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Electronic Theses, Treatises and Dissertations
The Graduate School
2005
A Qualitative Study: Algebra Honor
Students' Cognitive Obstacles as They
Explore Concepts of Quadratic Functions
Ali Eraslan
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THE FLORIDA STATE UNIVERSITY
COLLEGE OF EDUCATION
A QUALITATIVE STUDY: ALGEBRA HONOR STUDENTS’ COGNITIVE
OBSTACLES AS THEY EXPLORE CONCEPTS OF
QUADRATIC FUNCTIONS
BY
ALI ERASLAN
A Dissertation submitted to the
Department of Middle and Secondary Education
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded
Fall Semester, 2005
The members of the Committee approve the dissertation of Ali Eraslan defended on
October 24, 2005.
____________________________
Leslie Aspinwall
Professor Directing Dissertation
____________________________
Emanuel I. Shargel
Outside Committee Member
____________________________
Elizabeth Jakubowski
Committee Member
____________________________
Maria L. Fernández
Committee Member
Approved:
_____________________________________________________________________
Pamela S. Carroll, Chairperson, Department of Middle and Secondary Education
The Office of Graduate Studies has verified and approved the above named committee
members.
ii
For
Berkay and Nehir
iii
ACKNOWLEDGEMENTS
First of all I would like to express my sincere appreciation to my advisor, Dr.
Leslie Aspinwall, for his challenge, guidance, support, and inspiration in every step of
this dissertation.
I wish to express my special thanks to my committee members Dr. Emanuel I.
Shargel, Dr. Elizabeth Jakubowski, and Dr. Maria L. Fernández for their constructive
feedbacks and thoughtful suggestions.
I would like to thank my colleagues, Selcuk, Carla, and Rebecca for their valuable
comments in the process of data analyzing.
I thank the students and the classroom teacher who eagerly participated in the
study.
I also would like to thank the Ministry of National Education, Turkey, for its
generous financial support throughout my educational experience in the USA.
Finally, I want to thank my family members for their continued support, love and
encouragement.
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TABLE OF CONTENTS
List of Tables
List of Figures
Abstract
1. INTRODUCTION
Rationale and Conceptual Framework
Piaget and Constructivism in Mathematics Education
Research Questions
Significance of the Study
Summary
2. LITERATURE REVIEW
What is a Cognitive Obstacle?
Students’ Understanding of Mathematics
Cognitive Psychology and the Piagetian Theory
Students’ Understanding of and Difficulties with Functions
Students’ Understanding of and Difficulties with Quadratic
Functions
3. METHODOLOGY
A Multiple Case Study
Research Site
School and Classroom
Teacher and Teaching Style
Course chosen for the Study
Time Line
Participants
Research Tasks used for the Study
Questionnaire Tasks
Interviews Tasks
Methods of Data Collection
Clinical Interviews
Classroom Observations
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Documental Data
Data Analysis
Trustworthiness of the Study
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4. DATA ANALYSES AND FINDINGS
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The Case of Richard
The Case of Colin
Cross-Case Analyses
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5. MAJOR FINDINGS AND DISCUSSIONS
Richard’s Cognitive Obstacles
Colin’s Cognitive Obstacles
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6. CONCLUSIONS
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Implications for Teaching
Limitations of the Study
Issues for Further Study
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APPENDICES
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REFERENCES
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BIORAPHICAL SKETCH
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LIST OF TABLES
Table 2.1
Table 3.1
Table 3.2
Table 4.1
Table 4.2
Table 5.1
Schoenfeld’s (1989) levels of analysis and structure
Clinical interview procedures for research in mathematical thinking
Fundamental questions posed during the process of interviewing
Summary of Richard’s compartmentalization cases with the vertex
of the parabola
Richard’s identified obstacles in his cognitive structure
Colin’s identified obstacles in his cognitive structure
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LIST OF FIGURES
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 3.1
Figure 3.2
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 4.11
Figure 4.12
Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17
Figure 4.18
Figure 4.19
Figure 4.20
Figure 4.21
Figure 4.22
Figure 4.23
Figure 5.1
Figure 5.2
Figure 5.3
Haylock’s model of students’ understanding of mathematics
Sfard’s general model of concept formation
Herscovics’ analytical framework
Pirie & Kieren’s theory of the growth of mathematical
understanding
Vinner’s framework of concept image and concept definition
Structure of the Algebra-2-Honors Class
Components of Data Analysis: Interactive Model
Richard’s evoked image for the quadratic function
Translation task in the questionnaire
Translation task in the first interview out of two
Translation task in the second interview
Task of horizontal and vertical shift in the first interview out of
two
Richard’s graph of y = (x -3) 2
Richard’s graph of y = x2 -3
Task of horizontal and vertical shift in the second interview
Richard’s graph of y = (x +2) 2
Richard’s graph of y = x2 +2
Comparison task between a quadratic function and an absolute
value function
Richard’s graph of y = (x - 4) 2
Richard’s graph of y = | x - 4 |
Richard’s graph of y = (x - 2) 2
Richard’s graph of y = | x - 2 |
Determining task in the first interview
Richard’s graph for the x-intercepts
Determining task in the questionnaire
Interpretation task in the first interview
Task of using quadratic model in the quiz
Task of using quadratic models in the questionnaire
Richard’s answer for the quadratic model problem
Task of using quadratic models in the first interview
Colin’s evoked image for the quadratic function
Translation task in the questionnaire
Translation task in the first interview out of two
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Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 5.13
Figure 5.14
Figure 5.15
Figure 5.16
Figure 5.17
Figure 5.18
Figure 5.19
Figure 5.20
Figure 5.21
Figure 5.22
Figure 5.23
Figure 5.24
Translation task in the second interview
Colin’s graph for y = 3/2 (x+1) (x -3)
Colin’s graph for y = - x2 - 3x +5
Colin’s graph for y = (x -2)2 +5
Task of horizontal and vertical shift in the first interview
Colin’s graph of y = x2
Colin’s graph of y = (x -3)2
Task of horizontal shift in the second interview
Colin’s graph of y = x 2
Colin’s graph of y = (x +2)2
Determining task in the first interview
Colin’s graph for the axis of symmetry
Colin’s graph for the vertices of the parabolas
Determining task in the questionnaire
Interpretation task in the first interview
Colin’s graph for the y-intercept
Task of using quadratic model in the quiz
Task of using quadratic model in the questionnaire
Colin’s answer for the task of using quadratic model
Task of using quadratic model in the first interview
Colin’s graph for the quadratic model
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ABSTRACT
With the paradigm shift from a behavioral to a constructivist perspective in
teaching and learning of mathematics, students’ thought processes have become a major
focus for learning and students’ learning of the specific subject matter has been analyzed
and approached more qualitatively. In parallel to this development, the present study
attempted to describe two algebra-honor students’ cognitive obstacles in the learning of
quadratic functions. In particular, along with students’ concept image and definition for
the quadratic function (Tall & Vinner, 1981), five other aspects of quadratic functions
were examined to identify students’ cognitive obstacles surrounding quadratic functions.
These five aspects, adapted by Wilson (1994) who identified the most important aspects
of the function concept for deep understanding, were as follows: translating, determining,
interpreting, solving quadratic equations, and using quadratic models.
A multiple case study involving two algebra honor students was designed and
implemented. Two honor students under the pseudonyms of Richard and Colin were
purposely chosen and voluntarily participated in this study. Data were obtained from
one-on-one clinical interviews, students’ written work (a test, quiz, and questionnaire),
and classroom observations. The analysis particularly focused on identifying students’
cognitive processes as they worked on quadratic tasks during the interviews. The whole
data were analyzed through the lens of an integrated framework using Schoenfeld’s
(1989) level of mathematical analysis and structure and Tall and Vinner’s (1981)
framework of concept image and concept definition.
The study revealed the cognitive obstacles that Richard and Colin encountered
during the study of quadratic functions. In light of these obstacles, the following four
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assertions were made in this study: (1) one of the obstacles arises from a lack of making
and investigating mathematical connections between algebraic and graphical aspects of
the concepts, (2) another cognitive obstacle arises from the need to make an unfamiliar
idea more familiar, (3) a third cognitive obstacle arises from the disequilibrium between
algebraic and graphical thinking, and (4) the image of the quadratic formula or absolute
value function has a potential to create an obstacle to mathematical learning.
This study has important applications for classroom teaching. By identifying the
students’ cognitive obstacles based on the six aspects of quadratic functions, the study
indicates which obstacles are associated with certain aspects of quadratic functions.
Moreover, in light of these obstacles, it emphasizes the interrelation and complementary
aspect of algebraic and graphical thinking in an ongoing back-and-forth process in
learning and teaching of quadratic functions.
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CHAPTER I
INTRODUCTION
If we had to reduce all of educational psychology to just one principle, we
would say this: the most important single factor influencing learning is what the
learner already knows ( Ausubel, Novak, & Hanesian, 1978, p. 163).
Rationale and Conceptual Framework
Research on learning and teaching of mathematic has shifted its focus from a
behavioral to a constructivist perspective (Fischetti, Dittmer, & Diane, 1996). This new
focus has changed some traditional ideas on teaching and learning in mathematics
classrooms. Methods of student assessment, methods of teaching, and teachers’ beliefs
about how students learn have particularly been affected by the current constructivist
view.
Assessment methods used for what a student knows have changed. In the
traditional approach, teachers were using classroom tests or standard tasks to assess what
their students knew in their classrooms. These tests assessed simple, isolated knowledge
and skills transferred by teachers through instruction (Ginsburg, 1997). The primary
focus was whether or not a student got the right answer. The number of correct answers
was an indicator about what a student knew or how well he or she was doing. With the
constructivist view, students’ thinking and strategies have become central in the process
of learning (Simpson, 2002). Using more open-ended and problem-solving tasks as well
as allowing students to reflect on these tasks have started becoming routine in assessing
what students know in mathematics (Ginsburg, 1997). Teachers have come to see that
getting a right answer does not necessarily mean that students really understand the
mathematics underlying the answer. Therefore, they have started paying more attention
to figuring out the thought processes underlying students’ right and wrong answers.
1
With the reform efforts in mathematics education (National Council of Teachers
of Mathematics [NCTM], 1989, 1991; Mathematical Sciences Education Board [MSEB],
1988), method of teaching has moved from a direct instruction to a constructivist
approach; in other words, from drill and rote learning to learning with understanding
(Ginsburg, 1997). The reform documents provided “a more complete description of what
it means to learn with understanding and to teach for understanding” (Hiebert et al., 1997,
p. 3-4). This has had a major effect on teachers’ beliefs about how students learn. No
longer is knowledge ‘out there’ to be transmitted from the teacher to the student; nor are
students empty containers waiting to be filled; nor is the mind an unknown ‘black box’.
Instead, “knowledge is constructed by learners as they attempt to make sense of their
experiences. Learners, therefore, are not empty vessels to be filled, but rather active
organisms seeking meaning” (Driscoll, 1994, p. 360); and the mind is the place in which
human cognitive processes occur (Driscoll, 1994). This view has taken place as the
‘learning principle’ in the Principles and Standards for School Mathematics (NCTM,
2000) states that “students must learn mathematics with understanding, actively building
new knowledge from experience and prior knowledge” (p. 11).
The behaviorist tradition has been described as process-product research in which
“teacher behavior is the process; the pupil outcome is the product” (Norman, 1993, p.
161). Thus, it has been criticized for not having the capacity to handle complex cognitive
interactions that occur in the classroom environment (Ginsburg, 1997) and, more
importantly for the purpose of this study, for keeping out of students’ thought processes
(Norman, 1993). In behaviorist classrooms, students were passive receivers and teachers
were ones who talk students what to do and how to do it (Thompson, 1992). On the other
hand, in constructivist classrooms, students are active participants and teachers are
facilitators to help students figure out for themselves what to do, both individually and
collectively (Chrenka, 2001). In this environment, teachers are no longer authorities who
know all strategies and right answers to any problem. Rather, they provide “learning
opportunities that encourage students to construct their own learning” (Ginsburg, Jacobs,
& Lopez, 1998, p. 207).
The constructivist view makes students central in the process of learning and tries
to understand the process of how they think and make sense of the world (Even & Tirosh,
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2002). This view is built on an assumption that students construct their own knowledge
based on their existing knowledge and experiences; and this process occurs from the
interaction between a student’s existing knowledge and experiences and the new ideas or
situations he or she encounters (Airasian & Walsh, 1997). Therefore, each student has a
different way of understanding, and this understanding is also different from that of
teachers. The nature of understanding in this perspective is considered as “the personal
building and re-organization of one’s knowledge structures” (Pirie & Kieren, 1992, p.
243).
As a result of the effect of the constructivist view, students’ thought processes
became an important part of the learning and teaching of mathematics; and clinical
interviews, originally developed by Piaget, became a main tool for understanding these
processes in mathematics (Ginsburg, 1997). In light of Piaget’s work, Ginsburg (1981)
argued that the method of clinical interview is used for two different research purposes:
the discovery of cognitive processes and the identification or specification of cognitive
processes. With the method of clinical interview, we can understand “how children
construct their personal worlds, how they think, how their cognitive processes (at least
some of them) operate, how their minds function” (Ginsburg, 1997, p. 28). Therefore,
using this method and allowing students to think aloud as they work on given
mathematical tasks provides us an opportunity to uncover students’ thought processes,
conceptions and cognitive obstacles.
Uncovering students’ thinking processes simultaneously reveals important
information related to students’ difficulties called systematic errors, bugs, gaps, or
learning difficulties (Lehn, 1983; Baroody & Ginsburg, 1990). Similar to Baroody and
Ginsburg (1990), who mentioned a gap that prevents assimilation between formal
instruction and a learner’s existing knowledge, Herscovics (1989a) described an internal
conflict that might appear between the existing learner’s constructed knowledge and the
new conceptual schemata in the accommodation of new knowledge. He called this
internal conflict a ‘cognitive obstacle.’ The research on mathematical thinking done by
Ginsburg (1977) and Baroody & Ginsburg (1990) indicated that most mathematical
errors are not simple or random mistakes. On the contrary, errors in students’
conceptualizations “are all strong indicators of the existence of cognitive obstacles”
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(Herscovics, 1989a, p. 80) and “provide a window to the learner’s thinking” (Baroody &
Ginsburg, 1990, p. 56).
Therefore, in the present study, students’ errors consisting of
incomplete and inaccurate answers would be an opportunity for the researcher to analyze,
assess, and identify students’ cognitive obstacles.
In the literature, as the purpose of this study, there are many research studies that
attempted to identify students’ misconceptions with, and difficulties in, the concept of
function. The importance and centrality of the function concept in mathematics has been
specifically emphasized by more and more researchers in the last two decades. “The
concept of function is one of the most important concepts in mathematics, perhaps as
important as number itself (Cooney, 1996, p. 1),” “functions are one of the most powerful
and useful notions in all of mathematics (Romberg, Fennema, & Carpenter, 1993, p. 1),”
“few concepts in mathematics are as fundamental as the notion of function (Herscovics,
1989a, p. 75),” and “the concept of function is one of the core ideas in mathematics
(Schoenfeld, 1989, p. 110)” are just a few of examples. Basically what arithmetic means
for elementary school mathematics is about the same as what function means for
secondary school mathematics. Functions are foundations of the rest of the topics in the
secondary mathematics curriculum. They provide an excellent opportunity for students
to study various relationships that are central to the study of mathematics. The Principles
and Standards for School Mathematics (NCTM, 2000) emphasizes that “instructional
programs from pre-kindergarten through grade 12 should enable all students to
understand patterns, relations, and functions” (p. 294). Even though the function concept
is considered to be one of the most important notions in mathematics, it is difficult for
students to understand and has been the subject of extensive research in mathematics
education (Vinner, 1983; Dreyfus & Eisenberg, 1983; Matkovits, Eylon, & Bruckheimer,
1986; Vinner & Dreyfus, 1989; Tall & Bakar, 1992; Adams, 1997; Carlson, 1997;
Janvier, 1998; Even, 1998; Hitt, 1998; Clement, 2001).
With regard to the understanding of functions, most studies have been done in the
1980’s and early 1990’s. Particular attention has been given to the formal definitions and
images for the concept of function as well as linear functions as a special family (Vinner
& Dreyfus, 1989; Matkovits et al., 1986; Vinner, 1983; Dreyfus & Eisenberg, 1983; Tall
& Bakar, 1992). Most of them have focused on what students write down and largely
4
ignore what students think. On the other hand, the concept of quadratic function, an
important bridge between linear and polynomial functions in the mathematics curriculum
(Movshovitzs-Hadar, 1993), has drawn little attention in terms of students’ difficulties.
The only study that particularly focused on the concept of quadratic functions was
conducted by Zaslavsky (1997) in Israel. In this quantitative research, Zaslavsky
analyzed students’ notebooks and written responses to non-standard problems to identify
their conceptual obstacles. However, Davis (1990) pointed out that what a student thinks
is far more fundamental than what he or she writes. Also Pirie and Kieren (1994)
emphasized that understanding requires not only close examination of students’ written
work, but also involves a careful analysis of the students’ thought processes on given
mathematical tasks. In recent years, a few studies have been conducted on a single
quadratic task by employing interviews to collect data (Borgen & Manu, 2002; Zazkis,
Liljedahl, & Gadowsky, 2003; Sajka, 2003). However, interview techniques used in
these studies were questionable. Two students worked together on a given task and
discussed it with each other in 6.5 minutes in the study of Borgen & Manu (2002), while
an average student in Sajka’s work was not able to understand the non-standard task for a
long time in a 42-minute interview. In the study of Zazkis et al. (2003), a total of 41
participants were superficially interviewed rather than in-depth clinical interviews. The
previous studies used either a quantitative method that is not sufficient to figure out all
the ways students are thinking or the questionable interview alone on given quadratic
tasks. As a result, these studies were not able to provide a detailed and sufficient picture
of students’ cognitive processes in the learning of quadratic functions. Therefore, there
is a need for a better understanding of students’ cognitive processes by means of one-toone-in-depth clinical interviews described by Ginsburg et al. (1998) with regard to their
obstacles.
Piaget and Constructivism in Mathematics Education
Piaget is considered as the pioneer of the constructivist approach to cognition in
this century (von Glasersfeld, 1995). He has offered powerful insights into the human
mind and its development as well as our understanding of how children view the world
(Confrey, 1994). According to Piaget:
5
“…all knowledge is tied to action, and knowing an object or an event is to use it
by assimilating it to an action scheme…”
“…to know an object implies its incorporation in action schemes, and this true on
the most elementary sensorimotor level and all the way up the highest logicalmathematical operations” (cited in von Glasersfeld, 1995, p. 56).
The Piagetian theory emphasizes individual knowledge construction stimulated by
internal cognitive conflict as learners strive to resolve mental disequilibrium (Applefield,
Huber, & Moallem, 2001). The process of equilibration involves both assimilation and
accommodation. Herscovics (1989a) defined assimilation as the integration of things to
be known into some existing cognitive structure, and accommodation as changes in the
learner’s cognitive structure necessitated by the acquisition of new knowledge.
According to von Glasersfeld (1995), cognitive assimilation comes about when a learner
fits an experience into a conceptual structure it already has. In other words, the learner
filters and interprets new information in terms of his/her existing knowledge (Baroody &
Ginsburg, 1990). During this process, a gap or an internal conflict may appear between a
learner’s existing knowledge and formal instruction, and then prevent assimilation. As a
result, this may lead to learning difficulties in the learner’s mind (Herscovics, 1989a;
Baroody & Ginsburg, 1990).
In mathematics education, Noddings (1990) characterized constructivism as both
a cognitive position and a methodological perspective. From the methodological
perspective, the author framed the constructivists’ position with the following
assumptions: (a) human beings are knowing subjects, (b) they have a highly developed
capacity for organizing knowledge, and (c) their behaviors are purposive. And then, she
argued that these assumptions require us to conduct research studies that use methods of
ethnography, clinical interviews, or overt thinking.
From the cognitive position, constructivism assumes that “all knowledge is
constructed and the instruments of construction include cognitive structures that are
either innate or are themselves products of developmental construction” (Noddings, 1990,
p. 7). In particular, there is a base structure to begin construction (a structure of
assimilation). The purposive action leads to transformation of existing structures which
is a process of continual revision of structure (a process of accommodation). In order to
identify and describe the cognitive structures in all phases of construction, clinical
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interviews and prolonged observations help us to make inferences about the cognitive
structures that underlie the behavior (Noddings, 1990).
The following transition made by Noddings (1990) in light of Piaget’s work
presents an overall view of the present study. Piaget’s cognitive constructivism leads to
methodological constructivism. Then, methodological constructivism becomes
pedagogical constructivism for teachers. And then, pedagogical constructivism provides
such tools as clinical interviews and prolonged observations that help us to uncover
thinking patterns, systematic errors, misconceptions, or cognitive obstacles. In particular,
constructivism as characterized by Noddings provides a ground for this study in terms of
assumptions being made.
In summary, the researcher believes that “all knowledge is necessarily a product
of our own cognitive acts” (Confrey, 1990, p. 108) and learning is accommodating new
knowledge into the existing cognitive structure, resolving conflicts that occur between the
new knowledge and prior knowledge as the new knowledge is acquired and
accommodated. The internal conflicts in this process may lead to learning difficulties,
called cognitive obstacles by Herscovics (1989a), in the learner’s mind.
Research Questions
The purpose of this study is to identify algebra honor students’ cognitive obstacles
in the learning of quadratic functions. By using multiple case study method, I tried to
answer the following question: what cognitive obstacles do algebra honor students
experience as they explore concepts of quadratic functions? In particular, along with
students’ concept image and definition for the quadratic function (Tall & Vinner, 1981),
five other aspects of quadratic functions, adapted by Wilson (1994) who identified the
most important aspects of the function concept for deep understanding, are examined to
identify two algebra-honor students’ cognitive obstacles in the learning of quadratic
functions. So, a total of six aspects of quadratic functions in the present research are as
follows:
(a) concept image and definition for the quadratic function
(b) translating among multiple representations of quadratic functions (Appendix D1)
(c) determining the axis of the symmetry, the vertex and x-and y-intercepts of the
quadratic functions (Appendix D2)
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(d) interpreting quadratic functions represented by graphs, formulas, tables and
situation descriptions (Appendix D3)
(e) solving quadratic equations by factoring, completing the square, and using the
quadratic formula (Appendix D4)
(f) using quadratic models to solve problems presented in real-world situations
(Appendix D5)
Interpretation here refers to “the action by which a student makes sense or gains
meaning from a graph (or a portion of a graph), a functional equation, or a situation”
(Leinhardt, Zaslavsky, & Stein, 1990, p. 8), while translation refers to “ (a) the act of
recognizing the same function in different forms of representations, (b) identifying for a
specific transformation of a function in one representation its corresponding
transformation in another representation, or (c) constructing one representation of a
function given another one” ( Leinhardt et al., 1990, p. 16). As a result, an interpretation
task requires students to react to a given piece of data such as a graph, an equation, or a
data set, whereas a translation task requires them to determine an equation from a graph,
or vice-versa (Leinhardt et al., 1990). Overall, this is going to be both an examination of
how algebra honor students construct their own mathematical knowledge in the learning
of quadratic functions and an identification of what obstacles arise in these processes.
Significance of the Study
The concept of functions, in general, and of quadratic functions, in particular, is
an important component of Algebra classes in the secondary mathematics curriculum
(Zaslavsky, 1997). In order to teach students these concepts, “we need to know what our
students are thinking, how they produce the chain of little marks we see on their papers,
and what they can do (or want to do) with the material we present to them” (Noddings,
1990, p. 15). In other words, a key aspect of effective teaching is to know what and how
students are thinking (Dunham & Osborne, 1991). This brings us to know obstacles that
students face in the learning of these concepts.
The present study is based on a need for a better understanding of students’
cognitive processes involved in learning quadratic functions with respect to their
obstacles. In particular, this qualitative research involving case studies attempts to reveal
8
algebra honor students’ cognitive obstacles surrounding quadratic functions. Therefore,
it is essential to have a clear understanding of the processes of students’ thinking and
learning that operate on tasks related to quadratic functions. This helps us to figure out
why and how these obstacles occur as well as have a clear picture about how they hinder
students’ understanding of the concept. For any concept in mathematics, knowing about
what students’ obstacles are is vital to improve students’ understanding of that concept.
If obstacles are known before a lesson is taught, with carefully developed lesson plans
and modified teaching strategies, we can help students to overcome, or at least minimize
these obstacles in the process of learning. This could be done by providing learning
opportunities that encourage students either to build a new construction from an old one
or to modify it with the new construction. As a result, I believe that this qualitative work
may add valuable knowledge and more dimensions to the research on mathematical
understanding, particularly providing insights into the cognitive processes that algebra
honor students use in the learning of quadratic functions.
Summary
In this chapter, I described an overall view of this study. My curiosity about
students’ thought processes as well as the importance and centrality of the learning of
functions in general, and of quadratic functions in particular, brought me to feel the need
for doing this study. The following sections were provided in this chapter: rationale and
conceptual framework, Piaget and constructivism in mathematics education, research
questions and significance of the study.
In the following chapter, I review various literature related to functions in general,
and quadratic functions as a special family. This review also included two other
important issues discussed in this study: cognitive psychology and constructivist models
that explain learning in mathematics.
In chapter III, I describe the methodology used for this study to answer the
research questions. A multiple case study design involving two honor students is used to
conduct the study. Multiple data are collected from three different sources: clinical
interviews, classroom observations, and documental data (a quiz, test, and questionnaire).
For the data analysis, within-case and cross-cases analyses are employed among two case
9
study data bases (Yin, 1994). Trustworthiness of the study is also discussed in the third
chapter.
In chapter IV, I describe, analyze and interpret the findings for each case on the
basis of the research questions. Chapter V includes major findings and discussions.
Lastly, in chapter VI, the study is concluded with implications, limitations and issues for
future research.
10
CHAPTER II
LITERATURE REVIEW
The purpose of the present study is to reveal cognitive obstacles that Algebra-2honor students face as they explore the concept of quadratic functions. Therefore, the
section starts with the notion of cognitive obstacle in terms of how it is defined. Then,
the following three notions, which are foundations of the present study, are discussed: (1)
students’ understanding of mathematics from constructivist perspective, (2) cognitive
psychology, and (3) students’ understanding of functions in general, of quadratic
functions in particular.
In particular, a number of constructivist models that have been developed in the
last two decades are discussed in terms of what knowledge is and how learning occurs.
Then, an overview is provided with regard to how cognitive psychology, particularly the
Piagetian perspective influenced and shaped the research studies in mathematics
education. This is followed by a review of research on students’ understanding of
functions in general, and quadratic functions in particular. This overall review provides a
good foundation for the present study and locates it in related research in terms of
perspectives used as a lens to examine mathematics learning, research design, data
collection and analysis.
What is a Cognitive Obstacle?
The term cognitive obstacle is grounded in the concept of epistemological
obstacle. The notion of an epistemological obstacle was first introduced in the context of
the development of scientific thinking by the French philosopher Gaston Bachelard
(1938), as translated and cited in Herscovics (1989a), Bachelard described the term:
When one looks for the psychological conditions of scientific progress, one is
soon convinced that it is in terms of obstacles that the problem of scientific
knowledge must be raised. The question here is not that of considering external
obstacles, such as the complexity and transience of phenomena, or to incriminate
11
the weakness of the senses and of the human spirit; it is in the very act of
knowing , intimately, that sluggishness and confusion occur by a kind of
functional necessity. It is there that we will point out causes of stagnation and
even regression; it is there that we will reveal causes of inertia which we will call
epistemological obstacles (p. 61).
In the area of science, Bachelard reported several kinds of epistemological
obstacles: the tendency to rely on deceptive intuitive experiences, the tendency to
generalize, and the obstacles caused by natural language. Then, with respect to
individual learning experiences, Herscovics (1989a) replaced the term ‘cognitive
obstacle’ for obstacles that are encountered in the acquisition of new conceptual schemata
by the learner. The author explained this notion with the Piagetian learning theory,
“where the learner is confronted with new ideas that cannot be fitted into the learner’s
existing cognition, leading to an inability to cope adequately with the new information”
(Tall, 1989, p. 87-88). In particular, the term cognitive obstacle lies on processes of
assimilation and accommodation. Assimilation is the integration of things to be known
into some existing cognitive structure, while accommodation is changes in the learner’s
cognitive structure necessitated by the acquisition of new knowledge (Herscovics,
1989a). The author argued that some new ideas cannot be easily assimilated into the
learner’s existing cognition and new cognitive structures are needed. In the process of
accommodation, the learner faces major cognitive obstacles because of the difficulty of
changing the existing cognitive structure.
In summary, cognitive obstacles in the present study, similar to Herscovics
(1989a) and Zaslavsky’s (1997) definition, refer to obstacles that have a cognitive nature
and that can be explained in terms of the mathematical structures and concepts that
underlie students’ previous learning experiences and internal processing of these
experiences.
Students’ Understanding of Mathematics
There has been a variety of approaches to the study of mathematical
understanding. When I looked at the research on students’ understanding of
mathematics, I came across several theoretical perspectives that attempt to address either
what it means to understand a concept or how students make meaning of mathematics
presented to them. As the purpose of this study, some major theories or frameworks that
12
explain mathematical understanding with the lens of constructivism are discussed in the
following section. They could be called on when analyzing and interpreting students’
thinking processes.
In the first model, Skemp (1978) used the terms ‘relational’ and ‘instrumental’ to
distinguish two types of mathematical understanding. He defined ‘understanding’ as
ways of knowing by describing relational understanding as “knowing both what to do and
why” (p. 9), and instrumental understanding as “rules without reasons” (p. 9). Similar to
Skemp’s definition of mathematical understanding, Hiebert and Lefevre (1986) suggested
two types of mathematics knowledge: conceptual knowledge and procedural knowledge.
Whereas conceptual knowledge refers to knowledge that is rich in relationships,
procedural knowledge includes algorithms and memorized rules.
Another model for understanding in mathematics was discussed by Haylock
(1982). He stated that “to understand something means to make (cognitive) connections”
(p. 54). He also pointed out that the more connections the learner can make between the
new experience and previous experiences, the deeper the understanding occurs.
According to Haylock, if the new experience connects the previous unconnected
experiences, an important advance in understanding happens. In contrast, if the learner
fails to make connections between the new knowledge and prior knowledge, the new
knowledge exists in the learner’s cognitive structure as isolated and unconnected. This
situation is described as ‘rote learning’ by the author. It is argued that this type of
learning may be only used in a limited range of situations or be lost forever.
Haylock suggested four components of mathematical experience to analyze
learners’ connections in the process of doing mathematics: words, pictures, concrete
situations, and symbols. With this model, it is assumed that learners can demonstrate
some degree of understanding by making a suitable connection between two of these
categories of experience. The following figure shows his model of students’
understanding of mathematics:
13
Mathematical language
Pictures
Symbols
Concrete situations
(Actual or described)
Figure 2.1: Haylock’s model of students’ understanding of
mathematics (Haylock, 1982).
Sierpinska (1990) sees understanding as a series of acts and cognitive activity of
overcoming obstacles, claiming that “overcoming an obstacle and understanding are just
two ways of speaking about the same thing” (p. 28). According to her, they are two
complementary pictures of the unknown reality of the important qualitative changes in
the human mind. Epistemological obstacles look back to figure out what was wrong,
insufficient, in our way of knowing, whereas understanding looks forward to the new
ways of knowing (Sierpinska, 1990). She also pointed out that all understanding
occurred on previous beliefs, prejudgments, preconceptions, convictions, and
unconscious schemes of thought. Some acts of understanding may help to overcome
obstacles, while some may turn out to be new obstacles.
Based on cognitive scheme theory, Sfard (1991) presented a new theoretical
framework to investigate the role of algorithms in two different, but complementary
ways: ‘structural conception’ as objects and ‘operational conception’ as process. She
defined understanding as an “intricate interplay between operational and structural
operations” (p.1). In Sfard’s model, the operational conception is the first step in the
acquisition of new mathematical notions. Then transition from conceptual operations to
the structural level is accomplished in three hierarchical steps: interiorization,
condensation, and reification.
14
Interiorization is described in the Piagetian sense that, “a process has been
interiorized if it can be carried out through mental representations” (p. 18). The next
step, condensation, is “a period of “squeezing” lengthy sequences of operations into more
manageable units (p. 19). A person in this level thinks of a given process as a whole,
without going into details. The last step, reification, occurs when a person becomes
capable of conceiving the notion as a full object. Therefore, through the steps of
interiorization, condensation and reification, mathematical ideas derived from the
operational view of the notion become a structural view. Sfard’s model of concept
formation is presented in the figure as follows:
Concept B
Object B
Reification
Condensation
Concept A
Object A
Interiorization
Processes
on
A
Reification
Condensation
Interiorization
Concrete
Objects
Processes
on concrete
objects
Figure 2.2: Sfard’s general model of concept formation (Sfard, 1991).
Based on a specific study and the related cognitive literature, Schoenfeld (1989)
develop a four-level framework to analyze mathematical understanding from the
student’s point of view. As shown below in the figure, while the left column identifies
15
four levels of cognitive structure, the middle column illustrates the three top levels in
standard curriculum analyses and the right column shows the learner’s cognitive structure
for the subject matter.
In particular, level 1 reflects one’s general knowledge and perceptions of the
world that are organized at the macro level in large “chunks” called schemata. At this
level, the standard curricular knowledge related to the task is considered. The second
level “describes objects in the domain and their familiar properties” (p. 113). At this
level, the properties needed to work through the task are determined. At level 3, the
connections between different concepts are supported by the rich knowledge base. The
presence of this rich knowledge base also serves as the link between the manipulations
the algebraic world and the graphical world. The fourth is the level at which learners
illustrate limited understanding or misunderstandings about the concept being studied.
The traditional view has no fourth level. Schoenfeld (1989) emphasizes that although
this structure appears to be a linear process, this does not imply that these knowledge
structures are hierarchically organized or that knowledge is accessed in a hierarchical
fashion.
Table 2.1: Schoenfeld’s levels of analysis and structure (Schoenfeld, 1989).
LEVELS
Traditional view of
Subject matter
1. Knowledge Schemata
Macro-organization of knowledge, at the
schemata level
2. Entities and Entailments
Compiled knowledge… macro-entities
and entailments
3. Deep Connections
Fine-grained superstructure supporting
domain knowledge: conceptual atoms
(nodes) and connections.
4. Conceptual “Atoms”
The limited applications context out of
and across which individuals construct
the conceptual atoms (are indexed by
context) that are seen at the level 3
VOID
16
Our understanding of
Subject’s cognitive
structure
In light of Piaget’s theory, Herscovics (1989b) developed a model of
understanding that describes the construction of a conceptual scheme in two levels. The
model takes into account the learners’ action on his or her physical environment and
moves from an intuitive understanding of physical concepts to the understanding of the
emerging mathematical concept. The first level deals with the understanding of
preliminary physical concepts, while the second describes the understanding of the
emerging mathematical concept. In this model, the author explained that logico-physical
understanding that results from thinking about procedures applied to physical objects and
about the spatio-physical transformations of these objects. On the other hand, logicomathematical understanding that results from thinking to applied procedures and
transformations dealing with mathematical objects.
In particular, as shown in the figure, the understanding of preliminary physical
concepts involves three levels of understanding: intuitive understanding, procedural
understanding, logico-physical abstraction. And the understanding of the emerging
mathematical concept also involves three levels of understanding: procedural
understanding, logico-mathematical abstraction, and formalization.
UNDERSTANDING OF PRELIMINARY PHYSICAL CONCEPT
Intuitive
understanding
Logico-physical
procedural understanding
Logico-physical
abstraction
Logico-math’l
Procedural understanding
Logico-math’l
abstraction
Formalization
UNDERSTANDING OF EMERGING MATHEMATICAL CONCEPT
Figure 2.3: Herscovics’ analytical framework (Herscovics, 1989b).
Based on a constructivist perspective, Pirie and Kieren (1994) developed a theory
that explains mathematical understanding “as a whole, dynamic, leveled, but non-linear,
transcendently recursive process” (Pirie & Kieren, 1994, p. 166). This theory, as shown
17
below in the figure, consists of eight embedded circles that represent different levels of
understanding or the growth of understanding for a specific person in any specific topic.
These potential levels are as follows: primitive knowing, image making, image having,
property noticing, formalizing, observing, structuring, and inventising.
1. Primitive Knowing
8
7
2. Image Making
6
5
4
3. Image Having
3
2
1
4. Property Noticing
5. Formalising
6. Observing
7. Structuring
8. Inventising
Figure 2.4: Pirie-Kieren model of growth of mathematical understanding
(Pirie & Kieren, 1994).
The researchers described the procedure of one’s understanding with a feature of
folding back. The folding back process occurs as follows:
When faced with a problem or question at any level, which is not immediately
solvable, one needs to fold back to an inner level in order to extend one’s current,
inadequate understanding. This returned-to, inner level activity, however, is not
identical to the original inner level actions; it is now informed and shaped by
outer level interests and understandings. Continuing with our metaphor of folding,
we can say that one now has a ‘thicker’ understanding at the returned-to level.
This inner level action is part of recursive reconstruction of knowledge, necessary
to further build outer level knowing. Different students will move in different
ways and at different speeds through the levels, folding back again and again to
enable them to build broader, but also more sophisticated or deeper understanding
(p. 173).
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Tall and Vinner (1981) proposed a framework for concept image and definition.
They formulated a distinction between the mathematical concepts as formally defined
and the cognitive processes by which learners make individual meaning of the concepts.
By the concept image, the researchers refer to “total cognitive structure that is associated
with the concept, which includes all the mental pictures and associated properties and
processes” (p. 152). The cognitive structure that holds these images may be at variance
with the actual definition of the concept, a form of words used by the mathematical
community at large to specify that concept. When evoked simultaneously, the potentially
conflicting images can turn into cognitive conflict factors, at which time the learner either
rectifies the cause of the conflict, or the conflict manifests itself by a sense of unease.
The image that is at variance with the formal definition, the researchers claim, can
seriously impede the learning, “…for they [potential conflicts] cannot become actual
cognitive conflict factors unless the formal concept definition develops a concept image
which can then yield a conflict (p. 154). Learners in such cases can become secure with
their own images and “simply regard the formal theory as inoperative or superfluous” (p.
154).
Later in a particular concept, Vinner (1983) presented a simple model, shown
below in Figure 2.5, to explain cognitive processes that arise between the definition and
image of a function. By the phrases concept image and concept definition, Vinner refers
to “the set of all pictures that have been associated with concept in person’s mind,” and
“a verbal definition that accurately explains the concept in a non-circular way” (p. 293)
respectively. This model assumes the existence of two different cells in the cognitive
structure: one is for concept definition and the other is for concept image. As shown in
the figure, when a cognitive task is given, the concept image and the concept definition
cells are supposed to be activated. It is argued that either concept image or definition
might be empty at the beginning, but it is gradually filled after several examples and
explanations in the process of the learning of the function concept. According to Vinner
(1983), discrepancies between one’s concept definition and concept image lead to
obstacles in a learner’s mind. Later in another study, Vinner and Dreyfus (1989)
described the existence of “two different potentially conflicting schemes in [one’s]
cognitive structure (p. 357)” as the phenomenon of compartmentalization.
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An intellectual behavior (an answer)
Output
Concept definition
Concept image
Input
A cognitive task (identification or construction)
Figure 2.5: Vinner’s framework of concept image and concept definition
(Vinner, 1983).
In summary, the theoretical frameworks such as Skemp (1978), Sfard (1991), and
Hiebert and Lefevre (1986) looked at the understanding of mathematics with dual
approaches: instrumental versus relational, structural versus operational, or procedural
versus conceptual respectively. While Haylock (1982) emphasizes the cognitive
connections in understanding, Sierpinska (1990) sees understanding as cognitive actions
to overcome obstacles. In addition, the hierarchical framework developed by Herscovics
(1989b) can possibly provide a framework for discussing the kind of understanding that
has taken place. Since the Pirie-Kieren theory (1994) maps the growth of one’s
mathematical understanding over time, it does not serve the purpose of this study.
Therefore, I decided to use an integrated approach using Schoenfeld’s (1989) level of
mathematical analysis and structure and Tall and Vinner’s (1981) framework of concept
image and concept definition.
While Tall and Vinner’s (1981) framework makes it possible for me to connect
students’ understanding with their activated mental images, Schoenfeld’s (1989) level of
mathematical analysis and structure provides a method to investigate students’
mathematical understandings and the obstacles that may arise in these processes both
from the standard curricular perspective and a student’s perspective. Within the
20
cognitive literature, Schoenfeld’s dynamic, leveled, but non-linear framework seems to
be a good fit to describe students’ understanding of mathematics as they work on
mathematical tasks as well as to discus why students could or could not make the
necessary connections in their understanding of a particular concept. Thus, students’ data
are analyzed through the lens of these two frameworks: Tall and Vinner’s (1981)
framework of concept image and concept definition, and Schoenfeld’s (1989) level of
mathematical analysis and structure.
Cognitive Psychology and the Piagetian Theory
In the 1960’s and 70’s, as a result of a philosophical shift from behaviorism to
various forms of structuralism and cognitivism, research in cognitive psychology focused
on concept formation, complex problem-solving, and connection between cognitive
structure and behavior (Noddings, 1990). Cognitive processes in mathematical thinking
had an impact with the notion of metacognition (learner’s knowledge about his or her
own cognitive processes) in the 1970’s and articles about think aloud problem-solving
protocols written by Ericson and Simon in the early 1980’s (Schoenfeld, 1992). The aim
of cognitive psychology was to understand the fundamental processes of thinking and
learning that operate on different tasks (Chaiklin, 1989). The Piagetian constructivist
perspective was one of major forces behind this psychological shift. The Piagetian theory
is based on an assumption that “individuals do not perceive the world directly, but that
they perceive interpretations of it mediated by the interpretive frameworks they have
developed” (Schoenfeld, 1992, p. 346). On the basis of Piaget’s work, cognitivism
became known as constructivism (Noddings, 1990).
Two important elements of Piaget’s theory are assimilation and accommodation.
While assimilation refers to the integration of the things to be known into some existing
cognitive structure, accommodation refers to changes in one’s cognitive structure
necessitated by the acquisition of new knowledge (Hersovics, 1989a). In this view,
learning is considered as a dynamic equilibrium process including both assimilation and
accommodation, claiming that “cognitive change and learning in a specific direction take
place when a scheme, instead of producing the expected result, leads to perturbation, and
perturbation, in return, to an accommodation that maintains or re-establishes equilibrium”
(von Gleserfeld, 1995, p. 68). In other words, acquiring new knowledge necessitates a
21
series of transition and reconstruction. The purpose of these actions is to re-establish
cognitive equilibrium. In this process of accommodating new knowledge, however,
sometimes a disequilibrium or conflict may happen. Then this leads to learning
difficulties and inadequate understanding.
Research on students’ understanding of mathematics began focusing on
identification of these conflicts or obstacles. In order to discover these obstacles,
cognitive studies attempted to understand the processes of thinking and learning that
operate mathematical tasks in a specific content, such as algebra (Chaiklin, 1989). Some
of these studies were conducted on concept images and definitions in the learning of
functions (Vinner, 1983; Vinner & Dreyfus, 1989, Adams, 1997; Clement, 2001), the
concept of limit and continuity (Tall & Vinner, 1981; Sierpinska, 1987, Davis & Vinner,
1986), and the concept of quadratic function (Zaslavsky, 1997; Zazkis et al., 2003;
Borgen & Manu, 2002; Sajka, 2003). The following section provides a detailed overview
of research studies done in the area of functions in general, and of quadratic function in
particular.
Students’ Understanding of and Difficulties with Functions
In terms of understanding of the concept ‘function,’ previous research studies
have looked at various aspects of this concept. These aspects include graphing ( Dunham
& Osborne, 1991; Clement, 1989), definitions and images of function as well as the
relationship between them (Vinner, 1983; Even, 1993; Lloyd & Wilson, 1998; Vinner &
Dreyfus, 1989; Adams, 1997; Clement, 2001), translation of the word problems into
rules or equations (Wollman, 1983), representations of functions and relationship
between them (Even, 1998; Zaslavsky, 1997; Matkovits et al., 1986; Dreyfus &
Eisenberg, 1983), teachers’ teaching emphasis (Lloyd & Wilson, 1998; Wilson, 1994),
and use of technology (Hollar & Norwood, 1999; O’Callaghan, 1998; Quesada &
Maxwell, 1994; Ruthven, 1990).
In a quasi-experimental study, Vinner (1983) studied the understanding of a
function by applying a model for cognitive processes constructed using the notions of
concept image and concept definition. Concept image is described as “the set of all the
mental pictures (diagram, graph, symbolic form, etc.) associated with the concept in a
person’s mind” (p. 293), whereas concept definition is “a verbal definition that accurately
22
explains the concept in a non-circular way” (p. 293). A questionnaire including five
open-ended questions were administered to the total of 146 students in grades 10 and 11
in two high schools in Israel. Then, common aspects of students’ concept images of
functions were identified. These images include: (a) a function should be given by a
single rule, (b) the graph of a function should be continuous, (c) a function should be oneto-one, (d) a function must include some algebraic formula, (e) the graph of a function
should pass the vertical-line test, and (f) a function should be systematic--an arbitrary
correspondence is not considered as a function. In addition, students’ concept definitions
of functions were classified into four main categories: (1) the textbook definition
sometimes mixed with elements from the concept image cell, (2) the function is a rule of
correspondence, (3) the function is an algebraic term, a formula, an equation, an
arithmetical manipulation, and (4) some elements in the mental picture are taken as a
definition for the concepts.
Results of the study indicated that students have the idea that only patterned
graphs represent functions; others look strange, artificial, or unnatural. Constant
functions, functions given more than one rule, or functions consisting of an arbitrary
correspondence are not recognized as functions. Additionally, the work of Vinner
showed that cognitive conflicts might happen when students’ concept image and
definition were not consistent with each other. And this might lead to obstacles to
students’ performance when doing mathematics, such as incorrect or incomplete
responses to problems.
Another study conducted by Matkovits et al. (1986) investigated how students
understand the following components of the function concept: (a) classifying relations
into functions and non-functions, (b) identifying pre-image, image, and (preimage,
image) pairs for a given function, (c) identifying identical functions and transferring from
one representation to another, and (d) identifying functions satisfying some given
constraints. Fourteen problems related to the graphical and algebraic representation of
numerical functions were given to 400, students aged 14 and 15, in grade 9. Results of
the study showed that students have a restricted view of what graphs of functions should
look like. Constant function, piecewise function, and a function represented by a discrete
set of points are not viewed as functions. Also, students required that the elements of the
23
two sets be in one to one correspondence and examples of functions be linear. This is
consistent with the result of the study of Vinner (1983).
In addition, the researchers observed that transfer from graphical to algebraic
form is more difficult than vice-versa. Both in the algebraic and the graphical form, the
concept and representation of images and pre-images are only partially understood.
Similar research conducted by Vinner & Dreyfus (1989) looked at the concept
images held by 271 college students who were majoring in mathematics, physics
chemistry, biology, economics, agriculture, technological education, or industrial design,
and 36 junior high school teachers for the concept of a mathematical function. The
researchers compared the definitions that teachers and students gave for the concept of a
function. They used a questionnaire designed to exhibit the cognitive schemes for the
function concept. Then, they compared these schemes with the definition. Overall,
results of the study showed that many of the definitions and concept images are primitive
among all but the mathematics majors and teachers. In particular, students identified
only graphs that exhibit an obvious or straightforward pattern as graphs of functions. In
the study, students looked for a variety of patterns other than linear but still demanded
some form of ‘reasonableness’ such as symmetry, persistence, always increasing or
always decreasing. These results are consistent with the research of Markovits et al.
(1986) and Vinner (1983).
In a simple quantitative study, Dreyfus & Eisenberg (1983) attempted to
determine if college students are attracted to linearity, smoothness and periodicity. To do
this, a questionnaire booklet including thirty-four questions was given to eighty-four
college students enrolled in the course named introduction to mathematics in Israel. The
percentages of students’ responses to the problems given in graphical and algebraic forms
showed that students seemed to feel more comfortable with graphical exercises than with
algebraic ones. Like the previous three studies mentioned above, the students’ strong
attraction to linearity and smoothness was observed. In addition, the idea that only a
linear function can contain two points was very strong. The researchers explained that
such a case might result from the statement given to students that through two points in
the plane there exists one and only one straight line. Completely ignoring the possibility
24
of thought processes occurring in students’ minds, the researcher made strong
conclusions at the end of the study.
Dreyfus and Eisenberg (1982) also studied students’ intuitions on functional
concepts in three categories: representations of functions, sub-concepts of functions, and
levels of abstraction and generalization. The subjects were all junior high school students
in grades 7 to 9. Three questionnaire booklets consisting of multiple-choice questions
were distributed in 24 classes in 12 different schools. The intuitions of the students on
functional concepts were analyzed by means of a four-way ANOVA. The hypotheses
addressed in the study were decided based on statistical results. This was a typical
quantitative study that examines students’ mathematical intuitions on the basis of a
behaviorist approach.
The study of Tall & Bakar (1992) investigated students’ images of the concept of
function. The subjects were a total of 137 high school and first year university students
in the United Kingdom. Nine graph sketches were shown and asked whether they could
represent functions. Students responded by marking a box in each case and provided a
little explanation when they said no. Looking at the percentages of students’ responses
“yes” or “no”, the conclusions were drawn. In the study, students’ thoughts or ideas were
almost not taken into consideration other than one or two word explanations. Therefore,
the study was too weak to support its results in terms of data sources.
Students’ difficulties with the concept of function were also studied by Adams
(1997), who applied graphing calculators and a model of conceptual change at a
community college. The research sample consisted of 128 students in six treatment
classes and two control classes. For the treatment classes in the study, the researcher
created the environment based on the constructivist assumptions and let students engage
in this environment. This environment helped students facilitate cognitive
accommodations. It is assumed that learning is an interaction between previous and
existing knowledge with the outcome depending on the interaction. She concluded that
the students’ definition of function was dominated by the ordered pair representation and
the vertical line representation. Consequently, students viewed functions as collections
of points or ordered pairs.
25
In another study on the concept of functions, students’ flexibility in moving from
one representation to another and other aspects of knowledge and understanding were
studied by Even (1998). She collected data from 152 college mathematics students who
completed an open-ended questionnaire. After responding to the questionnaire, ten of
them were interviewed. She concluded that students deal with functions point-wise; for
example, they can plot and read points, but cannot think of a function as it behaves over
intervals or in a global way. Some of the findings of this study indicated that subjects
who can use a global analysis of changes in the graphic representation have a better and
more powerful understanding of the relationships between graphic and symbolic
representations than people who prefer to check some local and specific characteristics.
However, she explained that this does not necessarily mean that the use of a global view
of function leads to a better understanding of the meaning of graphs and functions in
general. In terms of a point-wise approach, results of Even’s study are supported by
Adams (1997).
In recent years, another study done by Clement (2001) aimed to explore a concept
image that is aligned with the mathematical definition. In a public high school known for
its academic excellence, she looked at thirty-five precalculus students’ responses to the
assessment items that were selected from previous research studies. Then five of them
were interviewed. She found that what ‘function’ means for students is an image of a
graph that passes the vertical line test. Also students tend to recognize functions most
often when the functions are familiar to them. That is, prototypes of functions are a
greater factor in determining functionality than even the vertical-line test.
In summary, the research studies reviewed in this section revealed that students
have a variety of misconceptions and difficulties in the learning of functions. These are
as follows: (a) students’ strong tendency towards linearity and one-to-one correspondence
(b) students’ pointwise focus to functions, (c) students’ demand for some kind of patterns
on the graphs of functions (increase, decrease, or symmetry), (d) students’ unwilling to
consider some graphs as graphs of functions (constant or piecewise functions), and (e)
difficulty with transformation from graphical form to algebraic form.
Another point needed to be addressed in this review is related to methods of data
collection. In terms of understanding of functions, some studies looked at the issue in a
26
simple way by requiring students to answer multiple choice or yes/no questions (Dreyfus
& Eisenberg, 1982; Dreyfus & Eisenberg, 1983; Tall & Baker, 1991). Others used
questionnaires consisting of open-ended questions (Vinner, 1983; Matkovits et al., 1986;
Vinner & Dreyfus, 1989), focusing mainly on students’ written responses. The recent
two studies only employed an interview process to figure out students’ thought processes
(Adams, 1997; Clement, 2001). This suggests that students’ thinking processes become a
major factor for learning. As a result, recently students’ learning of the specific subject
matter has been analyzed and approached more qualitatively, emphasizing student
conceptions, conceptual knowledge, and metacognitive abilities.
Students’ Understanding of and Difficulties with Quadratic Functions
The concept of function has been investigated by researchers for many years in
terms of students’ difficulties. Most of the research studies have particularly focused on
the formal definitions and images for the concept of function as well as linear functions
as a special family. The only study that specifically focuses on students’ obstacles in the
learning of quadratic functions has been done by Zaslavsky in 1997. In the study, to
reveal students’ conceptual obstacles, their conceptual understanding was examined in
light of both learning experiences and underlying mathematical features. By conceptual
obstacles, Zaslavsky refers to “obstacles that have a cognitive nature and that can be
explained in terms of the mathematical structures and concepts that underlie students’
earlier learning experiences” (p. 20). This quantitative study involved more than 800
students in grades 10th and 11th in 25 mathematics classes from 8 high schools in Israel.
Data were obtained from classroom observations, and examinations of students’
notebooks and written responses to two different sets of non-standard problems. Joint
consideration of the results of the wrong answers’ frequencies and the explanations that
were given to them led to the identification of the conceptual obstacles.
Results of the study showed that students treated the graph of quadratic function
as a picture rather than as a symbolic representation. For a given parabola that does not
show the y-intercept on the visible part of the graph, students inferred that such a point
does not exist. Also when asked to determine the number of the points on the graph, if a
given point was too distant from the graph, they made their decisions as if there was no
such point. The second obstacle has been identified in terms of the relation between a
27
quadratic function and a quadratic equation. Students treated a quadratic function as if it
were a quadratic equation. For example, students treated the function y = x2 + 2x -3 as a
representative of the function y = 2x2 + 4x -6, just as the corresponding equations x2 + 2x
-3 = 0 and 2x2 + 4x -6 = 0 respectively. Another obstacle is related to the analogy
between a linear function and a quadratic function. When asked to find the equation of a
quadratic function on the basis of three points on the graph, students first used two points
to calculate the slope between those points, and then inserted this value into the equation
as the leading coefficient.
The fourth obstacle appeared when one of the parameters in a quadratic function
is zero. Students rejected this sort of equation as an equation of quadratic functions. In
fact, it belongs to the quadratic family as a special case. The last obstacle was reported
with respect to overemphasis on only one coordinate of special points. For example,
when asked to decide if two parabolas have the same vertex, students made their
decisions based on the x-coordinate of the vertex, without considering the y-coordinate of
the vertex. Furthermore, students were observed to prefer translating form equations to
graphs on matching tasks involving parabolas. This finding is supported by Markovits et
al. (1986), who found translations from graphs to equations to be more difficult than the
reverse.
Zaslavsky’s study is limited in several ways. The first is related to the method of
data collection. The study used two different data sources to support its results: students’
written responses to non-standard problems and students’ notebooks. This means that the
study focused mainly on what students wrote down and completely ignored what students
thought. However, understanding requires not only close examination of students’
written work, but also involves a careful analysis of the students’ thought processes on
given mathematical tasks (Pirie & Kieren, 1994). Studying conceptual obstacles requires
that the researcher analyze students’ cognitive processes in terms of how they are
thinking, what assumptions they are making, and what knowledge they are applying.
This could have been done by means of one-to-one interviews with students.
The second problem for the study is the research instrument and the time it was
administered. The instrument consisted of non-standard problems and was administered
to students six months after the completion of the chapter. The work of Selden, Mason,
28
& Selden (1989) showed that not a single average college student solved a nonroutine
problem correctly. A nonroutine problem in the study refers to a problem that “the solver
does not begin knowing a method of solution” (p. 45), as opposed to an exercise.
Moreover, it is a possibility that throughout the six months period, students might have
difficulty remembering the issues being studied or they might have been affected by other
topics being studied between the time that the questionnaire was administered and the
time that the chapter was completed.
In recent years, a few studies have been conducted on a single quadratic task to
figure out student difficulties (Borgen & Manu, 2002; Zazkis et al., 2003; Sajka, 2003).
In terms of a single quadratic task, Zazkis et al. (2003) studied a horizontal translation of
a function, focusing on the example of the parabola y = (x-3)2 and its relationship to y=
x2. The researchers particularly attempted to investigate the difficulty presented by a
horizontal translation for both teachers and students. Participants in the study were 15
preservice secondary teachers, 16 practicing secondary teachers, and 10 students in
grades 11 and 12. All of them were volunteers joining this study and classified as having
‘above-average’ ability. In a clinical interview setting, all participants were asked to
predict, check, and explain the relationship between the graph of y = x2 and the graph of
y= (x-3)2. Data obtained from interviews were analyzed in terms of common trends in
explanations, attitudes toward perceived inconsistencies, and differences between the
groups of participants.
Findings of the study showed that students who did not sketch the graph of y= (x3) 2 correctly showed an instrumental understanding (Skemp, 1978). They talked about
memorized rules like “to do the opposite” or “move opposite to the sign of the number,”
rather than explaining the mathematical reason behind the graphical movement. On the
other hand, all teachers completed the task correctly and provided a variety of
explanations. Additionally, the researchers found no significant differences between
practicing and preservice teachers. It is interesting to note that a majority of teachers (18
out of 31) also mentioned a specific rule in their explanations, like students. Based on the
fact that such a big size study involved 41 participants and provided no information about
how long each interview takes, it could be argued that the Zazkis et al.’s (2003) study had
29
a more superficial approach to the thinking processes of the participants rather than indepth analyses.
Another study was conducted by Borgen and Manu (2002) to investigate students’
understanding of a calculus problem. The researchers asked two senior math students to
determine the stationary point of the quadratic function, y = 2x2 – x +1 and then decide if
it has a minimum or maximum. Two students worked together on the same problem and
discussed their ideas with each other as they were videotaped. The whole discussion on
the question lasted for about 6.5 minutes. Data obtained from students’ actions and
statements in the video and their written work were the basis for the analysis of this
study. The researcher employed Schoenfeld’s (1989) four levels of analysis and Pirie and
Kieren’s (1994) model to analyze the students’ mathematical understanding. Borgen and
Manu have acknowledged that even though they were looking at a particular situation
with a single question, they used the Pirie-Kieren theory, which is used for the growth of
student understanding in a particular time period.
The authors concluded that Janet, who took the initiative to solve the problem,
indicated a lack of understanding in terms of making connections with related concepts,
based on Schoenfeld’s levels of analysis. Additionally, the analysis by the Pirie-Kieren
theory showed that her inability to fold back to proper images created a disconnected
understanding and made it impossible for her to connect her different areas of
understanding. The visual image of a parabola opening upward was not enough evidence
for Janet to decide that the function had a minimum. Therefore, Borgen and Manu,
pointed out that her improper images at the formalized level led to obstacles that even
physical evidence could not overcome. Besides a very short discussion (6.5 min.)
between two students, analysis of this videotaped study was relying mainly on students’
written work. During the process, the researcher did not participate in the discussion by
posing questions or challenging the student’s thoughts. This approach prevented the
researcher from figuring out all the ways of why and how students are thinking and
limited the study’s results.
The work of Sajka (2003) aimed to investigate an average student’s understanding
of a functional equation. In an interview setting, the researcher asked a 16-year-old
student a non-standard problem. The task required the student to give an example of a
30
function f such that for any real numbers x, y in the domain the following equation holds:
f(x +y) = f(x) + f (y). The interview lasted 42 minutes and was recorded. Data were
analyzed by using the procept theory formulated by Gray and Tall (1994). The procept
theory consists of three components: “a process that produces a mathematical object, and
a symbol that represents either the process or the object” (Gray and Tall, 1994, p. 121).
Sajka acknowledged that the student had difficulty understanding the problem at the
beginning; and after a long conversation with the researcher, the student understood the
task content. In order to help the student understand the task, the researcher provided the
following two specific quadratic functions: f (x) = x2 - 2x +3 and f (x) = x2 +5. Then, the
interview turned out to be a study on a quadratic task.
The analysis revealed that the student had a very limited procept of function and a
misinterpretation of the symbols used in the functional equation. More specifically, the
author identified the students’ difficulties in three categories: (a) the intrinsic ambiguities
in the mathematical notation, (b) the restricted context in which some symbols occur in
teaching and a limited choice of mathematical tasks at school, and (c) the students’
idiosyncratic interpretation of mathematical tasks. Consistent with the result of the work
of Selden et al. (1989), the student was not able to understand the non-standard problem
for a long time. With the help of the researcher, the interview was completed. Since the
only data source is used in a limited manner, the study is too weak to support its results.
In summary, the reviewed studies with respect to students’ understanding of and
difficulties with quadratic functions revealed some learning problems in four categories:
(a) quadratic functions represented by graphs are taken as a picture, (b) quadratic
functions are perceived as quadratic equations, (c) difficulty of translation from graphical
form to algebraic form, and (d) over emphasizing on one coordinate of special points. As
opposed to Zaslavsky (1997), who used a quantitative approach in his study, some recent
studies (Borgen & Manu, 2002; Zazkis et al., 2003; Sajka, 2003) approached more
qualitatively by employing students’ interviews on a given single quadratic task.
However, the interviews techniques in these studies are problematic. Ginsburg (1981)
suggested using one-on-one clinical interviews to discover and identify students’
cognitive processes, conceptions and obstacles. He provided a procedure for research in
mathematical thinking (Ginsburg, 1981) and a general strategy for posing questions
31
(Ginsburg et al., 1998). In a study, unlike Ginsburg’s model, two students worked
together and discussed a task with each other (Borgen & Manu, 2002); an average student
worked a non-standard problem and had much difficulty understanding the content of the
problem in the study of Sajka (2003); and forty-one participants were interviewed in a
superficial way in the work of Zazkis et al. (2003). The present study, in light of
Ginsburg’s recommendations, uses one-to-one-in-depth interviews to discover and
identify students’ cognitive processes in the learning of quadratic functions.
32
CHAPTER III
METHODOLOGY
This chapter describes the research design, research site, participants, tasks used
for the study, methods of data collection and analysis, and trustworthiness of the study.
The methodology used in this qualitative research is a multiple case study involving two
algebra honor students. In general, qualitative research is defined by Creswell (1998) as
“an inquiry process of understanding based on distinct methodological traditions of
inquiry that explore a social or human problem” (p. 15) and described in detail by Denzin
& Lincoln (1994) as “multimethod in focus, involving an interpretive, naturalistic
approach to its subject matter…researchers study things in their natural settings,
attempting make sense of, or interpret, phenomena in terms of the meanings people bring
to them” (p. 2). The present study uses the method of qualitative research to reveal
cognitive obstacles that algebra honor students face when they explore concepts of
quadratic functions.
A Multiple Case Study
The methodology employed in the present study is a multiple case study involving
two algebra honor students in the Florida State University School [FSU-School]. A
multiple case study involves collecting and analyzing data from several cases (Merriam,
1998). In terms of number of cases in a study, Merriam (1998) stated that “the more
cases included in a study, and the greater the variation across the cases, the more
compelling an interpretation is likely to be” (p. 40). Compared with a single case study,
Herriott & Firestone (1983) claimed that the evidence from multiple cases is often
considered more compelling, and the overall study is therefore regarded as being more
robust.
33
Research Site
(a) School and Classroom
The FSU-School is a state-rated ‘A’ elementary, middle and high school in North
Florida (Florida Department of Education, 2003). It is a laboratory school that provides
research and development opportunities for educators. With approximately 1600
students, it is designed to represent the statewide population in Florida. An algebra-2honor class in the FSU-School was chosen based on consultation with the school’s
research director and the classroom teacher.
This classroom, as shown in the figure below, includes twelve pairs of student
desks arranged in a square. Each desk is shared by two students. There are a total of 20
students including 10 boys and 10 girls in the class. The Smart Board connected to the
computer for the teacher’s use is in the corner, and the overhead and projector are in the
front of the whiteboard so the teacher can use the transparencies she prepared for the
class. Class met five times a week in the morning at 8:00 am and each class lasted 50
minutes.
Smart
Board
Overhead &
Projector
Door
Door
Whiteboard
Door
Teacher’s table & Computer
Figure 3.1: Structure of the Algebra-2-Honor Class
34
(b) Teacher and Teaching Style
The class was taught by a teacher who has a total of 20 years teaching experience;
last seven years in this school teaching algebra-2 and algebra-2-honors. She has National
Board Certification and likes using technology in her classes.
A typical lesson starts out with homework checks by calling on students for
answers. Before collecting homework assignments, the teacher picks up one or two
problems that students struggled with the most and then solves them on the board. After
that, she introduces the new topic, and solves a couple of examples by either using the
overhead or using the whiteboard. Then, she gives students worksheets including
questions related to the new topic and the students work on them for a while. She then
assigns homework for the next class. In addition, at the end of each class, the teacher
spends some time discussing what has been done on that day’s class by making
connections with the next day’s homework assignments. Besides the regular
assignments, each week the teacher posts five to ten bonus-problems in her website. But,
this opportunity is rarely used by students. After completion of the chapter in the first
three weeks, the teacher reviewed all topics by using graphing calculators in the fourth
week, while the researcher completed the interviews with students. So, students’
understandings of the concepts and the obstacles that arose in these processes were not
affected by the use of graphing calculators.
(c) Course Chosen for the Study
The course, Algebra-2-Honors, is the continuation of the study of algebra for
students with above average skills in mathematics. However, it does not review those
skills taught in Algebra I. Students in this course had an 86-100 average in Algebra-IHonors or an average of 94-100 in Algebra I.
The chapter chosen for the research covers the Quadratic functions in Algebra-2
at FSU-School in Fall 2004. The course Algebra-2-Honors uses the textbook Florida
Prentice Hall Mathematics: Algebra 2 by Bellman, Bragg, Charles, Handlin, & Kennedy
(2004). The book covers the following chapters: (1) Tools of Algebra, (2) Functions,
Equations, Graphs, (3) Linear Systems, (4) Matrices, (5) Quadratic Equations and
Functions, (6) Polynomials and Polynomial Functions, (7) Radical Functions and
Rational Exponents, (8) Exponential and Logarithmic Functions, (9) Rational Functions,
35
(10) Quadratic Relations, (11) Sequences and Series, (12) Probability and Statistics, (13)
Periodic Functions and Trigonometry, and (14) Trigonometric Identities and Equations.
The chapter being studied is taught in the following order (p. xiii):
5.1: Modeling Data with Quadratic Functions
5.2: Properties of Parabolas
5.3: Translating Parabolas
5.4: Factoring Quadratic Expressions
5.5: Quadratic Equations
5.6: Completing the Square
5.7: The Quadratic Formula
5.8: Complex Numbers
The teacher approximately spent 100 minutes or two class-times on teaching each
subsection in the chapter of quadratic function. In the first three weeks, she completed
the concepts from 5.1: Modeling data with quadratic functions to 5.7: The quadratic
formula. In the last week, besides teaching the concept of complex number, she reviewed
the previous quadratic topics by using graphing calculators.
The teacher started with the chapter by giving some quadratic relations in the real
world such as pouring water, tossing an object into the air, or the path of the football in
the game. Following this discussion, she introduced students to the standard form, y =
ax2 +bx +c, of the quadratic function, and asked them to classify the functions in the
textbook as a quadratic, linear or constant. Later, she described relationships with
quadratic functions and graphs on a specific example. The axis of symmetry, vertex of a
parabola, and the maximum/ minimum value of the function were identified and
discussed, both algebraically and graphically. In this process, some students asked
questions such as “what is the difference between the vertex and the axis of symmetry,”
or “how can we find the axis of symmetry if we know the vertex?” In addition, the
teacher showed how to find a quadratic model with three given points. On the board, she
substituted the values of x and y into the standard form, y = ax2+bx+c, and solved the
system. She also explained that they would learn how to find that quadratic model easily
with graphing calculators in the final week. In another meeting, she explained properties
of the parabolas (e.g., a> 0, a <0, the x-coordinate of the vertex, or the y-intercept) and
36
demonstrated step by step how to graph quadratic functions by re-emphasizing the vertex,
axis of the symmetry, opening up/down, y-intercept, and finding smart points from both a
graphical and algebraic perspective. In the process, some students asked questions such
as “what happens if one of the coefficients, let’s say b, is missing,” or “what happens if
the function starts with 5x2 instead of x2 ?” In addition, substantial amount of time was
spent to demonstrate how to solve the problems (e.g., a rock club’s profit and landscape
design) represented in real world-situations in that week.
In the second week, the teacher introduced students to the vertex form, y = a (x h) 2 + k, of quadratic function and analyzed the properties of a parabola given in the
vertex from. For example, the vertex was (h, k); when h>0 and h <0, the graph shifted
right and left respectively; and when k>0 and k<0, the graph shifted up and down
respectively. Later, based on exercises in the textbook, she asked students to say the
vertices of the functions. A whole-class discussion was a primary teaching method in
that class. In another meeting, an important amount of time was spent to write the
equations of the given parabolas in vertex form. Certain questions from the textbook
were answered step by step on the board in an active discussion. In a new meeting, the
teacher introduced the topic of factoring quadratics and showed how to find common and
binomial factors of quadratic expressions as well as factoring special expressions (e.g., a
perfect square and difference of two squares). Later, she distributed worksheets to the
whole class and let students work on them. In another meeting, the section on solving
quadratic equations by factoring and finding square roots was studied by using the
method of direct teaching.
The third week started with the concept of the completing square. On the board,
the teacher demonstrated the process of finding the last term of a perfect square trinomial.
Later, students worked individually on their worksheets. In another meeting, she derived
the quadratic formula from the general form of a quadratic equation, ax2+bx+c = 0 on the
whiteboard, and then showed how to use it in order to solve quadratic equations. The rest
of that class, students were allowed to work on their worksheets. To review the previous
topics, similar to the chapter-test, a worksheet was given students to work with. The
following day, students’ questions related to the worksheet was answered by the teacher
in the class.
37
With regard to assessments, daily students assignments and weekly bonus
problems were posted in the teacher’ website and shown on the smart board everyday in
the class. At the beginning and end of the each class, the teacher reminded students of
homework assigned for the next class. Other than the first introductory class, homework
assignments were initially discussed and then collected by the teacher in each class
throughout the study. Particularly, questions that students struggled with the most were
answered on the board. In addition, at the end of the second week, a project assignment
was distributed to students who wanted to take an extra credit.
(d) Time Line
The study was conducted during the last 4 weeks out of the second 9 weeks in the
Fall semester, 2004. There was one quiz (Appendix G) and a chapter test (Appendix H)
during the study. While the quiz was given in the second week, the chapter-test and
questionnaire were administered at the end of the third week. All interviews were
conducted in the fourth week of the study, while the teacher was reviewing the previous
quadratic topics by using graphing calculators. So, the only time the teacher used
graphing calculators in the teaching of quadratic functions was in the final week of the
study. The section on complex numbers was also studied in that week.
Participants
The participants were two high school students, called under the pseudonyms of
Richard and Colin, in an algebra-2-honor class at FSU-School. During a four-week
investigation, all students in the class completed the study of quadratic functions. After
that, a purposeful sampling (Patton, 1990) strategy was used to select the interviewed
participants. Purposeful sampling was based on the assumption that “the investigator
wants to discover, understand, and gain insight and therefore must select a sample from
which the most can be learned” (Merriam, 1998, p. 61). LeCompte and Preissle (1993)
pointed out that criterion-based sampling, which is the same as what Patton (1990) calls
purposeful sampling, requires that “ the researcher establish in advance a set of criteria or
a list of attributes that the units for study must possess” (p. 69). In other words, for
purposeful sampling, it is essential for the investigator to set up some standards or bases
so that he or she can choose participants who match those criteria. Therefore, in order to
gain an in-depth understanding of students’ thinking processes, two students were
38
selected for interviews (Appendix F) on the basis of the following four criteria: (1)
examinations of students’ test, quiz, and assignments, (2) classroom observations, (3)
examinations of students’ responses to a questionnaire (Appendix E) administered before
the interviews, and (4) their personalities were open, talkative and comfortable with the
researcher.
Both students, Richard and Colin, fitting to the criteria above, voluntarily
participated and got paid $10 for each interview. I preferred not to choose more than two
students for this research because this allowed me to be able to conduct an in-depth
investigation for each case and make extensive comparisons between the cases.
According to Merriam (1998), findings from different participants help the researcher to
get more conceptually dense and potentially more useful knowledge.
Research Tasks Used for the Study
(a) Questionnaire Tasks
In light of classroom observations and students’ written work (a quiz, a test, and
homework), a questionnaire included 5 open-ended tasks (Appendix E) were designed to
reveal the possible difficulties related to the following five main aspects needed for deep
understanding of quadratic functions: (a) interpreting, (b) determining, (c) translating, (d)
solving quadratic equations, and (e) using quadratic models. In order to answer the tasks,
participants had to effectively utilize their graphical and algebraic thinking and
continuously move back and forth between the graphical and algebraic aspects of the
concepts.
Task 1(a) required students to consider the coefficients, a, b, and c in the standard
form, y = ax2 + bx + c, or the coefficients, a, h, and k in the vertex form, y = a (x - h) 2+k,
of quadratic function in terms of how each coefficient functioned in the graph of a
parabola. Task 1(b) was related to the axis of the symmetry. It required participants to
read the information from the graph and then related that information with the aspect of
the axis of the symmetry to find the point. Tasks 1(c) and 1(d) required students to
determine if certain points were on the graph of the parabola. These tasks could be
answered by substituting the points into the quadratic function after finding the algebraic
representation for the given parabola. Plotting points on the graph of the parabola may
39
not provide a definite answer because the graph might not necessarily represent exact
quantitative information.
Task 2 asked participants to graph the given quadratic function and determine the
axis of symmetry and max/min point. In order to find the axis of symmetry, participants
can use either an algebraic or graphical approach. In an algebraic approach, they can use
the formula of x = – b/2a without graphing the function; or in a graphical approach, the
line that passes through the max/min point of the parabola indicates the axis of symmetry.
The following task, Task 3, required participants to write an equation for the given
parabola. This type of task could be approached either by substituting the vertex into the
vertex form, y = a (x - h) 2 + k, of quadratic function, and then checking another point to
find the coefficient, a, or by solving a system with substitution or elimination after
substituting each of three points into the standard form, y = ax2+bx+c, of quadratic
function.
Task 4 required students to solve the given quadratic equation. This can be
solved by using three different strategies: the method of factoring, completing the square,
or using the quadratic formula. The next task, Task 5, asked students to use the quadratic
model to solve the problem given in a real-world situation. This could be solved by using
the quadratic formula after deciding the distance in the problem.
(b) Interviews Tasks
The interview tasks (Appendix F) consisted of a variety of open-ended questions
related to the following six main aspects needed for deep understanding of quadratic
functions: (a) concept image and definition, (b) translating, (c) determining, (d)
interpreting, (e) solving quadratic equations, and (f) using quadratic models. These six
aspects included the whole knowledge components that were supposed to be acquired by
the students in the chapter of quadratic functions.
These tasks were adapted or developed based on analyses of students’ written
responses to the test, quiz, and questionnaire as well as field notes obtained from
classroom observations. Five of the tasks in the interviews were adapted from previous
studies in the literature. The remaining tasks were developed in light of the analyses of
prior interviews. In this process, the following considerations were taken into account:
(a) the textbook language and format was used in the designed and adapted tasks in order
40
to avoid students’ misunderstanding of tasks, (b) for new tasks, the students’ textbook
and workbook were used as main sources, (c) no non-routine task or problem was
administered to students, and (d) interviews were conducted after completion of the
chapter to be able to answer the question of whether the students possessed the
knowledge to answer the tasks.
Methods of Data Collection
Three different methods were used to collect data for this study: clinical
interviews, classroom observations, and documental data including students’ test and
quiz, as well as a questionnaire administered before the interviews. More specifically,
the following data collection procedures were employed in the study. In the first step, in
order to construct a questionnaire (Appendix E) and develop interview questions
(Appendix F) at the end of the chapter, detailed field notes were taken by the researcher
on a daily basis during the study of quadratic functions in the classroom. This process
also provided an opportunity for the investigator to learn the language spoken in the
classroom and develop an atmosphere of trust with students. In addition to field notes, a
test, a quiz and assignments that were given to students in regular class meetings were
collected and analyzed to reveal potential difficulties for the following step. Then, based
on classroom observations and examinations of students’ documents obtained from the
prior step, a questionnaire was constructed and administered to all students in the
classroom. The questionnaire also included tasks from the literature review with regard
to students’ difficulties about quadratic functions.
In the final step, two students were selected to be interviewed based on the
analysis of prior classroom observations and their written responses to the questionnaire
in the previous step. In the selection process, aspects of the students’ personalities such
as being open, talkative and comfortable were also considered. Interviews with each
student continued until the researcher was convinced that the reason for the students’
errors, mistakes or incomplete answers was cognitively guided, and not a simple or
random error.
(a) Clinical Interviews
The main data source in this study was the one-on-one clinical interview as
originally developed by Piaget. In Piaget’s words, it is:
41
“…much more interesting to try to find the reasons for the failures. Thus, I
engaged my subjects conversations patterned after psychiatric questioning, with
the aim of discovering something about the reasoning process underlying their
right, especially wrong answers” (cited in Ginsburg, 1997, p.30).
Merriam (1998) described interviewing as the best technique to use when
conducting intensive case studies of a few selected students. The process of the clinical
interview was described by Ginsburg (1997), saying that, “the examiner begins with
some common questions but, in reaction to what the child says, modifies the original
queries, asks follow-up questions, challenges the child’s response, and asks how the child
solved various problems and what was meant by a particular statement or response” ( p.
2). One advantage of using this method of interviewing, as Ginsburg et al. (1998) stated,
“it can allow you to enter the child’s mind—that is, to gain some understanding of
children’s constructions of mathematics” (p. 16). The National Council of Teachers of
Mathematics (NCTM, 1989) also encouraged teachers to conduct flexible interviews to
assess and improve student learning of mathematics in the classroom.
In order to discover or identify cognitive processes of students’ thought in
mathematics, Ginsburg (1981) suggested using the clinical interview method as an
instrument. The author provided the following procedures to discover or identify
cognitive activities (structures, processes, or thought patterns):
Table 3.1: Clinical interview procedures for research in mathematical thinking
(Ginsburg, 1981)
Procedures for Discovering Cognitive
Process
Procedures for Identifying Cognitive
Process
1. Begin with an open-ended
mathematical task
1. Facilitate rich verbalization which may
shed light on underlying process
2. Ask further questions in a contingent
manner
2. Check student answers and clarify
ambiguous statements
3. Request a good deal of reflection on
the problem solving steps taken and the
reasons for them
3. Test alternative hypotheses concerning
underlying process
42
For this study, a semistructured interview technique was employed. In this
technique, tasks and questions are planned in advance, but the researcher follows up on
the student’s responses in a flexible way (Ginsburg et al., 1998). Therefore, a list of tasks
and questions was prepared before the interview process. Then they were administered to
each participant during the interviews. Tasks and questions were seeking specific
cognitive structures in the learning of quadratic functions. The tasks were clearly written
and their solutions were explored by the researcher prior to interviews. Students’
textbook, workbook and previous research literature were primary sources in the process
of constructing interview tasks and questions. These tasks and questions both engaged
students in thinking and encouraged them to describe tasks as fully as possible.
Depending on the student’s response to questions, I asked a series of more focused and
direct questions to gain deep insights into student’s thinking.
In order to uncover students’ thinking on a given mathematical task, Ginsburg et
al. (1998) suggested posing the following types of questions during the process of
interviewing:
Table 3.2: Fundamental questions posed during the process of interviewing
(Ginsburg et al., 1998)
Exploration of Students’
Observing and Challenging
Justification of students’
Ideas
of Students’ Ideas
Ideas
How did you figure it out?
What would happen if…?
How could you prove
Explain how did you do it?
If you wanted to show …what
that?
Tell me how you did that?
would you need?
Is there another way you
How else could you figure that
Why did you not use…?
can check…?
out?
Does that always work? Why
How would you show it?
How do you know …?
or why not?
Why do you think that is
How did you decide…?
How does this relate to…?
true or correct?
What do you mean?
Which way do you like better?
How could you make sure
Do you see a pattern? What is
Why?
you are really right about
it? Explain.
Which method is better,…why
that?
do you think so?
43
In the present study, all interviews with students took place in the teacherplanning room, and they were audiotaped and transcribed. After that, the transcripts were
reviewed in terms of what was said, what weaknesses or gaps in their knowledge were
revealed, and which responses to a particular task should be questioned further.
(b) Classroom Observations
Descriptive field notes from each class were taken by the researcher during the
study of quadratic functions. During the observations, particular attention was given to
interactions between student-student and student-teacher in terms of questions posed,
responses given, materials and activities used to teach, tactical approaches to instruction,
and the teaching-learning sequence.
After each day of observation, in light of the descriptive field notes and informal
discussions with the students and teacher inside and outside the classroom, I wrote down
my ideas and impressions about what was going on in the class. This particularly
included comments on interactions among students and teacher and raised questions
about the process of meaning-making.
(c) Documental Data
Documental data consisted of students’ test, quiz, and questionnaire given to the
whole class before the interviews. During the study of quadratic functions, regular
classroom instruction also led to documents in the form of students’ homework
assignments and worksheets. These documents were copied and analyzed in terms of
students’ incorrect and incomplete answers to questions. They were then used as one of
the main sources in the construction of the questionnaire tasks for the next step.
Data Analysis
Data gathered from interviews, classroom observations, and students’ written
work (a test, quiz, questionnaire and homework assignments) were the basis for the
analysis of this study. Analysis and collection of the data were an ongoing and
simultaneous process throughout the study. Since this was a multiple case study, the
within-case and cross-case analyses were employed to analyze data. For the within-case
analysis, each case first was treated as a complete case itself. As shown in the figure
below, the researcher moved among the following four streams: data collection, data
reduction, data display, and conclusion drawing/verification. This process allows the
44
researcher to condense more and more data into a more and more coherent understanding
of what, how, and why (Miles & Huberman, 1994).
Data
Collection
Data
Display
Data
Reduction
Conclusions:
Drawing/verifying
Figure 3.2: Components of Data Analysis: Interactive Model
(Miles & Huberman, 1994)
After the analysis of each case was completed, the cross-case analysis began. The
aim of the cross-case analysis is “to build a general explanation that fits each of the
individual cases, even though the cases will vary in their details” (Yin, 1994, p. 112). All
information for each student was brought together in a case study data base (Yin, 1994):
interview transcripts, field notes, and documental data including students’ questionnaire,
quiz, test, worksheet, and assignments. Yin (1994) argued that the case study database
increases the reliability of the entire study because it helps other researchers to review the
evidence directly and not be limited to the written reports. Since tentative major
categories and themes were identified for each student during the study, all these themes
were further revised and synthesized within and across the data sources. In general, a
major focus in the whole process of category construction was students’ obstacles
consisting of incorrect or incomplete answers in their documental data and interview
transcripts, as well as questions and responses about the process of meaning-making in
classroom observations. Particular attention was given to students’ statements about their
45
strategies, justifications, and interpretations on given mathematical tasks in the
interviews.
In the final phase of the analysis, all categories and themes for each student were
compared with each other in terms of whether they have something in common. This
process helps the researcher to answer the question of “do these findings make sense
beyond this specific case?”, and more importantly, “to develop more sophisticated
descriptions and more powerful explanations” (Miles & Huberman, 1994, p. 172-173).
Trustworthiness of the Study
Data do not speak for themselves; there is always an interpreter, or a translator.
One cannot observe or measure a phenomenon or event without changing it, even
in physic where reality is no longer considered to be single-faceted. Numbers,
equations, and words are all abstract, symbolic representations of reality, but not
reality itself ( Merriam, 1998, p. 202).
According to Lincoln and Guba (1985), “reality is a multiple set of mental
constructions…made by humans; their construction are in their minds, and, they are, in
the main, accessible to the humans who make them” (p. 295). Therefore, through
observations and interviews, the researcher could access interpretation of the reality,
which is closer to the reality (Merriam, 1998).
Creswell (1998) asked the following question, “How do we know that the
qualitative study is believable, accurate, and ‘right’? (p. 193). In other words, an
essential requirement for qualitative research is to convince the reader with respect to the
validity, reliability, and objectivity of the study or in the Lincoln & Guba’s terms (1985),
credibility, transferability, dependability and confirmability of the study. Creswell
(1998) presented eight verification procedures to increase trustworthiness of the study.
They are as follows: (1) prolonged engagement and persistent observation, (2)
triangulation, (3) peer review or debriefing, (4) negative case analysis, (5) clarifying
researcher bias, (6) member checks, (7) rich, thick description, and (8) external audits.
The author suggested that researchers be engaged in at least two of them in any
qualitative study. The following five verification procedures were involved in the present
study.
46
The first is prolonged engagement and persistent observation in the field. This
includes “building trust with participants, learning the culture, and checking for
misinformation that stems from distortions introduced by the researcher or informants”
(Creswell, 1998, p. 201). Lincoln and Guba (1985) suggested researchers be involved
with the site long enough to detect and take account of distortions and not to be “a
stranger in a strange land” (p. 302). Thus, throughout the study of quadratic functions,
observations were done on a daily basis in the research. The second is a thick and rich
description of the study in terms of participants and settings. This allows readers “to
transfer information to other settings and to determine whether the findings can be
transferred because of shared characteristics (Creswell, 1998, p. 203). The third is
triangulation. Researchers in this procedure use “multiple and different sources,
methods, investigators and theories to provide collaborating evidence (Creswell, 1998, p.
202). Thus, the present qualitative study obtained multiple evidence from three different
sources to explain students’ obstacles involved in learning quadratic functions: classroom
observations, clinical interviews, and documental data. The fourth is a member check.
This involves “taking data, analyses, interpretations, and conclusions back to the
participants so that they can judge the accuracy and credibility of the account” (Creswell,
1998, p. 203). Therefore, in this study, interview transcripts were presented to students to
look for their agreement on the content of the transcripts. They were asked, “is there any
content or section in the transcripts that they want to revise, change or delete?” Then,
these agreed upon transcripts became the basis of the analyses in this research study. The
last procedure is a peer review (or debriefing). This involves “asking colleagues to
comment on the findings as they emerge” (Merriam, 1988, p.169). With this procedure,
“the inquirer’s biases are probed, meanings explored, the basis for interpretations
clarified” (Lincoln & Guba, 1985, p. 308). So, three of my colleagues in the department
of mathematics education participated at this part of the study. They were PhD students
who had both teaching experience and a great deal of knowledge about mathematical
research and methodology. The researcher’s analyses and interpretations for each case
were presented to them to get their suggestions and considerations on emerging findings
by means of previously scheduled meetings. This process provided a good opportunity to
test emerging assumptions derived from the data by pushing me to reconsider my
47
position. In other words, this really helped me to avoid or minimize the possibility of
misinterpreting the students’ real intentions.
48
CHAPTER IV
DATA ANALYSES AND FINDINGS
Since the purpose of this research study was to identify algebra honor students’
cognitive obstacles surrounding quadratic functions, this chapter involves an examination
of how students construct their own mathematical knowledge in the learning of quadratic
functions and an identification of what obstacles arise in these processes. Description of
findings for each case starts with an examination of each student’s (1) concept image and
definition for the quadratic function (Tall & Vinner, 1981). Then, it is followed by five
other aspects of quadratic functions adapted by Wilson (1994): (2) translating among
multiple representations of quadratic functions, (3) determining the axis of the symmetry,
the vertex and x-and y-intercepts on the quadratic functions, (4) interpreting quadratic
functions represented by graphs, formulas, tables and situation descriptions, (5) solving
quadratic equations by factoring, completing the square, and using quadratic formula, and
(6) using quadratic models to solve problems presented in real-world situations. Even
though these six aspects of quadratic functions are identified as separate areas of
investigation, all of them are interconnected in the learning of quadratic functions.
In the present study, two students from an Algebra-II-Honors class in the FSUSchool were interviewed as they attempted to answer particular quadratic tasks. To
protect students’ real identification, they were pseudonymously called Richard and Colin.
Each student was interviewed twice and each interview lasted for about 50 minutes. The
following section starts with Richard, who is followed by Colin. Each section consists of
an introduction of the student, the analyses of the student’s audiotaped interviews and
written work, and summary. In the analysis part, as recommended by Merriam (1988),
interview sessions were narrated in terms of what has been done and said in their natural
sequence of their occurrence. Then, the patterns discovered in the data were highlighted.
Lastly, interpretive comments on students’ responses were made by using (or
49
highlighting) direct quotes from transcripts and written statements from the documents as
supporting evidence.
THE CASE OF RICHARD
Introduction
Richard was an African-American, 15 year-old, tenth-grade male student who has
previously taken algebra-1 and geometry. During the study, he seemed to be curious,
talkative and confident with his mathematical knowledge. When asked about his favorite
subjects, he said: “my favorite subject is math and Spanish [sic], because I like speaking
another language, in math like, always I want to be a computer analyst, or a web
designer, I know that they have lots of math in it.” He attended all classes and actively
participated in classes by asking questions and responding to the teacher’s questions, but
did not turn in any of the homework assigned by the teacher. In general, he individually
worked on the questions and had a very limited contact with the student sitting next to
him in the class.
He likes mathematics, claming that “I like math because it only has one answer, is
not more than one answer. I like figuring out stuff, it’s like a puzzle, or put stuff
together.” He describes mathematics “like computers, it’s like…everything’s based on,
everything goes down to like…anything you have to do technologically…all has to go
down numbers.” He believes that “if you are good at figuring out the numbers, you can
basically do anything in the world pretty much.” So, mathematics is quite important to
him. He also likes playing basketball and has been learning martial arts. During the
study, he was very comfortable working and communicating with the researcher and
showed no hesitation to speak his mind both in and outside the class.
In terms of his performance on written instruments before the interviews, Richard
scored 95 out of 100 in the Quiz-A (Appendix G) administered in the second week of the
study (the average score = 80). Other than a few calculation errors, he correctly
responded to the all questions. He found the vertices from the quadratic functions given
in either the standard form or the vertex form, identified smart points to graph the
parabolas besides the vertex, solved the manufacturing problem, and found the x-values
after factoring out the quadratic equations. In the Chapter-Test-A (Appendix H),
however, he made a couple of mistakes and scored 90 out of 100 (the average score =
50
88). He could not correctly find the vertex from the quadratic function, y = 3x2 -9x. The
vertex’s location and the opening direction of the parabola were not correct at the figure
in his test. In addition, at one particular situation (the question #6), after writing the
quadratic equation for the problem, he left the equation that way without finding the xvalues at the end. He also left two questions blank in his test. After completion of the
test, however, Richard and many other students in the class complained about not having
enough time to answer the whole questions. Thus, the limited time had an effect on
students’ responses to the questions in the given test.
Data Analysis
In the following section, Richard’s thought processes on particular quadratic tasks
are analyzed with respect to his incorrect or incomplete responses, and then their possible
causes are speculated in light of evidence obtained from the student’s data.
1. Concept Image and Definition for the Quadratic Function
By demonstrating a distinction between the formal definition that a student holds
for a given concept and the way that he or she thinks about the concept, Vinner (1983)
concluded that revealing the concept images of students gives us a better understanding
of students’ “knowing what caused them to act as they acted” (p. 297). Therefore,
before starting any particular task, I wanted to know about students’ images and
definitions of quadratic functions at the opening of each student interview. These
images are also used to make connections to students’ responses to the tasks throughout
the chapter.
When I asked Richard, “If you put in your sentences, what a quadratic function
is,” his answer was: “Immediate thing, I think quadratic function is …formula like
negative b minus and plus square root b square minus four a c over two a [(–b ± √b2 4ac) / 2a], like you use a parabola, in that stuff…we’ve currently learned.” Then, I asked
him to give an example of a graph of a quadratic function, he provided the following
graph shown in Figure 4.1 and explained while working on the graph:
Quadratic function would be like…you have like, a starting point, I’ve almost
started at the origin, then you have a maximum point appears the vertex [pointed
to the maximum on the parabola], and show like, progress over time [pointed to
the second half of the parabola]
51
Figure 4.1: Richard’s evoked image for the quadratic function
When the concept of quadratic function is first heard, it generated two different
mental images in Richard’s mind; one is a ‘quadratic formula’; the other is a ‘parabola.’
He seems to be connecting the concept to the mental images that he holds about quadratic
function, but these images are not coherently related to the formal definition given in the
textbook, in which “a quadratic function is a function that can be written in the standard
form f (x) = ax2 +bx +c, where a ≠ 0” (Bellman, Bragg, Charles, Handlin, & Kennedy,
2004, p. 234). At this point, it could be argued that the student uses these two mental
images as a definition for the concept of quadratic function in light of the conclusion
made by Vinner (1983), who reported that “some elements in the mental picture are taken
as a definition for the concepts” (p. 300).
In addition, the graph in Figure 4.1 showed that his evoked concept image, “the
portion of the concept image which is activated at a particular time” (Tall & Vinner,
1981, p. 152), related to quadratic function contained a downward parabola and some
associated properties such as “a starting point,” “a maximum point,” “a vertex,” or
“progress over time [symmetry].” Thus, his evoked image of quadratic function is
consistent with the student’s own definition which is a ‘parabola.’ It appears that, besides
the parabola, the student develops another concept image, a quadratic formula [(–b ± √b2
-4ac) / 2a], in his mind. However, this image is not activated in the cognitive structure
when his concept image is evoked. Since the student’s observations of his experience are
quite selective (Vinner, 1983), it seems that the cognitive process that formed the concept
52
image shown in Figure1.1 ignored other part of the student’s experience which is a
‘quadratic formula’, perhaps because a quadratic formula does not have a graphical
aspect.
In connection with this issue, Zaslavsky (1997) found that the relation between
quadratic functions and quadratic equations impedes students’ understanding of quadratic
functions. In her study, students conceived that the quadratic functions of y = x2 +2x -3
and y = 2x2 +4x -6 were the same as their corresponding equations of x2 +2x -3 = 0 and
2x2 +4x -6 = 0 respectively. In fact, the equations, x2 +2x -3 = 0 and 2x2 +4x -6 = 0, are
equivalent; however, their corresponding functions, y = x2 +2x -3 and y = 2x2 +4x -6, are
different for any given value of x, except for the x-coordinates of their intersection points.
Therefore, in an effort to be able to figure out how the student constructs mathematical
meaning for the concepts of quadratic function and quadratic equation, I asked Richard to
write down any example for a quadratic function and quadratic equation respectively. He
provided the following two examples: “y = ax2 +bx +c” and “y = x2 +2x +1” and then
explained:
I am not completely sure wording like, differences between the formula and like
the function. I mean, yes you can use that too, like any time you have a quadratic
formula, like you have to use…I am trying to…as far as my understanding like…
function is just like y equals whatever x or is it x equals negative b plus and
minus [x= -b ±], all that stuff, I can’t tell the difference.
When particularly asked what he was thinking about the difference between a quadratic
function and a quadratic equation, he said:
Usually I think function is like f of x equals whatever, like whatever f, x’s, y, so
when I think the function, I think y equals ax square plus bx plus c [y=
ax2+bx+c]…here, I am like an equation, formula like x equals negative b plus and
minus [x = –b ± ] that’s what I usually think. Sometime people use it differently.
First, both of the examples provided by the student represent a quadratic function.
While “y = ax2 +bx +c” is the standard form of quadratic function, “y = x2 +2x +1” is a
specific quadratic function, not a quadratic equation. “A quadratic equation is ax2 +bx +c
= 0, where a ≠ 0” (Bellman, Bragg, Charles, Handlin, & Kennedy, 2004, p. 263). The
claims—“differences between the formula and like the function” and “I am like an
equation, formula like x equals negative b plus and minus”—indicate that he seemed to
53
be interchangeably using the terms of quadratic equation and formula (Vinner, 1983).
He was also not sure if there was a difference between them, claiming that, “I can’t tell
the difference” and “sometimes people use it differently.”
In summary, his initial mental images included a quadratic formula and a
parabola. And then, he again described his understanding of the quadratic function as a
formula explained as “y equals whatever x’s” and then a quadratic formula described as
“x equals negative b plus and minus, all that stuff.” So, his image of quadratic formula
[x = (–b ± √b2 -4ac) / 2a] was activated again in the cognitive structure along with y = ax2
+bx +c. On the follow-up question, he again claimed that “The function, I think y equals
ax square plus bx plus c [y= ax2+bx+c]” and then, “formula like x equals negative b plus
and minus [x = –b ±] that’s what I usually think.” So, it appears that there is an ongoing
interaction between two images of y = ax2+bx+c (standard form) and x = (–b ± √b2 -4ac)
/ 2a (quadratic formula) in the student’s mind. When the concept is heard, it has been
observed that each time, it yields an image of quadratic formula in his cognitive structure.
So, when these two images are evoked simultaneously in the student’s mind, it seems to
cause an actual cognitive conflict, called a cognitive conflict factor by Tall and Vinner
(1981), for him for the concept of quadratic function. In other words, his concept image
of quadratic formula, which is not coherently related to the concept definition of
quadratic function, seems to cause a conflict in the student’s cognitive structure. Later
on, when the student meets the concept represented in a broader context such as in
calculus, he may not be able to cope when confronted with such tasks or optimization
problems. In the present study, this conflict is named under the category of ‘the
quadratic formula as an image is a potential cognitive conflict.’ This could be a result of
the teaching method used by the teacher who gave examples by employing a quadratic
formula for a long period of time while working with the general notion of quadratic
functions. This is also supported by the study of Tall and Bakar (1992), who reported
that teaching the concept through examples leads to mental prototypes which give
erroneous impressions of the general idea of the concept. In other words, the student has
been exposed to quadratic formula far more than to the standard form during the learning
of quadratic functions.
54
2. Translating among Multiple Representations of Quadratic Functions
This aspect of quadratic functions is analyzed under the two subsections: (a)
translating a quadratic function from graph to algebraic form, and (b) a horizontal and
vertical shift of a quadratic function.
(a) Translating a Quadratic Function from Graph to Algebraic Form
The questionnaire given prior to the interview showed that 46% of the students, 7
out of 15, were not able to find an algebraic representation for the given parabola.
Richard was one of them. The task in the questionnaire required him to write an equation
for the parabola shown below in Figure 4.2.
y
1
x
(-1,0)
2
(5,0)
(2,-3)
Figure 4.2: Translation task in the questionnaire
In order to translate the graph of the parabola to an algebraic form, he substituted
the vertex, (2, -3) into the standard form, y = ax2+bx+c, of quadratic function, as written
down: “-3 = A (4) + B (2) + C” and then, “don’t know where to start.” This type of task
could be approached either by substituting the vertex into the vertex form, y = a (x - h) 2 +
k, of quadratic function, and then checking another point to find the coefficient, a, or by
solving a system with substitution or elimination after substituting each of three points
into the standard form, y = ax2+bx+c, of quadratic function. In the given questionnaire,
however, the student only focused on the vertex, (2, 3) and used it with the standard form,
y = ax2+bx+c, of quadratic function rather than the vertex form, y = a (x - h) 2 + k.
55
To be able to figure out the student’s cognitive processes on this task and identify
if he has any obstacles to move from graphical to algebraic representation, the first
interview included the following task shown in Figure 4.3. The task includes a
downward parabola with the vertex in the first quadrant. I asked him to find an algebraic
representation for the given parabola.
y
(3, 6)
6
4
2
(5, 0)
0
(1, 0)
3
x
6
Figure 4.3: Translation task in the first interview out of two
At the beginning, Richard stayed silent for a while, and then he was asked about
what he was thinking, he said:
I am trying to think about where to start…uhmm…like you see three points
there…when I see like is the point, one point five and two point five on the
parabola… trying to think like, am I supposed to find an equation…or, like…
am I supposed to find y-intercept without the equation…I don’t know where to
start … usually like my mind…when we get the slope and a point, or just two
points, you can find the slope…since the parabola has infinite amount slope, I
don’t know where to start.
Here the student seemed to fold back to the previous experience or knowledge in
order to extend his current, inadequate understanding, claiming that “usually like my
mind…when we get the slope and a point, or just two points, you can find the slope.” It
is obvious that he was looking for a slope and point [the point-slope form] or two points
to find the slope. This suggests that he tries to generalize the idea of finding equations
for linear functions to the present situation. However, having no specific slope or having
56
more than two points on the parabola seems to be confusing him with regard to decide
how to start to answer the question. At this point, the student’s existence schema (or
conceptual structure) on linear functions and the internal processing of his experiences
appears to be creating an obstacle to write an algebraic representation for the given
parabola. Then, the interview continued as I asked the following question which
containing a hint to open up his ideas: “do you know the standard form or vertex form of
the quadratic function?” He answered my question by writing down, “a (h-x) + k” and
explained that “That is the vertex, vertex form.” Then, he substituted the vertex; 3 for x
and 6 for k and then got: “vertex = a (h-3) +6.” When asked about ‘a’ in the formula, he
said:
I can find out that is negative because parabola opens down. And I don’t know
like…what exactly coefficient be, I don’t know exactly what the number a is, I
just know that opens down, because it’s negative…so I am guessing that negative
quantity h minus three, plus six.
First the student tried to write down the vertex form, y = a (x - h) 2 + k, of
quadratic function, but made two errors on the formula, probably because of incorrect
recollection. He misplaced x and h in the formula and forgot the square over the
parenthesis. Then, he substituted the vertex into the quadratic function, y = ax2+bx+c,
but looked unsure about what he was doing at that time. In addition, although he
determined the sign of the coefficient, a, he was not able to find the value of that
coefficient. Afterward, the student wanted to give it a try again and claimed that:
I am trying to figure out like how to get this equation from three points… okay
let’s see if I do the vertex, that would be six equals… a times nine, nine a, plus
three b, plus c, equals six…[wrote down y = ax2+bx+c, then got: 6 = 9a+3b+3],
that’s what I got to stop before when you gave us the questionnaire, I don’t know
what to do.
As mentioned in the excerpt, he performed the same procedure just as his
questionnaire: substituting the vertex, (h, k) into the standard form, y = ax2+bx+c, of
quadratic function. It is important to notice that he only used the vertex to substitute into
the standard form of quadratic function, even though he claimed that he wanted to get the
equation “from three points.” At this point, it seems that he does not know how to use
three given points with the standard form, y = ax2+bx+c, to write a quadratic function.
57
Two days later in the second interview, in order to find out if there is any pattern
to his constructs for this type of translation, I presented him with the graph of a quadratic
function shown in Figure 4.4, which is similar to the previous task, except that it opens
upward. I asked him to find an algebraic representation for the given parabola.
y
1
(3,0)
(-1,0)
2
x
-2
(1,-6)
Figure 4.4: Translation task in the second interview
At this time, Richard substituted the vertex, (1,-6) into the vertex form, y = a (x h) 2 + k, of quadratic function, but his equation of vertex form did not include the
coefficient, a, in front of the parenthesis and he wrote: “y = (x -1)2 -6.” He then
transformed the vertex form into the standard form of “y = x2 -2x -5” and claimed that
“should be a here, so equation y equals ax square minus two x minus five.” He placed ‘a’
in front of x2 and wrote: “y = a x2 -2x -5.” Finally, he substituted the point (3, 0) into the
function and got: “a = 11/9.” By replacing ‘a’ into the equation, he wrote the whole
equation as “y = 11/9 x2-2x-5.” When asked to prove that this corresponds to the given
graph, he substituted the point (-1, 0) into the function and got: “0 = -16/9.” Then he
said, “I think that is incorrect!” and then explained that “This is wrong equation… uuhm,
wrong equation...because zero is not equal ...I mean it’s just something that we haven’t
learned yet, but like parabola I don’t know.” When asked about what he was thinking
58
about where the problem was, he stated, “I can’t figure out what I do wrong...I am sure
probably something like very simple, I just messed up.” In this attempt, the student
correctly substituted the vertex, (1,-6) into the vertex form, y = a (x-h) 2 + k, of quadratic
function. Then, he transformed the vertex form into the standard form, y = ax2+bx+c.
By doing so, he took away the coefficient, a, from the vertex form, and then put it back in
front of x2 in the standard form. Actually, he should have distributed ‘a’ inside the
parenthesis in the vertex form. This may suggest that he perceives the coefficient, a, as if
it is independent of the whole formula in the vertex form. In other words, probably he
assumes that the coefficient, a, in the vertex form can be taken away and put back in as
the leading coefficient in the standard form after the transformation has been completed.
So, at this point, it seems that this assumption led him to fail to be able to find an
algebraic representation for the given parabola.
In all attempts, he consistently picked up the vertex first and substituted that into
either the standard, y = ax2+bx+c or vertex form, y = a (x-h) 2 + k. He chose the standard
form twice and the vertex form once with the vertex. But, when he picked up the vertex
form, he could not correctly write the quadratic function, and then proceeded to work on
the standard form. In his last attempt, after picking up the vertex form, he immediately
transformed it into the standard form of y = ax2 +bx +c. When he realized that his answer
was not correct, he said, “It’s just something that we haven’t learned yet… I don’t know.”
This may suggest that he was not confident about what he was doing at that time. All
these attempts suggest that he has a clear tendency towards using the standard form, y =
ax2 +bx +c over the vertex form, y = a (x-h) 2 + k. In fact, each form of quadratic
function provides different graphical information related to the location of the parabola.
While the standard form of y = ax2 +bx +c directly indicates the location of the yintercept, (0, c), the vertex form of y = a (x -h) 2 + k directly indicates the location of the
vertex, (h, k). However, the tasks presented in Figure 4.3 and Figure 4.4 do not provide
information about what exactly the y-intercepts are, but the vertices. So, using the
vertices, (h, k), with the vertex form, y = a (x -h) 2 + k, could have been a more effective
and simpler approach to find an equation of the quadratic function. This reminded me of
the study of Vinner and Dreyfus (1989), who reported that “sometimes, a given situation
does not stimulate the scheme that is the most relevant to the situation; instead, a less
59
relevant scheme is activated (p. 357)” when the researchers explained an irrelevant
situation as an indication of the compartmentalization phenomenon in their study. At
this point, however, more evidence is needed to make such a conjecture. Thus,
subsequent tasks will have to confirm this supposition and to determine the possible
reason that leads to the problem.
On the other hand, all his attempts also indicated that he had a tendency towards
using the standard form, y = ax2 +bx +c. This preference might have been affected by
Richard’s image of quadratic function. We already knew that none of his images
included the vertex form of quadratic function as an example. On the contrary, he has a
strong image of the standard form, y = ax2 +bx +c. So, since learners’ images play an
important role in their actions (Vinner, 1983; Vinner & Dreyfus, 1989), this might have
caused an obstacle by letting Richard pick up the vertex, (h, k) with the standard form, y
= ax2 +bx +c, instead of the vertex form, y = a (x-h) 2 +k. Also, his actions on translating
the vertex form into the standard form and placing the coefficient, a, just in front of x2 in
the standard form after taking away it from the vertex form is consistent with this
supposition being made.
In summary, all attempts showed that Richard was confused about when and how
to use the vertex, (h, k) with the vertex form, y = a (x-h) 2 +k, of quadratic function in
translating from graphical to algebraic representation. With these tasks, he mostly
attempted to use the vertex (h, k) with the standard form, y = ax2 +bx +c, although using
the vertex form, y = a (x-h) 2 +k, could have been a more effective and simpler approach.
It seems that the confusion or a lack of clarity about when and how to use the vertex, (h,
k) with the vertex form, y = a (x-h) 2 +k, of quadratic function causes an obstacle in the
student’s mind. In the present study, this obstacle is named under the category of ‘a lack
of understanding about when and how to use the vertex, (h, k) with the vertex form, y = a
(x-h) 2 +k, of quadratic function.’ A well designed set of activities or examples that
emphasize about when and how to use the vertex with the vertex form, y = a (x - h) 2+k,
may help to strengthen this understanding.
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(b) A Horizontal and Vertical Shift of a Quadratic Function
The literature revealed that students really did not know the mathematical reason
behind a horizontal and vertical shift of a quadratic function (Zazkis et al., 2003).
Therefore, in order to reveal students’ thoughts on this issue, I asked Richard the
following task, as shown in Figure 4.5, adopted from the study of Zazkis et al. (2003). In
the task, the graph of y = (x-3) 2 is a horizontal shift of y = x2, while the graph of y = x2 -3
is a vertical shift of y = x2.
Task: Compare the graph of y = x2 relative to the graph of y = (x - 3)2.
Then, compare y = (x - 3)2 to the graph of y = x2 - 3
Figure 4.5: Task of horizontal and vertical shift in the first interview out of two
He first graphed y = x2. And then, while graphing y = (x -3) 2 shown below in
Figure 4.6, he explained that “first I want to write this down until like trinomial, so it
would be easier to look at…[wrote down x2 +9], I mean I just look at that, I think…the
vertex is like…three, zero [(3, 0)], then opens up because it’s positive.” He then added to
that, “this graph translates to the right, shifted to the right, like three, that one [pointed to
y = x2 ] just stays on the origin.” At this part, he first wanted to transform the y = (x -3) 2
into the standard form of y = ax2 +bx +c because he thought “it would be easier to look
at.” So, he squared y = (x -3)2 and got: “x2 +9.” And then, he changed his mind and
directly wrote the vertex, (3, 0) from the first form of the function, y = (x -3) 2. At the
end, he graphed the parabola of y = (x -3)2 as shifting to the right by 3 units on the xaxis. It is worth mentioning here that his first action on this task was to transform the
vertex form, y = a (x - h) 2+k, of the quadratic function into the standard form, y = ax2
+bx +c; evidence is compelling that he has a tendency to use the standard form over the
vertex form. Therefore, at this point, this is named under the category of ‘having a
tendency toward using the standard form over the vertex form’ in this study.
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Figure 4.6
Richard’s graph of y = (x -3) 2
Figure 4.7
Richard’s graph of y = x2 -3
Lastly, he graphed y = x2-3 shown in Figure 4.7, and claimed that “That one…x
square minus three…I think this is the same as this one [pointing to y = (x - 3)2], just
shifted down…because of y-intercept, where the vertex is zero, negative three [(0,-3)]…I
think, opens up because positive a…that’s what I think.” At this part, he directly wrote
the vertex, (0, -3) from the function and graphed y = x2-3 as shifting down by 3 units. He
then made a quick comparison with y = (x - 3)2, claiming that “this is the same as this
one…just shifted down.” Before probing further questions on the task, a fire alarm went
off in the building in which the interview took place, and the interview was not
completed at that time.
Two days later in the second interview, in an effort to specifically figure out why
the student thought that “graph translates to the right, shifted to the right” or “just shifted
down,” I presented him the following task shown in Figure 4.8, which is similar to the
first one, except that it is horizontally shifted to the left and vertically shifted upward.
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Task: Compare the graph of y = x2 relative to the graph of y= (x +2) 2.
Then, compare y = (x + 2) 2 to the graph of y = x2 +2
Figure 4.8: Task of horizontal and vertical shift in the second interview
After graphing y = x2, he drew y = (x +2) 2 in a different Cartesian coordinate
system, as shown in Figure 4.9. When asked to explain the procedure he employed, he
said:
When I found the vertex, I did the opposite, which is added to x term before
squared, opposite of two [2] which is negative two [-2], which is x value of the
vertex and y value is zero, since there is no adding like c for the intercept, so
the vertex negative two, zero[(-2,0)], vertex of parabola is shifted to the left on
the x-axis.
Figure 4.9
Richard’s graph of y = (x +2) 2
Figure 4.10
Richard’s graph of y = x2 +2
Then, when an explanation was requested about how he figured out the way the graph
moved, he claimed:
Because we just learned that, I mean, I don’t really remember the origin of how
that comes to be, but we learned that in the parenthesis and x-term has not squared
yet, you find the opposite of this number [pointing to the + 2], and then that
would be the x-value of the vertex, if you find that, you could just graph that
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point.
After that, the student explained while working on the graph of y = x2 +2, as shown in
Figure 4.10 above:
Positive two is y-value in the vertex, x is zero, so x is zero, y equals two, that
determines that the vertex because there is like no change in the x square, there is
no like x plus something in the parenthesis, x square plus two, like I don’t have to
shift to the left or right, I just have to find y-intercept and then, this opens up
because the coefficient of x square is positive.
I then asked him to compare the graphs of y = (x +2) 2 and y = x2 +2, he claimed:
Because in this equation [pointed to y = (x + 2) 2], x plus two squared, x
wasn’t squared yet, like after you square, it would be like x square plus four x
plus two …[pause]…okay, well in this [pointed to y = (x + 2) 2 ], we were
taught that we found the opposite, like x’ s in the parenthesis and another number,
but this [pointed to y = x2+2] x square already by itself its own term, two [2]
is not added to x, two [2] is added to x square [pointed to y = x2+2], that’s why
this one doesn’t shift to the left or right, shift up.
Lastly, when I posed the question, “do you know why this is happening,” he said, “not
exactly I mean, I know how to figure out if that happens, but I don’t know why.”
In summary, the claims—“I did the opposite, which is added to x term before
squared,” “we learned that in the parenthesis and x-term has not squared yet, you find the
opposite of this number,” and “we were taught that we found the opposite”—indicate that
he seems to be performing a memorized rule to find the vertex without necessarily
understanding in a coherent way. While comparing the vertical and horizontal shift, his
focus was just on the outside clues on the functions; for example, “there is like no change
in the x square,” “there is no like x plus something in the parenthesis,” “like x’s in the
parenthesis and another number,” or “x square already by itself its own term.” When an
explanation is requested on why the vertex of the parabola is shifted to the left, he said, “I
don’t really remember the origin of how that comes to be, but we learned that in the
parenthesis and x-term has not squared yet, you find the opposite of this number.” For
the vertical shift of y = x2 +2, he stated that “x square already by itself its own term, two
[2] is not added to x, two [2] is added to x square, that’s why this one doesn’t shift to the
left or right, shift up.” Here, the student presented a memorized and rule-bound
procedure for the horizontal and vertical shift of the quadratic function without knowing
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why they happened. On the task, he relied mainly on his memory for the cognitive work.
The claim, “I know how to figure out if that happens, but I don’t know why,” suggests
that he has an instrumental understanding, relying on memorized rules (Skemp, 1993). In
other words, Richard performed the procedure that took him to the correct answer, but he
could not make any cognitive connection about the way the graphs moved. Thus, he did
not reveal an understanding of why the parabola horizontally shifted on the x-axis or
vertically on the y-axis. Therefore, in the present study, this is named under the category
of ‘a lack of understanding of why a parabola horizontally or vertically shifts.’ In order
to provide more evidence on this supposition as well as to reveal a connection observed
in the class about the vertex form of quadratic function and an absolute value function, I
presented Richard the following task shown in Figure 4.11. The task asks him to
compare the graphs of two functions: one is a quadratic function given in the vertex form,
y = a (x - h) 2+k; the other is an absolute value function.
Task: How are the graphs of y = (x - 4)2 and y = | x - 4 | similar?
How are they different?
Figure 4.11: Comparison task between the vertex form of quadratic
function and an absolute value function
He immediately sketched the following two graphs shown in Figure 4.12 and
Figure 4.13. When an explanation was requested for y = (x - 4)2 in Figure 4.12, he
claimed:
Fist you find the vertex which is four, zero [(4, 0)], because opposite of negative
four [-4] and then y would be zero [0] because there is no number outside the
parenthesis. Once you find the vertex that opens, like you can find random points
like first then plug it in, get x, and then find the mirror point of that, which I did
here, I used x is two [2] and y is four [4] and found the mirror point of that.
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Figure 4.12
Richard’s graph of y = (x - 4) 2
Figure 4.13
Richard’s graph of y = | x - 4 |
Then, I asked him how to graph y = | x - 4 | in Figure 4.13, he explained his
procedure:
Absolute function, to find the vertex, it’s pretty much the same way as you find
the vertex for quadratics, except like instead of having parenthesis, absolute value,
like kind of defined some quantity which is added or subtracted from x, and to
find the opposite of that, which is adding four [4] and then value outside of the
absolute value or parenthesis, be y for the vertex, which is zero [0] here, so vertex
also four, zero [(4, 0)].
Lastly, when asked to compare these two graphs, he said:
When you compare, both have the same vertex, the same absolute value, like it
has two slopes, negative one [-1] slope and positive one [+1] slope, going off the
vertex [showed y = | x - 4 | ], this [pointed to y = (x - 4)2] has more than
one slope for each point, that is absolute value, but it causes a curve going up with
and has a minimum point.
He then claimed that “I see the same way when the equation is vertex form” when
particularly asked about his understanding of between the absolute value function and the
vertex form, y = a (x - h) 2+k, of quadratic function. Then, I posed the question: “do you
think this always works?” his response was: “yes, if there is vertex form and absolute
value. But I mean if it is not, y equals parenthesis, h minus k squared, when it is not that, I
have to either move or rearrange into that form for me to find the vertex easily, or do
other method.” After providing new functions of y = (x - 2)2 and y = | x - 2 |, I asked him
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to prove that this works on these examples. He said, “These two have the same vertex of
two, zero [(2, 0)] because outside value is zero [0], like adding zero [0] to those…[wrote
down y = (x - 2)2 +0 and y = | x - 2 | + 0]” and then explained his procedure while
working on the functions shown in Figure 4.14 and Figure 4.15:
Pretty much the same as the intercept, vertex for both would be two, zero [(2, 0)],
and after finding point using this equation, okay, I’ll do… x’ s one [1] and y is
one [1], and mirror point of that would be three, one [(3,1)]...so this parabola…
[graphed y = (x - 2) 2 in Figure 4.15], and absolute value, there is no coefficient
for the absolute value, so it would be positive one…[graphed y = | x – 2 | in
Figure 4.16], and so you have to, both positive and negative absolute value, go up,
one-on-one.
Figure 4.14
Richard’s graph of y = (x - 2) 2
Figure 4.15
Richard’s graph of y = | x - 2 |
In this part, he first graphed y = (x - 4)2 by finding the vertex, (4, 0), claiming that
“because opposite of negative four [-4] and then y would be zero [0] because there is no
number outside the parenthesis.” Here when he was presented with the task that requires
for a horizontal shift, he once again used the same rule-based strategy and outside clues
to find the vertex and then graph the function. Evidence is compelling that Richard’s
understanding of horizontal shift includes no cognitive connection rather than a
memorized rule. Second, while working on y = | x - 4 |, he said that “absolute function,
to find the vertex, it’s pretty much the same way as you find the vertex for quadratics,
except like instead of having parenthesis, absolute value.” So, it is clear that the student
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perceives these two tasks the same way in terms of finding the vertex from the function.
To him, there is no difference between the two “except like instead of having parenthesis,
absolute value.” For y = | x - 4 |, the claim, “some quantity which is added or subtracted
from x, and to find the opposite of that, which is adding four [4] and then value outside of
the absolute value or parenthesis, be y for the vertex, which is zero here” clearly shows
that the student’s approach to find the vertex of an absolute value function is similar to
his approach to find the vertex of a quadratic function: a memorized, rule-based
procedure. This connection between two functions is also acknowledged by the student,
saying that “I see the same way when the equation is vertex form.” According to him,
this is always true “if there is vertex form and absolute value.” If he did not have a vertex
form, he claimed that “I have to either move or rearrange into that form for me to find the
vertex easily, or do other method.”
In summary, the student’s actions and explanations on this task presented a
connection between the vertex form of a quadratic function and an absolute value
function. While working on these two different functions, he applied the same specific
procedure to find the vertices and graph the functions. Applying this automatic
procedure enabled him to get the correct answer without analyzing properties of the
mathematical concept. In other words, the mathematical concept is not thought about and
manipulated through its properties, but rather, the task is answered by using a specific
procedure or algorithm. This could be a result of students’ intensive practice on the same
type of questions with a specific procedure. A well designed set of non-routine examples
or activities that do not allow students to follow an automatic procedure may encourage
them to use different strategies as well as help them make cognitive connections between
the vertex form of quadratic function and an absolute value function.
3. Determining the Axis of Symmetry, Vertex, and x-and y-Intercepts of Quadratic
Functions
The questionnaire given prior to the interview indicated that 60 % of the students,
9 out of 15, made mistakes on graphing and determining tasks. Therefore, in order to
examine Richard’s cognitive processes on determining tasks such as the axis of
symmetry, the vertex, and x-and-y intercepts, the following task shown in Figure 4.16
was adopted from the study of Zaslavsky (1997). The task asks the student to compare
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two parabolas given in the standard form. With this task, the student tries to determine
whether two quadratic functions have the same axis of symmetry, the vertex, and x-and
y-intercepts.
Task: The equations of two parabolas are given by
y = a x2 + b x + 1
y = a x2 + b x + 4
Answer the following questions about these two parabolas.
a) Do the two parabolas have the same axis of symmetry?
b) Do the two parabolas have the same vertex?
c) Do the two parabolas have the same x-intercepts?
d) Do the two parabolas have the same y- intercept?
Figure 4.16: Determining task in the first interview
This task could be answered by using either a graphical or algebraic method.
From the graphical approach, the second parabola is similar to the first one, except that it
is just vertically shifted by 3 units. Therefore, these two graphs do not have any common
points except for the axis of symmetry. From the algebraic approach, the axis of
symmetry for both parabolas is the line of x = - b/2a, so they are the same. Since the
vertex of both parabolas is the point (-b/2a, f (-b/2a)), and the value of f (-b/2a), which is
the y-coordinate of the vertex, in the second parabola will always be 3 greater than the
first one [e.g., (-b/2a, y), (-b/2a, y+3)], so the vertex will be different. The x-and-y
intercepts will also be different because of different constant values in the quadratic
functions.
When asked to determine whether both parabolas have the same axis of
symmetry, he said, “yes” and then explained: “Because only thing changes y-intercept,
y-intercept changes that means that the vertex will be going up and down but, is not
moving from left to right…that means that axis of symmetry stay the same position.” He
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correctly answered the question by taking graphical considerations into account. He
visualized the graphs as a whole to answer the question rather than to use the algebraic
formula of x = -b/2a. Without using any algebraic calculation, he used a mental construct
about the graphs of quadratic functions. At the end of the task, he claimed that “x of the
vertex is the same that’s what axis of the symmetry goes to x.” He also correctly
articulates the situation and combines aspect of his images to decide if they have the
same axis of symmetry.
In terms of the vertices of the parabolas, he claimed: “Uhhm…but that would be
all letters…[wrote down -b/2a, f (-b/2a) for the first parabola], and then it would be…
[wrote down -b/2a, f (-b/ ) for the second parabola without completing.]” He correctly
wrote the formula of vertices for two parabolas. Unlike the first part of the task, he
preferred to use the algebraic approach at this part. The claim, “but that would be all
letters” may suggest that he thinks the vertices are not known because of being
represented as letters. When asked about what all that meant to him, he said:
That’s pretty much the same thing. But I mean, if you don’t have the x, you can’t
really figure out the function of x, until you get like until get the value. This is
some other way we haven’t learned yet, or I mean, hold down…what was the
question, do they have the same vertex?
At this point, he seems to be confused because of not having specific values for the
vertices. And then he continued:
Ohh, okay, no, they don’t have the same vertex, wait…hold on…they do have
the same vertex…because both equations, negative b over two a … I mean
constants don’t have anything to do with it…this only has to do if they are skinner
or wider probably… okay they have the same vertex, they have the same vertex,
because negative b over two a, they are, a and b are the same…they have the
same vertex.
At the end of the task, he said as his conclusion, “I think they have the same
vertex though, because constant has nothing to do with vertex. It is just b and a that affect
vertex.” At this part, in an algebraic approach, he correctly wrote the formulas of vertices
for each parabola, and then made his decision by only focusing on the x-coordinate of the
vertex. Here it is evident that the x-coordinate of the vertex, -b/2a, alone is used for the
purpose of deciding if the vertices are the same. In fact, the vertex of a parabola is
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determined by its two coordinates; just x-coordinate is not sufficient. The claims—“both
equation, negative b over two a,” “they are, a and b are the same,” and “It is just b and a
that affect vertex”—demonstrated the student’s path of thinking to justify his answer.
Although he considered the constants in the quadratic functions, he took them out of the
decision-making process, claiming that “constants don’t have anything to do with it,”
“constant has nothing to do with vertex,” and “this [constant] only has to do if they are
skinner or wider probably.” This is also supported by the work of Zaslavsky (1997), who
reported that “the vertex of a parabola may seem as if it is determined by one of its
coordinates” (p. 33). In the tasks, subsequent to this, the student did not present
additional evidence to support this finding. So, this was not categorized as an obstacle in
the present study.
It is important to note that at the very beginning of this task, his answer to the axis
of symmetry included the claim, “the vertex will be going up and down, but is not
moving from left to right.” This suggests that he seemed to have a correct vision of
vertices for the parabolas as a whole. However, this visual image of the parabola was not
sufficient evidence for him to make his decision with regard to the vertices. Later on, in
the second part of the task, he chose to use an algebraic approach for answering the
question related to the vertex of the parabola, but he failed to be able to correctly answer
it. At this point, it appears that in two different situations, two different schemes were
activated in Richard’s mind, and this led him to reach two different responses: one was
correct, the other not. And the student also did not notice the contradiction of his two
responses. Compared with the first part of the interview, this inconsistent algebraic
approach to the task could be explained by the study of Vinner and Dreyfus (1989), who
concluded that “several cognitive schemes, some even conflicting with each other, may
act in the same person in different situations that are closely related in time” (p. 365).
They explained that when a person has conflicting schemes in his cognitive structure, the
person appears to behave either inconsistently or in an irrelevant way. This is because
different situations stimulate different schemas. This inconsistent behavior, as defined by
Vinner and Dreyfus (1989), is named under the category of the phenomenon of
compartmentalization in the present study. At this point, evidence is mounting that he
71
has two different, potentially conflicting schemes in his cognitive structure. His
performance on the subsequent tasks will provide additional evidence on this issue.
On the other hand, one can argue that the student’s response might have been
affected by his prior learning experiences of linear functions. In order to determine the xand y-intercepts in a linear or quadratic function, students need to decide no more than
one coordinate because the other coordinate is already known, which is zero. Therefore,
one coordinate is enough to differentiate and make decision on that. It is reasonable to
assume that Richard generalizes the idea of identifying these special points based on just
one coordinate to the situation here for the vertices of parabolas. Having such a tendency
to decide the vertex based on its x-coordinate is consistent with the study of Zaslavsky
(1997), who found that the vertex of a parabola is determined by one of its coordinates.
When Richard was asked if the parabolas had the same x-intercepts, after a long
pause, he said as working on the graph shown in Figure 4.17: “You know I wasn’t even
thinking about having x-intercept, first I was just thinking about like some like this
[pointing to the upper parabola], and then some like this [pointing to the lower parabola],
then from right this [pointing to the whole figure].”
Figure 4.17: Richard’s graph for the x-intercepts
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Then, he added to that, “what I mean…uhm…they will have x-intercepts
automatically if a negative, but it’s positive, opens up.” When I asked him why he was
thinking that way, he said:
Because if it is negative, it goes down, when it goes down, like open down, both
side of the parabola, like they both two x’s, they go forever… [drew an imaginary
∩-shape with a pen that intercepts on the x-axis], but if they open up, the only
way they will be able touch that the intercepts, if the vertex is below the x-axis or
if the vertex is on x-axis.
Then, he described his procedure on finding the x-intercepts: “y is zero, and y is
zero [replaced zero with two y’s in the functions], the one [1] and four [4] like affect x
would be equaled to, so I mean…one [1] and four [4] change that, so they have different
x-intercepts…they have different x-intercepts because of constant.” At the end, he again
emphasized that “I don’t think they have the same x and y-intercepts because constants
are different…only thing different would be x and y because.”
His initial thought about
the situation of the parabola was corrected by the student, who claimed that “If they open
up, the only way they will be able touch that the intercepts, if the vertex is below the xaxis or if the vertex is on x-axis.” Then, he correctly substituted 0 for y and decided that
the x-intercepts would be different, claiming that “one [1] and four [4] like affect x would
be equaled to.”
It is interesting to note that the Figure 4.17 in which Richard provided a graph for
the x-intercepts indicates that the student once again seems to have a correct graphical
construct in his mind for the vertices of the parabolas as a whole. The graphs in the
figure clearly show that these two graphs do not have the same vertex. However, he is
still unaware about the contradiction of his responses to the previous task. In this
situation, like in the task of axis of symmetry, a graphical scheme was activated in his
cognitive structure. But, we knew that he did not use this scheme when answering the
question about the vertices of the parabolas. Thus, this inconsistent behavior described
the compartmentalization phenomenon by Vinner and Dreyfus (1989) was observed
again in his cognitive structure.
In the questionnaire given prior to the interview, Richard wrote, “dunno [sic]”
for the questions that asked him to decide if the points, (-2.5, 1.5) and (-5, -12) were on
the graph. The task in the questionnaire is shown below in Figure 4.18.
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y
3
1
x
-3
-1
1
-2
-4
Figure 4.18: Determining task in the questionnaire
In order to figure out what he meant by the “dunno,” he was asked to determine
certain points if they were on the parabola. The task in the interview is shown below in
the figure, which is previously used for the translation task.
y
(3, 6)
6
4
2
(5, 0)
0
(1, 0)
3
x
6
Figure 4.3: Translation task in the first interview out of two
For the fist point of (1.5, 2.5), he did not immediately know what to do and said,
“Is it asking me like, to plug it into the vertex form?” And then he continued, “I don’t
know, I don’t know the equation I got to plug them into…figure out like if it is true or if
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something is wrong, correct the parabola.” It is evident that he was looking for an
equation of the quadratic function to determine if the point was on the parabola. For the
second point, (-1, -18), which was not visible part of the graph, he claimed, “I don’t
know, I am sorry [frustration]…negative eighteen equals, uhmm, would be a, negative
one squared, a…plus negative b, plus c…[wrote down: -18 = a + -b + c]” He then
attempted to substitute the point into the standard form, y = ax2 +bx +c, of quadratic
function even if it was not correct. His action on this task and his claim, “I don’t know
the equation I got to plug them into…figure out like if it is true” suggest that he seems to
know what to do to decide if the point is on the graph. However, since he could not find
the algebraic representation for the given parabola, he was not able to decide if the point
was on the parabola. Unlike the study of Zaslavsky (1997), who found that students in
her study conceived the graph of quadratic function as if it represented precise
quantitative information (actually, it may not be) and made their decisions by plotting
points on the graph, Richard showed no tendency to plot the points; on the contrary, he
consistently looked for the algebraic representation for the given parabola.
With regard to the y-intercept on the graph, he claimed that “If you had the
equation, you just figure out y, x’ s are zero, so usually just c, but I don’t know what c
is…I don’t know what the equation is.” He correctly explained the algebraic procedure
about how to decide the y-intercept if he had the equation. He also showed no tendency
to extend either the graph or coordinate axes to make a prediction about the y-intercept.
At this point, however, it is unclear whether Richard has a graphical construct in his mind
for the y-intercept of the parabola. So, the following task will provide additional
evidence on this part of the task.
4. Interpreting Quadratic Functions Represented by Graphs, Formulas, Tables and
Situation Descriptions
Analyses of the questionnaire indicated that 46 % of the students, 7 out of 15,
could not correctly answer the interpretation task in which they were asked to read or
gain information from a graph of quadratic function. Therefore, to be able to figure out
students’ thought processes on this task, I used the following task, as shown in Figure
4.19, adopted from the study of Zaslavsky (1997). Based on a parabola in the fourth
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quadrant, I asked Richard to interpret the coefficients, a, b, and c in the standard form, y
= ax2 + bx + c, of quadratic function.
y
x
Figure 4.19: Interpretation task in the first interview
After a short time, he said, “okay, a is negative…b is uhmm…c is negative.”
When an explanation is requested, he claimed that “because y-intercept is below the xaxis and like c is the y-intercept, so c is negative.” For the coefficient, a, he said,
“because it opens down.” When asked about ‘b’, he first wrote, “-b/2a” and then
claimed:
b has to be positive, wait, wait, wait!…no b is negative…because in order to get
the vertex, x part of the vertex is negative b over two a, a is negative, b has to be
negative, canceled out both negative, positive x because of the forth quadrant
which is positive axis.
But, he looked unsure about his answer at that point. So, I asked how sure he was, he
again wrote, “-b/2a” and explained that:
b has to be negative…because b is negative and…wait, wait, wait!…negative b,
so b has to be positive…I am sorry, I forgot the negative b over two a, b has to be
positive, if b is positive, then that would be negative here [pointing to the
numerator of –b/2a], if a is negative in this picture, it would be negative b over
negative two a, those are canceled out to make positive x and in vertex, vertex has
positive x since it is positive side of x-axis…That’s my explanation [laugh].
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In this part, Richard had no problem to interpret the coefficients, a and c, in the
quadratic function of y = ax2 + bx + c. For the coefficients, a and c, he correctly gained
the information from the graph. However, he struggled to make his decision for the
coefficient, b, which is related to the vertex of the parabola. In an algebraic approach, he
tried to interpret the formula of ‘–b/2a’ with combination of the graph of the quadratic
function in the figure. In his fist attempt, without considering the negative sign in front
of the formula of –b/2a, he said, “b has to be negative.” Then, when asked about how
sure he was, he changed his mind and said, “b has to be positive” because “I forgot the
negative b over two a.” In the second attempt, he noticed his mistake and corrected it.
At the end, his statement of “that’s my explanation” seemed to imply that this was his
final answer and would not change it.
At the very beginning, the claims—“b has to be positive, wait, wait, wait...no b is
negative,” and “wait, wait, wait…negative b, so b has to be positive”—indicate that he
was confused about whether it was positive or not. While the student used the graphical
information to answer the question for the coefficients, a and c, he employed the formula
of –b/2a to make decision for the coefficient, b. In fact, in a graphical approach, if the
vertex of the parabola is on the right side of the y-axis, then the x-coordinate of the vertex
always has to be positive. Next, since the parabola opens downward, the coefficient, a, is
negative and then b has to be positive. Note that with regard to the vertex of the
parabola, each time he wrote the formula of –b/2a before interpreting it in light of the
parabola in the figure. So, this first and immediate action on this task suggests that an
algebraic scheme was activated in his cognitive structure as opposed to a graphical
scheme.
So far, it was documented that in terms of the vertex of the parabola, different
situations stimulated different schemes in Richard’s mind. As shown in Table 4.1 below,
in five different situations, a graphical or algebraic scheme was activated in the student’s
cognitive structure. In two situations, he provided a correct vision for the vertices of the
parabolas when answering the questions related to the axis of symmetry and the xintercepts of the parabola. However, in other three situations, he employed the algebraic
approach for the vertex of the parabola and could not correctly answer the questions or, at
least was confused on them. Additionally, even though all these cases occurred in a short
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time in the same session, the student did not notice the contradiction of his two different
responses. In light of these evidences, it seems reasonable to assume that these
inconsistent or irrelevant behaviors are the indication of the compartmentalization
phenomenon described by Vinner and Dreyfus (1989). In other words, Richard’s
performance on the tasks presented so far suggest that he has two different, potentially
conflicting schemes in his cognitive structure: one is an algebraic scheme; the other is a
graphical scheme.
Table 4.1: Summary of Richard’s compartmentalization cases with
the vertex of parabola
The situations in which Richard
showed an evidence related to the
vertex of the parabola
1. A translation from graph to
algebraic form
2. Axis of symmetry
Activated
Scheme
Observed
Behavior
Result
Algebraic
Irrelevant
Failure
Graphical
Inconsistent
Correct
3. The vertex of the parabola
Algebraic
Inconsistent
Incorrect
4. x-intercepts of the parabola
Graphical
Inconsistent
Correct
5. Interpretation of the coefficient, b,
in y = ax2 +bx +c
Algebraic
Inconsistent
Confused
In terms of the y-intercept, he said, “y has to be negative” and explained that by
saying “because below the x-axis.” When asked how sure he was, he claimed that:
I am sure y has to be negative…because if you give me a positive y, then that
would be above the x-axis. Because if it is positive y, I am like sounds
completely idiot [laugh], there is a positive y above the x-axis, that means all
these equals positive thing… but b is positive, a is negative. And negative c, since
c is the y-intercept…you go negative thing, why c is negative.
Then, I asked him, “Do you think each quadratic function has to have yintercept?” he claimed that “yes, y-intercept is c…c has to be negative because it opens
down, so whenever reaches the y-intercept, it’s going to be way below the x-axis, it’s
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going to be negative.” This last statement clearly answered the question about if the
student has a graphical construct in his mind with regard to the y-intercept. In addition,
the claim, “whenever reaches the y-intercept,” indicates that he knows that the parabola is
going to intercept with the y-axis somewhere below the x-axis, unlike the study of
Zaslavsky (1997), who reported that students considered the only visible part of the graph
and inferred that the y-intercept did not exist when it was not a visible part of the graph.
5. Solving Quadratic Equations by Factoring, Completing the Square or Using the
Quadratic Formula
The questionnaire given prior to the interview indicated that 88 % of the students,
13 out of 15, were able to correctly solve the quadratic equation of 3x2 +7x-6= 0 by using
either factoring or the quadratic formula. No one chose the method of completing the
square, probably because of the leading coefficient of three. For this task, Richard used
the method of factoring and correctly found the x values of 2/3 and -3. He also
explained, as written down: “I know how to solve using the quadratic formula but
factoring is better because it’s quicker and you can do it in your head.”
In order to figure out the student’s thought processes in a detailed manner, I asked
him to solve the quadratic equation of 7x = 15 - 2x2, although he showed no obstacle on a
similar task given earlier in the questionnaire. He first moved all terms in one side and
wrote: “2x2+7x-15 = 0.” Then, he said, “two [2] times five [5] is ten [10], minus three [3]…okay that would be plus five [+5] and minus three [-3]” and wrote: (2x-3) (x+5) and
got: “x = 3/2 or -5.” When asked about any other method that he could use, he claimed:
Yeah! whole bunch, you can do the completing the square, I mean…two x square
plus seven x , and move the fifteen over, equals fifteen. Divide everything
through, x square….equals fifteen half… [wrote down 2x2 +7x =15, then x2 +
7/2x + 49/16 = 15/2], that’s another way you can do it. You can do the quadratic
formula, which would be easier, that would be negative seven plus and minus
square root of forty nine minus, four times negative fifteen, times two over
four…[ wrote down -7 ± √ 49 - 4 (-15)(2)/ 4 ]
When an explanation was requested for his preference, he said:
I can do factoring, I can do that, next thing I go to...I can give that... a is already
one [1]…I had to divide by two [2], I did right here…I’ll do that… if I don’t…
like, if it comes with the fraction, I would do it by the quadratic formula, that’s
easier.
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At this part, the student again showed no obstacle in terms of solving quadratic
equation. The claim, “if it comes with the fraction, I would do it by the quadratic
formula, that’s easier” suggests that he seems to know when to use quadratic formula
effectively for solving quadratic functions. Therefore, I decided not to go further on this
task.
6. Using Quadratic Models to Solve Problems Presented in Real-world Situations
In the quiz (Appendix G), Richard correctly answered the manufacturing problem
given in a real world situation, and got: x =1000 units and P =$500. The problem in the
quiz is as follows:
Task: Dalco Manufacturing estimates that its weekly profit, P, in hundreds
of dollars, can be approximated by the formula P = -2x2 +4x+3, where x is
the number of units produced per week, in thousands.
a) How many units should the company produce per week to earn the
maximum profit?
b) Find the maximum weekly profit.
Figure 4.20: Task of using quadratic model in the quiz
Later, the questionnaire given prior to the interview indicated that 43 % of the
students, 5 out of 15, were not able to solve a problem presented in a real-world situation
by using the quadratic model. The task in the questionnaire is shown in Figure 4.21.
Task: On wet concrete, the distance d in feet needed to stop a car traveling
at speed s in miles per hour given by the function d = 0.055 s2 +1.1s. At
what maximum speed should a car be traveling on wet concrete to stop a
traffic light 44 ft ahead?
Figure 4.21: Task of using quadratic models in the questionnaire
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In this task, students are expected to use the quadratic formula to find the
maximum speed after deciding that the distance, d in the model is 44 ft. Richard’s
response for the task is as follows:
Figure 4.22: Richard’s answer for the quadratic model problem
As shown above, Richard first translated the quadratic model into the quadratic
equation by writing down: “0.055s2 + 1.1s – 44 = 0.” Then, he used the quadratic
formula to correctly get the answer of “20 mph.” At this point; however, it is unclear
whether if the student really understood the task or just applied an algorithm previously
learned. In other words, did he correctly interpret the coefficients, d and s, in the model?
Did he have a correct vision for the task as a whole? How did he know that his answer
was correct? We, as mathematic educators, knew that getting a correct answer does not
always mean that one really understands mathematics underlying the answer. So, in
order to figure out Richard’s thoughts on this task, I presented a similar task shown below
in Figure 4.23. The task asks the student to use the quadratic model to solve the problem
presented in a real-world situation.
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Task: Suppose a basketball player throws the ball towards the basket. The
equation h (t) = -16t2 +20t + 6 gives the height h, in feet, of a basketball as a
function of the time t, in seconds.
a) What can you learn by finding the graph’s intercept with h-axis?
b) What can you learn by finding the graph’s intercept(s) with t-axis?
c) What is the maximum height the ball reaches?
d) At what time does the ball hit the ground?
Figure 4.23: Task of using quadratic models in the first interview
The student fist divided the h (t) = -16t2 + 20t + 6 by 2 and got: “-8t2+10t+3” and
then explained that “I am trying to get a half of the whole thing…maybe it would make
easier for me.” He then substituted the values of coefficients, -8, 10, and 3, into the
quadratic formula of x = (-b ± √ b2 - 4 ac) /2a. When asked what he was doing, he said,
“I am trying to find what the high is…it’s like…I don’t know what t is…at the maximum,
wait, hold on.” I then asked him what the‘t’ represents in the model, he claimed that “t is
time in seconds…I am doing this in a hard way…hold on.” He then gave up working on
the quadratic formula and wrote: “-20/-32” and got: “5/8.” After that, he substituted the
value, 5/8, into the quadratic function of h (t) = -16t2 + 20t + 6. At that time, when asked
what he was trying to do, he said:
I’m making t, it is like x …and that means…I found vertex so I am trying to find
out what high the ball is, at the maximum…like the maximum highest ball,
maximum amount of seconds…not maximum amount of seconds…what…how
much time does take the ball to get the highest point, that is why I am trying to
find vertex.
I then asked him what the graph’s intercept with h-axis tells us, he said that “what
high the ball starts at, before, like there is no time, like before time starts what high ball is
at.” Lastly, when asked about the t-axis, he claimed that “t-axis which how you…like
how much time for the ball to get when…how much time take the ball get to, like…no
high like, zero high, like ground level.”
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On this task, the student started answering the question just as he did in the
questionnaire. After dividing the quadratic model by 2, he immediately substituted the
values of coefficients into the quadratic formula, x = (-b ± √ b2 - 4 ac) /2a. Later on,
when he realized his mistake, he said, “I am doing this in a hard way.” And he then
proceeded in finding the vertex from the model. By using the algebraic formula of ‘b/2a’, he first found the x-coordinate, 5/8, of the vertex, and then substituted this value
into the quadratic model, h (t) = -16t2 +20t + 6, to find out “how much time does take the
ball to get the highest point.” The claims— “time starts what high ball is at,” “how much
time take the ball get to, like…no high like, zero high,” and “how much time does take
the ball to get the highest point”— for the intersection point with h-axis, the t-axis, and
the vertex respectively indicate that he has a good understanding of what the coefficients
mean in the model.
In summary, other than an initial confusion about how to start the task in the
interview section, Richard showed a clear understanding of how to use the quadratic
models to solve the problems presented in real-world situations. In working with the
quadratic models, he employed the quadratic formula to find the maximum speed in the
questionnaire, whereas he used the vertex to get the maximum height in the interview.
The interview session also showed that he has a correct vision for the model as a whole
and a good understanding of what the coefficients represent in the model. Therefore, I
decided not to probe more questions on this task.
Summary of the Analysis of the Case of Richard
This chapter attempted to identify obstacles that arose in the meaning-making
process in which Richard constructed his mathematical knowledge based on six aspects
of quadratic functions: concept image and definition, translating, determining,
interpreting, solving quadratic equations, and using quadratic models. The following is a
summary of the examination of Richard’s cognitive processes on these aspects.
(1) Concept Image and Definition for the Quadratic Function
It has been observed that when the concept of quadratic function is heard, it
yields an image of quadratic formula [x = (–b ± √b2 -4ac) / 2a] along with the standard
form, y = ax2+bx+c, of quadratic function in his cognitive structure. Each time, these two
images evoke simultaneously and led an actual cognitive conflict in Richard’s mind. In
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other words, his concept image of quadratic formula, which is not coherently related to
the concept definition of quadratic function, causes a conflict in the student’s cognitive
structure (Tall & Vinner, 1981). This is significant because when the concept is
represented in a broader context such as in calculus, he may not be able cope when
confronted with such tasks or optimization problems.
(2a) Translating a Quadratic Function from Graph to Algebraic Form
Richard struggled to translate quadratic functions from graph to algebraic
representation. In particular, he was confused about when and how to use the vertex, (h,
k) with the vertex form, y = a (x - h) 2 + k, of quadratic function. In all attempts, he
consistently picked up the vertex, (h, k) first and substituted that into the standard form, y
= ax2+bx+c except for one case. When he picked up the vertex form, he could not
correctly write the quadratic function, and then proceeded to work on the standard form.
All his attempts, however, he failed to find an algebraic representation for given
parabolas. This finding is parallel with the result obtained by Zaslavsky (1997), who
reported that students in her study showed a preference to translate quadratic functions
from algebraic forms to graphs rather than from graphs to algebraic forms.
(2b) A Horizontal and Vertical Shift of a Quadratic Function
With respect to the horizontal and vertical shifts, Richard did not reveal an
understanding of why the graph of the parabola horizontally or vertically shifts on the xand y-coordinate axes respectively. Instead, he performed a memorized rule focusing on
the outside clues. He did not attempt to make any cognitive connection about the way the
graphs moved. This is also consistent with the study of Zazkis et al. (2003). In addition,
the analysis between the vertex form, y = a (x - h) 2 + k, of quadratic function and an
absolute value function indicated that Richard uses the same strategy to find the vertices
and graph the functions while working on these two different functions. For these
functions, his strategy includes a memorized, rule based procedure without analyzing
properties of the mathematical concept. In other words, the mathematical concept is not
thought about and manipulated through its properties, but rather, the task is answered by
using a specific algorithm.
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(3) Determining the Axis of Symmetry, Vertex, and x-and y-intercepts of Quadratic
Functions
In terms of the axis of symmetry, Richard used a mental construct about the
graphs of quadratic functions without taking any algebraic action. For the vertices of the
parabolas, however, Richard employed an algebraic approach by focusing only on the xcoordinate of the vertices, which is -b/2a. He also considered the constants in the
quadratic functions, but he took them out of the decision-making process, claiming that
“constant has nothing to do with vertex.” Regarding the x-and y-intercepts, Richard
correctly answered the task by taking graphical and algebraic considerations into account.
To determine if certain points were on the graph of the parabola, Richard consistently
sought to find an algebraic representation of quadratic functions. Unlike Zaslavsky’s
(1997) study, Richard showed no interest to plot the points on the x-and y-coordinates
axes.
Another important observation at this part is the compartmentalization
phenomenon (Vinner & Dreyfus, 1989). The analyses showed that Richard has
conflicting schemes, an algebraic and graphical, in his cognitive structure. Related to the
vertex of the parabola, either a graphical or algebraic scheme was activated in Richard’s
mind at five different situations. While he had a correct vision for the vertices of the
parabolas in two situations, he could not correctly answer the questions when using the
algebraic schemes in other situations. In spite of the fact that all the cases occurred in a
short time, the student did not notice the contradiction of his responses. A similar
situation was reported in the work of Zaslavsky (1997), but it was for the axis of
symmetry, not for the vertex of the parabola.
(4) Interpreting Quadratic Functions Represented by Graphs, Formulas, Tables and
Situation Descriptions
Richard correctly read the information from the graph of the parabola to
determine coefficients, a, b, and c of y = ax2+bx+c. He used the given graph of the
parabola to interpret the coefficients, a and c. For the coefficient, b, however, the student
employed an algebraic approach by considering the formula of -b/2a. With regard to the
y-intercept of quadratic function, Richard correctly interpreted the graph in a correct
graphical construct and claimed that the parabola would intercept with the y-axis in
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somewhere below the x-axis, unlike the study of Zaslavsky (1997), who reported that
students considered the only visible part of the graph and inferred that the y-intercept did
not exist.
(5) Solving Quadratic Equations by Factoring, Completing the Square or Using the
Quadratic Formula
Richard easily solved the given quadratic equations. In two different tasks, he
first used the method of factoring and then the quadratic formula. He showed a clear
understanding of when and how to use these different methods in the process of solving
quadratic equations.
(6) Using Quadratic Models to Solve Problems Presented in Real-world Situations
Richard showed a clear understanding of how to use the quadratic models to solve
the problems given in real-world situations. In three different tasks, he used the quadratic
formula or the vertex at the maximum point to solve the problems. He also correctly
interpreted the coefficients presented in the model.
On the basis of these six aspects of quadratic functions, the obstacles identified
from Richard’s data are listed in the table below:
Table 4.2: Richard’s identified obstacles in his cognitive structure
(1) The quadratic formula as an image is a potential cognitive conflict
(2) A lack of understanding about when and how to use the vertex, (h, k) with
the vertex form, y = a (x - h) 2 + k, of quadratic function
(3) A lack of understanding of why a parabola horizontally or vertically shifts
(4) The phenomenon of compartmentalization: Having two different, potentially
conflicting algebraic and graphical schemes in the cognitive structure
(5) Having a tendency toward using the standard form over the vertex form
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THE CASE OF COLIN
Introduction
Colin was a Caucasian-American, 15 year-old, tenth grade male student who has
previously taken algebra-1 and geometry. In terms of his performances on these courses,
he said, “I think I had some B’s consistently for algebra-1 and geometry I dropped it
somewhere around C.” During the study, he attended all classes, but was not active,
mostly stayed quiet in the class. He only turned in two homework assignments. The only
problem he has with mathematics is that “it has only one answer, it’s not much room to
be…not around but creative I guess you can say, you know it’s hard to go about getting
another answer for me, I have just only one way to do, I have to stick with that.” And
this situation bothers him, claiming that “if you are off one point, or did not carry one, it’s
wrong, there is no, there is no half-way really, it’s nice to know that there is one definite
answer but, getting it a kind of hard.”
He described mathematics as a “kind of equation you know every since
kindergarten I have always been too close to, it’s just last year I was [the ]only time [I ]
had geometry, so it’s not so much shape-based things even though it can be a big part of
math does [sic].” In terms of his favorite subjects, he said, “I like bands, it is not a really
school subject but I do enjoy it, and English.” He is a member of the school band and
also likes to read as an extra curricula activity. During the study, he was comfortable
working and communicating with the researcher and showed no sign of intimidation. He
looked confident with his mathematics knowledge while working on the mathematical
tasks. He often communicated with the student sitting next to him in the class.
In terms of his performance on written instruments before the interviews, Colin
scored 58 out of 100 in the Quiz-A (Appendix G) administered in the second week of the
study (the average score = 80). He could not correctly graph any of the parabolas given
in either the standard form or vertex form. The vertices that he found for the parabolas
were not correct in those questions. Besides the vertex, he did not attempt to find smart
points to graph the parabolas. In addition, he left blank the last question that required
him to use the quadratic model to solve the manufacturing problem given in a real-world
situation. On the other hand, he correctly factored out the mathematical expressions and
solved the given quadratic equations in the quiz. At the end of the third week, in the
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Chapter Test-A (Appendix H), Colin scored 75 out of 100 (the average score = 88). He
again failed to graph the given parabolas in the test. In the first question, he transformed
the vertex form, y = - (x -1)2 - 2, into the standard form, y = - x2 +2x -3, before graphing
the parabola. In both of the graphing tasks, however, he could not correctly locate the
parabolas’ vertices on the x-and y-coordinate axes. For the rest of questions in the test,
he mostly worked on the questions related to the aspect of solving quadratic equations.
In those questions, he employed either the method of quadratic formula or the method of
factoring. He was only able to solve one problem (the problem #3) out of five. He also
left six questions blank in his test. Like Richard, Collin also complained about the
limited time of answering the questions.
Data Analysis
In the following section, Colin’s thought processes on particular quadratic tasks
are analyzed with respect to his incorrect or incomplete responses, and then their possible
causes are speculated in light of evidence obtained from the student’s data.
1. Concept Image and Definition for the Quadratic Function
Vinner (1983) concluded that revealing the concept images of students gives us a
better understanding of students’ “knowing what caused them to act as they acted” (p.
297). Therefore, before starting any particular task, I wanted to know about students’
images and definitions of quadratic functions. These images are also used to make
connections to students’ responses to the tasks throughout the chapter.
When I asked Colin, “If you put in your sentences, what a quadratic function is,”
his answer was: “mostly just kind of math homework right now… going to the space
program or something, if I do, you know I am sure I will be using quadratic functions,
things on that level.” Next, I asked him to give an example of a graph of a quadratic
function, he then explained as working on the following graph in Figure 5.1:
Quadratic function for me is you know just…I don’t know it just depends I guess
you could say, you know either a curve or may be a point or something, right now
kind of all stuff like parabolas, and then of course, what Ms. X [the teacher] was
talking about what matrix would you have, different equations, plug into the
matrix, it can be any of that, of course, it can also be for me that an absolute value
and a point, so really whenever quadratic function, you could have the curve,
sometimes all imagine the points and all of that, it’s just kind of blending in my
mind as quadratic function, anything like that a kind of melting in there, I can see
the differences if you put into the equation, I know how to do it but, if you just
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quadratic function in general thing, all kind of comes to my mind.
Then, by pointing to the graph of absolute value function in the Figure 5.1, I asked him
why he thought that this was a quadratic function, his response was:
It’s this just happened you know, if I know, I do an equation, putting on the graph,
that’s just quadratic function right now, maybe later if I am going to trigonometry,
before now I was not in geometry not so much , but right now, this is all I am
seeing.
Figure 5.1: Colin’s evoked image for the quadratic function
When the concept of quadratic function was first heard, it did not yield any form
of definition in the student’s mind. Then, when I asked him for an example, it generated
several different mental images in his cognitive structure: a ‘curve’, a ‘point’, a
‘parabola’, an ‘equation’, and an ‘absolute value.’ Colin did not specifically single out
any of these images, claiming that “all of that, it’s just kind of blending in my mind as
quadratic function.” Therefore, the student seems to be connecting the concept to the
mental images that he holds about quadratic function, but these images are not coherently
related to the formal definition given in the textbook, in which “a quadratic function is a
function that can be written in the standard form f (x) = ax2 +bx +c, where a ≠ 0”
(Bellman, Bragg, Charles, Handlin, & Kennedy, 2004, p. 234). Even though the
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student’s cognitive processes did not yield a dominant image in his mind, the claim, “if I
know, I do an equation, putting on the graph, that’s just quadratic function right now,”
suggests that he seems to be looking for a sort of relation or rule such as being
represented as an equation and a graph. This particular connection is also mentioned in
the Vinner’s (1983) study. The student also believes that this is something that may
change when he studies different mathematical subjects, claiming that “maybe later, if I
am going to trigonometry, before now I was not in geometry not so much, but right now,
this is all I am seeing.”
On the other hand, the graph in Figure 5.1 showed that his evoked concept image,
“the portion of the concept image which is activated at a particular time” (Tall & Vinner,
1981, p. 152), related to quadratic function contained three different graphs: a parabola, a
graph of an absolute value function, and a part of the graph of inverse function, 1/x.
Thus, his evoked image of quadratic function is consistent with the student’s own
definition which is a ‘parabola’ and an ‘absolute value function.’ At this point, it appears
that, besides the parabola, the student develops another concept image, an absolute value
function, in his mind.
In connection with this issue, Zaslavsky (1997) found that the relation between
quadratic functions and quadratic equations impedes students’ understanding of quadratic
functions. Therefore, in an effort to be able to figure out how the student constructs
mathematical meaning for the concepts of quadratic function and quadratic equation, first
I asked Colin to write down any example for a quadratic equation. He wrote, “x2 +16x
+64 = 0.” Then, I asked him for a quadratic function, he claimed that “quadratic, you got
to y equals and then factoring, and you know x minus blank times x plus blank… that’s
sort of quadratic function.” He then continued to his explanation:
When you break it down [wrote down: (x+8) (x+8) = y], when you do that, I
am not still used doing x plus 8 squared, but when you do that, that to
me, kind of link together, because you can do, you can, going to do y equals
form, and break that down into x plus a whatever equals y, it’s just all.
When specifically asked if he knew the difference between a quadratic equation and a
quadratic function, he said:
No, not really, I mean…quadratic function is mostly…equation you are trying to
get the…quadratic function always seems to lead to the graph, so if you are going
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to establish a quadratic function, it’s going to be on the graph, but in an equation
you just came up with an answer, right away on a piece of paper you don’t need to
graph to see how it looks like.
At this part, in terms of knowing the difference between a quadratic equation and
a quadratic function, he said, “no, not really”; however, he provided the examples of
“x2 +16x +64 = 0” and “y = (x +8) (x +8)” for a quadratic equation and a quadratic
function respectively. He then added to that, “in an equation you just came up with an
answer, right away on a piece of paper you don’t need to graph to see how it looks like”
for a quadratic equation, and “going to do y equals form, and break that down into x plus
a whatever equals y, it’s just all” for a quadratic function. Here all of his actions and
explanations indicated that he knows what a quadratic equation and a quadratic function
are, even if he initially could not put them into his sentences.
In summary, his initial concept images included a ‘curve’, a ‘point’, a ‘parabola’,
an ‘equation’, and an ‘absolute value.’ And then, it has been observed that his evoked
concept image also included an image of an absolute value function and a parabola in his
cognitive structure. It seems that when the concept of quadratic function is heard, his
image of absolute value function is activated along with a parabola. According to Tall
and Vinner (1981), at different times, if two images evoke simultaneously in a person’s
cognitive structure, it may cause a sense of conflict or confusion, called a potential
conflict factor, for that person on the concept being studied. Therefore, Colin’s concept
image of absolute value function, which is not coherently related to the concept definition
of quadratic function, seems to contain the seeds of future conflict in the student’s mind.
Later on, when the student meets the concept represented in a broader context such as in
calculus, he may not be able to cope when confronted with such tasks or optimization
problems. In the present study, this conflict is named under the category of ‘the absolute
value function as an image is a potential conflict factor.’
It is important to note that unlike the case of Richard, whose image of quadratic
formula is called as a cognitive conflict factor, Colin’s image of absolute value function
is called as a potential conflict factor. This is because the concept in the case of Colin did
not develop an image that yields a cognitive conflict in the student’s mind. In other
words, the image of absolute value function was activated in his cognitive structure
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without causing any cognitive conflict. Provided examples and explanations indicated
that Colin had a clear understanding of what a quadratic equation and a quadratic
function were, unlike Richard, who had the image of quadratic formula that led to a
cognitive conflict in his mind.
2. Translating among Multiple Representations of Quadratic Functions
This aspect of quadratic functions is analyzed under the three subsections: (a)
translating a quadratic function from graph to algebraic form, (b) translating a quadratic
function from algebraic to graphical form, and (c) a horizontal and vertical shift of a
quadratic function.
(a) Translating a Quadratic Function from Graph to Algebraic Form
The questionnaire given prior to the interview showed that 46% of the students, 7
out of 15, were not able to find an algebraic representation for the given parabola below
in Figure 5.2. Colin was among them.
y
1
x
(-1,0)
2
(5,0)
(2,-3)
Figure 5.2: Translation task in the questionnaire
In the questionnaire, he directly wrote the equation of quadratic function from the
graph, as written down: “y = 4/2 (x +5) (x -1).” Here, he seems to be using the
multiplicative form, y = a (x - x1) (x - x2), of quadratic function. This form uncovers
graphical information related to the location of the x-intercepts, (x1, 0) and (x2, 0). In
addition to that, there are two other ways to write an algebraic representation for this
parabola. The first is to substitute the vertex, (h, k) into the vertex form, y = a (x- h) 2 +
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k, of quadratic function, and then checking another point to find the coefficient, a. The
second is to solve a system with substitution or elimination after substituting each of
three points into the standard form, y = ax2+bx+c, of quadratic function. In the given
questionnaire, it seems that the student picked up the x-intercepts, -1 and 5, and then
placed these values into the multiplicative form, y = a (x - x1) (x - x2), of quadratic
function. However, he has got y = 4/2 (x +5) (x -1) instead of y = 1/3 (x - 5) (x +1). He
also did not provide any information about how he got the value, 4/2, for the coefficient
in the function.
To be able to figure out Colin’s cognitive processes on this task and identify if he
has any obstacles translating from graphical to algebraic representation, the first
interview included the following task shown in Figure 5.3. The task includes a
downward parabola with the vertex in the first quadrant. I asked him to find an algebraic
representation for the given parabola.
y
(3, 6)
6
4
2
(5, 0)
0
(1, 0)
3
x
6
Figure 5.3: Translation task in the first interview out of two
At the beginning, Colin stayed silent for a while, and then he said, “I wish I
studied before coming here…that’s weird.” When asked about what he was thinking, he
claimed, “I don’t know, going backwards from the parabola…but I have really no idea.”
Then, to open up his ideas, I asked the following question contained a hint, “do you know
the standard form or vertex form of the quadratic function?” After long pause, he wrote,
“y = x – h 2 +k” and said, “it doesn’t really help, no.” Lastly, I asked him if he had any
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idea what could be an equation of quadratic function for this parabola, his response was:
“no, it’s kind of blank in my head.” At this part, unlike the questionnaire, the student was
not able to apply any form of quadratic function to write an algebraic representation for
the parabola. With a hint, he seems to be trying to write the vertex form of quadratic
function, but he wrote, “y = x – h 2 +k” instead of y = a (x – h) 2 +k. After that, he
stopped and claimed that “it’s kind of blank in my head.”
Three days later in the second interview, in order to get more information related
to the student’s thoughts on this task, I presented him with the graph of a quadratic
function shown in Figure 5.4, which is similar to the previous task, except that it opens
upward. I asked him to find an algebraic representation for the given parabola.
y
1
(3,0)
(-1,0)
2
x
-2
(1,-6)
Figure 5.4: Translation task in the second interview
At this time, he first wrote, “x 2” and said, “okay…x square, positive.” Then
when asked why it is positive, he claimed that “Because it opens up, and then x, one x, so
plus x,” and wrote, “x2 +1x.” Next, when I asked him how to get the “plus one x,” he
explained:
It’s over one [1] to the right, which means going to the right, means is positive,
that would make it plus one x [+1x ], plus x, of course it’s also six [6 ] below
the y-axis, so that is minus six [-6 ], which is lower, subtracted, x square plus x
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minus six [wrote down: x2 +1x - 6 ]
At the end, he got the quadratic function as “x2 +1x – 6.” Then, he again claimed
that “it’s over to the right, vertex over to the right on the x-axis, one [1] over [showed the
points, (0,-6) and (1,-6) on the Cartesian axes], so you could write, plus one x [+1x].” So,
I asked him to prove that this is the correct function that represents the given parabola.
After staying silent for a while, on the graph in Figure 5.3, he slowly connected three
points, (-1, 0), (3, 0), and (1,-6) with straight lines. When an explanation is requested, he
claimed that “I just want to establish, I thought I was going to establish slope there that
would help me…hold down, I can’t get exact slope, I just want to get the distance
between these points, negative one, zero [(-1,0)], one, negative six [(1,-6)], that one help
to prove it.” Then, when asked how that would help you to prove it, he said, “Right, it
would not help me,” and then added to that, “I just really can’t remember [frustration]…I
can’t say.”
In summary, in order to translate the graph into an algebraic form, the student first
used the multiplicative form, y = a (x - x1) (x - x2), of quadratic function in the
questionnaire. He then tried to write the vertex form, y = a (x – h) 2 +k, in the first
interview. In both attempts, however, he was not able to translate the parabola into an
algebraic form. In the questionnaire, without taking consideration of the negative signs
in the multiplicative form, he directly placed the x-intercepts, -1, and 5, into the y = a (x x1) (x - x2), and got: “y = 4/2 (x +5) (x -1)” instead of y = 1/3 (x-5) (x +1). He did not
provide any information in terms of how he got the value, 4/2, as the leading coefficient
in the function. In the first interview, he was not able to do anything other than writing
down, “y = x – h 2 +k,” and claiming that “it’s kind of blank in my head.” In the last
attempt, he first wrote, ‘x2’, and said, “because it opens up.” He then wrote, ‘x2 +1x’, and
claimed that “it’s over one to the right, which means going to the right, means is positive,
that would make it plus one x.” Later, he added more to that, “It’s over to the right,
vertex over to the right on the x-axis, one over, so you could write, plus one x.” Lastly,
he completed the whole quadratic function as ‘x2 +1x -6’, and explained that “it’s also six
below the y-axis, so that is minus six, which is lower, subtracted.”
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Throughout this process, he only considered the vertex, (1,-6), to write the whole
function. Similar to the idea of going from the vertex, (h, k), to the vertex form, y = a (x
– h) 2 +k, of quadratic function, the student substituted the vertex, (1, -6) into the standard
form, y = ax2+bx+c, of quadratic function. In this process, he inserted the values, 1, and 6, of the vertex in place of b and c respectively. In other words, he seems to be
considering the coefficients, b and c in the form of y = ax2+bx+c as the x-and ycoordinate of the vertex respectively, although, in fact, the point of (-b/2a, f (-b/2a))
indicates the vertex in the standard form. In addition, he automatically considered the
value of the leading coefficient, a, of y = ax2+bx+c as ‘1’ in the process, actually it was
1/3. Therefore, he appears to be misinterpreting the coefficients a, b, and c in the
standard form, y = ax2+bx+c, of quadratic function. At this point, however, additional
evidence is needed to make such an assumption. Thus, the following tasks will have to
confirm this assumption and determine the possible reason that leads to the problem.
(b) Translating a Quadratic Function from Algebraic to Graphical Form
In the quiz (Appendix G), Colin was not able to graph parabolas given in either
the standard or the vertex form. Later in the chapter-test (Appendix H), he again was
unable to translate quadratic functions from algebraic to graphical forms. In both
attempts, the student could not correctly locate the parabolas’ vertices on the x-and ycoordinate axes. In the questionnaire, as opposed to the quiz and chapter-test, the
quadratic function was presented in the multiplicative form, y = 3/2 (x +1) (x -3).
However, as shown below in Figure 5.5, the student once again failed to graph the given
function. Similar to the quiz and chapter-test, the vertex was not correctly located in the
x-and y-coordinate system. He marked the vertex at (0,-1) instead of (1,-6). In addition,
the mathematical expression of “(1.5x +1.5) (1.5x - 4.5)” in the figure suggests that he
seems to be using the method of factoring for y = 3/2 (x +1) (x -3), even though the
function has already been factored.
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Figure 5.5: Colin’s graph for y = 3/2 (x+1) (x -3)
In an effort to be able to figure out the student’s thought processes on this task, in
the first interview, I presented y = - x2 - 3x +5, and asked him to graph the parabola.
After a short silence, he explained while sketching the parabola on the coordinate system:
“yes, starts…it’s something like that…it’s negative so…x square [x2] is negative so opens
down, minus three x [-3x]…up plus five [+5]…plus five moves it up on the y-axis.” The
student’s graph is shown below in Figure 5.6. When an explanation is requested for
graphing the part of ‘-3x’, he first said, “over to the left on the x,” and then continued his
explanation: “when you subtract because…negative x squared minus three x [-x2 -3x], so
instead of going up where the number positive, it’s going right after the negative, from
zero over to the left.” To challenge his ideas, I asked him what happens if the sign of
‘3x’ becomes positive, he claimed that “it will move to the right, of course it was just
plus x and minus x, only moves over one to the right plus x, one to the left minus x, it was
not there, all go up direct.”
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Figure 5.6: Colin’s graph for y = - x2 - 3x +5
In summary, the student first decided the opening direction of the parabola,
claiming that “it’s negative so…x square is negative so opens down.” He then seemed to
be determining the x-coordinate of the vertex, saying that “minus three x…over to the left
on the x.” Lastly, for the y-coordinate of the vertex, he said: “up plus five…plus five
moves it up on the y-axis,” and marked the point, (-3, +5) as the vertex, and then
completed the whole graph of y = - x2 - 3x +5 in the Figure 5.6 above. In this attempt, he
used the coefficients, -3 and 5 in the form of y = - x2 - 3x +5 as the x-and y-coordinate of
the vertex respectively. In other words, the student seems to be considering the point, (3, +5) as the vertex of y = - x2 - 3x +5. In general, probably he assumes that the point, (b,
c) in the standard form of y = ax2 +bx +c indicates the location of the parabola’s vertex.
In fact, the vertex of a parabola in the standard form is determined by the point of (-b/2a,
f (-b/2a)), not the point (b, c). His performance on this task provided further evidence
that he considered the point (b, c) as the vertex of the standard form, y = ax2+bx+c, of
quadratic function.
In the same task, to be able to find out how he acts if the function is in the vertex
form, y = a (x - h) 2 +k, I presented y = (x -2)2 +5, and then asked him to graph it. He
explained while working on the parabola shown in Figure 5.7: “x square [x2]…minus four
x [-4x], and then negative two times negative two four, plus five, plus nine [+9] so it
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would be…[wrote down x2 -4x +9], x square, helping up…it would be up here towards
nine, like that [graphed the parabola with the vertex at (4, 9)]”
Figure 5.7: Colin’s graph for y = (x -2)2 +5
At this time, the student immediately transformed the vertex form, y = (x -2)2 +5,
of the quadratic function into the standard form of y = x2 - 4x +9. Then, he once again
used the point (-4, 9) as the vertex of y = x2 - 4x +9 while graphing the parabola. Thus,
evidence is compelling that he considers the point (b, c) as the vertex of the standard
form, y = ax2+bx+c, of quadratic function. This obstacle is named under the category of
‘misinterpreting the point (b, c) as the vertex of the standard form, y = ax2+bx+c, of
quadratic function’ in the present study. Here, it is also worth mentioning that an
immediate transformation from the vertex form, y = a (x - h) 2 +k, to the standard form, y
= ax2+bx+c, may suggest that he is more comfortable using the standard form rather than
the vertex form. In other words, he seems to have a tendency toward using the standard
form over the vertex form. At this point, however, more evidence is needed to support
this assumption. Subsequent tasks will provide additional evidence on this issue.
(c) A Horizontal and Vertical Shift of a Quadratic Function
The literature review revealed that students really did not know the mathematical
reason behind a horizontal and vertical shift of a quadratic function (Zazkis et al., 2003).
Therefore, in order to reveal students’ thoughts on this issue, I asked Colin to do the
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following task, as shown in Figure 5.8, adopted from the study of Zazkis et al. (2003). In
the task, the graph of y = (x-3) 2 is a horizontal shift of y = x2, while the graph of
y = x2 -3 is a vertical shift of y = x2.
Task: Compare the graph of y = x2 relative to the graph of y = (x - 3)2.
Then, compare y = (x - 3)2 to the graph of y = x2 - 3
Figure 5.8: Task of horizontal and vertical shift in the first interview
The student first worked on y = x2. He explained as sketching the function shown in
Figure 5.9:
If you directly do y equals x square, and then if you have x is one [1], y would be
one [1], of course x two [2], y would be four [4]…that sort of curve into the right,
some x three [3], y nine [9], that get much higher before gets further out with y
equals x negative three.
Figure 5.9: Colin’s graph of y = x2
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Second, the student squared y = (x -3)2 and wrote it in the standard form, “y = x2 -6x +9.”
By pointing to y = x2 -6x +9, he claimed that:
You graph that…that’s the starting point [marked a point below the x-axis], first
of all, y equals x square [y = x2] there, then y equals x square here [pointed to the
y = x2-6x+9] except you know what, difference stuff so…it would start off like
this. When you throw negative six plus nine [-6x +9], that would change, so you
add nine [9], of course you don’t know what x is, so it’s basic graph like the first
one was, then you have to add nine [9], and immediately starts up there, and then
depends on what negative six x [-6x ] is, it bring back down and raise up,
dependent on that.
At the end, as shown below in Figure 5.10, Colin drew the graph for the y = (x-3)2.
Figure 5.10: Colin’s graph of y = (x -3)2
When asked if he knew the vertex of this graph, he said, “Uhm…no.” I then
asked him to explain the whole process step by step. He said, “minus three [-3], because
you are relating to y, it subtracts 3 from y, it’s just minus three, would be like the first one
[referring to y = x2] except the three, it’s going to be down here, it’s going to be right
there, down three [marked a point under the x-axis].” When asked how you could prove
that this was the graph of y = (x -3)2, he claimed that “you can’t, I can’t do it
mathematically to prove it, but if you just subtract three [3], because of relating to what x
equals, if you are subtracting that relating to y, it’s just going to be…three down.” Lastly,
when I asked him to graph y = x2 -3, he immediately said: “Uhm…x square minus three
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[x2 -3], that’s what I was doing. That is that…that [showed the graph of y = (x-3)2] is y
equals x square minus three [y = x2-3].” So, I returned to the previous function and asked
him again what would be a graph of y = (x-3)2. His response was: “I am not sure about
that…x minus three squared, that’s the equation…I don’t know what to do with it.”
In summary, the student first graphed y = x2 by marking the points of (1, 1), (2,
4), and (3, 9) on the x-and y-coordinate axes. However, as shown above in the Figure
5.9, he only drew the right-half of the parabola. Second, for y = (x-3)2, he immediately
transformed it into the standard form of quadratic function and got: “y = x2 -6x +9.” He
then compared y = x2 with y = x2 -6x +9, claiming that “y equals x square [y = x2] there,
then y equals x square here [pointed to y = x2-6x+9] except you know what, difference
stuff so…when you throw negative six plus nine [-6x +9], that would change.” After
that, he explained how ‘-6x+9’ would change the graph: “you have to add nine [9], and
immediately starts up there, and then depends on what negative six x [-6x] is, it brings
back down and rises up, dependent on that.” At this point, it seems that he was
considering the part, ‘-6x +9’, of the quadratic function to locate the vertex of y = x26x+9. It is also important to note that as happened in the previous section, the student
again immediately transformed the vertex form of y = (x -3)2 into the standard form of y
= x2 -6x+9. This provided further evidence that he has a tendency to use the standard
form, y = ax2 +bx+c, of quadratic function rather than the vertex form, y = a (x -h)2 +k.
A short time later, another explanation was brought up by the student, saying that
“minus three, because you are relating to y, it subtracts 3 from y,” and “ if you just
subtract three, because of relating to what x equals, if you are subtracting that relating to
y, it’s just going to be…three down.” Here, it seems that he was considering the graph of
y = x2 -3 rather than y = (x -3)2. At the end, when asked to graph a parabola for y = x2 -3,
he realized his mistake and claimed that “that’s what I was doing. That is that…that
[pointed to the graph of y = (x -3)2] is y equals x square minus three.” For y = (x -3)2,
however, he could not take any action, saying that “I don’t know what to do with it.” At
this part, he seemed to be confusing the graphs of y = (x -3)2 with y = x2 -3. When he
determined that the graph shown above in Figure 5.10 corresponds to y = x2-3, he was
not able to come up with any type of graph for y = (x -3)2. It appears that the student
struggles to work with the vertex forms of quadratic functions.
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Three days later in the second interview, in order to find out if there is any pattern
in his constructs for the vertex form or horizontal shift of a quadratic function, I presented
him the following task shown in Figure 5.11, which is similar to the previous task. I
asked him to graph and compare the following parabolas.
Task: Compare the graph of y = x2 relative to the graph of y= (x +2) 2.
Then, compare y = (x + 2) 2 to the graph of y = x2 +2
Figure 5.11: Task of horizontal shift in the second interview
He explained as working on y = x2 shown below in Figure 5.12.
The graph y equals x square …if it’s started zero, y being x square, if x is zero [0],
x square would be zero [0]…so x goes over one [1], y would be one [1] as well.
Of course once it gets the two [2], y would be four [4]…it will get bigger very
quickly…it would go up…negative one [-1], still be one [1], so it would be right
there, negative two [-2] still be four [4], so it would look like that…it would open
up.
Figure 5.12
Colin’s graph of y = x 2
Figure 5.13
Colin’s graph of y = (x +2)2
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He first located the vertex at (0, 1), and then extended it towards the origin, (0, 0),
claimed that “It should be on zero, I am sorry…I drew that incorrectly, it’s going to be on
zero, so it hits zero.” After that, for y = (x +2)2, he squared it and wrote: “x2 + 4x +4”
and then explained while working on the graph shown above in Figure 5.13: “x square so
is positive, opening up…so it started right there, that’s the bottom vertex [marked the
point (4, 4)], opens up…like that.” When asked how to come up with this graph, he said,
“x square is positive so opens up, plus four x [+4x], moves to the right because it’s
positive, plus four [+4] y moves up on the y-axis.” For the vertex, he stated that “four,
four [(4, 4)], on the right, vertex is four, four [(4, 4)] because it’s four over four.” Lastly,
when I asked him what he was thinking about the y-intercept of the parabola, he
explained while trying to extend the parabola and the y-axis on the graph in Figure 5.13
above:
Y-intercept…y would extend up as this extends over…once the parabola going
up, going up to the left, it eventually hit the y-axis, that part would be the
intercept [showed the intersection point]…some of them eventually get there…
like this, this one y equals x square, vertex is at the origin, that’s already on the yaxis, and this one [pointed to the parabola] keeps extending on, eventually hit the
y-axis, but some of them may never get there, just infinitely get smaller and
smaller, never actually meet the y-axis, just keep getting closer.
At this point, it has been observed again that when he was presented with the
vertex form, y = (x +2)2, of quadratic function, he immediately transformed it into the
standard form, y = x2 + 4x +4, and then used the point, (4, 4) as the vertex to graph the
function. Right now, it is evident that he perceives the point (b, c) as the vertex of the
standard form, y = ax2+bx+c, of quadratic function. In addition, his first action on y = (x
+2)2 was to transform it into the standard form, y = x2 + 4x +4; so evidence is compelling
that he has a tendency to use the standard form over the vertex form. Therefore, this is
named under the category of ‘having a tendency toward using the standard form over the
vertex form,’ in the present study.
Another important point needed to be addressed is that for the y-intercept, the
claims—“some of them eventually get there,” and “but some of them may never get there,
just infinitely get smaller and smaller, never actually meet the y-axis, just keep getting
closer,”—suggest that he seems to be referring to an asymptotic behavior about the y-
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axis. In other words, he assumes that some quadratic functions may have a vertical
asymptote, although, in fact, none of them has an asymptote. So, it could be argued that
the student is considering the y-axis as a possible vertical asymptote for some quadratic
functions. This assertion is also supported by the study of Zaslavsky (1997), who
reported that “the graph of a quadratic function may seem as if it has vertical asymptotes”
(p. 30). At this point, however, additional evidence is needed to make such an assertion.
Thus, tasks in the fourth section (interpretation aspect of the quadratic function) will have
to confirm this assertion and to determine the possible reason that causes the problem.
Overall, it has been documented from the student’s data in the subsections a, b,
and c that he uses a specific strategy for any type of translation among quadratic
functions. Colin completes the whole translation process in three steps. While
translating from a graph to an algebraic form (the subsection-a), he first assumes that the
value of leading coefficient, a, of y = ax2+bx+c is 1. He then writes “x2” or “-x2” based
on the opening direction of the parabola: up or down respectively. Second, he assumes
that the coefficient of x, b, is the x-coordinate of the vertex of y = ax2 +bx +c. By
checking the location of the x-coordinate of the vertex on the x-and y-coordinate axes, he
determines the value of ‘b’ and then writes the function as “y = x2 +bx.” In the last step,
similar to the previous step, he made his decision with the value of ‘c’ by checking the
location of the y-coordinate of the vertex on the x-and y-coordinate axes. The
assumption here is that the coefficient, c, is the y-coordinate of the vertex of y = ax2 +bx
+c.
While translating from an algebraic form into a graph (the subsection-b), Colin
uses the same assumptions mentioned above. He first decides the opening direction of
the parabola by checking the sign of the leading coefficient, a: up or down. At this point,
he assumes that the value of the leading coefficient, a, of y = ax2+bx+c is 1. Second, he
locates the x-coordinate of the vertex on the x-and y-coordinate axes by checking the
value of ‘b’ in the form of y = ax2 +bx +c. In the final step, he locates the y-coordinate of
the vertex by checking the value of ‘c’ in the form of y = ax2 +bx +c. As a result, he
graphed the whole parabola by using the point (b, c) as the vertex of y = ax2 +bx +c.
When the quadratic function was presented in the vertex form, y = a (x - h)2 +k (the
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subsections b and c), he directly transformed it into the standard form, y = ax2 +bx +c,
and then applied the same procedure explained above.
3. Determining the Axis of Symmetry, Vertex, and x-and y-Intercepts of Quadratic
Functions
The questionnaire given prior to the interview indicated that 60 % of the students,
9 out of 15, made mistakes on tasks including graphing and determining aspect of
quadratic functions. Therefore, in order to specifically examine Colin’s cognitive
processes on determining tasks such as the axis of symmetry, the vertex, and x-and-y
intercepts, the following task shown in Figure 5.14 was adopted from the study of
Zaslavsky (1997). The task asks the student to compare two parabolas given in the
standard form. With this task, the student tries to determine whether two quadratic
functions have the same axis of symmetry, the vertex, and x-and y-intercepts.
Task: The equations of two parabolas are given by
y = a x2 + b x + 1
y = a x2 + b x + 4
Answer the following questions about these two parabolas.
a) Do the two parabolas have the same axis of symmetry?
b) Do the two parabolas have the same vertex?
c) Do the two parabolas have the same x-intercepts?
d) Do the two parabolas have the same y- intercept?
Figure 5.14: Determining task in the first interview
This task could be answered by using either a graphical or algebraic method.
From the graphical approach, the second parabola is similar to the first one, except that it
is just vertically shifted by 3 units. Therefore, these two graphs do not have any common
points except for the axis of symmetry. From the algebraic approach, the axis of
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symmetry for both parabolas is the line of x = - b/2a, so they are the same. Since the
vertex of both parabolas is the point (-b/2a, f (-b/2a)), and the value of f (-b/2a), which is
the y-coordinate of the vertex, in the second parabola will always be 3 greater than the
first one [e.g., (-b/2a, y), (-b/2a, y+3)], so the vertex will be different. The x-and-y
intercepts will also be different because of different constant values in the quadratic
functions.
When asked to determine whether both parabolas have the same axis of
symmetry, he explained by graphing the parabolas shown in Figure 5.15:
It’s not the same line on the graph…if you did the graph…first one just was
here... splitting half like that of the symmetry…that was bx plus one [bx+1]
[pointing to the left parabola]...then bx plus four [bx+4] would not be on the same
line, that would be like that [pointing to the right parabola], wherever would be,
we have line through like that, right in the middle, it would not be the same xaxis line, that’s what the symmetry line goes through…so that would move
somewhere else…location would be different.
Figure 5.15: Colin’s graph for the axis of symmetry
For y = ax2+bx +1, he graphed the parabola on the left above, claiming that “that
was bx plus one [bx+1].” Then, he located the graph of y = ax2+bx+4 next to the first
one and said “bx plus four [bx+4] would not be on the same line, that would be like that.”
At the end, on the basis of these parabolas, the student explained the reason why they do
not have the same axis of symmetry, claming that “it would not be the same x-axis line,
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that’s what the symmetry line goes through…so that would move somewhere else…
location would be different.” His actions and explanations suggest that he seems to be
using the constants, 1 and 4, as if they were the axis of symmetry of quadratic functions,
y = ax2+bx +1, and y = ax2+bx+4 respectively. In fact, the axes of the symmetry for both
parabolas are independent of the constants, 1 and 4, and the same value of x = - b/2a. It
is also interesting to note that while he was sketching the first graph opening down, he
graphed the second one opening up. He seems to be trying to place the vertices of the
parabolas in different spots where the difference between the two is 3. So, as happened
in the previous tasks, Colin again struggled to graph the quadratic functions on this
aspect.
In terms of the vertices of the parabolas, after a long pause, he claimed, “Yes,
they have the same vertex,” and then explained by graphing the parabola shown below in
Figure 5.16:
They got the same vertex, one [1] and four [4] a just kind of move slope around, it
changes the two other points that you can find…the question is always asked to
find the other two points besides the vertex, so the other two points just would
move out or something. This is the first one [pointing to the lower-left
highlighted point], then it would be question of the same point of symmetry, same
vertex, it would just be out a little bit further like that…you know that would not
happen…these two points would not be the same spot [showing the two bottom
highlighted points on the graph].
Figure 5.16: Colin’s graph for the vertices of the parabolas
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Unlike the graph in Figure 5.15, which is the graphical construct activated in the
student’s mind for the two parabolas, Colin claimed that “they got the same vertex.” The
parabolas in Figure 5.15 clearly show that they do not have the same vertex, even though
they are not exact representations of the given functions. At this part, his focus was
mostly on the constant values of one and four, and their effect on finding the other two
points besides the vertex, claiming that “one [1] and four [4] a just kind of move slope
around, it changes the two other points that you can find.” At this point, he is not aware
about the contradiction of his responses to the task. In the same situation, he also did not
use the activated graphical scheme to answer the question related to the vertices of the
parabolas. This could be an indication of the compartmentalization phenomenon
described by Vinner and Dreyfus (1989). At this point, however, more evidence is
needed to support such a supposition.
Afterward, when Colin was asked if the parabolas had the same x-intercepts, he
said:
I think x-intercepts would be the same, because vertex…vertex is changing, but
it’s getting wider…I am not exactly sure they would be the same…I don’t think,
we haven’t done enough…yeah intercepts are the same…I would say if they do
change, I would say x’ s are the same, but y isn’t the same…the biggest
difference is that this one [pointing to y = ax2+bx +4 ]is wider than the other
[referring to y = ax2+bx +1], shape is different.
Shortly after claiming that “x-intercepts would be the same,” another claim, “I am
not exactly sure they would be the same…I don’t think, we haven’t done enough”
suggests that he seems to be unsure about his response on the issue. He also assumes that
the coefficients of one and four affect the size of the graph, claiming that “the biggest
difference is that this one [pointing to y = ax2+bx +4] is wider than the other [referring to
y = ax2+bx +1].” In fact, the value of the leading coefficient, a, of y = ax2+bx +c
determines how wide the graph would be.
On the statement of “vertex is changing, but it’s getting wider,” I asked him what
he meant by that. After a short silence, he said:
It gets bigger, it’s a different spot right, because it does not get any higher
anything, so x-intercepts would change, but y-intercept would stay the same,
because it is not moving up and down anything, the same vertex…I am trying
to convince myself that there is only one that’s going to change somehow…it
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looks like…both of them would change, x-intercepts and y-intercepts would
change…yeah, both of them would change…does not matter whether getting
higher or lower, or the vertex getting up and down, because it’s just getting wider,
so both would change yes.
At the beginning of this part, he stated, “x-intercepts would change, but yintercept would stay the same, because it is not moving up and down anything, the same
vertex.” Later on, he changed his mind and said that “x-intercepts and y-intercepts would
change…does not matter whether getting higher or lower, or the vertex getting up and
down.” These claims suggest that he seems to be struggling with the graphical construct
in his mind for the parabolas as a whole. The justification for these different responses is
explained by the student, saying that “I am trying to convince myself that there is only
one that’s going to change somehow.” At the end of the interview, when asked how sure
he was in his final response, he said, “now if you asked me the question, I would say both
would change yes.”
In summary, throughout this aspect of quadratic function, Colin preferred to use
graphical approaches to answer the tasks, even though he struggled to correctly graph the
functions. At this part, two different graphical schemes in his cognitive structure have
been observed. The first one is shown in the Figure 5.15 activated for answering the
question related to the axis of symmetry. The second one is described with Colin’s final
claims of “x-intercepts and y-intercepts would change,” and “[they got] the same vertex.”
Here it seems that Colin’s approach to how to graph a function still affects the way he
answers the tasks. In his first graphical scheme shown in Figure 5.15, he used the part of
‘bx+1’ and ‘bx+4’ to graph the functions of y = ax2+bx +1, and y = ax2+bx +4
respectively. At this time, however, unlike the previous strategy used in the translation
tasks, he seems to be using ‘1’ and ‘4’ as the x-coordinates of the vertex rather than the ycoordinates. As a result, he drew two parabolas whose vertices were on the vertical lines
of x = 1 and x = 4, assuming that they were the axes of symmetry. In the second scheme,
with a hint, he was able to make his decision related to the x-and y-intercepts. However,
he was still unsure about the vertices of the parabolas. It seems that not knowing how to
get the vertex of a parabola in an algebraic manner, or as identified in the previous
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section, not knowing it correctly, impede his understanding of the effects of the
coefficients on a graph of the parabola in general, the location of the vertex in particular.
In the questionnaire given prior to the interview, after marking the point on the xand y-coordinate axes, Colin wrote, “it is on the parabola, although it is not highlighted,”
for the question that asked him to decide if the point, (-2.5, 1.5) was on the graph. The
task in the questionnaire is shown below in Figure 5.17.
y
3
1
x
-3
-1
1
-2
-4
Figure 5.17: Determining task in the questionnaire
The student here seems to be using a method of ‘eye measurement’ to make his
decision if the point of (-2.5, 1.5) is on the parabola. By plotting the point on the graph,
he claimed that “it is on the parabola,” in fact, it was not on the parabola. For another
point, (-5, -12), he wrote, “no, it is not…the slope is 3/1, so the parabola crosses the -5
line at (-5,-6).” At this part, the student first found a slope of the parabola as if it was a
linear function. Then, he again appears to be employing the eye measurement technique,
claiming that “no, it is not [on the parabola]” because “the parabola crosses the -5 line at
(-5,-6).” The student failed once again on this task because actually the point of (-5, -12)
was on the parabola. In fact, such an approach used by the student may not provide a
definite answer because the graph in the figure does not necessarily represent exact
quantitative information. So, probably he assumes that the graph of quadratic function
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represents precise quantitative information and the method of eye measurement could be
effectively used in such situations. This assumption is also supported by the study of
Zaslavsky (1997), who found that students in her study determined whether certain points
were on a graph based only on the method of eye measurement. At this point, however,
more evidence is needed to make such an assumption in the present study.
In order to figure out if there is any pattern in the student’s thoughts about this
issue, Colin was asked to determine if certain points were on the graph of parabola during
the interview session. The task in the interview is shown below in the figure, which is
previously used for the translation task.
y
(3, 6)
6
4
2
(5, 0)
0
(1, 0)
3
x
6
Figure 5.3: Translation task in the first interview out of two
For the first point of (1.5, 2.5), he said, as plotting it on the x-and y-coordinates
axes: “yeah, one point five and two point five…x before y…that would be right there, the
point is just there, also on the parabola.” At the end, he marked the point on the parabola.
When asked how he could prove it, he stated:
If you know the equation first of all, that would help…you know the slope, I just
assume slope is two [2] over one [1], then not one exactly be right, but it should
be…one point five [1.5] should be half way between there, it should be between
one [1] and two [2] on the x-axis, at three [3] on the y-axis, so it should be right
there, from what computer crew established that, it’s between two [2] and three
[3] on the y.
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Here Colin first located the x-coordinate, 1.5, of the point on the x-axis, claiming
that “it should be between one [1] and two [2] on the x-axis.” He then placed the ycoordinate, 2.5, of the point on the y-axis, saying that “it’s between two [2] and three [3]
on the y.” It became apparent from the data that the student located the point on the
coordinate axes on the basis of the eye measurement method. Evidence is mounting that
the student conceives the graph of quadratic function as if it represents precise
quantitative information.
For the second point of (-1, -18), which was not visible part of the graph, he
claimed:
Okay, I go to negative one [-1], down if it is there, if it is at one, zero [(1, 0)], and
go down to, you go down here to zero, negative two [(0, -2)], then if you , slope
still two [2] , you go down two [2] more, that would be negative one, negative
four [(-1, -4)], so negative one, negative eighteen [(-1, -18)] couldn’t be on the
parabola because you have already hit at negative one, negative four [(-1, -4)], if
you went down one more step, negative two, negative six [ (-2 -6)] , so to get to
negative eighteen [-18] would be far beyond negative four [-4].
At this part, the student first found a slope on the parabola, claiming that “if it is
at one, zero [(1, 0)], and go down to, you go down here to zero, negative two [(0,-2)],
then if you, slope still two [2].” By considering this slope, he extended the graph going
through the point of (-1,-4). At the end, he made his decision by claming that “negative
one, negative eighteen [(-1,-18)] couldn’t be on the parabola because you have already hit
at negative one, negative four [(-1, -4)].” Once again, the student only worked on the
graph to determine if the point, (-1,-18) was on the parabola. In light of the slope, he
decided that the graph would have an intersection on the y-axis. He then extended the
parabola and intersected it with the point of (-1,-4). At this point, evidence is compelling
that he conceives the graph of quadratic function as if it represents exact quantitative
information In the present study, this is placed under the category of ‘conceiving the
graph of quadratic function as if it represents precise quantitative information.’
In addition, in order to figure out how he constructs the concept of slope on the
parabola, I asked him to explain the way he got the slope of two. His response was: “I
took this point, the vertex point, went down to the nearest intersection between two lines
I can tell…because there it’s just going through, x-axis line, y-axis line and in there going
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between both of them, and two [2] down one [1].” Then, when asked to compare the
slopes on the curve and line, he claimed that “on the curve, it could change after a while,
but if you get, the closest I could figure was to the slope of two [2], but of course it could
keep getting larger and larger, change.” Here, similar to the way to get a slope on a
straight line, the students found the slope on the parabola. In a short time, however, he
explained that “the closest I could figure was to the slope of two [2]…it could keep
getting larger and larger, change.” Therefore, it seems that he has a correct construct for
the concept of slope. The claims also suggest that he knew that there is not a specific,
fixed slope on the parabola.
4. Interpreting Quadratic Functions Represented by Graphs, Formulas, Tables and
Situation Descriptions
The analysis of the questionnaire indicated that 46 % of the students, 7 out of 15,
could not correctly answer the interpretation task in which they were asked to read or
gain information from a graph of quadratic function. Therefore, to be able to figure out
students’ thought processes on this task, I used the following task, as shown in Figure
5.18, adopted from the study of Zaslavsky (1997). Based on a parabola in the fourth
quadrant, I asked Colin to interpret the coefficients, a, b, and c in the standard form, y =
ax2 + bx + c, of quadratic function.
y
x
Figure 5.18: Interpretation task in the first interview
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After reading the task, he said, “I would say that ‘a’ is a negative number.” When
an explanation is requested, he claimed:
Because the parabola is opening down… b is positive number because it moves to
the right of y-axis which means moves on the x-axis…[pointed to the vertex
point], and then c is negative number because moves down on the x-axis,
downwards into the fourth quadrant…well the parabola itself is in the fourth
quadrant that helps us.
I then asked him what he meant by “the parabola itself is in the forth quadrant,”
he explained that “it’s positive and negative, so coordinates would be…not necessarily x,
which is plus x, minus x…[wrote: (+x, -x)], it’s positive number on the x-axis and
negative number on the y-axis.” After that, when asked about the y-intercept, he said,
“Intercept…I don’t think it would have an intercept point,” and then explained the reason
for that: “well, I don’t know the numbers so it’s hard to say but, just the way opening
downward and the fourth quadrant, I don’t think it would ever hit the y-axis…it would
not hit the y-axis.” He then extended the y-axis and the parabola, as shown below in
Figure 5.19, but he did not let them touch by leaving a little space between them.
Figure 5.19: Colin’s graph for the y-intercept
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On the question of why he was thinking that way, he said, “I can’t do without a,
b, and c…I can’t,” and then continued his explanation, “If I had…what a, b, and c were, I
might be able to do it, but without them.” To be able to make sure, by pointing to the
parabola, I asked him again if the graph had intersected the y-axis, he stated that “it
always keeps pushing out, but it wouldn’t…it just keeps getting smaller and smaller, not
really never hit zero…it would continue to getting closer and closer to zero, never, never
actually touching it.”
In summary, Colin had no problem interpreting the coefficient, a, claiming that “a
is negative number…because the parabola is opening down.” For the coefficients, b and
c, however, he made his decision by focusing only on the vertex of the parabola, saying
that “so coordinates would be…not necessarily x, which is plus x, minus x… [wrote: (+x,
-x)], it’s positive number on the x-axis and negative number on the y-axis.” It is obvious
here that, as identified before in his cognitive structure, he interprets the coefficients, b
and c as if they were x-and y-coordinates of the vertex respectively. In addition, along
with the graph in Figure 5.20, he claims, “I don’t think it would ever hit the y-axis…it
would not hit the y-axis,” and “it just keeps getting smaller and smaller, not really never
hit zero…it would continue to getting closure and closure to zero, never, never actually
touching it.” This suggests that he once again seems to be considering the y-axis as a
vertical asymptote of the parabola, although the graph of a quadratic function does not
have any asymptote. Therefore, this is placed under the category of “considering the yaxis as a vertical asymptote of the quadratic function,” in the present study. Since
evoked concept images play an important role in a learner’s actions (Vinner, 1983;
Vinner & Dreyfus, 1989), this particular action and explanations could be a result of
Colin’s evoked image of inverse function, 1/x, shown in Figure 5.1. As seen in the graph
of 1/x, the function approaches the y-axis, but never touches it. Similarly, the function
also gets closer and closer to the x-axis, but it never touches it, either.
5. Solving Quadratic Equations by Factoring, Completing the Square or Using the
Quadratic Formula
The questionnaire given prior to the interview indicated that 88 % of the students,
13 out of 15, were able to correctly solve the quadratic equation of 3x2 +7x-6= 0 by using
either factoring or using the quadratic formula. No one chose to complete the square,
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probably because of the leading coefficient of three. For this task, Colin used the
formula, x1, 2 = (-b±√b2-4ac) /2a, and correctly found the x values of 2/3 and -3 after
inserting the coefficients a, b, and c into the quadratic formula.
In order to figure out the student’s thought processes in a detailed manner, I asked
him to solve the quadratic equation of 7x = 15 - 2x2, although he showed no obstacle on a
similar task given earlier in the questionnaire. After reading the task, he explained while
working on the equation:
The first thing I noticed that there is no y equals there…so if I subtract seven x,
get everything all in one side, I would have to write zero equals negative two
x square, minus seven x, plus fifteen…[ wrote down: 0 = - 2x2-7x+15], what I
think y equals…it’s very, so foreign to put, to have to put zero equals for the
quadratic equation…it should be y I think…if y is supposed to be there, it would
be written there y plus seven x equals fifteen minus two x square…[ wrote down:
y +7x = 15-2x2 ]
Later, he said, “So, it’s y negative two x square, minus seven x…[wrote: y = - 2x27x+15], then open down.” I then asked him to explain what he was trying to do. He
claimed:
I am trying to graph it but that’s not what I should be doing it… it’s almost like I
just need to graph it first, one reason to another I can’t say but, it’s not exactly
working, I am not mentally making the equation work.
At this point, he seems to be confused about what to do with the equation. He
first tried to graph the equation instead of solving it. When I asked him to explain what
“solving an equation” means, his response was: “to find answers with the x, you want to
find out, you want to reduce as much as possible, essentially that’s not exactly what you
are doing as reducing it, but you are simplifying.” He then continued his explanation:
Simplifying…negative two x [-2x]…it’s like that… [wrote down: (-2x….) (x…)]
to get plus fifteen [+15], seems like I have to have minus three [-3], minus five [5], but that one wouldn’t work…I think the connection between negative seven x
[-7x], and fifteen [15] is eight [8], there is a difference eight [8] between seven
[7] and fifteen [15], but I can’t make that work, the context of that…all I think
is…three [3] and five [5], but if you subtract three [3] and five [5] to get negative
seven [-7], you wouldn’t get negative seven [-7], make it eight [8], so I am not
sure how to solve this one.
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After that, when asked if he knew any other method to solve this equation, he
said:
To simplify into that, no other method like that, no other method to put into
parenthesis…I can’t remember any names for simplifying in other way…it’s so
strange minus seven x plus fifteen [-7x+5]… wait, wait, two x, minus six, of
course plus five [+5], plus three [+3], so negative two x [-2x] times x equals
negative two x square [-2x2], three x [3x], wait…negative ten x [-10x] plus
three x [3x], yes five [5] times two [2], negative seven x [-7x], three [3] times five
[5] fifteen [15]…that’s correct for factoring.
At the end, he wrote, “y = (-2x+3) (x+5)” and claimed that “that’s the solution,
simplifying.” At this part, after getting all terms on one side, he first substituted ‘y’ for
‘0’ and then tried to graph the equation. Probably, his intention to graph the equation led
him to make that replacement. Later on when he realized his mistake, he said, “I am
trying to graph it but that’s not what I should be doing.” Then, for the term of ‘solving an
equation,’ the claim, “to find answers with the x,” suggests that he knows what that term
means. He also correctly factored the equation out as “y = (-2x+3) (x+5).” At the end,
however, he claimed, “that’s the solution, simplifying,” without finding the x-values.
In summary, other than forgetting to determine the x-values at the end in the
interview session, he showed a good understanding of how to solve a quadratic equation.
In the questionnaire, he used the quadratic formula to find the x-values, while he
employed the method of factoring in the interview. Therefore, I decided not to continue
probing more questions on this type of task.
6. Using Quadratic Models to Solve Problems Presented in Real-world Situations
In the quiz (Appendix G), the only question that Colin did not attempt to answer
was the manufacturing problem given in a real-world situation. The problem in the quiz
is shown below in Figure 5.20. This problem could be solved by finding the vertex of the
quadratic model of P = -2x2 +4x+3. Since the model represents a downward parabola, it
has a maximum point, and the y-coordinate of the vertex gives the maximum profit per
week.
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Task: Dalco Manufacturing estimates that its weekly profit, P, in hundreds
of dollars, can be approximated by the formula P = -2x2 +4x+3, where x is
the number of units produced per week, in thousands.
c) How many units should the company produce per week to earn the
maximum profit?
d) Find the maximum weekly profit.
Figure 5.20: Task of using quadratic model in the quiz
Later on, the questionnaire given prior to the interview indicated that 43 % of the
students, 5 out of 15, were not able to use the quadratic model to solve the problem
presented in a real-world situation. The task in the questionnaire is shown below in
Figure 5.21.
Task: On wet concrete, the distance d in feet needed to stop a car traveling
at speed s in miles per hour given by the function d = 0.055 s2 +1.1s. At
what maximum speed should a car be traveling on wet concrete to stop a
traffic light 44 ft ahead?
Figure 5.21: Task of using quadratic model in the questionnaire
In this task, students are expected to use the quadratic formula to find the
maximum speed after deciding that the distance, d in the model is 44 ft. Colin’s response
for the task is as follows:
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Figure 5.22: Colin’s answer for the quadratic model task
As shown above, Colin on the task first wrote, “d = 44,” and then inserted ‘20’
for‘s’ in the d = 0.055s2 +1.1s. At the end, he found, “22 +22 = 44,” and wrote, “the car’s
maximum speed is 20 miles per hour.” The procedure he followed seems to be a
checking process to prove if the answer of 20 mph was correct rather than using the
model to find the answer of 20 mph. He did not provide any other information in terms
of how to get the car’s maximum speed, although it was the correct answer. In other
words, there was no indication on the paper that he used the quadratic model to solve the
problem. Therefore, in order to find out if Colin would attempt to use the model given in
the task, I presented a similar task shown below in Figure 5.23. He was asked to use the
quadratic model to solve the problem given in a real-world situation.
Task: Suppose a basketball player throws the ball towards the basket. The
equation h (t) = -16t2 +20t + 6 gives the height h, in feet, of a basketball as a
function of the time t, in seconds.
a) What can you learn by finding the graph’s intercept with h-axis?
b) What can you learn by finding the graph’s intercept(s) with t-axis?
c) What is the maximum height the ball reaches?
d) At what time does the ball hit the ground?
Figure 5.23: Task of using quadratic model in the first interview
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After a long pause, he drew the following graph shown below in Figure 5.24, and
then claimed that “I can’t understand why I don’t understand…why I am just doing this.”
Then when I asked him what he could learn with the intercept of h-axis, he first labeled
the x-and y-coordinate plane as‘t’ and ‘h’ respectively, and then stopped.
Figure 5.24: Colin’s graph for the given quadratic model
I then asked the same question in a different way: “What kind of information you can get
with the h-intercept? His response was:
It’s going to let me know what point…high and time is [sic] going to tell me what
time the basketball reaches …intersection with h- axis would be the starting from
lower point…if it hits h-axis at all, you make sure it’s not zero on the h or t-axes,
then it wouldn’t intercept with h-axis of course, but it would start there, because if
you, you start the time as soon as gets off the ground, it starts there and go up
then, start zero go up there.
Lastly, when asked what he was thinking about the vertex, he claimed:
Well…maximum high would be the vertex, even the graph almost like
shooting the basketball, because it goes up and arc, of course, it may just not
straight out, but…if you shoot the ball, it will go up, and start back down.
At this part, the claims—“high and time is [sic] going to tell me what time the
basketball reaches,” “intersection with h- axis would be the starting from lower
point…it’s not zero on the h,” “maximum high would be the vertex,”—suggest that he
seems to have a correct graphical construct in his mind. However, he once again did not
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use the quadratic model, h (t) = -16t2 +20t + 6, to find the maximum height or the time of
the ball in the air.
In summary, Colin initially was frustrated with the task and claimed that “I can’t
understand why I don’t understand.” He then sketched the parabola in the figure above,
and described it as “almost like shooting the basketball.” Lastly, he tried to interpret the
coefficients, t and h, in the quadratic model. However, he did not attempt to use the
model to find the maximum height or the time of the ball in the air. So far in three
different tasks, Colin failed to use the quadratic model to solve the problems. Instead, he
left the problem blank in the quiz; he checked his answer to prove whether it was correct
in the questionnaire; and he just graphed a downward parabola in the interview. So, all
his actions and explanations suggest that he does not know how to use quadratic models
to solve the problems given in real-world situations. Therefore, this obstacle is named
under category of “failure to use the quadratic model to solve problems given in realworld situations,” in the present study. Such a case might result from the teaching
strategy used by the teacher. In the classroom, the teacher solved the similar problems by
finding the vertex from the model without having any, or a very limited discussion with
the students in terms of how the problem related to the quadratic model or how the
problem fits the model. Not making necessary connections between the problem and the
model used for it might lead to isolated and unconnected knowledge in the student’s
cognitive structure; and this later, turns out to be lost at some point (Haylock, 1982).
When the student meets the similar task again, he may not know what to do with the
quadratic model in the new situation.
Summary of the Analysis of the Case of Colin
This chapter attempted to identify obstacles that arose in the meaning-making
process in which Colin constructed his mathematical knowledge based on six aspects of
quadratic functions: concept image and definition, translating, determining, interpreting,
solving quadratic equations, and using quadratic models. The following is a summary of
the examination of Colin’s cognitive processes on these aspects.
(1) Concept Image and Definition for the Quadratic Function
It has been observed that Colin’s evoked concept image includes an image of an
absolute value function and a parabola in his mind. When the concept of quadratic
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function is heard, his image of absolute value function is activated along with a parabola.
When these two images are evoked simultaneously in Colin’s cognitive structure, it
causes a sense of conflict or confusion (Tall & Vinner, 1981), but it does not yield a
cognitive conflict because given examples and explanations indicate that Colin still has a
clear understanding of what a quadratic equation and a quadratic function are. Therefore,
Colin’s concept image of absolute value function, which is not coherently related to the
concept definition of quadratic function, contains the seeds of future conflict in his mind.
This is important because when the concept is represented in a broader context such as in
calculus, he may not be able cope when confronted with such tasks or optimization
problems.
(2a) Translating a Quadratic Function from Graph to Algebraic Form
While translating from a graph to an algebraic form, Colin used a new strategy
based on misinterpretation of the coefficients, b and c in the form of y = ax2+bx +c as if
they were x-and y-coordinates of the vertex. He first assumes that the value of the
leading coefficient, a, of y = ax2+bx+c is 1. He then writes “x2” or “-x2” based on the
opening direction of the parabola: up or down respectively. Second, he assumes that the
coefficient of x, b, is the x-coordinate of the vertex of y = ax2 +bx +c. By checking the
location of the x-coordinate of the vertex on the x-and y-coordinate axes, he determines
the value of ‘b’ and then writes the function as “y = x2 +bx.” In the last step, similar to
the previous step, he made his decision with the value of ‘c’ by checking the location of
the y-coordinate of the vertex on the x-and y-coordinate axes. Here the assumption is
that the coefficient, c, is the y-coordinate of the vertex of y = ax2 +bx +c.
(2b) Translating a Quadratic Function from Algebraic to Graphical Form
While translating from an algebraic form into a graph, Colin uses the same
strategy developed in the previous section 2a. He first determines the opening direction
of the parabola by checking the sign of the leading coefficient, a: up or down. He also
assumes that the value of leading coefficient, a, of y = ax2+bx+c is 1. Then, he locates
the x-coordinate of the vertex on the x-and y-coordinate axes by checking the value of ‘b’
in the form of y = ax2 +bx +c. In the final step, he locates the y-coordinate of the vertex
by checking the value of ‘c’ in the form of y = ax2 +bx +c. So, he drew the parabola by
using the point (b, c) as the vertex of y = ax2 +bx +c.
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(2c) A Horizontal and Vertical Shift of a Quadratic Function
When the quadratic function is presented in the vertex form, y = a (x - h)2 +k ,
Colin directly transformed it into the standard form, y = ax2 +bx +c, and then applied the
same strategy mentioned in the sections 2a and 2b.
(3) Determining the Axis of Symmetry, Vertex, and x-and y-intercepts of Quadratic
Functions
In terms of the axis of symmetry, Colin drew two graphs of parabolas, but he was
not able to correctly graph these functions. He then used the constants, 1 and 4, as if they
were the axes of symmetry of quadratic functions, y = ax2+bx +1, and y = ax2+bx+4
respectively. For the vertices of the parabolas, Colin could not find the vertices of the
parabolas in an algebraic manner. He mostly focused on the constant values and their
effects on finding the other two points, besides the vertex, to draw the parabola.
Regarding the x-and y-intercepts, he took graphical considerations into account, but his
graphical construct for the parabolas as a whole challenged him to determine if they had
the same x-and y-intercepts. To determine if certain points were on the graph of the
parabola, like Zaslavsky’s (1997) study, Colin attempted to plot the points on the x-and
y-coordinates axes. He used the eye measurement technique, and failed in his all
attempts. He assumes that the graph of quadratic function represents exact quantitative
information and the method of eye measurement can be effectively used in such
situations.
In addition, the compartmentalization phenomenon (Vinner & Dreyfus, 1989)
occurred once for the vertices of the parabolas, but no additional case was observed
throughout the study.
(4) Interpreting Quadratic Functions Represented by Graphs, Formulas, Tables and
Situation Descriptions
Colin correctly read the information from the graph of the parabola for the
coefficients, a, b, and c of y = ax2+bx+c. Colin used the given parabola to interpret the
coefficients, a, b, and c. For the y-intercept of quadratic function, however, Colin
considered the y-axis as a vertical asymptote of the parabola, although the graph of a
quadratic function does not have any asymptote. This finding is also parallel with the
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work of Zaslavsky (1997), who reported that “the graph of a quadratic function may seem
as if it has vertical asymptotes” (p. 30).
(5) Solving Quadratic Equations by Factoring, Completing the Square or Using the
Quadratic Formula
Other than forgetting to determine the x-values at the end in the interview session,
Colin correctly solved the quadratic equations. For the first task, he used the quadratic
formula to find the x-values, while he employed the method of factoring in the second
task.
(6) Using Quadratic Models to Solve Problems Presented in Real-world Situations
Colin failed to use the quadratic models to solve the problems. In three different
tasks, he did not attempt to use the quadratic models given in the problems. Instead, he
left problem blank; he checked his answer to prove if it was correct; or he just provided a
graph representing the real-world situation. He provided no evidence in terms of how to
use these models to solve the problems given in real-world situations.
On the basis of these six aspects of quadratic functions, the obstacles identified
from Colin’s data are listed in the table below:
Table 5.1: Colin’s identified obstacles in his cognitive structure
(1) The absolute value function as an image is a potential conflict factor
(2) Misinterpreting the point (b, c) as the vertex of the standard form,
y = ax2+bx+c, of quadratic function
(3) Conceiving the graph of quadratic function as if it represents precise
quantitative information
(4) Considering the y-axis as a vertical asymptote of the quadratic function
(5) Failure to use the quadratic model to solve problems given in real-world
situations
(6) Having a tendency toward using the standard form over the vertex form
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Cross-Case Analyses
In chapter IV, the within-case analyses were employed to analyze the data
obtained from Richard and Colin. Findings for each student were summarized and
presented in the tables at the end of the each case. In this section, on the basis of six
aspects of quadratic functions, these findings for each student were compared to each
other in terms of whether they have something in common.
(1) Concept Image and Definition for the Quadratic Function
With regard to concept image and definition, both Richard and Colin developed
an image that led to an obstacle in their minds. Richard’s evoked image was the
quadratic formula [x1, 2 = (–b ± √b2 -4ac) / 2a], while Colin’s evoked image was the graph
of an absolute value function. However, Colin’s image of absolute value function was
called a potential conflict factor, unlike Richard, whose image of quadratic formula was
called a cognitive conflict factor (Tall & Vinner, 1981). This is because the concept in
the case of Colin did not develop an image that yielded a cognitive conflict in his mind.
In other words, the image of absolute value function was activated in his mind without
causing a cognitive conflict. Given examples and explanations indicated that Colin had a
clear understanding of what a quadratic equation and a quadratic function were, unlike
Richard, whose image of quadratic formula led to a cognitive conflict in his mind.
(2) Translating among Multiple Representations of Quadratic Functions
As reported in Zaslavsky’s (1997) study, both Richard and Colin struggled to
translate quadratic functions from graph to algebraic form in this study. On these tasks,
Richard showed a lack of understanding when and how to use the vertex, (h, k) with the
vertex form, y = a (x - h) 2 + k, of quadratic function, while Colin developed a new
strategy based on misinterpretation of the coefficients, b and c, in the form of y = ax2+bx
+c as if they were x-and y-coordinates of the vertex. Colin later used this strategy in
other types of translation tasks as well. With regard to the horizontal and vertical shifts,
like Zazkis et al. (2003) study, Richard did not reveal an understanding of why the graphs
of parabolas horizontally or vertically shift on the x-and y-coordinate axes. Instead, he
performed a memorized, rule-based strategy without making any connection about the
way the graphs moved. On the other hand, Colin immediately transformed the given
vertex form, y = a (x - h) 2 + k, of quadratic functions into the standard form, y = ax2+bx
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+c, and then used his strategy developed on the basis of considering the point (b, c) as the
vertex of y = ax2+bx +c.
In addition, the analysis of Richard’ data on the vertex form, y = a (x - h) 2 + k, of
quadratic function and an absolute value function indicated that he uses the same strategy
to find the vertices and graph the functions while working on these two different
functions. His strategy for these functions included a similar rule based procedure
without going thorough the properties of the mathematical concept.
(3) Determining the Axis of Symmetry, Vertex, and x-and y-intercepts of Quadratic
Functions
For the axes of symmetry, Richard correctly visualized the graphs as a whole
without using any algebraic calculation. On the other hand, Colin drew two graphs of
parabolas, but he was not able to correctly locate the axis of the vertices on the x-and y
coordinate axes. He used the constant values as if they were the axes of symmetry of
quadratic functions. In terms of the vertices of the parabolas, similar to the study of
Zaslavsky (1997), Richard employed an algebraic approach by focusing only on the xcoordinate, -b/2a, of the vertices. He considered the constants in the quadratic functions,
but he took them out of the decision-making process, claiming that “constant has nothing
to do with vertex.” On the other hand, Colin focused on the constant values and their
effects on finding the other two points, besides the vertex, to draw the parabola.
Therefore, both of them failed on this task. Regarding the x-and y-intercepts, Richard
correctly answered the task by taking graphical and algebraic considerations into account.
Colin, however, struggled for a long time. He later took graphical considerations into
account, but his graphical construct for the parabolas as a whole challenged him to
determine if they had the same x-and y-intercepts.
To determine if certain points are on the graph of the parabola, unlike Zaslavsky’s
(1997) study, Richard consistently sought to find an algebraic representation for the given
parabolas. On the other hand, Colin, similar to the work of Zaslavsky (1997), directly
used the eye measurement technique to make his decisions on these points. Each time,
Colin plotted the points on the parabolas, assuming that the graph of a quadratic function
represents exact quantitative information.
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Another important observation at this part is the compartmentalization
phenomenon (Vinner & Dreyfus, 1989). The analyses revealed that Richard has
conflicting schemes (algebraic and graphical) that lead to an obstacle in his cognitive
structure. Related to the vertex of the parabola, either a graphical or algebraic scheme
was activated in Richard’s mind at different situations. This led him to reach two
different (correct and incorrect) responses for the same task in a short time. A similar
case was mentioned in the work of Zaslavsky (1997), but it was for the axis of symmetry,
not for the vertex of the parabola. On the other hand, in the case of Colin, this situation,
the compartmentalization phenomenon, for the vertex of the parabola occurred only once;
and no additional case was observed throughout the study.
(4) Interpreting Quadratic Functions Represented by Graphs, Formulas, Tables and
Situation Descriptions
Both Richard and Colin correctly read the information from the graph of the
parabola for the coefficients, a, b, and c of y = ax2+bx+c. Richard, like Colin, used the
given parabola to answer the questions for the coefficients, a and c. For the coefficient,
b, however, Richard employed an algebraic approach by applying the formula of -b/2a,
while Colin interpreted the location of the parabola’s vertex on the parabola.
With regard to the y-intercept of quadratic function, unlike the study of Zaslavsky
(1997), Richard correctly interpreted the graph in a correct graphical construct and
claimed that the parabola would intercept with the y-axis in somewhere below the x-axis.
Colin, on the other hand, parallel with Zaslavsky’s (1997) work, considered the y-axis as
a vertical asymptote of the parabola, even though the graph of a quadratic function does
not have any asymptote.
(5) Solving Quadratic Equations by Factoring, Completing the Square or Using the
Quadratic Formula
Richard and Colin were able to correctly solve the given quadratic equations. In
two different tasks, they both used the same strategies in the process: the method of
factoring and quadratic formula. However, by using the method of factoring to solve the
quadratic equation, Colin was initially confused. He wanted to substitute 0 for y at the
beginning of the process. After the completion of factoring, he left the equation that way
without finding the x-values at the end.
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(6) Using Quadratic Models to Solve Problems Presented in Real-world Situations
Richard showed a clear understanding of how to use the quadratic models to solve
the problems given in real-world situations. In three different tasks, he correctly solved
the problems and interpreted the coefficients in the model. On the other hand, Colin
failed to use the quadratic models to solve the problems. In three different tasks, he did
not attempt to use the quadratic models presented in the problems. He provided no
evidence in terms of how to use these models to solve the problems given in real-world
situations.
In addition, Richard and Colin’s strong tendency towards using the standard form,
2
y = ax +bx+c, of quadratic function over the vertex form, y = a (x - h) 2 + k has been
observed throughout the study. When they were presented with the vertex form, y = a (x
- h) 2 + k, of quadratic function, they both immediately transformed it into the standard
form, y = ax2+bx+c before going to the next step.
In summary, Richard is a smart, articulate student with a good deal of
mathematics knowledge. He employed both graphical and algebraic strategies while
working on quadratic tasks during the study. He was comfortable with the tasks
represented either graphically or algebraically. In working on quadratic tasks, Richard
mostly synthesized these two approaches. He enriched his algebraic approach with
graphical structures as well as his graphical approach with algebraic symbols or formulas
in an ongoing back-and-forth process. However, when a disequilibrium occurred in this
process, it led to certain obstacles in his cognitive structure. Therefore, most of his
obstacles arose as a result of the disequilibrium between his algebraic and graphical
thinking. For example, for the vertex of the parabola, in five different situations, either a
graphical or algebraic scheme was activated in his cognitive structure. While he was
employing a graphical approach in two situations, he used an algebraic approach in other
three situations. And his responses to the task fell into two different categories. In
addition, he could not relate the vertex, (h, k) with the vertex form, y = a (x - h) 2 + k, of
quadratic function in translating from the graph into the algebraic form.
On the other hand, Colin mostly took graphical considerations into account and
worked on provided figures, or parabolas to answer the tasks. Because of his lack of
algebraic structure of the concepts, he could not make connections between his graphical
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and algebraic strategies. Therefore, most of Colin’s obstacles arose as a result of his lack
of making connections between algebraic and graphical aspects of the concepts. For
example, he thought that the y-axis was a vertical asymptote for the graph of a parabola.
He also only used the eye measurement technique to determine if certain points were on
the graph. During the interviews, he mostly looked confused and unsure about his
responses to the tasks.
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CHAPTER V
MAJOR FINDINGS AND DISCUSSIONS
In chapter IV, I attempted to describe, analyze and interpret the mathematical
constructions of two honor students, Richard and Colin, in the learning of six aspects of
quadratic functions: concept image and definition, translating, determining, interpreting,
solving quadratic equations, and using quadratic models. In these processes, certain
obstacles were identified in the students’ cognitive structures. The obstacles that each
student encountered as working on quadratic tasks were listed at the end of each case. In
this chapter, these cognitive obstacles which emerged from the whole data are discussed
in detail for each student.
Richard’s Cognitive Obstacles
(1) The quadratic formula as an image is a potential cognitive conflict
Analyses showed that when the concept of quadratic function is heard, it yields an
image of quadratic formula along with a parabola in Richard’s cognitive structure. In
different times, he described his understanding of quadratic function both as a function,
“I think y equals ax square plus bx plus c [y= ax2+bx+c]” and a formula “x equals
negative b plus and minus, all that stuff [x = (–b ± √b2 -4ac) / 2a].” Each time, his image
of the quadratic formula of x = (–b ± √b2 -4ac) / 2a was activated in the cognitive
structure along with y = ax2 +bx +c. So, it appeared that there was an ongoing interaction
between the two images in the student’s mind (Vinner, 1983). Having two simultaneous
evoked images for the concept caused an actual cognitive conflict in Richard’s mind (Tall
& Vinner, 1981). In other words, his concept image of quadratic formula, which is not
coherently related to the concept definition of quadratic function, causes a conflict in
Richard’s cognitive structure. This is critical because when the concept is presented in a
broader context such as in calculus, he may not be able cope when confronted with such
tasks or optimization problems. This could be a result of the teaching method used by the
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teacher who gave examples by employing quadratic formula for a long period of time
while working with the general notion of quadratic functions. This is also supported by
the work of Tall and Bakar (1992), who concluded that teaching the concept through
typical examples leads mental prototypes that give erroneous impressions of the general
idea of the concept. In other words, the student has been exposed to the quadratic
formula, x = (–b ± √b2 -4ac) / 2a, far more than to the standard form, y = ax2 +bx +c in
the learning of quadratic functions. A well-designed set of tasks or examples that
emphasize the distinction between the two might help students to reconstruct or modify
their mental images related to the concept.
(2) A lack of understanding about when and how to use the vertex, (h, k) with
the vertex form, y = a (x - h) 2+k, of quadratic functions
In translating the graph of a parabola into an algebraic representation, the analyses
indicated that Richard consistently picked up the vertex, (h, k) first and substituted that
into the standard form, y = ax2 +bx +c except for one case. But, in all attempts, he failed
to write an algebraic representation for given parabolas, like the study of Zaslavsky
(1997) who reported that students her study worked in the same direction from algebraic
to graphical when asked to determine which of four parabolas correspond to a given
equation and which of four equations correspond to a given parabola.
Since the vertex form itself, y = a (x - h) 2+k, directly indicates the location of the
vertex, (h, k), employing the vertex, (h, k) with the vertex form, y = a (x - h) 2+k, could
have been a more effective and simpler approach in these situations. However, he could
not properly connect his algebraic thinking with provided graphical information. It
seemed to be lacking of making connections between graphical and algebraic thinking in
terms of when and how to use the vertex, (h, k) with the vertex form, y = a (x - h) 2+k, of
quadratic functions. In addition, presenting the vertex form, y = a (x - h)2 +k, as just
another form of quadratic function without sufficiently emphasizing or revealing the
underlying thought might have contributed to this obstacle as well. A well designed set
of activities or examples that emphasize about when and how to use the vertex, (h, k)
with the vertex form, y = a (x - h) 2+k, by utilizing of the student’ algebraic and graphical
thinking may help to strengthen this understanding.
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(3) A lack of understanding of why a parabola horizontally or vertically
shifts
With regard to horizontal and vertical shifts, Richard did not reveal an
understanding of why the graphs of parabolas shift horizontally or vertically. Instead, he
performed a memorized, rule-bound procedure that took him to the correct answer
without knowing why. In other words, he could not make any connection about the way
the graphs shifted. He showed an instrumental understanding relying on memorized rules
(Skemp, 1993). This is also consistent with the work of Zazkis et al. (2003) who
concluded that students really do not know the mathematical reason behind a horizontal
and vertical shift of a quadratic function. This could be explained by a lack of making
(cognitive) connections between algebraic and graphical aspect of the concepts.
(4) The phenomenon of compartmentalization: Having two different,
potentially conflicting algebraic and graphical schemes in the cognitive structure
In terms of the vertex of the parabola, different situations stimulated different
schemes in Richard’s mind. In five different situations, either a graphical or algebraic
scheme was activated in the student’s cognitive structure. In two situations, Richard
provided a correct vision for the vertices of the parabola when answering the questions
related to the axis of symmetry and the x-intercepts of the parabola. However, in other
three situations, when employing the algebraic approach for the vertex of the parabola, he
could not come up with the correct answer in two situations and was confused with it on
the third situation. Even though all these cases occurred in a short time, the student did
not notice the contradiction of his two responses. According to Vinner and Dreyfus
(1989), this is an indication of the compartmentalization phenomenon; in other words,
Richard’s graphical and algebraic schemes are conflicting with each other in his cognitive
structure. A similar case was reported in the work of Zaslavsky (1997), but it was for the
axis of symmetry, not for the vertex of the parabola.
This may have been caused by a disequilibrium between Richard’s algebraic and
graphical thinking. The study revealed that Richard does not have a tendency one way or
another and indeed can employ both algebraic and graphical thinking while working on
mathematical tasks. Throughout the study, he simultaneously used graphical and
algebraic strategies on given tasks in a back-and-forth process. He showed no difficulty
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working with the tasks represented either graphically or algebraically. However, when a
disequilibrium occurred in the process of algebraic and graphical thinking, he shifted his
focus to either the algebraic or graphical approach and used one of these approaches
separately in that situation. So, different situations stimulated different schemes and
Richard’s responses fell into two different categories.
This is crucial for teachers in the learning of any particular concept because
knowing about the conflicting schemes activated in the same person requires teachers to
be more sensitive to students’ reactions in the learning process. In light of this
knowledge, teachers can be open and more prepared to communicate with students in
terms of confronting, discussing, and dealing with these conflicting schemes.
(5) Having a tendency toward using the standard form over the vertex form
As shown in the analyses of his work, Richard presents a clear preference for the
standard form, y= ax2+bx+c, over the vertex form, y = a (x - h) 2 + k. Throughout the
study, whenever Richard is presented with the vertex form of quadratic function, he
immediately transformed it into the standard form. This is also supported by the work of
Zaslavsky (1997) who reported that students in her research preferred the standard form,
y= ax2+bx+c over the vertex form, y = a (x - h) 2 + k.
This tendency could be a result of the learning sequence of the concept. At the
very beginning of the chapter, students were first introduced to the standard form, y =
ax2+bx+c, of a quadratic function. Right after that, they were routinely given tasks that
required transformation from the expressions in the parentheses to the standard form;
students initially opened or squared the parentheses and put together the similar terms
(e.g., x2’s, x’s, and constants) to determine if functions were a quadratic, linear, or
constant. This sequence, recognizing the standard form first and then converting the
expressions in the parentheses into the standard form, created a routine in students’ work
with quadratic functions. As a result, the standard form of a quadratic function became a
predominant form over the vertex form, y = a (x - h) 2 + k, which was considered as an
expression in the parenthesis; and it was supposed to be converted into the standard form.
This immediate transformation process prevents students from using the vertex form on
certain tasks (e.g., translating a graph into the algebraic form) in which the vertex form
clearly provides a more effective and simpler approach than the standard form. This
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could be avoided by an alternative sequencing including a simultaneous introduction of
both the standard form and vertex form of a quadratic function at the beginning of the
chapter. This also helps students to understand the mathematics behind the vertex form.
Colin’s Cognitive Obstacles
(1) The absolute value function as an image is a potential conflict factor
The analyses showed that when the concept of quadratic function is heard, Colin’s
image of absolute value function is activated along with a parabola. Since these two
images are evoked simultaneously in Colin’s cognitive structure, it causes him a sense of
conflict or confusion for the concept of quadratic function (Tall & Vinner, 1981). In
other words, his concept image of absolute value function, which is not coherently related
to the concept definition of quadratic function, contains the seeds of future conflict in his
mind (Tall & Vinner, 1981). At these situations, Colin’s image of absolute value
function is activated in his mind without causing a cognitive conflict. Given examples
and explanations indicated that Colin has a clear understanding of what a quadratic
equation and a quadratic function are. But, later on when the concept is presented in a
broader context such as in calculus, he may not be able to cope when confronted with
such tasks or optimization (max/min) problems. So, a well-designed set of tasks or
examples that emphasize the distinction between the two might help students to
reconstruct or modify their mental images related to the concept.
(2) Misinterpreting the point (b, c) as the vertex of the standard form, y = ax2+bx+c,
of quadratic functions.
The analyses indicated that Colin uses a particular strategy to translate quadratic
functions from one form to another. His approach includes three steps. While translating
from a graph to an algebraic form, he first assumes that the value of leading coefficient,
a, of y = ax2+bx+c is 1. He then writes “x2” or “-x2” based on the opening direction of
the parabola: up or down respectively. Second, he assumes that the coefficient of x, b, is
the x-coordinate of the vertex of y = ax2 +bx +c. By checking the location of the xcoordinate of the vertex on the x-and y-coordinate axes, he determines the value of ‘b’
and then writes the function as “y = x2 +bx.” In the last step, similar to the previous step,
he made his decision on the value of ‘c’ by checking the location of the y-coordinate of
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the vertex on the x-and y-coordinate axes. The assumption here is that the coefficient, c,
is the y-coordinate of the vertex of y = ax2 +bx +c.
While translating from an algebraic form into a graph, Colin uses the same
assumptions mentioned above. He first decides the opening direction of the parabola, up
or down, by checking the sign of the leading coefficient, a. At this point, he again
assumes that the value of leading coefficient, a, of y = ax2+bx+c is 1. Second, he locates
the x-coordinate of the vertex on the x-and y-coordinate axes by checking the value of ‘b’
in the form of y = ax2 +bx +c. In the final step, he locates the y-coordinate of the vertex
by checking the value of ‘c’ in the form of y = ax2 +bx +c, and then completes the whole
translation process. When the quadratic function is presented in the vertex form, y = a (x
- h)2 +k , he directly transforms it into the standard form, y = ax2 +bx +c, and then applies
the same strategy explained above.
In these processes, his approach, a connection built between the point (b, c) and
the standard form of y = ax2 +bx +c, is similar to the idea used between the vertex, (h, k),
and the vertex form, y = a (x - h)2 +k, of quadratic function. As documented in some
students’ homework assignments, students generally tend to think that they may
transform the standard form, y = ax2 +bx +c, of quadratic function into the vertex form by
writing, “y = x (ax +b) +c.” Then, they use this form, y = x (ax +b) +c, just as the vertex
form, y = a (x - h)2 +k, of quadratic function. In other words, they relate the vertex (h, k)
in the form of y = a (x - h)2 +k to the point (b, c) in the form of y = x (ax +b) +c as if the
point (b, c) is the vertex of quadratic function. For example, the function of y = x2+5x+6
is first transformed into the vertex form as “y = x (x +5) + 6.” Then, the point (5, 6) in
the form of y = x (x +5) + 6 is taken as the vertex of y = x2+5x+6 by relating to the vertex
form of quadratic function. This could be attributed to the student’s tendency to make an
unfamiliar idea more familiar. This could have been avoided by using a well-designed
set of examples or activities that emphasize the difference between y = x (ax +b) +c and y
= a (x - h)2 +k as well as emphasize or reveal the underlying thoughts that generated the
vertex form, y = a (x - h)2 +k ,of quadratic function.
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(3) Conceiving the graph of quadratic function as if it represents precise
quantitative information
Colin uses the eye measurement technique to determine if certain points are on
the parabola. He assumes that the graph of quadratic function represents precise
quantitative information; and the method of eye measurement could effectively be used in
such situations. Actually, the graph of a parabola does not necessarily represent exact
information other than highlighted points on the graph. So, such an approach may not
provide a definite answer. This is also mentioned in the study of Zaslavsky (1997) who
found that students in her study determined whether certain points were on a graph based
only on the method of eye measurement.
Here Colin’ lack of algebraic understanding of the concepts led him to take
graphical considerations into account and work on the given graphs to answer the tasks.
In other words, he could not use algebraic components of the concept in his graphical
approach; and this prevented him from developing a synthesis of algebraic and graphical
strategies. So, Colin’s graphical understanding needs to be enriched by appropriate
algebraic strategies to enhance his understanding of the concept and overcome the
obstacle that arose in this process.
(4) Considering the y-axis as a vertical asymptote of the quadratic function
The analyses showed that Colin assumes that some quadratic functions may have
a vertical asymptote, even though the graph of a quadratic function does not have any
asymptote. This might have been due to a combination of Colin’s evoked image of
inverse function, 1/x, and lack of algebraic understanding of the concept. Since he was
lacking of algebraic components of the concept, he worked on the given parabola with an
image of1/x that gets closer and closer to the y-axis without touching it. This thought
process seemed to be creating an obstacle to him. This is also consistent with the work of
Zaslavsky (1997) who reported that “the graph of a quadratic function may seem as if it
has vertical asymptotes” (p. 30). This could be avoided by teaching Colin how to
manipulate the algebraic symbols or formulas with graphical strategies in the process of
answering this type of quadratic tasks.
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(5) Failure to use the quadratic model to solve problems given in real-world
situations
The analyses from Colin’s data indicated that he does not know how to use
quadratic models to solve the problems given in real-world situations. In attempting to
solve these problems, it is necessary to use the quadratic models given in the problems.
However, in three different tasks, he failed to use these quadratic models to solve the
problems. Instead, he left the problem blank in the quiz; he checked his answer to prove
whether it was correct in the questionnaire; and he provided a downward parabola with
correct interpretations of the coefficients of the model in the interview.
Such a case might result from a combination of Colin’s lack of algebraic
understanding and the teacher’s teaching strategy. In similar problems, a strategy used
by the teacher included a procedure of finding the vertex from the model without having
any, or a very limited discussion with students in terms of how the problem was related to
the quadratic model or how the problem fitted the model. Not making connections
between the problem and the model used for it led to isolated and unconnected
knowledge in the student’s cognitive structure. Later on, this turns out to be lost at some
point (Haylock, 1982). Colin’s lack of algebraic understanding contributed to this
process by preventing him from making connections between algebraic and graphical
aspects of the concept. When the student was presented with the similar task later on, he
did not know what to do with the quadratic model given in the problem. This could have
been avoided by utilizing the student’s graphical and algebraic thinking and developing a
rich synthesis of them in the learning process.
(6) Having a tendency toward using the standard form over the vertex form
The analyses indicated that Colin has a clear tendency to use the standard form,
y= ax2+bx+c, quadratic function over the vertex form, y = a (x - h) 2 + k. Throughout the
study, when he came cross with the vertex form of quadratic function, he immediately
transformed it into the standard form. This is also consistent with the study of Zaslavsky
(1997) who reported that students in her research preferred the standard form over the
vertex form, y = a (x - h) 2 + k.
As mentioned in the case of Richard, this could be a result of the learning
sequence of the concept. At the very beginning of the chapter, students were first
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introduced to the standard form, y = ax2+bx+c, of a quadratic function. Right after that,
they were routinely given tasks that required transformation from the expressions in the
parentheses to the standard form; students initially opened or squared the parentheses and
put together the similar terms (e.g., x2’s, x’s, and constants) to determine if functions
were a quadratic, linear, or constant. This sequence, recognizing the standard form first
and then converting the expressions in the parentheses into the standard form, created a
routine in students’ work with quadratic functions. As a result, the standard form of a
quadratic function became a predominant form over the vertex form, y = a (x - h) 2 + k
which was considered as an expression in the parenthesis. This could be avoided by an
alternative sequencing including a simultaneous introduction of both the standard form
and vertex form of a quadratic function at the beginning of the chapter. This also helps
students to understand the mathematics behind the vertex form.
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CHAPTER VI
CONCLUSIONS
The over-arching, ultimate end of the whole enterprise is to promote/improve
students’ learning of mathematics (Niss, 1999, p. 5).
In this qualitative study, two algebra-honor students’ cognitive obstacles in the
learning of quadratic functions have been investigated during a four-week study. Each
student, Richard and Colin, was purposely selected and interviewed twice. In addition to
interviews, the students’ written work consisting of a test, quiz, questionnaire, and
homework assignments were examined with regard to their incorrect or incomplete
answers. The students were also observed on a daily basis in the classroom throughout
the study. Data obtained from interviews, the students’ written work and classroom
observations were analyzed based on the following six aspects of quadratic functions:
concept image and definition, translating, determining, interpreting, solving quadratic
equations, and using quadratic models.
The study revealed the cognitive obstacles that both algebra-honor students,
Richard and Colin, encountered during the study of quadratic functions. Note that no
single piece of data was considered in the process of analysis unless it could be
triangulated. Richard’s identified obstacles are as follows:
(1) The quadratic formula as an image is a potential cognitive conflict
(2) A lack of understanding about when and how to use the vertex, (h, k) with the
vertex form, y = a (x - h) 2 + k, of quadratic function
(3) A lack of understanding of why a parabola horizontally or vertically shifts
(4) The phenomenon of compartmentalization: having two different, potentially
conflicting algebraic and graphical schemes in the cognitive structure
(5) Having a tendency toward using the standard form over the vertex form
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Colin’s identified obstacles are as follows:
(1) The absolute value function as an image is a potential conflict factor
(2) Misinterpreting the point (b, c) as the vertex of the standard form, y =
ax2+bx+c, of quadratic function
(3) Conceiving the graph of quadratic function as if it represents precise
quantitative information
(4) Considering the y-axis as a vertical asymptote of the quadratic function
(5) Failure to use the quadratic model to solve problems given in real-world
situations
(6) Having a tendency toward using the standard form over the vertex form
In light of these obstacles, the following four assertions are made in this study:
(1) One of the cognitive obstacles arises from a lack of making and investigating
mathematical connections between algebraic and graphical aspects of the concepts.
In the case of Richard, this situation appeared to be a lack of understanding about
when and how to use the vertex, (h, k) with the vertex form, y = a (x - h) 2+k, of quadratic
functions as well as not knowing about why the graph of a parabola horizontally or
vertically shifts. On the other hand, in the case of Colin, this situation appeared to be in a
different form. Because of Colin’s lack of algebraic understanding of the concepts, he
mostly took graphical considerations into account and worked on provided figures, or
parabolas to answer the tasks. Therefore, this presented different obstacles to Colin. For
example, he thought that the y-axis was a vertical asymptote for the graph of a parabola.
In addition, he only used the eye measurement technique to determine if certain points
were on the graph. When confronted with the problems given real world situations, he
never attempted to use the quadratic models to solve these problems. He provided no
evidence to relate the model with the problem in an algebraic manner. This idea that
underlies the students’ cognitive obstacles is also mentioned by the work of Arcavi
(2003) and Schoenfeld, Smith and Arcavi (1993).
141
(2) Another cognitive obstacle arises from the need to make an unfamiliar idea more
familiar
In this study, Colin uses a particular strategy to translate quadratic functions from
one form to another. His approach includes three steps. While translating from a graph
to an algebraic form, he first assumes that the value of leading coefficient, a, of y =
ax2+bx+c is 1. He then writes “x2” or “-x2” based on the opening direction of the
parabola: up or down respectively. Second, he assumes that the coefficient of x, b, is the
x-coordinate of the vertex of y = ax2 +bx +c. By checking the location of the xcoordinate of the vertex on the x-and y-coordinate axes, he determines the value of ‘b’
and then writes the function as “y = x2 +bx.” In the last step, similar to the previous step,
he made his decision on the value of ‘c’ by checking the location of the y-coordinate of
the vertex on the x-and y-coordinate axes. The assumption here is that the coefficient, c,
is the y-coordinate of the vertex of y = ax2 +bx +c.
While translating from an algebraic form into a graph, Colin uses the same
assumptions mentioned above. He first decides the opening direction of the parabola, up
or down, by checking the sign of the leading coefficient, a. At this point, he again
assumes that the value of leading coefficient, a, of y = ax2+bx+c is 1. Second, he locates
the x-coordinate of the vertex on the x-and y-coordinate axes by checking the value of ‘b’
in the form of y = ax2 +bx +c. In the final step, he locates the y-coordinate of the vertex
by checking the value of ‘c’ in the form of y = ax2 +bx +c, and then completes the whole
translation process. When the quadratic function is presented in the vertex form, y = a (x
- h)2 +k , he directly transforms it into the standard form, y = ax2 +bx +c, and then applies
the same strategy explained above.
In these processes, Colin’s approach, a connection built between the point (b, c)
and the standard form of y = ax2 +bx +c, is similar to the idea used between the vertex,
(h, k), and the vertex form, y = a (x - h)2 +k, of quadratic function. As documented in
some other students’ homework assignments, students tend to think that they can
transform the standard form, y = ax2 +bx +c, of quadratic function into the vertex form by
writing, “y = x (ax +b) +c.” Then, they use the form of y = x (ax +b) +c just as the vertex
form, y = a (x - h)2 +k, of quadratic function. In other words, they relate the vertex (h, k)
in the form of y = a (x - h)2 +k to the point (b, c) in the form of y = x (ax +b) +c as if the
142
point (b, c) is the vertex of quadratic function. For example, the function of y = x2+5x+6
is first transformed into the vertex form as “y = x (x +5) + 6.” Then, relating with y = a
(x - h)2 +k, the point (5, 6) in the form of y = x (x +5) + 6 is taken as the vertex of y =
x2+5x+6. Therefore, Colin’s thought processes on this obstacle could be attributed to his
tendency to make an unfamiliar idea more familiar. This idea is also consistent with
Davydov’s theory (1990), Hershkowitz, Schwarz and Dreyfus’ (2001) perspective, and
Hazzan and Zazkis’s (2005) theory.
(3) A third cognitive obstacle arises from the disequilibrium between algebraic and
graphical thinking
Throughout the study, it has been observed that Richard uses both algebraic and
graphical approaches while working on quadratic tasks. He does not have a tendency one
way or the other in those situations. In other words, he effectively uses both algebraic
and graphical strategies on different mathematical tasks in a good deal of mathematical
knowledge. However, when a disequilibrium occurred between his algebraic and
graphical thinking, his focus shifts to either an algebraic or graphical approach and then
he applies one of the approaches separately in a given situation. If this situation is not reequilibrated, that leads to inconsistent or irrelevant behaviors in his responses that fell
into two different categories.
For example, in the case of Richard, in five different situations, either a graphical
or algebraic scheme was activated for the vertex of the parabola in the student’s cognitive
structure. In two situations, he applied a graphical approach for the vertices of the
parabolas when answering the questions related to the axis of symmetry and the xintercepts of the parabola. However, in the other three situations, he employed an
algebraic approach for the vertex of the parabola and could not correctly answer the
questions or, at least was confused about them. According to Tall and Vinner (1981), this
is the indication of the compartmentalization phenomenon. In other words, this occurs as
a result of the student’s conflicting algebraic and graphical schemes in his mind. The
present study showed that there is an ongoing back and forth interaction between
algebraic and graphical thinking; these two thoughts are interrelated and it is hard to
distinguish between them. This idea is also supported by the work of Zazkis, Dubinsky,
& Dautermann (1996), Krutetskii, (1976), and Presmeg (1986a, 1986b).
143
(4) The image of the quadratic formula or absolute value function has a potential to
create an obstacle to mathematical learning.
In the case of Richard, the study revealed that the student’s evoked images
included both the quadratic formula, x = (–b ± √b2 -4ac) / 2a and the standard form, y =
ax2 +bx +c, of quadratic function. In other words, whenever the concept is heard, these
two images are evoked simultaneously in his mind. So, his image of quadratic formula,
which is not coherently related to the concept definition of quadratic function, caused an
actual cognitive conflict (Tall & Vinner, 1981).
On the other hand, in the case of Colin, an image of absolute value function is
activated along with a parabola without causing a cognitive conflict. So, his image of
absolute value function, which is not coherently related to the concept definition of
quadratic function, contained the seeds of future conflict in his mind (Tall & Vinner,
1981). This is crucial because when the concept is presented in a broader context such as
in calculus, he may not be able cope when confronted with such tasks or optimization
(max/min) problems.
Learning and Instruction
These results suggest that both students’ graphical and algebraic thinking need to
be utilized and integrated in order to construct a deep mathematical understanding of the
concepts. Traditionally, algebraic and graphical thinking are considered as two separate
thought processes as well as different learning styles in mathematic curricula. Thus,
curriculum developers and textbook authors provided graphical tools to support a
person’s graphical learning (Eisenberg & Dreyfus, 1991). In the same way, it also
entailed providing algebraic tools to a person who has an algebraic learning style in an
assumption that students like to work with strategies they are most comfortable with.
The present study revealed that this traditional perspective emphasizing two separate
thoughts and learning styles in teaching and learning of mathematics can cause obstacles,
even to honor students. Therefore, in light of the findings of this study, the pedagogical
approach in new curricular or mathematics textbooks can focus on the interrelation and
complementary aspect of algebraic and graphical thinking in an ongoing back-and-forth
process in learning quadratic functions. There is a need to continuously relate algebraic
thinking with graphical strategies and graphical thinking with algebraic strategies for any
144
particular concept. In particular, a learner needs algebraic thinking in determining what
to graph, and also that learner constructs a graphical image that determines what
algebraic symbols to write. In the classroom, it is the teacher’s responsibility to create an
environment in which students can utilize both their algebraic and graphical thinking and
develop a rich synthesis of them. This is also emphasized by the Principles and
Standards for School Mathematics (NCTM, 2000) that calls for instructional programs
that provide students opportunities that enable them to recognize, use and understand
connections among mathematical ideas.
In addition, this study revealed that the selection of a problem or mathematical
task is crucial. In order to monitor students’ thinking, identify their knowledge structure,
and address the obstacles that emerge in these processes, teachers have to consider
discussing their students’ thought processes while working on mathematical tasks as well
as to include assessments consisting of well-designed tasks, problems or activities that
encourage students to reveal their more versatile thinking or deep mathematical
understanding. This is supported by the Principles and Standards for School
Mathematics (NCTM, 2000) which in the problem-solving standard, calls for classroom
teachers to choose meaningful problems and mathematical tasks.
With regard to its theoretical contribution, the present study revealed two algebra
honor students’ cognitive obstacles surrounding quadratic functions through the lens of
an integrated approach using Schoenfeld’s (1989) level of mathematical analysis and
structure and Tall and Vinner’s (1981) framework of concept image and concept
definition. This approach provided an effective tool to analyze the students’
understanding. While Schoenfelds’ table helped me to describe students’ understanding
from both the standard curricular perspective and the students’ perspective as they work
on mathematical tasks, Tall and Vinner’s framework made possible for me to connect the
students’ understanding with their activated mental images. In this manner, this
combination proved to be an effective means of relating to these students’ understanding
and the obstacles that arose in these processes.
Implications for Teaching
This qualitative study revealed two algebra-honor students’ cognitive obstacles
surrounding quadratic functions. Therefore, this has direct implications for classroom
145
teaching. These obstacles inform mathematics teachers about weaknesses and strengths
of students’ thought processes in the learning of quadratic functions. In particular, they
indicate which obstacles are associated with certain aspects of quadratic functions.
Taking into account these obstacles in a lesson being taught can help teachers to improve
students’ understanding of the concept. By developing new lesson plans and teaching
strategies, teachers may help students overcome, or at least minimize these identified
obstacles in the learning process.
The present study suggests a new perspective related to algebraic and graphical
thinking that is somewhat different from the pervious one. Thus, this is also a call for
curriculum developers and textbook authors to implement this perspective to see if it
makes any difference. A new curriculum that emphasize the interrelation and
complementary aspect of algebraic and graphical thinking in an ongoing back-and-forth
process may change the nature of understanding and the type of the obstacles that may
arise in the process. In the classroom, teachers can create an environment that
encourages students to utilize both their algebraic and graphical thinking in accordance
with one another. This enables students to construct a deeper understanding of the
concepts (Eisenberg & Dreyfus, 1991; Haylock, 1982). Note that teachers do things
differently when given new textbooks to deliver their lessons (Remillard, 2000).
The next implication is related to students’ assessment. This research showed the
importance of in-depth analysis of students’ thinking. In assessing students’
understanding of a particular concept, teachers can discuss with students their thought
processes on given tasks and activities that reveal students’ more versatile thinking or
deep mathematical understanding. In this way, they can monitor their students’ thinking,
identify their knowledge structure, and address their cognitive obstacles that might
emerge in these processes. Such analyses are not accessible to teachers through the
written work or direct observation. Otherwise, assessing students’ general level of
understanding with the similar tasks previously solved in the classroom may not provide
opportunities for teachers to reach students’ high level thinking and identify their
obstacles that may arise in these processes.
Another implication is connected with the phenomenon of compartmentalization.
As shown in the case of Richard, several conflicting schemes may act in students’
146
cognitive structure in different situations that are closely related in time. So, this informs
teachers about students who may hold conflicting schemes in their mind. Knowing about
these conflicting schemes activated in the same person makes teachers more sensitive to
students’ reactions in the learning process. Teachers can be open and more prepared to
communicate with students in terms of confronting, discussing, and dealing with these
conflicting schemes. With the help of teachers, students may reconstruct their
inconsistent or irrelevant behaviors with regard to their responses that fall into two
different categories, and develop a deeper understanding of related concepts.
Limitations of the Study
This qualitative study intended to identify algebra honor students’ cognitive
obstacles surrounding quadratic functions. In particular, by selecting two students from
an algebra-2-honor class, the researcher examined students’ mathematical constructions
under the six aspects of quadratic functions, and attempted to identify their obstacles that
arose in these processes. Data were obtained from one-on-one interviews, students’
written work (the quiz, test and questionnaire) and classroom observations. During the
interviews, the students’ thought processes were carefully monitored as they were
working on different quadratic tasks. Thus, the findings of this research study are
dependent to a great extent on this method of inquiry which gave the researcher an
opportunity to explore the students’ thinking. In other words, if this research had been
conducted in a regular classroom with average students, in small groups as they work
together, or in a computer lab environment where mathematics software or graphing
calculators were available, these findings, or the students’ cognitive obstacles, could have
been different.
In the present study, a thick (or rich) description was provided so that other
researchers could decide if the findings of this research were applicable or transformable
to their own situations. The obstacles identified depend, to a large extent, on each student
as an individual because they are intrinsically related to each student’s level of
understanding and previously built knowledge structure. They also depend on the
researcher who is making the decision when the student struggles or makes mistakes on
given mathematical tasks. In addition, the researcher acknowledged that after completion
of the interviews, taking only the transcripts back to the students without including the
147
researcher’s analyses and conclusions led to a limitation on the procedure of member
checks, even though the students agreed with what had been written in the transcripts.
Issues for Future Research
This qualitative study revealed which obstacles are associated with certain aspects
of quadratic functions. So, continued research is necessary to determine instructional
designs for overcoming these identified obstacles as well as developing students’ deep
mathematical understandings regarding quadratic functions.
This study has also shown that two honor students, Richard and Colin, have an
incorrect or inappropriate image of quadratic function in their mind: a quadratic formula
and an absolute value function respectively. Knowing what other images, correct or
incorrect, a student might have will be useful for teachers to promote students’
understanding.
Another important issue is related to the phenomenon of compartmentalization.
As shown in this study, students may have conflicting schemes that may block their
understanding of particular concepts and lead to an obstacle in their cognitive structure.
So, it is worthwhile to research the following questions:
(1)
How often does the compartmentalization phenomenon occur in
mathematics learning?
(2)
Does this happen to only honor or high-achiever students?
(3)
Are these cases limited to the concepts that can be both graphically and
algebraically represented?
Lastly, the present study did not focus on the effect of use of technologies (i.e.,
computers, graphing calculators) in the learning of quadratic functions. Thus, it would be
worthwhile to find out how technology supported environment changes the nature of
these six main quadratic aspects, the development of students’ constructions, and the
nature of cognitive obstacles that may arise in these processes.
148
APPENDIX A
APPROVAL OF THE HUMAN SUBJECTS COMMITTEE
149
APPENDIX B
150
APPENDIX C
151
APPENDIX D
FIVE ASPECTS OF QUADRATIC FUNCTIONS IN THE TEXTBOOK
D.1: Translating Among Multiple Representations of Quadratic Functions
Graph each function. (Bellman et al., 2004, p. 251)
1. y = (x-1)2 + 2
2. y = (x+3)2 - 4
5. y = 2(x-2)2 + 5
4. y = 2(x +1)2
7. y = -3(x +7)2 - 8
8. y = - ½ (x-2)2 + 1
9. y = (x-5)2 – 3
10. y = (x+2)2 - 3
11. y = -(x-7)2 + 10
12. y = -4(x+8)2 – 6
Write the equation of each parabola in vertex form (p. 251)
y
(0, 4) y
4
(1, 3)
2
(2,1)
-3
(0, 0 )
y
-2
0
2
x
x
-3
(2, 0)
-1
x
4
1
3
(-2, 0)
6
-4
y
0
(-3,-1)
-2
-4
(-4, 4)
-4
152
x
Graph each function (p. 244)
1. y= -x2 +1
2. y= -x2 -1
3. y= 2x2 +4
4. y= 3x2 - 6
5. y= -1/3x2 -1
6. y= -5x2 +12
7. y= 1/2x2 +3
8. y= 1/4x2 -3
9. y= -2x2 +3/4
Write the equation of each parabola in vertex form (p.252)
0
Y
-4
(0,4)
y
-2
2
4 x
(1,-2)
-2
2
x
-4
-2
0
(2 ,0)
4
6
(17)
(18)
y
(-2, 4)
Y
3
2
1
x
-5
-4
0
-2
2
(2, 1)
0
2
4
(3,-2)
-2
(-3,-2)
(19)
(20)
153
x
D.2: Determining the axis of the symmetry, vertex and x-and y-intercepts of the
quadratic functions
Graph each function. Label the vertex and the axis of symmetry (Bellman et al., 2004,
p.244).
10. y= x2 +2x +1
11. y= -x2 +2x +1
12. y= x2 +4x +1
13. y= x2 +6x +9
14. y= -x2 -3x +6
15. y= 2x2 +4x
16. y= 4x2 -12x +9
17. y= -6x2 -12x -1
18. y= -3/4x2 +6x +6
19. y= 3x2 -12x +10
Graph each function. If a>0 find minimum value. If a<0 find the maximum value
(p.244).
22. y= -x2 + 2x +5
23. y= 3x2 - 4x -2
24. y= -2x2 - 3x +4
25. y= 1/3x2 + 2x +5
26. y= -x2 - x + 6
27. y= 2x2 + 5
Write each function in vertex form. Find its maximum and minimum value? (p. 294)
17. y= x2 +x -12
18. y= -x2 +2x +2
19. y= 2x2 +8x -3
20. y= -0.5x2 +5
154
D.3: Interpreting quadratic functions represented by graphs, formulas, tables, and
situation descriptions
Determine whether each equation is linear or quadratic. Identify the quadratic, linear,
and constant terms (Bellman et al., 2004, p.237).
1. y= x +4
2. y= 2x2 - (3x -5)
3. y = 3x (x -2)
4. f (x)= x2 – 7
5. y= (x -2) + (x +5)
6. g (x)= -7 (x -4)
7. h (x) = (3x)(2x) +6
8. y= x (1 -x) - (1 -x2)
9. f (x)= -x (2x +8)
Identify the vertex and the axis of symmetry of each parabola (p. 237)
y
y
y
4
2
x
-3
-1
1
3
2
-2
x
-2
-3
-1
1
3
2
-2
-2
(10)
(11)
155
(12)
Match each function with its graph. (p. 245)
37. y= x2 + 4x +1
38. y= -x2 -4x +1
39. y= -1/2x2 - 2x +1
y
y
4
4
4
2
2
2
0
-5
-3
y
x
1
-1
0
-5
-3
-1
x
1
0
-5
-3
-2
-2
-2
-4
-4
-4
(A)
(B)
x
1
-1
(C)
For each parabola, identify points corresponding to P and Q (p. 237).
y
y
8
8
P
y
Q
2
-8
6
P
4
2
-4 Q
2
2
6
-6
2
P
Q
-2
-4
-2
2
x
6
-8
(13)
(14)
156
(15)
4
D.4: Solving quadratic equations by factoring, completing the square, and using the
quadratic formula
Solve each equation by factoring. Check your answer (Bellman et al., 2004, p.266).
1. x2 +6x+ 8 = 0
2. x2 +18 = 9x
3. 2x2 -x = 3
4. x2 -10x+ 25 = 0
5. 2x2 +6x = -4
6. 3x2 = 16x+ 12
Solve each equation by factoring (p. 267)
36. x2 +6x +5 = 45
37. x2 -11x +24 = 0
38. 2x2 -5x -3 = 0
39. x2 +2x = 6 -6x
40. 6x2 +13x +6 = 0
41. 2x2 +8x = 5x +20
Solve each quadratic equation by completing square (p.281)
13. x2 -3x = 28
14. x2 -3x = 4
15. x2 +6x+41 = 0
16. x2 -2x = -2
17. w2 -8w -9 = 0
18. x2 +6x = -22
19. x2 +4 = 0
20. -x2 -2x = 5
21. 6x- 3x2 = -12
22. 2p2 = 6p-20
23. 3x2 -12x +7= 0
24. 4c2+10c= -7
Solve each equation using the quadratic formula (p. 289)
1. x2 -4x +3 =0
2. x2 +8x +12 =0
3. 2x2 +5x -7 =0
4. 3x2 +2x -1 =0
5. x2 +10x = -25
6. 2x2 +3x -5 =0
7. x2 =3x -1
8. x2 +6x -5 =0
9. 3x2 -4x -2 =0
10. 8x2 -2x -3 =0
11. x (x -5) = -4
157
12. 9x2 +12x -5 =0
D.5: Using quadratic models to solve problems presented in real-world situations
28. Revenue: A model for a company’s revenue is R= -15 p2 +300p +12,000, where p is
the price in dollars of the company’s product. What price will maximize revenue? Find
the maximum revenue? (Bellman et al., 2004, p. 244).
29. Physics: The equation for the motion of a projectile fired straight up at an initial
velocity of 64ft/s is h= 64t-16t2, where h is the height in feet and t is the time in seconds.
Find the time the projectile needs to reach its highest point. How high it will go? (p. 244)
30. Manufacturing: The equation for the cost in dollars of producing automobile tires is
C= 0.000015x2 -0.003x + 35, where x is the number of tires produced. Find the number
of tires that minimizes the cost. What is the cost for that number of tires? (p. 244)
54. A rock club’s profit from booking local bands depends on the ticket price. Using past
receipts, the owner find that the profit p can be modeled by the function p= -15t2 +600t
+50, where t represents the ticket price in dollars.
a. what price yields the maximum profit?
b. what is the maximum profit ?
c. Open-Ended: What price would you pay to see your favorite local band?
How much profit would the club owner make using that ticket price? (p.245)
14. Physics: For a model rocket, the altitude h, in meters, as a function of time, in
seconds, is given by h= 68t – t2. Find the maximum height of the rocket. How long does
it take to reach the maximum height? (p. 296)
158
APPENDIX E
159
160
APPENDIX F
INTERVIEW QUESTIONS - I AND II
(1) In your words, what is a quadratic function?
a. Give an example of a graph of a quadratic function?
b. Why do you think this is a quadratic function?
c. Are the y= x2 +4 and y= 3x2 – 16x the quadratic functions? Why?
d. What is the difference between quadratic function and quadratic equation?
(2) Write an equation for the parabola shown below.
a. Is the point (1.5, 2.5) on the parabola?
b. Is the point (-1, -18) on the parabola?
c. What is the y-intercept (s) point?
y
(3, 6)
6
4
2
(5, 0)
0
(1, 0)
3
x
6
161
(3) Answer the following questions for y = x2 – 3x +2
a. Graph the parabola?
b. Determine if the parabola has a maximum or minimum value?
c. Label the vertex and the axis of symmetry.
d. Solve the quadratic equation: 7x = 15 - 2x2
e. Is there another way to solve this equation?
(4) The equations of two parabolas are given by:
y = a x2 + b x + 1
y = a x2 + b x + 4
Answer the following questions about these two parabolas.
a. Do the two parabolas have the same axis of symmetry? Why?
b. Do the two parabolas have the same vertex? Why?
c. Do the two parabolas have the same x-intercepts? Why?
d. Do the two parabolas have the same y- intercept? Why?
162
(5) Compare the graph of y = x2 relative to the graph of y = (x - 3)2
Then compare y = (x - 3)2 to the graph of y = x2 - 3
(6) Suppose a basketball player throws the ball towards the basket. The equation
h (t) = -16t2 +20t + 6 gives the height h, in feet, of a basketball as a function of the time t,
in seconds.
a. What can you learn by finding the graph’s intercept with h-axis?
b. What can you learn by finding the graph’s intercept(s) with t-axis?
c. What is the maximum height the ball reaches?
d. At what time does the ball hit the ground?
163
(7) The graph of the parabola y = ax2 + bx + c is shown below. What can you say about
the coefficients of a, b, or c in the formula of quadratic function?
y
x
(8) Compare the graph of y = x2 relative to the graph y = (x+2) 2
Then compare y = (x+ 2)2 to the graph of y = x2 + 2
164
(9) How are the graphs of y = (x - 4)2 and y = | x - 4 | similar? How are they different?
(10) Write an equation for the parabola shown below
y
1
(3,0)
(-1,0)
2
x
-2
(1,-6)
165
(11) Graph the following parabolas?
a. y = - x2 - 3x +5
b. y = (x - 2)2 +5
y
x
166
APPENDIX G
QUIZ
Group-A
167
Group-B
168
APPENDIX H
CHAPTER TEST
Group-A
169
170
Group-B
171
172
REFERENCES
Adams, T.L. (1997). Addressing students’ difficulties with the concept of function:
applying graphing calculators and a model of conceptual change. Focus on
Learning Problems in Mathematics, 19 (2), 43-57.
Airasian, P. W., & Walsh, M. E. (1997). Constructivist cautions. Phi Delta Kappan, 78,
444-449.
Applefield , J. M., Huber, R, & Moallem, M. (2001). Constructivism in theory and
practice: Toward a better understanding. The High School Journal, 84(2), 35-53.
Arcavi, A. (2003). The role of visual representations in the learning of mathematics.
Educational Studies in Mathematics, 52, 215-241.
Ausubel, D.P., Novak, J. D., & Hanesian, H. (1978). Educational Psychology: A
cognitive view. New York, NY: Holt, Rinehart, and Winston.
Baroody, A. J., & Ginsburg, H.P. (1990). Children’s mathematical learning: A cognitive
view. In Davis R. B., Maher, C. A., & Noddings, N. (Eds.), Constructivist views
on the teaching and learning of mathematics (pp. 51-64). Reston, VA: NCTM.
Bellman, A. E., Bragg, S. C., Charles, R. I., Handlin, W. G., & Kennedy, D. (2004).
Florida Prentice Hall Mathematics: Algebra 2. Upper Saddle River, NJ: Pearson
education.
Borgen, K. L., & Manu, S. S. (2002). What do students really understand? Journal of
Mathematical Behavior, 21, 151-165.
Carlson, M. P. (1997). Obstacles for college algebra students in understanding functions:
what do high performing students really know? AMATYC Review, 19(1), 48-59.
Chaiklin, S. (1989). Cognitive studies of algebra problem solving and learning. In S.
Wagner, & C. Kieran (Eds.), Research issues in the learning and teaching of
algebra (pp. 93-114). Reston, VA: NCTM.
Chrenka, L. (2001). Misconstructing constructivism. Phi Delta Kappan, 82(9), 694-695.
173
Clement, L. L. (2001). What do students really know about functions? Mathematics
Teacher, 94(9), 745-758.
Clement, J. (1989). The concept of variation and misconceptions in Cartesian graphing.
Focus on Learning Problems in Mathematics, 11(2), 77-87.
Confrey, J. (1990). What constructivism implies for teaching. In Davis, R. B., Maher, C.
A., Noddings, N. (Eds.), Constructivist views on the teaching and learning of
mathematics (pp. 107-122). Reston, VA: NCTM.
Confrey, J. (1994). A theory of intellectual development. For the Learning of
Mathematics, 14(3), 2-8.
Cooney, T. J. (1996). Mathematics, pedagogy, and secondary education: Developing a
topic across the curriculum: functions. Portsmouth, NH: Heinemann.
Creswell, J. W. (1998). Qualitative inquiry and research design: Choosing among five
traditions. Thousand Oaks, CA: Sage.
Davydov, V. V. (1990). Soviet Studies in Mathematics Education. In Kilpatrick, J., &
Teller, J. (Eds), Types of generalization in instruction: logical and psychological
problems in the structuring of school curricula, Reston, VA: NCTM
Davis, R. B. (1990). Discovery learning and constructivism. In Davis, R. B., Maher, C.
A., & Noddings, N. (Eds.), Constructivist views on the teaching and learning of
mathematics (pp. 93-106). Reston, VA: NCTM.
Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable
misconception stages. Journal of Mathematical Behavior, 5, 281-303.
Denzin, N. K., & Lincoln, Y.S. (1994). Handbook of qualitative research. Thousand
Oaks, CA: Sage.
Dreyfus, T., & Eisenberg, T. (1982). Intuitive functional concepts: a baseline study on
intuitions. Journal for Research in Mathematics Education, 13, 360-380.
Dreyfus, T., & Eisenberg, T. (1983). The function concept in college students’: linearity,
smoothness, and periodicity. Focus on Learning Problems in Mathematics, 5, 119132.
Driscoll, M. P. (1994). Psychology of learning for instruction. Needham Heights, MA:
Allyn and Bacon.
Dunham, P. H., & Osborne, A. (1991). Learning how to see: Students’ graphing
difficulties. Focus on Learning Problems in Mathematics, 13 (4), 35-49.
174
Eisenberg, T., & Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W.
Zimmermann, & S. Cunningham (Eds.), Visualization in Teaching and Learning
Mathematics (pp. 127 – 138). Washington, DC: MAA.
Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge:
prospective secondary teachers and the function concept. Journal for Research
in Mathematics Education, 24(2), 94-116.
Even, R. (1998). Factors involved in linking representations of functions. Journal of
Mathematical Behavior, 17(1), 105-121.
Even, R. & Tirosh, D. (2002). Teacher knowledge and understanding of students’
mathematical learning. In L. D. English (Ed.), Handbook of International
Research in Mathematics Education (pp. 219-240). Mahwah, N J: Lawrence
Erlbaum.
Fischetti, J. C., Dittmer, A. E. K., & Diane, W. (1996). Shifting paradigms: Emerging
issues for educational policy and practice. The Teacher Educator, 31, 189-201.
Florida Department of Education (2003). Grading Florida public schools 2002-2003.
Electronic version. http://www.myfloridaeducation.com.
Ginsburg, H.P. (1977). Children’s arithmetic: the learning process. New York, NY: Van
Norstand.
Ginsburg, H. P. (1981). The clinical interview in psychological research on mathematical
thinking: Aims, rationales, techniques. For the Learning of Mathematics, 1(3), 411.
Ginsburg, H. P. (1997). Entering the child’s mind: The clinical interview in
psychological research and practice. New York, NY: Cambridge university press.
Ginsburg, H. P., Jacobs, S.F., & Lopez, L.S. (1998). The teacher’s guide to flexible
interviewing in the classroom: Learning what children know about math.
Needham Heights, MA: Allyn and Bacon.
Gray, E., & Tall, D. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of
simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116140.
Haylock, D. W. (1982). Understanding in mathematics: making connections.
Mathematics Teaching, 98, 54-56.
Hazzan, O., & Zazkis, R. (2005). Reducing abstraction: the case of school mathematics.
Educational Studies in Mathematics Education, 58, 101-119.
175
Herscovics, N. (1989a). Cognitive obstacles encountered in the learning of algebra. In S.
Wagner, & C. Kieran (Eds.), Research issues in the learning and teaching of
algebra (pp. 66-86). Reston, VA: NCTM.
Herscovics, N. (1989b). The description and analysis of mathematical processes. In C.
Maher, G. Goldin, & R. Davis (Eds.), Proceedings of the eleventh PME-NA
conference (pp. 9-36). New Brunswick, NJ: Rutgers Centers for Mathematics,
Science, and Computer Education.
Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context:
Epistemic actions. Journal for Research in Mathematics Education, 32, 195-222.
Herriott, R. E., & Firestone, W. A. (1983). Multisite qualitative policy research:
Optimizing description and generalizability. Educational Researcher, 12, 14-19.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics:
An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural
knowledge: the case of mathematics (pp. 1-28). Hillsdale, NJ: Lawrence Erlbaum.
Hiebert, J., Carpenter, T.P., Fennema E., Fuson K.C., Wearne, D., Murray, H., Olivier A.,
& Human, P. (1997). Making sense: Teaching and learning mathematics with
understanding. Portsmouth, NH: Heinemann.
Hitt, F. (1998). Difficulties in the articulation of different representations linked to the
concept of function. Journal of Mathematical Behavior, 7(1), 123-134.
Hollar, J.C., & Norwood, K. (1999). The effects of a graphing-approach intermediate
algebra curriculum on students' understanding of function. Journal for the
Research of Mathematics Education, 30(1), 220-226.
Janvier, C. (1998). The notion of chronical as an epistemological obstacle to the concept
of function. Journal of Mathematical Behavior, 17(1), 79-103.
Krutetskii, V. A. (1976). The Psychology of Mathematical Abilities in Schoolchildren.
In J. Kilpatrick, & I. Wirszup (Eds.). Chicago: The University of Chicago Press
LeCompte, M. D., & Preissle, J. (1993). Ethnography and qualitative design in
educational research. New York, NY: Academic press.
Lehn, K.V. (1983). On the representation of procedures in repair theory. In Ginsburg
(Ed.), The development of mathematical thinking (pp. 201-251). New York, NY:
Academic press.
Leninhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing:
tasks, learning, and teaching. Review of Educational Research, 60, 1-64.
176
Lincoln, Y. S., & Guba, E.G. (1985). Naturalistic inquiry. Newburry Park, CA: Sage.
Lloyd G. M., & Wilson M. S. (1998). Supporting innovation: The impact of a teacher's
conceptions of functions on his implementation of a reform curriculum. Journal
for Research in Mathematics Education, 29(3), 248-274.
Mathematical Sciences Education Board (1988). A framework for revision of the K-12
mathematic curriculum. Washington, DC: National Research Council.
Matkovits, E., Eylon, B. S., & Bruckheimer, M. (1986). Functions today and yesterday.
For the Learning of Mathematics, 6(2), 18-28.
Merriam, S. B. (1988). Case study research in education: a qualitative approach. San
Francisco, CA: Jossey-Bass.
Merriam, S. B. (1998). Qualitative research and case study applications in education. San
Francisco, CA: Jossey-Bass.
Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded
sourcebook. Thousand Oaks, CA: Sage.
Movshovitzs-Hadar, N. (1993). A constructive transition from linear to quadratic
functions. School Science and Mathematics, 93(6), 288-298.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation
standards for school mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics. (1991). Professional standards for
teaching mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics. (2000). Principles and Standards for
School Mathematics. Reston, VA: NCTM.
Niss, M. (1999). Aspects of the nature and state of research in mathematics education.
Educational Studies in Mathematics, 40, 1-24.
Noddings, N. (1990). Constructivism in mathematics education. In Davis, R. B., Maher,
C. A., & Noddings, N. (Eds.), Constructivist views on the teaching and learning
of mathematics (pp. 7-18). Reston, VA: NCTM.
Norman, F. A. (1993). Integrating research on teachers’ knowledge of functions and
their graphs. In Romberg, T.A., Fennema, E., & Carpenter, T. P. (Eds.),
Integrating research on the graphical representations (pp. 150-187). Hillsdale, NJ:
Lawrence Erlbaum.
177
O'Callaghan, B. R. (1998). Computer-intensive algebra and students’
conceptual knowledge of functions. Journal for Research in Mathematics
Education, 29 (1), 21-41.
Patton, M. Q. (1990). Qualitative Evaluation Methods. Thousands Oaks, CA: Sage.
Pirie, S., & Kieren, T. (1992). Watching Sandy’s understanding grow. Journal of
Mathematical Behavior, 11, 243-257.
Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we
characterize it and how can we represent it? Educational Studies in Mathematics,
26, 165-190.
Presmeg, N. C. (1986a). Visualization and mathematical giftedness. Educational Studies
in Mathematics, 17, 297-311.
Presmeg, N. C. (1986b). Visualization in high school mathematics. For the Learning of
Mathematics, 6, 42-46.
Quesada, A.R, & Maxwell, M.E. (1994). The effects of using graphing calculators to
enhance college students' performance in precalculus. Educational Studies in
Mathematics, 205-215.
Remillard, J. T. (2000). Can curriculum materials support teacher’s learning? Two
fourth-grade teachers’ use of a new mathematics text. Elementary School
Journal, 100, 331-350.
Romberg, T.A., Fennema, E., & Carpenter, T. P. (1993). Integrating research on the
graphical representations. Hillsdale, NJ: Lawrence Erlbaum.
Ruthven, K. (1990). The influence of graphic calculator use on translation
from graphic to symbolic forms. Educational Studies in Mathematics
Education, 21, 431-450.
Sajka, M. (2003). A secondary school student’s understanding of the concept of
function-a case study. Educational Studies in Mathematics, 53, 229-254.
Schoenfeld, A. H. (1989). Exploring the process problem space: notes on description and
analysis of mathematical processes. In C. Maher, G. Goldin, & R. Davis (Eds.),
Proceedings of the eleventh PME-NA conference (pp. 95-120). New Brunswick,
NJ: Rutgers Centers for Mathematics, Science, and Computer Education.
Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving,
metacognition, and sense making in mathematics. In D. A. Grows (Ed.),
Handbook of research on mathematics teaching and learning (pp. 334-370). New
York: Macmillan.
178
Schoenfeld, A. H., Smith, J., & Arcavi, A. (1993). Learning: the microgenetic analysis of
one student’s evolving understanding of a complex subject matter domain. In R.
Glaser (Ed), Advances in instructional psychology (pp. 55-176). Hillsdale, NJ:
Lawrence Erlbaum.
Selden, J., Mason, A., & Selden, A. (1989) Can average calculus students solve
nonroutine problems? Journal of Mathematical Behavior, 8, 45-50.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on
process and objects as different sides of the same coin. Educational Studies in
Mathematics, 22, 1-36.
Sierpinska, A. (1987). Humanities students and epistemological obstacles related to
limits. Educational Studies in Mathematics, 18, 371-397.
Sierpinska, A. (1990). Some remarks on understanding mathematics. For the Learning
of Mathematics, 10 (3), 24-36.
Simpson, T. L. (2002). Dare I oppose constructivist theory? The Educational Forum,
66(4), 347-354.
Skemp, R. (1978). Relational understanding and instrumental understanding.
Mathematics Teaching, 77, 20–26.
Tall, D. (1989). Different cognitive obstacles in a technological paradigm. In S.
Wagner, & C. Kieran (Ed.), Research issues in the learning and teaching of
Algebra (pp. 87-92). Reston, VA: NCTM.
Tall, D., & Bakar, M. (1992). Students’ mental prototypes for functions and graphs.
International Journal of Math, Education, Science, and Technology, 23 (1), 39–
50.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with
particular reference to limits and continuity. Educational Studies in Mathematics,
12 (2), 151-169.
Thompson, A. G. (1992). Teachers’ beliefs and conceptions: a synthesis of the research.
In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and
Learning (pp. 127-146). New York: MacMillan
Vinner, S. (1983). Concept definition, concept image, and the notion of function.
International Journal of Mathematics Education in Science and Technology,
14(3), 293-305.
Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function.
Journal for Research in Mathematics Education, 20(4), 356-366.
179
Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning.
Washington, D.C: The Falmer.
Wilson, M.R. (1994). One preservice secondary mathematics teacher's evolving
understanding of mathematical functions: The impact of a course integrating
mathematical content and pedagogy. Journal for Research in Mathematics
Education, 25(4), 346-370.
Wollman, W. (1983). Determining the sources of errors in a translation from sentence to
equation. Journal for Research in Mathematics Education, 14(3), 169-181.
Yin, R. K. (1994). Case study research: Design and methods. Thousand Oaks, CA: Sage.
Zazkis, R., Dubinsky, E., & Dautermann, J. (1996). Coordinating visual and analytic
strategies: A study of students’ understanding of the group D4. Journal for
Research in Mathematics Education, 27(4), 435-457.
Zazkis, R., Liljedahl, P., & Gadowsky, K. (2003). Conceptions of function translation:
Obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437450.
Zaslavsky, O. (1997). Conceptual obstacles in the learning of quadratic functions. Focus
on Learning Problems in Mathematics, 19(1), 20-45.
180
BIOGRAPHICAL SKETCH
Ali Eraslan was born August 25, 1970, in Yozgat, and grew up in Ankara,
Turkey, where he attended public schools. He began undergraduate studies Fall 1989 at
the Gazi University, where he majored in Mathematics Education. He completed his
Bachelor of Science Degree in June 1993. He then taught algebra and geometry courses
in public middle and high schools for five years in Turkey.
In 1999, he was awarded with a full scholarship by the Turkish Ministry of
National Education to pursue a graduate degree in the area of mathematics education in
the USA. In the Spring of 2000, he began graduate studies in Mathematics Education at
Oregon State University where he received a Master of Science Degree Fall 2001. In the
Summer of 2002, he moved to Tallahassee, Florida and continued his doctoral studies in
Mathematics Education at the Florida State University. He earned his Doctor of
Philosophy Degree in Fall 2005.
181