here are some similarities and
differences. Try to see them. Try to consider
the elements that are used or not used in the
sets. Ata.
ACT -
Knowledge of
Students’
Thinking
Connecting prior
knowledge to
new knowledge-
Supporting
Provides
background
knowledge
When a
student
reminded the
subset
number, the
teacher
reminded
previous
knowledge
and
emphasized
the focus and
the purpose of
the current
discussion. So
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 110
he prevented
the discussion
to go to an
undesired
way.
Stud. (Ata)
Teacher, for instance some elements of the
relations are common, for instance (a,1) is
common for all but (c,4) is not.
Ersin
Not common for all, okay. Let’s look the
elements of domain set. In relation β
1
,
Doğuş?
The teacher
interfered to
direct the
discussion.
...
Ersin
Yes, it is not used in relation β
2.
Ok. We
have a set that includes people. And there is
another set that includes meals. I want you
to match the elements of these two sets but
we have two conditions. The first condition
is that everyone will eat a meal. And one
person will not eat more than one meal.
First try to do in your notebook. Then...
ACT -
Knowledge of
Instructional
Strategies and
Representations
Real Life
Examples -
Supporting
Demonstrates
teacher-selected
real life
examples
(without
endorsing the
adoption of a
particular
example)
The teacher
gave a more
spesific
relation
example and
put in a real
life example.
Stud.
Do we have to write the question?
Ersin
No, not necessary. You don’t need to write
the sets. I only want to see the diagram. The
set of people is, “{Yaşar, Soner, Okan,
Hakan}”. These friends will eat something
[checking students’ notebooks]. Yes this
relation is one of them. [For another
student] Yes. Consider the conditions Ata.
There are two conditions. Yes Tuğçe. Okay,
nearly everyone drew similar diagrams. I
will draw one. Set A consists of
Yaşar,Soner,Okan and Hakan. Set B
consists of meals. We can put first capitals.
Kebab, Bean, Meatball, Spinach, Patato,
Celery, Wrap. Yaşar likes bean. Then, could
Soner also choose bean?
ACT -
Knowledge of
Instructional
Strategies and
Representations
Representations
- Eliciting
Wait for and
listen to
students’ detailed
descriptions of
the
representations
in their solution
methods
The teacher
gave
feedbacks by
checking the
students’
notebooks.
Smartboard
Teacher Ersin who shows an approach of teaching by focusing on students
thoughts in general has listened to his students and encouraged them to explain their
thoughts in detail. He paid attention to determining the current knowledge of his
students when entering into a new concept (function concept). He provided a
Berna Tataroğlu Taşdan, “dem Çelik -
“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
CONTENT KNOWLEDGE İN THE CONTEXT OF SUPPORTİNG M“THEM“TİC“L THİNKİNG
European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 111
discussion platform within the classroom and managed the discussions well. He did not
give short and strict feedbacks such as right/wrong regarding the answers received
from the students. He avoided the discussion going out of context through small
interventions. In addition, he made use of different representations and real-life
examples.
The situation where the students have a misconception about If an element
remains uncovered in the range set during matching, then this matching is not a function.In
Teacher Gökhans lesson and Teacher Gökhans approach towards this situation are
shown in the below section.
Source
of
expression
Expression
Basic
Components of
the framework
Sub-
components
of the
framework
Descriptions/
Notes
Board
ACT -
Knowledge of
Instructional
Strategies and
Representations
Representations -
Eliciting
Demonstrates
teacher-selected
representations
(without
endorsing the
adoption of a
particular
representation)
The teacher gave
examples for
correspondence
via Venn
diagrams.
Gökhan
Does the second one represent a
function?
ACT -
Knowledge of
Instructional
Strategies and
Representations
Problems -
Eliciting
Wait for and
listen to
students’
descriptions of
their solution
methods
Stud.
(together)
No.
Stud. 1
Teacher, one to two …[cannot be
understood]
Gökhan
Okay come to the board and please tell
us what you are thinking.
ACT -
Knowledge of
Instructional
Strategies and
Representations
Problems -
Eliciting
Encourages
students to
explain their
responses in
detail
The teacher
appreciated the
student’s answer
and called him to
the board.
Stud. 1
This one [student is showing the diagram
in the right of the board] does not
represent…
Gökhan
Listen your friend.
The teacher
warned the
students not to
speak.
Stud. 1
In our rule each element [pointing the
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European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 112
conditions of being a function written on
the board]
Gökhan
Ertuğrul is saying that this doesn’t
represent a function. How many of you
agree with this idea? [Few students raise
their hands] Others don’t agree, true?
ACT -
Knowledge of
Instructional
Strategies and
Representations
Problems -
Eliciting
Shares students’
solutions with
all class
The teacher
asked other
students’ if they
agree or disagree
with the friend’s
idea.
Stud.
It doesn’t represent.
Gökhan
It doesn’t represent.Why?
ACT -
Knowledge of
Instructional
Strategies and
Representations
Problems -
Eliciting
Encourages
students to
explain their
responses in
detail
The teacher
asked “why” to
deepen the
student’s
thought.
Stud.
(together)
Because it isn’t matched with something.
Gökhan
That’s right.
Stud. 2
There mustn’t be any unmatched
elements in the first set.
Gökhan
What did you say? Please repeat it
loudly.
ACT -
Knowledge of
Instructional
Strategies and
Representations
Problems -
Eliciting
Shares students’
solutions with
all class
When a student
told one
condition of
being a function,
the teacher
tended towards
this student’s
thought.
Stud. 2
There mustn’t be any unmatched
elements in first set, but there may be in
the second.
Gökhan
Good. Actually this [the
diagram]represents a function. What did
we say? How do the elements of the first
set be?
ACT -
Knowledge of
Students’
Thinking
Connecting prior
knowledge to new
knowledge -
Supporting
Provides
(students to
remember)
background
knowledge
ACT -
Knowledge of
Students’
Thinking
Knowing
students’
misconceptions -
Eliciting
Uses students’
misconceptions
(that are
determined
through their
explanations) for
lesson’s content
The teacher
realized that the
students have
some
misconceptions,
then he reminded
conditions of
being a function.
…
Gökhan
Then what did we say? How many pairs
does each element in the first set
ACT -
Knowledge of
Provides
(students to
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“ CONCEPTU“L FR“MEWORK FOR EX“MİNİNG M“THEM“TİCS TE“CHERS’ PED“GOGİC“L
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European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 113
include?
Students’
Thinking
Connecting prior
knowledge to
new knowledge -
Supporting
remember)
background
knowledge
ACT -
Knowledge of
Students’
Thinking
Knowing
students’
misconceptions -
Eliciting
Uses students’
misconceptions
(that are
determined
through their
explanations) for
lesson’s content
Stud.
(together)
One
In this section, Teacher Gökhan recognized that there were misconceptions so he
listened to the students thoughts in order to understand the cause of these
misconceptions. He did not remind the students of the conditions of being a function
directly and enables the students to examine if the matching in the given Venn diagram
complies with the conditions of being a function or not.
Regarding Problems-Supporting interconnection; a negative finding for Teacher
Özge about the sub-component of encouraging students to analyze, compare and
generalize mathematical concepts when they face a problem has been shown below as
an example.
Source
of
expression
Expression
Basic
Components of
the framework
Sub-
components
of the
framework
Descriptions/
Notes
Board
Teacher asked
students to
explain which
relation
represent a
function.
Özge
… How can I understand that there is an
unmatched element in the domain set or not?
If there is an element that makes the function
in the A undefined, then this element will be
unmatched. Namely you got an element from
ACT -
Knowledge of
Instructional
Strategies and
Representations
Encourages
students to
analyze,
compare, and
generalize
Negative:
There was a
suitable
environment in
the classs for
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European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 114
the domain set. If it doesn’t have an image
[demonstrating on the Venn diagram] then
you say that it is not a function. Is there a x
value that makes this undefined?
Problems -
Extending
mathematical
concepts for
encountered
problem
analyzing the
mathematical
concepts but the
teacher could
not use this
opportunity.
She played an
active role
instead of
engaging
students in a
discussion.
Stud.
-2
Stud.
Yes.
Özge
-2. Okay. Is -2 an element of the domain set?
Stud.
(together)
Yes.
Özge
Then, when I substitue -2 for x and can not
find an image…
This situation has been experienced in the fifth lesson of Teacher Özge when she
was examining if the expressions given algebraically indicate a function or not.
However, she played an active role and started to analyze the function concept by
herself without waiting for the answers during this examination. This has been
evaluated as a negative finding within the scope of the study.
Discussion
The purpose of this article is to propose a conceptual framework that helps researchers
examine mathematics teachers PCK in the context of supporting students
mathematical thinking. In this study, PCK has been examined in the context of
supporting mathematical thinking and has been interconnected to ACT and a new
framework has been obtained. ACT framework which is composed of three components
(eliciting, supporting, extending) has been interconnected to two components
knowledge of students thinking and knowledge of instructional strategies and
representations) of PCK. Instructional examples included within the framework
suggested as a result of the interconnection have become the indicators regarding PCK
of mathematics teachers in the context of supporting mathematical thinking.
Since the knowledge of a teacher regarding the thinking/mathematical thinking
of students is considered necessary for an effective teaching, this subject has been the
focus of many studies. “n, Kulm & Wu  suggested that knowledge of students
mathematical thinking helps teachers to enhance their own knowledge of content and
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European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 115
curriculum, prepare lessons thoroughly, and teach mathematics effectively. They also
highlighted that without knowledge of students thinking, teaching cannot produce
learning it may instead be like playing piano to cowsa Chinese idiom “n, Kulm &
Wu, 2004). Jenkins (2010) found that the structured interview process is a way to
develop prospective teachers knowledge of students mathematical thinking. In order
to find an answer to the question how can teacher educators reliably assess growth in
teachers’ PCK?Norton et al.  have examined school teachers understandings of
students mathematical thinking in their studies with regard to teachers development
of PCK. For this purpose, they have developed video-based prediction assessment
instruments and have experienced these. Unlike studies which examine PCK of teachers
in the context of how they support students mathematical thinking during their
teaching process and which focus on students thinking An, Kulm & Wu, 2004; Jenkins,
 Kılıç, ,  Lee,  Norton, McCloskey & Hudson, 2011; Sleep & Boerst,
 Yeşildere-İmre & “kkoç, , PCK of teachers in the context of supporting
mathematical thinking has been examined within the scope of a more concrete
framework in this study.
Similar to our study, Cengiz, Kline & Grant (20 have also considered students
thinking and Mathematical Knowledge for Teaching (MKT) all together. According to
the results of the study, MKT matters in the way teachers pursue student thinking.
Similar to this result and in the way that validates our assumption at the beginning of
the study, we have also found in this study that PCK of a teacher is important in
supporting/developing students mathematical thinking. However, unlike the study of
Cengiz, Kline & Grant (2011), our study has suggested a new framework by
interconnecting two frameworks (beyond examining mathematical thinking within the
scope of PCK).
The suggested framework has set forth teaching components that focus on
students thinking. These components predict that the teacher pays attention to the
prior knowledge, misconceptions, thoughts and questions of students, to take the
individual differences into consideration, to configure the lesson in accordance with
students thoughts, to enable them to explain their thoughts, to make use of different
representations, to switch between these representations and to give place to real-life
examples, problems that require high-level thinking and analogies for an effective
teaching. These components show similarity with the practices listed by An, Kulm and
Wu  for an effective teacher attends to students’ mathematical thinking. According to
the Kulm, Capraro, Capraro, Burghardt & Ford (2001), an effective teacher attends to
students mathematical thinking preparing instruction according to students needs,
delivering instruction consistent with students levels of understanding, addressing
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European Journal of Education Studies - Volume 2 │ Issue 5│ 2016 116
students misconceptions with specific strategies, engaging students in activities and
problems that focus on important mathematical ideas, and providing opportunities for
students to revise and extend their mathematical ideas (as cited in An, Kulm & Wu,
2004, p. 148).
The framework suggested in this article could help researchers in examining the
teaching and PCK of mathematics teachers in the context of supporting mathematical
thinking and could also enable the researchers (who focus on PCK development of
teachers) to see the PCK development of teachers clearer. In fact, examples given from
research Tataroğlu Taşdan,  where this framework has been used as an analytical
tool in data analysis have provided the readers with an opinion about how this
framework can be used in practice. Since teachers knowledge has a complex structure
by nature (Fennema & Franke, 1992), it is not easy to monitor the development of this
knowledge. In the research given as an example, PCK development of mathematics
teachers has been examined and it has been seen that the noted framework is beneficial
for the researcher in monitoring the mathematics teachers PCK development. When the
findings of the research have been examined for each component within the theoretical
scope of the study; it has been found that PCK of the mathematics teachers who
participated in the context of knowledge of students thinking component has improved
most in the sub-component of determining the misconceptions of students. The real-life
examples sub-component of knowledge of instructional strategies and representational
have been found as the component which all the teachers have improved the most.
When the findings of the same study have been considered within the scope of ACT; it
has been found that the participant mathematics teachers are more successful in
eliciting and supporting steps of the model. In their studies, Fraivillig, Murphy and
Fuson (1999) have also found that teachers are more successful in the supporting steps
and less successful in eliciting and extending steps. They have indicated the source of
this difference as the differences in pedagogical skills of teachers required for eliciting,
supporting and extending. The source of this difference between the findings of the two
studies may be the differences in teachers education, curriculum etc. of different
countries.
Conclusion
It is thought that this study has made a contribution to the field by examining PCK in
the context of supporting mathematical thinking and showing through which practices
could a mathematics teacher be able to support the students mathematical thinking
throughout his/her instruction. The suggested framework is a useful tool for the
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researchers and teacher educators who are dealing with teachers knowledge focusing
on students mathematical thinking and a guide for the teachers. “s Fraivillig, Murphy
and Fuson (1999) have suggested for ACT, this developed framework can serve as a
beneficial pedagogical tool in pre-service and in-service teacher education. In addition,
the framework is an analytical tool that can be used for monitoring mathematics
teachers PCK development.
The limitations of ACT Model and PCK which form the basis of this framework
are also the limitations of the framework suggested in this study. Fraivillig (2001) has
evaluated “CT model as Although eliciting, supporting, and extending describe elements of
effective instruction, the art of teaching is much too complex to be captured by these three
components. The multi-dimensional structure of PCK and the complex structure of
teaching make it difficult to define the teaching process through explicit components.
For this reason, it should be considered that the framework suggested in this study may
not work at all times. Besides, the framework has focused on only two components of
PCK knowledge of students thinking and knowledge of instructional strategies and
representations). Examining other components of PCK within the scope of this
framework and the effects of this on the mathematics teaching process focusing on
students mathematical thinking can be examined in the further studies.
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