THE EFFECTS OF GRAPHING CALCULATORS AND A MODEL FOR
CONCEPTUAL CHANGE ON COMMUNITY COLLEGE
ALGEBRA STUDENTS' CONCEPT OF FUNCTION
THOMASENIA LOTT ADAMS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
This dissertation is dedicated to my parents,
Pickens and Tennie Ruth Lott,
whose love and faith I was never without
to Larry Vanoy Adams, my mate and friend.
I am deeply indebted to a host of people who provided a variety and
substantial amount of support for me before and during the completion of this
I am especially indebted to my doctoral committee, particularly my chair,
Dr. Mary Grace (Eleanore) Kantowski. She provided a model fitting for the
future mathematics educator. To Dr. David Miller, I owe great thanks for his
statistical and professional advice. To Dr. John Gregory, Dr. Don Bernard, and
Dr. Li-Chien Shen, I owe thanks for their academic guidance.
I am forever in debt to Dr. Israel Tribble of the Florida Endowment Fund
for Higher Education for supporting me as a McKnight Doctoral Fellow
financially and emotionally. He, his staff, and all of the McKnight Fellows
provided motivation unconditionally.
To my friend and colleague, Juli K. Dixon, who provided relief from the
challenges, I owe the effort to strive for peace.
I owe all that I am to those who made me what I was before I began the
doctoral program. To my parents and to my siblings, Larry, Rhoda, Alie Ruth,
Jacqueline, Jerridene, Malcolm, and (in memory of) Christopher, for teaching
me how to learn, I owe a continuous effort to improve self and empower others
through education. To my husband, Larry, who was all that I needed him to be, I
owe all the joys of tomorrow.
TABLE OF CONTENTS
I DESCRIPTION OF THE STUDY
Introduction .. 1
Statement of the Problem 4
Justification of the Study 6
Theoretical Framework 8
Significance of the Study 20
Organization of the Study 21
II REVIEW OF THE LITERATURE
Development of the Concept of Function 22
Representations of the Concept of Function . 27
Difficulties with the Concept of Function 33
Computers and Calculators 51
Graphing Calculators 62
Conceptual Change Theory in Mathematics
III RESEARCH DESIGN AND METHODOLOGY
Research Objective. 78
Pilot Study 82
Research Population and Sample 84
Ill RESEARCH DESIGN AND METHODOLOGY Cont'd.
Instructional Materials . 93
Design of the Study. 95
Analyses for Domain / Range / Scale Instrument 99
Analyses for Ide tification / Construction / Definition
Test for Interactions. 103
Exploratory Analyses 105
Classroom Observations 110
Limitations of the Study 111
A Conceptual Change Assignment. 127
B Domain / Range / Scale Instrument 130
C Identification / Construction / Definition Instrument 137
D Student Information Sheet. 141
E Casio fx 7000G Graphing Calculator Profile 144
F Instructor Information Sheet 145
G Examples of Academically Acceptable and
Unacceptable Function Definitions 147
H Examples of Function Definitions by Category 148
I Observation Form 149
BIOGRAPHICAL SKETCH 169
Abstract of Dissertation Presented to the Graduate School
of the University Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE EFFECTS OF GRAPHING CALCULATORS AND A MODEL FOR
CONCEPTUAL CHANGE ON COMMUNITY COLLEGE
ALGEBRA STUDENTS' CONCEPT OF FUNCTION
Thomasenia Lott Adams
Chairperson: Dr. Eleanore L. Kantowski
Major Department: Instruction & Curriculum
Three treatment groups and a control group were compared on two
dependent variables regarding their understanding of the concept of function.
During the unit of study for the concept, Treatment Group I students used
graphing calculators and participated in a conceptual change assignment.
Treatment Group II students used graphing calculators only during study of the
unit. Treatment Group III students participated in the conceptual change
assignment only during the unit. Treatment Group IV served as a control group.
Regarding students' understanding and application of the function concepts,
domain and range, and their understanding of the concept of scale, the results
of the covariate analysis revealed a significant treatment interaction effect. The
least square means procedure indicated differences between Treatment
Groups I and II, between Treatment Groups I and III, and between Treatment
Groups III and IV. The group mean for students who used calculators only was
significantly higher than the group mean for students who used calculators and
participated in the assignment. The group mean for students who participated
in the assignment only was significantly higher that the group means for a)
students who used graphing calculators and participated in the assignment and
b) students in the control group.
Regarding students' ability to identify, construct, and define function, the
results of the covariate analyses revealed a significant effect regarding the
factor of conceptual change assignment. The group mean for students who
participated in the assignment was significantly lower than the group mean for
students who did not participate in the assignment.
Exploratory analyses revealed that the students' definitions of the
concept of function were dominated by the ordered pair representation of the
concept. This point-wise view of functions was further emphasized through the
students' images of the concept of function.
Classroom observations of the treatment and control groups revealed
additional information regarding the effect of the graphing calculator on
DESCRIPTION OF THE STUDY
One objective of reform in mathematics education has been to make the
mathematics curriculum stronger. Although many mathematical concepts have
gone through cycles of emphasis, deemphasis, and reemphasis in mathematics
education, focus on the concept of function has been constant throughout the
New Mathematics movement, the introduction of technology into mathematics
education, and the effort to empower learners and educators through the
application of the National Council of Teachers of Mathematics' (NCTM)
Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989).
The concept of function has been, and continues to be, upheld by the
mathematics education community as a fundamental and unifying principle of
mathematics (Brieske, 1973; Bruckheimer & Gowar, 1969; Herscovics, 1989;
Hirsch, Weinhold, & Nichols, 1991; Krause, 1984; Papakonstantinou, 1993;
A significant purpose of education is to assist students in acquiring
concepts (Leinhardt, Zaslavsky, & Stein, 1990). Thus, students' acquisition and
understanding of the concept of function should be considered by mathematics
educators. Understanding of this concept has always been of great practical
value. Dresden (1927) proposed that functions were a part of our everyday
experiences in which dependent relationships could always be found.
Moreover, the value of the concept of function in academia is exemplified by
calculus (Thorpe, 1989), because the study of calculus is essentially the study
of functions and graphs of functions (Fey, 1984).
The power of technology and its application in mathematics can be
realized through the use of graphing calculators as tools for learning. The
graphing calculator is ideal for developing the concept of function and for
focusing on its graphical representation. A renewed consideration of the
concept of function has resulted from the use of graphing calculators in
instruction. Two main concepts related to the concept of function, domain and
range, are especially important when using the graphing calculator.
The domain is one of the most important but often neglected aspects of the
concept of function, although a function is never completely defined until a
domain for the function is defined (Mancill & Gonzalez, 1962). The elements of
the domain of the function are the only elements to which the function can be
applied (Buck, 1970), and if a domain is not specified, then one can assume
"that it is the largest collection of numbers for which the function has real
values" ( Mancill & Gonzalez, 1962, p. 180). The range of the function follows
upon clarification of the domain and the rule for the function (Orton, 1971).
A person's definition of the concept of function may "include such ideas of
a function as an action, an operation, a rule, a relationship, a graph, or a
picture" (Confrey, 1981a, p. 8), while the mathematical community supports a
definition involving the formal relationship between a given domain and a given
range. The definition of the concept of function used for the purpose of this
study is described as the Dirichlet-Bourbaki definition and is stated as follows:
a function is a relationship or correspondence between two sets, the domain
and the range, such that any element of the domain is related to or corresponds
to one and only one element of the range.
One who fully understands the concept of function will show some
evidence of understanding the concepts of domain, range, and the relationship
between the domain and range (Markovits, Eylon, & Bruckheimer, 1983). By
neglecting to be attentive to the concepts of domain and range, learners of
mathematics fail to emphasize what Ayers, Davis, Dubinsky, and Lewin (1988)
described as "the key feature of function." This feature involves the
transformation of elements in the domain of the function to elements in the
range of the function. The nature of this transformation depends on the
relationship between the domain and the range. Moreover, inclusion of the
concepts of domain and range is a necessity and an asset because, "by
separating the function into its elements, we tend to make it a unity" (Hamley,
1934, p. 30).
When graphing a function whether using paper and pencil or a
mechanical graphing device, appropriate domain, range, and scale for the
axes must be chosen in order to obtain a useful visual description of the
concept. By emphasizing the importance of choosing a domain, range, and
scale for the axes for graphing the function, the learner can be led to an
understanding of the relationship between the domain and range of the
The concept of function is important in all levels of mathematics curricula,
including community college mathematics. Community colleges, among other
educational roles, serve as bridges between secondary schools and four-year
colleges or universities. Community college faculty often assist students to
overcome their educational deficiencies and prepare them for a college or
university. The associate degree received by community college graduates is
representative of a students' readiness for admittance to a college or university
or for employment (American Association of Community and Junior Colleges,
1984). By emphasizing the concept of function and the related concepts of
domain and range in mathematics curricula at the community college,
mathematics educators have the potential to greatly affect community college
algebra students' concept of function.
The concept of function is part of the foundation of mathematics. The
concepts of domain and range, which are directly related to the concept of
function, are the foci of the current study. Emphasis on the two concepts, at all
levels of mathematical learning, implies emphasis on the concept of function.
For the purpose of preparing graphs representing the concept of function,
one needs to be aware of the importance of the two concepts and the role they
play in choosing an appropriate set of axes for graphing functions. Otherwise
an insufficient representation of the function may result. This is especially
important when employing technological tools for graphing functions.
Statement of the Problem
Algebra students typically pay little attention to choosing appropriate
domain, range, and scale for the axes when graphing a function. Because the
graph of the function represents the relationship between the domain and
range of the function, lack of consideration of a reasonable domain, range, and
scale for the axes for graphing functions perpetuates students' difficulties with
the concept of function. In addition, various groups of learners of mathematics
have shown evidence of difficulties with the concept of function which have
impeded their understanding of the concept. Such difficulties include students'
inability to identify, construct, and define function (Vinner & Dreyfus, 1989).
These difficulties are possibly the result of students' misunderstanding of the
concepts of domain and range and the functional relationship between the two
The purpose of this study was to examine the effects of using graphing
calculators and a conceptual change assignment during instruction on the
students' concept of function. The emphasis was on the students' selection of
appropriate domain, range, and scale for the axes for graphing a function and
their identification, construction, and definition of function. This study was
designed to test the following null hypotheses as related to college algebra
students in the community college:
1. Students' use of the graphing calculator during instruction will not
affect their concept of function regarding their (a) application of
the concepts of domain and range, (b) selection of appropriate
domain, range, and scale for the axes for graphing functions,
(c) identification, construction, and definition of function.
2. Students' participation in a conceptual change assignment will not
affect their concept of function regarding their (a) application of
the concepts of domain and range, (b) selection of appropriate
domain, range, and scale for the axes for graphing functions,
(c) identification, construction, and definition of function.
3. Students' use of the graphing calculator during instruction and the
students' participation in a conceptual change assignment will not
interact to affect their concept of function regarding their
(a) application of the concepts of domain and range (b) selection
of appropriate domain, range, and scale for the axes for graphing
functions, (c) identification, construction, and definition
Justification of the Study
The current body of research on the concept of function and students'
misconceptions of the concept is significantly deficient. Researchers have
focused on the concept of function held by teachers (e.g., Even, 1989, 1990,
1993; Stein, Baxter, & Leinhardt, 1990; Vinner & Dreyfus, 1989); by college and
university students (e.g., Dreyfus & Eisenberg, 1983; Vinner & Dreyfus, 1989);
and by middle and high school students (e.g., Dreyfus & Eisenberg, 1982,
1984; Karplus, 1979; Markovits, Eylon, & Bruckheimer, 1988; Orton, 1971;
Papakonstantinou, 1993; Thomas, 1969; Wagner, 1981). Research in this area
pertaining to the community college student is essentially nonexistent.
Research results obtained from the other samples can not adequately be
applied without some knowledge of community college students' concept of
function. One major factor is the unique population of students that attend
community colleges and their distinct educational needs (Cohen & Brewer,
1987; Deegan, Tillery, & Associates, 1988). Another issue of focus is the status
of the concept in the community college algebra curriculum and the attention
given to emphasizing the concept.
Several major studies concerning students' difficulties, misconceptions and
inconsistencies regarding mathematics and science concepts were conducted
in schools, colleges, and universities outside of the United States (e.g., Dreyfus
& Eisenberg, 1982, 1983, 1984; Hand &Treagust, 1991; Movshovitz-Hadar &
Hadass, 1990; Vinner & Dreyfus, 1989). One cannot assume that the results
are completely generalizable to populations of students in the United States.
There is a need to support and/or refute researchers' findings regarding
students' difficulties and misconceptions and to determine whether or not
community college students exhibit misconceptions similar to those described
in the literature regarding the concept of function.
There are research studies which were conducted to examine the use of
technology to teach functions and graphs (Leinhardt, Zaslavsky, & Stein, 1990).
NCTM (1980, 1989) advocated the use of computers to teach functions and
graphs; Fey (1984) posited that this recommendation would bring functions
and graphs to the forefront of the algebra curriculum. However, the graphing
calculator has not been fully exploited in educational research nor in
instructional processes as a tool for enhancing conceptual understanding.
Used as part of instruction, the graphing calculator can provide students with
an opportunity to explore the graphical representations of functions.
Specifically, graphing calculators should be used in the learning of
mathematics at the community college level (Demana & Waits, 1988). Since
Demana and Waits (1990) have found that students with access to technology
in the learning process have success in learning algebraic concepts, it is
necessary to provide additional research regarding how students approach
graphing (Goldenberg & Kliman, 1988), and the benefits of employing graphing
technology (Fey, 1989; Ruthven, 1990). Shumway (1990) proposed that the
use of "supercalculators" (i.e., scientific calculators with graphing capabilities) in
research pertaining to the development of mathematical concepts is drastically
needed. Educators (e.g., Wilkinson, 1984) have suggested that technological
tools, such as computers and calculators, can be used to assist students in
developing mathematical concepts.
The purpose of the current study was to respond to the need for an
increase in attention to conceptual understanding in mathematics education
(Rosnick & Clements, 1980), and the use of technology for the learning and
teaching of algebra. The next section of this chapter is a description of the
theoretical framework for the present study.
Research focusing on conceptual understanding of mathematics must be
supported by a theory of learning which can be applied to the learners'
development of mathematical concepts and encompasses principles which are
directly applicable to mathematics education. In addition to a theory of
learning which can provide for these two conditions, this research must be
supported by a theory which can be used to acknowledge difficulties learners
might have in the process of developing concepts and to indicate a framework
for dealing with misconceptions that might arise throughout the process of
developing concepts. Moreover, there should be some indication that the two
theories can be applied simultaneously for the benefit of concept development.
Thus, the theoretical framework for the current research study will be discussed
with reference to Jean Piaget's theory of genetic epistemology and Leon
Festinger's theory of cognitive dissonance as applied within the context of the
theory of conceptual change.
Skemp (1987) described a concept as an "idea" and the name of a
concept as a "sound." For example, the concept of function can be identified by
the written or spoken word "function". This word is just an identifier for the
concept. The concept itself is an idea, something which does not have a truth
value (Bourne, Dominowski, Roger, Loftus, & Healy, 1986). Some researchers
(e.g., Confrey, 1981a; Larcombe,1985) have proclaimed that mathematical
concepts exist only in the sense that they become reality in the individual's
mind. Thus the term has a twofold meaning. According to Gilbert and Watts
(1983), the term concept applies to one's personal knowledge concerning an
idea and it also applies to the publicly accepted (sometimes formal) explanation
of the idea.
Skemp (1979, 1987) and Tennyson and Park (1980) suggested that
concepts for the most part always contain within themselves other concepts.
Cohen (1983) and Larcombe (1985) advocated that higher order concepts
were built upon these lower order concepts. These lower order concepts
should not be dismissed because one needs them as reference in defining the
larger concept (Buck, 1970). Each concept needs to be defined and
understood individually (Wilson, 1990). For instance, the definition of the
concept of function itself is composed of some words that are themselves
concepts, such as domain and range. For this reason, it can be described as a
lexical definition (Vinner, 1976).
Development of Concepts
Jean Piaget (1896-1970) described his approach to the learner's
development of concepts as genetic epistemology, indicating his concern with
how the learner formed knowledge and the meaning of the knowledge to the
learner (Piaget, 1981). Piaget's use of "the term genetic referred to
developmental growth rather than biological inheritance" ( Hergenhahn, 1976,
p. 269). Other researchers (e.g., Confrey, 1991) proposed that Piaget's
approach indicated that concepts develop in the mind of the learner as the
learner participates in instructional experiences and attempts to solve
problems. Piaget (1981) asserted that a learner can acquire mathematical
knowledge from one of two types of learning experiences: the "physical" and
the "logico-mathematical". The physical experience involves the learner acting
on physical objects in order to obtain characteristics about the object. The
logico-mathematical experience involves the gathering of information from the
actions carried out by the learner and not from the physical properties of the
object (Piaget, 1975). He hypothesized that mathematical knowledge, being
abstract in nature, was acquired through the logico-mathematical experience
It is the logico-mathematical experience which has implications for
research conducted regarding the concept of function. The concept, being an
idea, is not representable by physical objects, yet the representations of the
concept can be acted upon by the learner. Piaget (1975) proposed that "mental
or intellectual operations, which intervene in the subsequent deductive
reasoning processes, themselves stem from actions: they are interiorised
actions and once this interiorisation ... is sufficient, then logico-mathematical
experience in the form of material actions is no longer necessary and
interiorised deduction is sufficient" (p. 6). Commenting on Piagetian theory and
instruction in mathematics, Penrose (1980) wrote that operations are also
actions, thus operations involving the representations of the concept of function
can be conceived as actions. A basic principle of Piaget's theory of learning is
that when a learner knows an object, such as a mathematical concept, the
learner is able to act upon it and interact with it (Renner & Phillips, 1980; Piaget,
1981), thus performing operations involving the concept. "The important
Piagetian idea is activity, not necessarily physical, concrete activity. Important
for learning are active engagement and commitment, not necessarily actions on
things. Piaget's theory provides the theoretical underpinnings for an active
approach to education" (Ginsburg, 1992, p. 359).
The actions performed by a learner on a mathematical object should reflect
not a memorization or an adoption of actions carried out by the teacher, but at
some point in time, a "reinvention" of the concept by the learner (Piaget, 1975).
This reinvention is not a discovery by the learner, but an indication that the
learner knows the concept by performing original actions on the representation
of the concept (Ginsburg, 1992). There are a variety of non-physical actions
that a learner can perform during the development of a concept. For example,
Bourne, Dominowski, Roger, Loftus, and Healy (1986) proposed that a concept
can be obtained by associating common characteristics of examples of the
concept and by forming and testing hypotheses regarding the concept. It
should be kept in mind that due to student differences, students' learning of
concepts may vary (Cohen, 1983; Confrey, 1981b), because each student might
perform different actions while trying to acquire a concept. For instance,
Confrey (1981a) proposed that a person's definition of a concept, that is the
personal or private definition for understanding the concept, is often formulated
as a way to make sense of personal experiences and observations. For the
learner, this is a "reinvention" of the concept for application. Confrey (1981a)
stressed that the goal of mathematics education has been to transform the
learner's personal definition into an academically accepted definition, in such a
way that the new definition represents the learner's concept. This new,
academic definition might adhere to the "classical view of concept" described
by Gilbert and Watts (1983) as "the view that all instances of a concept share
common properties and that these properties are necessary and sufficient to
define the concept" (p. 65).
Skemp (1987) suggested that in determining whether or not one had a
concept, the main condition was whether or not the person behaved
accordingly when presented with new examples of the concept. Piaget (1964,
1981) argued that to have acquired an object, such as a mathematical concept,
was to act accordingly when given operations to perform with understanding of
the operations involved. Gilbert and Watts (1983) found that these actions
could be "linguistic and non-linguistic, verbal and non-verbal" (p. 69).
Historically, the mastery of a concept was described as the demonstrated
ability to recognize or identify the definition, description, or illustration of a
concept or the appropriate usage of the word or words naming the concept"
(Butler, 1932, p. 123).
The classical view of the development of concepts encompasses the idea
of misconception, defined by Gilbert and Watts (1983) as an error in one's
conceptual structure. The researchers proposed that when a learner treats a
non-example as an example, then the learner has a misconception. Gilbert
and Watts (1983) also characterized "the actional view of concept" as the
recognition of students' misconceptions "as being natural developmental
phenomena- personally viable constructive alternatives- rather than the result
of some cognitive deficiency, inadequate learning, 'carelessness' or poor
teaching" (p. 67). Misconceptions may be the result of the learner's intuition
(actional view) surrounding the concept or some factor of formal instruction
(classical view) (Leinhardt, Zaslavsky, & Stein,1990; Vinner, 1990) or a
combination of both. Gilbert and Watts (1983) proposed "that there are no (and
can be no) straightforward rules to conceptualisation that work for all
situations" (p. 69). Hence, this researcher proposes that consideration of both
views in instruction is necessary to attend to students' learning of mathematics.
There are researchers (e.g., Larcombe, 1985; Linn & Songer, 1991;
Steffe, 1990) who believe that emphasis should not be placed on what students
do not understand, but rather on what they understand. However,
misconceptions cannot be dismissed as common errors (Herscovics, 1989).
One major consequence of ignoring students' misconceptions in instruction is
that there is not any encouragement for "the students to impose meaning on the
new material by attaching it to what is already known or altering what is already
known" (Leinhardt, 1988, p. 122). Nor does instruction and curricula that take
misconceptions into consideration have to be negative in nature. Students'
misconceptions or inconsistencies can be used to enhance learning (Hewson &
Hewson, 1984; Piaget, 1973), but there is a lack of research on this in
mathematics education (Tirosh, 1990). Johnson and Johnson (1979)
proclaimed that if students' misconceptions are handled properly, then
conceptual understanding can be enhanced.
Driver (1981) adapted the term 'alternative framework'(actional view)
instead of 'misconception' or 'inconsistency'(classical view) and suggested
several reasons for focusing attention on students' ideas which were not
comparable to formal, acceptable ideas. SHe suggested that such a focus would
assist one in developing curriculum materials, teaching strategies, instructional
activities, and opportunities for unrestricted learning. "Indeed identifying those
areas which present particular learning problems may be an important first step
in developing teaching programmes" (Driver & Erickson, 1983, p. 54).
Conceptual Change Theory
Strike and Posner (1985) and Posner, Strike, Hewson, and Gertzog (1982)
suggested that learning is a conceptual change process. They employed the
terms "assimilation" and "accommodation" to describe the process. These
terms were first used by Piaget (Boden, 1979), and were applied as follows:
"Assimilation occurs whenever an organism uses something from its
environment and incorporates it. The system accommodates to the object; the
changes that occur in it depend on the characteristics of the object" (Phillips,
1981, pp. 13-14). These processes are inclusive of Piaget's theory that
learning occurs through physical and/or mental activity. "Piaget's description of
how knowledge is acquired and changes developmentally is explicitly placed
within the individual through the complementary processes of assimilation and
accommodation" (Mason et al., 1983, p. 229).
During recent decades, science education researchers have attempted to
incorporate Piaget's terms for application in the teaching of science concepts.
Strike and Posner (1985) and Posner, Strike, Hewson, and Gertzog (1982),
asserted that assimilation occurs when students use previous knowledge to
deal with new knowledge (Hergenhahn, 1976; Kaplan, Yamamoto, & Ginsburg,
1989; Linn & Songer, 1991), during which a major conceptual change is not
required. At this stage, new knowledge becomes a part of the learners
conceptual structure (Piaget, 1977). Accommodation (actional view) occurs
when the learner replaces old knowledge or reorganizes old knowledge to
"accommodate" new knowledge. Hergenhahn (1976) posited that assimilation
can be equated with recognition, while accommodation can be equated with
learning. Phillips (1981) stated that "without assimilation, there can be no
accommodation, and vice versa; and accommodation is a change in structure,
the function of which is to make possible the assimilation of some stimulus
pattern that is not entirely familiar" (p. 19).
Research projects pertaining to problems caused by conflicts in students'
knowledge of mathematical and scientific concepts have usually been
implemented to focus on the theory of conceptual change (Dreyfus, Jungwirth,
& Eliovitch, 1990; Driver & Scanlon, 1988; Posner, Strike, Hewson, & Gertzog,
1982), which includes applicable frameworks of cognitive dissonance, cognitive
conflict, conceptual conflict, and applications of conflict teaching ( Movshovitz-
Hadar & Hadass, 1990; Steffe, 1990; Tirosh, 1990; Tirosh & Graeber, 1990a,
1990b; Wilson, 1990). A model of conceptual change, the theory of cognitive
dissonance was introduced by Leon Festinger in 1957 (Festinger, 1957).
"Cognitive dissonance was defined as a motivational state that impels the
individual to attempt to reduce and eliminate it. Because dissonance arises
from inconsistent knowledge, it can be reduced by decreasing or eliminating
the inconsistency" (Wicklund & Brehm, 1976, p. 1). The underlying assumption
of the theory of cognitive dissonance is that the learner, once realizing that a
conflict exists within his/her conceptual structure or between his/her
conceptualization and reality, will proceed to eliminate the conflict. The conflict
thus acts as a catalyst for conceptual change.
Piaget's models of assimilation and accommodation were also
accompanied by the act of equilibration. According to Hergenhahn (1976),
"since there is an innate need for harmony (equilibrium), the organism's mental
structures change in order to incorporate these unique aspects of the
experience, thus causing the sought-after cognitive balance" (p. 271).
Conceptual change can be the result of new knowledge conflicting with
the old; Festinger (1957) found that exposure to information was one of the
most common ways dissonance was created. He proposed that the most direct
way to reduce dissonance was by "focusing on and directly dealing with
dissonant relations" (Wicklund & Brehm, 1976, p. 98). Making the learner
aware of misconceptions has been proposed as an important step in any
strategy involving cognitive dissonance (Nussbaum & Novick, 1981).
A major goal of research has been to provide a link between theory and
practice. If positive change is to occur, acknowledgement of the misconceptions
that students have pertaining to the concept of function must be coupled with
possible strategies to alleviate the problems, that is, there are advantages for
considering the classical and actional view of concept. Hewson and Hewson
(1984) adopted a model for conceptual change, which emphasized that
learning is an interaction between previous and existing knowledge with the
outcome depending on the interaction (actional). Accordingly, there are four
criteria for conceptual change to occur when based on conflicts in students'
knowledge. First, both ideas ( in conflict) must be understood by the learner.
Next, the student must compare both ideas and find them to be in conflict.
According to Johnson and Johnson (1979), when the student has two
conflicting ideas such as previous knowledge in mind which may conflict with
new information, the student experiences conceptual conflict. There is almost
no way to avoid this because "students will filter new input through their existing
conceptions (for good or ill)" (Brophy, 1986, p. 324).
Because it is possible for the two conflicting ideas to exist in the same
mind, students may need help in identifying the misconception (Fischbein,
1990). At the time that the student realizes that an alteration regarding the
concept in question is needed, then cognitive conflict results (Dreyfus, 1990;
Dreyfus, Jungwirth, & Eliovitch, 1990).
The third criteria involves the act of resolution. Resolution occurs either by
compartmentalization or assimilation. Scwharzenberger (1982) suggested that
compartmentalization meant that students have different, conflicting ideas in
different mental compartments. Most people will want to resolve the conflict as
soon as possible (Movshovitz-Hadar & Hadass, 1990). If the learner cannot
avoid the conflict, then the learner must take action (Skemp, 1979).
Finally, there are several teaching strategies which can be used to assist
students to alleviate their misconceptions. For example, the strategy of
diagnosis should always be carried out as a prerequisite to instruction (Hewson
& Hewson, 1984). "Once the existing, mistaken concepts are recognized then
an appropriate teaching strategy can be developed, for example by producing
empirical evidence which contradicts the students' beliefs" (Head, 1982, p.
636). The goal is to create conflict that the student will have to resolve in order
to promote accommodation of the given concept.
Nussbaum and Novick (1981, 1982) proposed a similar model which
supports a learners accommodation of concepts. The researchers suggested
that accommodation is a process which can be prepared for, but not scheduled
or guaranteed. They proposed that all "one can do is to try to characterize it
and to look for instructional strategies that may facilitate its occurrence"
(Nussbaum & Novick, 1982, p. 186). The model they suggested for facilitating
cognitive accommodation involves three major objectives. The first objective is
to create a learning situation that encourages learners to examine his/her
conceptions prior to instruction of a topic, express these conceptions orally and
written. The teacher's role involves assisting the learner in expressing
conceptions in a definitive manner and promoting an environment where
learners can debate on the conceptions.
Secondly, one should attempt to create a situation where conflict is born
between the students' misconceptionss and some academic truth or reality.
"The information must be presented in such a way as to challenge or stimulate
the student, for it is through this process of conflict that he integrates the new
material" (McMillan, 1973, p. 36). "Piaget's theory is that cognitive
development is promoted when there is a moderated degree of discrepancy
between the child's cognitive structure and the new event which he encounters"
(Ginsburg, 1992, p. 360).
The last objective requires one to support the learners' accommodation of
the concept. "The teacher's job is not to reinforce correct answers, but to
strengthen the learner's own process of reasoning" (Mason et al., 1983, p.
238). "According to Piaget, failure of previous knowledge for assimilation of an
experience causes accommodation or new learning" (Hergenhahn, 1976, p.
Berlyne (1965) suggested that "different forms of conceptual conflict are
readily applicable to different educational subject matters" (p. 78). Thus
strategies offered by science educators may need adjusting before applying
them to topics in mathematics. For example, Berlyne explained that "surprise"
tactics may be better for science subjects where students can be surprised by
occurring phenomena that they thought impossible. ( This is what Nussbaum
and Novick (1982) described as a "discrepant event.") On the other hand,
Berlyne suggested that "doubt" may be better for some topics in mathematics.
Cognitive dissonance has been characterized as a focus on changing a
person's attitude (Berlyne, 1965; Nussbaum & Novick, 1982) rather than the
person's conceptions. Some researchers (e.g., Hewson & Hewson, 1984) have
chosen to describe the phenomenon as conceptual conflict rather than
cognitive dissonance or cognitive conflict. Nussbaum and Novick (1982)
suggested that conceptual conflict should be used when referring to academia.
Conceptual conflict was described by Berlyne (1965) as "conflict due to
discrepant thoughts, beliefs, or attitudes" (p. 77).
Cognitive dissonance, cognitive conflict, and conceptual conflict are all
concerned with the need of a person to resolve dissonance or conflict between
two conceptions (Nussbaum & Novick, 1982). "In general, all of the theories
utilize the same dynamic: if a person encounters information, facts, or
experiences which makes him feel uncertain, which creates a sense of
imbalance, incongruity or conflict, then that person will change his behavior or
"knowledge" to integrate the information or experience into already existing
norms" (McMillan, 1973, p. 35). Thus in the forthcoming review of the
literature, the terms) that the various researchers used will be reported without
Burton (1984) contended that there were two ways of approaching
cognitive conflict: aggressively and passively. In the aggressive mode, the
student will actively seek methods to overcome the conflict, often by confronting
it directly. In the passive mode, the student might accept failure and neglect to
solve the conflict.
Significance of the Study
This study was supported by the need in mathematics education to provide
research data pertaining to students' understanding of the concept of function,
more specifically, their understanding of the concepts of domain and range,
and their ability to select appropriate domain, range, and scales for the axes
when graphing functions. The use of the graphing calculator in instruction
requires that students using the tool consider how they select the domain ,
range, and scale for the axes when graphing functions. One must be aware of
the effects that these elements have on the visual representation of the concept.
In addition to this concern, the conceptions that the students have prior to
instruction with the graphing calculator may be in conflict with new knowledge
presented to them, and if so this conflict should be capitalized to enhance
students' understanding of the concept. Moreover, this study included an
attempt to validate and extend previous research regarding students'
identification, construction, and definition of function.
There is an increased need to examine the uses of the graphing calculator
in mathematics instruction and to provide for the visibility of the graphing
calculator in the research literature as a tool for conceptual understanding. It is
evident from the literature that research regarding conceptual understanding
involving the graphing calculator is scarce.
Because very little research in mathematics education is conducted on the
community college level, it is imperative that research results are not quickly
applied to levels of instruction where the research is rarely conducted. This
portion of the study was not simply a replication, but it was an extension of
previous research. This extension included the use of a community college
sample, the use of a homogeneous sample (community college algebra
students), and the factors of graphing calculator use and concept assignment
participation regarding the concept of function and specifically the concepts of
domain and range.
Organization of the Study
A review of relevant literature regarding the concept of function, the use of
technology in teaching the concept of function and other related concepts, and
the application of conceptual change theory in mathematics education is
presented in Chapter II. The design and methodology of the study is reported in
Chapter Ill. In Chapter IV, results of the analyses and limitations of the study
are provided. A summary of the results, implications, and recommendations for
future research are presented in Chapter V.
REVIEW OF THE LITERATURE
A review of the relevant literature is presented in this chapter. The areas
under discussion are development of the concept of function, representations of
the concept of function, difficulties students have with the concept of function,
computers and calculators, graphing calculators and the application of
conceptual change theory in mathematics education research studies.
Development of the Conceot of Function
The concept itself was in use long before the word "function" entered into
the language of mathematics (Boyer, 1946; Miller, 1928). History has not
credited one person, place, or time with creating the concept of function
(Thomas, 1969). Because it is one of the oldest mathematical concepts,
records of its history are not always reliable (Miller, 1928). Mention of some of
the major contributors to the concept of function since the 17th century follows.
It was Descartes (1596-1650), through his emphasis on the graph of a
geometric curve as a representation of an algebraic function, who influenced
the connection between algebra and geometry ( Sidhu, 1981; Sobel &
Maletsky, 1988). This connection was a catalyst for the development of the
concept of function (Confrey, 1981b; Kleiner, 1989). Dines (1919) suggested
that this union of disciplines encouraged an increase of progress in the
individual fields. The presence of the concept of function in mathematics has
made possible an integration between algebra and geometry and an
understanding of trigonometry (NCTM 1989). The work of Fermat (1601-1665)
and Descartes (1596-1650), regarding their interest in the roots of equations in
two unknowns, provided the way for "variable and function of a variable" to
surface in mathematics (Boyer, 1946).
Leibniz (1646-1716) introduced the term "function" into mathematics in
1673 (Ponte, 1992). He was attempting to describe the characteristics of a
curve such as points on the curve and slope of the curve (Hamley, 1934;
Kinney, 1922). Bernoulli (1667-1748) and Euler (1707-1783) developed similar
definitions for function, including Bernoulli's definition of function as an
expression composed of variables and constants (Kleiner, 1989). Euler labeled
an equation or formula consisting of variables and constants as a function
(Eves, 1981; Reeves, 1969). At the beginning of the twentieth century a
function was commonly defined as an expression or formula (Hight, 1968).
Euler went further by classifying functions as algebraic or transcendental (Miller,
1928), single-valued or multivalued, and implicit or explicit (Kleiner, 1989).
Following the Eulerian concept of function (Boyer, 1946), Fourier (1768-1830)
was the first mathematician to develop new areas with his investigations of heat
flow (Eves, 1976, 1981; Ponte, 1992). For instance, he formed a general
equation for the motion of heat in a conductor (Bell, 1937). All of these events
helped to increase the strength of the concept. In some cases, the concept
became more constrained by certain parameters in its definition, but in other
cases the definition broadened the concept as did Dirichlet's (1805-1859)
Nicholas (1966) suggested that some agreement should take place
regarding the definition of function. There were many variations of Dirichlet's
definition for the concept of function. More important than the statement of his
definition of the concept were the doors which his definition opened in
mathematics. His definition did not rely on the existence of an algebraic
expression (Reeves, 1969). "A function, then, became a correspondence
between two variables so that any value of the independent variable, there is
associated one and only one value of the dependent variable" (Ponte, 1992).
His meaning for the term included discontinuous functions (Kleiner, 1989),
Bernoulli's formula concept, Euler's equation concept, and Descartes'
geometric curve concept. Furthermore, this definition ignored the restriction
that the variables had to represent real numbers (Dines, 1919). Dirichlet's
definition of the modern concept of function has dominated the mathematics
curricula of today (Dreyfus, 1990). The two components of this definition of the
concept of function were "arbitrariness" and "univalence" (Even, 1989, 1990,
1993). The arbitrary nature of functions eliminates the restriction that functions
have to be defined by one expression, graph, or defined on specific sets of
elements. The latter component, univalence, requires the restriction that each
element of the domain of the function corresponds to one and only one element
of the range. "The development of advanced analysis created the need to deal
with differentials of orders higher than one and, therefore, to distinguish
independent from dependent variables. In such a case, it became too difficult to
work with multivalued symbols, and the univalence requirement was added to
the definition of a function" (Even, 1993, p. 96). This correspondence between
the domain and the range can either be a one-to-one or a many-to-one
Set theorists continued to expand the concept of function by defining it as
a relationship between any two arbitrary sets of elements (Bennett, 1956;
Johnson & Cohen, 1970; Thomas, 1969). These sets could be numerical or
non-numerical (Ponte, 1992). Bourbaki gave the definition of function in terms
of a set of ordered pairs, that is, as a subset of a given Cartesian product
(Kleiner, 1989). In some instances, the concept has continued to be presented
in this way (Janvier, 1987). The definition of the modern concept of function
has often been described as the Dirichlet-Bourbaki definition.
The concept of function has a history marked by efforts to define and
redefine the concept and its definition ( Bennett, 1956; Eves, 1981; Hight,
1968). This growing and changing of the concept has been but one example of
the conceptual change theory discussed by Confrey (1981b). She proposed
that this view "portrays mathematics as having competing theories which are
the results of attempts to solve outstanding problems" (p. 248). One particular
problem that encouraged conceptual change was the Vibrating String problem
(Miller, 1928), which influenced the evolution of the concept of function (Dines,
1919). The Vibrating String problem was used to implement an extension of the
concept to include functions not previously included (Ponte, 1992). Examples
of these functions would be piecewise functions and functions drawn freely by
hand and not given by an algebraic expression (Kleiner, 1989). The Vibrating
String problem can be examined by fixing the ends of a string at two arbitrary
points on the Cartesian coordinate system then releasing the string to vibrate.
Any still shot of the vibration represents the graph of a function (Confrey,
1981b). The concept of function was a result of growth, change, and conflicting
ideas and theories among mathematicians.
The notion that the concept of function was important enough to be
included in the mathematics curricula of the United States can be attributed to
Felix Klein (Georges, 1926; Hamley, 1934), then a professor of mathematics at
the University of Gottingen (Reeves, 1969). He proposed this in an 1893
speech for the International Congress of Mathematicians in Chicago (Hamley,
1934). The first people to reinforce this proposal were David Eugene Smith
and E. R. Hedrick (Reeves, 1969), often publishing articles in the Mathematics.
Teacherto show their support for the concept (Hedrick, 1922; Smith, 1928). The
National Committee on the Reorganization of Mathematics in Secondary
Education also took on the concept of function as a major focus (Longley,
Later emphasis on the concept of function was often dominated by the idea
of "functional thinking" (Booher, 1926; Georges, 1929; Lennes,1932).
According to Breslich (1932) and Hamley (1934), a person who is able to think
functionally should be able to recognize relationships between variables,
determine the underlying theme of the relationships, express relationships
algebraically, and recognize the effects of changing variables. The educators
believed that these activities characterized the application of the concept of
function. The concept of function has entered into all realms of academic
disciplines especially mathematics (Breslich, 1928), and in any other context
where relationships have arisen (Boyer, 1946; Carver, 1927). The principle of
functionality was described as the relationship of dependence (Breslich, 1932;
Kinney, 1922 ). Hedrick was one of the first proponents of functionality. He
proposed that presenting a definition for the concept of function was not in itself
adequate, but what was needed was attention to relationships (Hedrick, 1922).
Functional thinking involved thinking in terms of quantitative relationships
(Georges, 1926; Hamley, 1934), described as "quantitative reasoning" by Fey
(1990). The identification, analysis, and usefulness of relationships were the
foci of functional thinking ( Breslich, 1940), because understanding
relationships was considered important to informed thinking (Booher, 1926).
Fey (1990) suggested that "quantitatively literate young people need a flexible
ability to identify critical relations, ... to express these relations in effective
symbolic form, to use computing tools to process information and to interpret the
results" (p. 65). It is the acquisition of the concept of function which would
provide a basis from which learners could begin to reason quantitatively.
Representations of the Concept of Function
Janvier (1983) proposed that a representation of a concept is composed of
three units: written symbols, real objects, and mental images. The link between
the three elements is completed by verbal and non-verbal language processes.
The researcher presented the concept of function as an example of a concept
which does not have a real object representation, but yet remains a powerful
The concept of function can be represented in multiple ways, such as by
graphs, equations, tables, and arrow diagrams (Dreyfus & Eisenberg, 1982;
1984; Even, 1990; Langer, 1957; Lennes, 1932; Markovits, Eylon, &
Bruckheimer, 1988; Schwarz, Dreyfus & Bruckheimer, 1990; Smith, 1972; Stein,
Baxter, & Leinhardt, 1990; Thorpe, 1989). Ponte (1992) proposed that the most
important representations of the concept were the numerical (tables and
computations), graphical (Cartesian), and algebraic (equations)
representations. Dufour-Janvier, Bednarz, and Belange (1987) developed
several reasons for emphasizing multiple representations of a concept, all of
which have implications for the concept of function. Some concepts, such as
the concept of function, are closely related to their representations. Each
representation presents a different view of the function (Friedlander, Markovits,
& Bruckheimer, 1988), but the nature of the function is explicit in its
representation, if it is clearly understood.
The representations are of the same concept. In the case of function, for
example, an algebraic expression, a table of values, and a graph can represent
the same function. Together these representations can be used to reinforce the
concept (Goldenberg, 1988).
Emphasis can be placed on trouble spots with certain representations.
Tirosh (1990) found that student may be able to successfully complete a task
using one representation of the concept of function, but when given the same
task, but a different representation to use, the student may be unsuccessful. For
example, it may be possible for a student to examine the roots of a function
more closely by the graph of the function rather than by the equation of the
function. The domain of a function may be easily determined by the function
given as a set of ordered pairs, a table, a mapping, an algebraic equation, or by
the graph of the function.
Multiple representations can be used to break up the monotony of working
in the same setting for a concept. Hence, different problems can be developed
concerning the different representations.
These reasons supports Confrey's (1992) theory of an "epistemology of
multiple representations". She suggested "that it is through the interweaving of
our actions and representations that we construct mathematical meaning" (p.
149). Thus, she also lends support to Piaget's theory of genetic epistemology.
The emergence of graphing calculators in education has encouraged
educators to consider the advantages of emphasizing the graphical
representation of function. Yerushalmy (1991) suggested that "only one topic
in a traditional algebra course utilizes visual-graphic representation in addition
to the symbolic one: investigating functions" (p. 42). Functions and graphs
should be central topics in algebra, one reason is because they are at the heart
of elementary calculus ( Fey, 1984; Hamley, 1934). Graphs and graphing
should be encountered by and introduced to students as early as possible in
the mathematics curriculum (Booher, 1926; Kinney, 1921; Kinney & Purdy,
1952). Georges (1926), Lennes (1932), and Buck (1970) suggested that
compared to other representations of the concept of function, the graphical form
of functions was most useful for students. From information obtained by
observing the graph, the learner can depict various characteristics of the
relationship, such as the possible values of the independent variable, that may
not be as obvious through another representation. From the graph, the learner
may also be able to determine the domain and range of the function. The
learner can use the graphing calculator to examine the effects of changing the
domain of a function or to examine the behavior of a function by restricting its
domain. The graph has the potential of enhancing the concept of function
(Clement, 1989; Lennes, 1932; Reeves, 1969), and is an accessible medium
with which this can be done.
Learners using the graphing calculator will find it necessary to select
appropriate domain, range, and scale for the axes in order to provide a useful
graph of a function. Thus the students' concept of function is enhanced through
realization that the domain dictates the resulting range and that the scale of the
axes dictates the visual properties of the graph of the function. Researchers
(e.g., Ayers, Davis, Dubinsky, & Lewin, 1988; Demana & Waits, 1991; Fey,
1990) are now beginning to emphasize the importance of scaling for providing
appropriate graphs of functions. The consideration of the scales of the axes on
which the graph appears reflects the importance one puts on viewing an
appropriate graph of a function. If one is not careful to choose a reasonable
scale for the two axes, "critical features" of the graph can be overlooked
(Goldenberg & Kliman, 1988).
Students should not only think functionally, but they should also think
graphically (Hamley, 1934) and realize that the graph is not the end of the
concept of function (Breslich, 1928; Fischbein, 1987). The purpose of the
graph is to increase the power of the concept of function and the function which
it symbolizes (Buck, 1970; Hamley, 1934; Kinney, 1922). "The graph
representing a function is also an intuitive model of that function and the
function in turn, is the abstract model of a real phenomenon" (Fischbein, 1987,
p. 121). The graph of a function is a representation of the quantitative
relationship between variables (Blank, 1929; Fey, 1990; Linn, Layman, &
Nachmias, 1987; Sidhu, 1981). Although a graph of a function is a pictorial
representation of the algebraic representation of the function (Georges, 1929;
Stein, Baxter, & Leinhardt, 1990), the resulting graph is controlled by the
design of the coordinate system on which the learner displays the graph.
The desired outcomes of focusing on graphs of functions is to increase
students' ability to interpret graphs (Clement, 1989), produce graphs, and to
identify properties of functions from their graphs (Kissane, 1989). Using the
graphing calculator, the graph of a function can be produced in the viewing
rectangle by inputting the algebraic representation of the function and
appropriate domain, range, and scale values. The choice of the values are
completely controlled by the learner (Waits & Demana, 1988). The learner has
the option of altering the values at any time that the graph of the function is not a
sufficient representation of the function. With pencil and paper, this would mean
erasing and/or starting over on a new set of axes, causing students to become
frustrated and discouraged while experimenting with different domains, ranges,
and scales of axes for graphing functions. Yerushalmy (1991) posited that the
lack of an efficient method of changing the scales of axes for graphing functions
has caused students to only relate "prototypical graphs" to their algebraic
Sidhu (1981) suggested that "a graphic representation appeals to the
aesthetic sense" (p. 271), and this may help students understand the abstract
concept of function (Leitzel, 1984). Graphs provide opportunities of
visualization in algebra, which is very important, and in particular should be a
part of the course content (Vinner, 1989). In this sense, the focus is on the
"noun" of visualization, characterized by Bishop (1989) as being "the product,
the object, the 'what' of visualization, the visual images" (p. 7).
Dreyfus (1990) proposed that "direct visual processing help students form
more complete concept images of functions" (p. 122). When students are able
to view images of functions, they become more active in the learning process
(Demana & Waits, 1991; Waits & Demana, 1988), and gain more conceptual
understanding of functions and their graphs (Dion, 1990). With the graphing
calculator, "the power of visualization can be exploited" (Demana & Waits,
1990, p. 28). Yet this cannot happen if students are not aware of the implication
of choosing appropriate domain, range, and scale for the axes for graphing
functions. This objective is one that is currently omitted form the traditional
mathematics curricula (Burrill, 1992). The current emphasis on pencil and
paper graphing does not provide students the opportunity to explore the
implications of choosing various domains, ranges, and scales for the axes
for graphing functions. In particular, teachers of traditional mathematics
curricula encourage students to only consider graphing functions with "nice"
domains and ranges (x and y from -10 to 10) and scales of axes equal (Hector,
1992). Unfortunately, the fact that "students learn about functional behavior as
they search for an appropriate domain and range to give a full view of the
function" (Hector, 1992, p. 132) has been neglected.
In the case where the visual representation is most useful, it is impractical
to suggest that all of the work should or could be done with pencil and paper.
The graphing calculator could easily be used to provide instruction which
focuses on the graphical representation of function. Bishop (1989) suggested
that visualization is both a 'personal' and individual' activity and the
personalization characteristics (i.e., portable, hand-held) of the graphing
calculator could increase the effectiveness of these factors. However, use of the
graphing calculator calls for emphasis on producing sufficient graphs of
functions. This in turn calls for a need to consider the limitations of the viewing
rectangle of the graphing calculator. Dick (1992) proposed that lack of
choosing sufficient domains, ranges, and scales can cause the "graphical
behavior" of the function to be hidden.
Difficulties with the Concept of Function
The concept of function, which should be presented to students as early
as possible (Eves, 1976,1981; Goals for School Mathematics, 1963), has
perhaps been the most difficult mathematical concept for students to master
(Dreyfus, 1990). Many teachers have had experiences where students were
simply not acquiring a mathematical concept presented to them (Malone &
Dekkers, 1984). Students have often experienced difficulties in their attempts to
acquire the concept of function (Dreyfus & Eisenberg, 1984; Henderson, 1970).
An awareness of these difficulties and students' misconceptions are important
(Confrey, 1981a; Tirosh & Graeber, 1990a, 1990b).
Many students have been hindered by a lack of intuitive understanding of
the concept of function caused by misconceptions and inconsistencies (Dreyfus
& Eisenberg, 1984; Herscovics, 1989; Leinhardt, Zaslavsky, & Stein,1990;
Markovits, Eylon, & Bruckheimer, 1988; Wagner, 1981). This realization was a
result of research which examined difficulties students had with the concept of
function. Topics on the concept of function, conflicts, and inconsistencies
regarding students' learning of mathematics have been addressed by the
International Group for the Psychology of Mathematics Education (IGPME)
(Nesher & Kilpatrick, 1990). Students have exhibited a variety of difficulties with
the concept of function which have highlighted inconsistencies in their
knowledge, in regards to the concept. More research is needed regarding the
concept of function (Even, 1989; Fischbein, 1990), and researchers have
begun to make inquiries regarding the concept in order to aid mathematics
curricula and instruction.
Stein, Baxter, and Leinhardt (1990) conducted an extensive study to
examine teacher subject matter knowledge of function and graphing to
determine how one's subject matter knowledge would be a factor during
instruction in the elementary grades. A portion of the study was reported as a
one-subject study, which focused on the teacher's knowledge of function and
graphing and instruction for fifth grade students. The measures, interview and
card sorting tasks, were designed by the researchers and field-tested for the
study. Reliability measures were not reported for either. The interview
consisted of open-ended questions. The card sorting task has been used
previously in mathematics education research. Collis (1971) employed the
card sorting task in a study designed to follow concept formation in
mathematics for a sample of eighth grade students in a convent school. The
results from the task were used to determine what the subjects chose as
"fundamental categories" by allowing the subjects to sort the cards freely.
Likewise, the volunteer subject in Stein, Baxter, and Leinhardt's (1990) study
was instructed to categorize the cards which depicted graphical and algebraic
representations of functions. The task was validated by a group of mathematics
educators. Grouping or sorting possibilities included function representation,
mathematical relationships, functions versus nonfunctions, and a combination
of the aforementioned.
Upon analyzing the subject's sorted cards and noting that the subject did
not group together the different representations of the same function, the
researchers concluded that the subject did not have mastery over the different
representations of the same function. In particular, the subject in the study was
lacking the connection between the graphical and the algebraic representation
of function. This finding has been corroborated by the findings of many other
researchers (e.g., Ayers, Davis, Dubinsky, & Lewin, 1988; Dreyfus & Eisenberg,
1983, 1984; Dunham & Osborne, 1991; Even, 1989, 1990; Fey, 1984; Graham
& Ferrini-Mundy, 1990; Hector, 1992; Kreimer & Taizi, 1983; Leinhardt,
Zaslavsky, & Stein, 1990; Markovits, Eylon, & Bruckheimer, 1990; Schwarz,
Dreyfus, & Bruckheimer, 1990). Because the development of the concept of
function was nurtured with the connection of algebra and geometry, one might
want to assume that establishing the relationship between the algebraic and
graphic representations of the concept would not be difficult for some students.
However, Even (1989) and Even, Lappan, and Fitzgerald (1988) observed that
students who could recognize a given function in its algebraic representation
could not always recognize the same function in its graphical representation.
Stein, Baxter, and Leinhardt (1990) considered the fact that many teachers
in the elementary grades are not required to obtain training in mastering or
instructing the concept of function and graphing. The subject was an 18-year
veteran of education and described by administrators at his school as an
excellent teacher, but there was no indication of how those conclusions were
reached, nor of his overall mathematical education or ability. Because the
concept of function has begun to be a focus in some elementary grades across
the nation (Even, 1989), it might be informative to examine the novice teacher's
subject matter of the concept of function and graphing. The researchers
acknowledged this as well. An important implication of this study and other
studies involving teachers and future teachers as subjects is that when teachers
do not exhibit a strong ownership of the concept of function, then one should
not expect that the students under instruction will develop a strong concept of
function (Cronbach, 1942).
Misconceptions surrounding the concept of function of a sample of ninth
and tenth grade students were analyzed by Markovits, Eylon, and Bruckheimer
(1988). The instrument used to measure the students' performance consisted of
what the researchers considered to be sufficient elements of the concept of
function. They determined that students were more successful when working
with the graphical representation rather than with the algebraic representation
of function. The researchers did not suggest that the algebraic representation
should be dismissed from instruction, rather it was recommended that more
work should be done with the graphical representation early in instruction.
They believed that this might help to increase students' success with the
abstract, algebraic representation. Dunham and Osborne (1991) reported on
the response of 400 precalculus students to two questions of algebraic
inequality between two different sets of functions. The students were asked to
solve the inequalities algebraically and graphically. The researchers found that
the students had the most difficulty with the graphical representation. One
explanation for this occurrence offered by the researchers involved the
students' "limited and superficial" experience with typical functions.
Questionnaires and individual interviews have been used by many
researchers (e.g., Even, 1989, 1990, 1993; Markovits, Eylon, & Bruckheimer,
1988; Thomas, 1969) to study students' concept of function. In a study
regarding subject matter knowledge and the teaching of function, Even (1989,
1990, 1993) administered questionnaires to 162 subjects, all of whom were
prospective secondary teachers, and conducted interviews with ten of the
participants. The instrument was developed for the study; both questionnaire
and interview components were pilot tested and validated by an expert panel of
educators. Neither instrument was designed to measure performance. The
questionnaire involved the subjects' response to function concepts and
comments on students' work. The interview was implemented to probe for
further information from the subjects. One difficulty which the subjects
exhibited was a rejection of function as a function because of an incorrect
assumption that functions should always be represented by formulae, as was
also found by Tall and Vinner (1981), Dreyfus and Eisenberg (1983), and
Vinner and Dreyfus (1989). These students also held the misconception that
functions should have precise, predictable graphs. In other studies, it is
reported that students have difficulties with functions that did not have the kinds
of graphs that they found acceptable, such as constant and discontinuous
functions (Dreyfus & Eisenberg, 1982; Hornsby & Cole, 1986; Maurer, 1974),
and circular functions (Herman, 1988). Even (1990, 1993) found that students
who could produce the graph of a discontinuous function were still uncertain
about the graph as representative of a function because of the visible holes in
the graph. In addition, she reported that the subjects did not understand the
vertical line test, nor the property of univalence. The graphing calculator could
play a role in giving students the opportunity to work with functions graphically.
Students need the opportunity to develop individual images of a concept
(Bishop, 1989), and the graphing calculator can be exploited by students who
need the opportunities to examine various functions. Students can not only
work with functions provided by instruction, but they are free to graph randomly
Markovits, Eylon, and Bruckheimer (1983) found after a series of studies
that students are restricted by an overbearing linear image of function. From a
larger study involving ninth grade students, the researchers were able to
determine that even after changing the context of a problem involving function,
the students were still guided by the linearity of functions. The researchers
suggested that this phenomenon is due to the fact that the linear function is the
simplest function and it is emphasized greatly in school, causing the students'
images of functions to be restricted. Along with the linear image,
Papakonstantinou (1993) found that students also tend to present a quadratic
image of function. Tall (1989) suggested that "by presenting mathematics to a
learner in a simplified context, we inadvertently present simplified regularities
which become part of the individual concept image. Later these deeply
ingrained cognitive structures can cause serious cognitive conflict and act as
obstacles to learning" (p. 37).
Other students have also shown the acceptance of the misconception that
all functions must be linear. For a sample of high school students and for a
sample of 84 college students, Karplus (1979) and Dreyfus and Eisenberg
(1983) respectively, found that the property of linearity is one which students
strongly depend on when completing graphs of functions or when producing
graphs of functions. In their study of college students' concept of function
regarding properties of linearity, smoothness, and periodicity, Dreyfus and
Eisenberg (1983) administered a questionnaire to college students enrolled in
a mathematics course at two Israeli universities to examine the students'
concept of function. Other important difficulties noted by the researchers
included difficulties by students in determining if a given relation was or was not
a function and in using the concept of function to solve problems. Regarding
their discussion of the analysis of the results, the researchers used such
expressions as "students seem to feel" and "some students feel"; since these
are subjective observations, it is questionable as to whether the conclusions
based on the observations are replicable.
Tall and Vinner (1981) defined concept definition as the collection of
words or phrases used to denote the concept. Tennyson and Park (1980)
suggested that a concept definition should contain all the critical attributes of
the concept in question. Tall and Vinner (1981) and Tall (1990) also defined
concept image as the image accompanying the concept, including mental
images and characteristics of the concept. The students' idea of what a function
looks like (concept image) is determined by the image created by examples
from experiences and not necessarily by an instructional definition (concept
definition) ( Friedlander, Markovits, & Bruckheimer 1988; Tall 1981). The idea
of concept image is comparable to Skemp's (1979) use of the term "object-
concept". He posited that the object-concept is the most basic image of a
concept which allows one to recognize whether or not a given object represents
the concept. Regarding the graph of a function, the most basic image
would most likely be different for each person, because the concept image the
person would have in mind at any given time would be based on his/her
personal definition and/or image of the concept and not necessarily the
mathematically accepted definition (Tall & Vinner, 1981).
A serious error in students' thinking is a conflict between concept definition
and concept image (Tall, 1986). This reflects a major inconsistency in the
students' conceptual system (Scwharzenberger, 1982; Tirosh, 1990). Students
may or may not be able to produce a concept definition; they may indeed have
acquired the concept, but yet be unable to provide evidence of this acquisition
(Henderson, 1970; Van Engen, 1953). For instance, they may not be able to
give a verbal definition (Skemp, 1979, 1987), and in any case, the concept
image may not be comparable (Even, 1990; Tall & Vinner, 1981; Tirosh, 1990;
Wilson, 1990). In regards to the conflict between two cognitive elements, the
cognitive dissonance theorist implied that conflict "can be eliminated by
changing one of those elements" (Festinger, 1957, p. 18). In application of
Festinger's recommendation for changing the element by adding a new
cognitive element to "reconcile" the conflicting elements, the graphing
calculator as a cognitive tool can become the new element which can assist
students in developing images for the concept of function. The number of
images of the concept of function that the student can create with the graphing
calculator is infinite, thus insuring that a variety of examples are available for
examination to help the student develop sufficient images of the concept.
However, the number of images of the concept of function that the student can
create is not as important as the variety and quality of the images.
The subject in Stein, Baxter and Leinhardt's (1990) study dismissed the
critical characteristic of function that one and only one range element can be
assigned to each domain element. Wilson (1990) found that even for those
students who are successful at a sorting task, it is possible that they may not be
able to provide a definition of function which describes their conditions for
sorting. Likewise, Papakonstantinou (1993) and Vinner and Dreyfus (1989)
found that even when students could provide the Dirichlet-Bourbaki definition of
function, they were in some cases unable to use this definition to identify
relations which were functions. Also, students may master formal mathematical
knowledge such as definitions (Cronbach, 1942), but yet be unable to apply this
knowledge in a problem situation (Fischbein, 1987). Dreyfus (1990) found this
to be one of the major conflicts students have regarding the concept of function.
Vinner and Dreyfus (1989), Vinner (1990), and Linn and Songer (1991) would
describe this phenomena as a compartmentalization of the students'
knowledge. When students can provide a definition of the concept of function,
they are using information which satisfies the condition of providing a sufficient
definition. The same students can possibly not rely on that information as a
means of identifying functions. Instead, they may rely on their images of the
concept. Therefore, asking a learner to produce a definition of the concept of
function does not guarantee that the learner is able to apply the definition. This
difficulty may be attributed to the fact that different students may have different
mental images of function and may construct these images in different ways,
depending on the images that are emphasized in instruction (Hershkowitz,
Arcavi, & Eisenberg, 1987).
Results by Vinner and Dreyfus (1989) were reported from a seven-item
questionnaire administered to a sample of 271 first year Israeli college students
and 36 Israeli junior high school teachers. There were no reliability or validity
estimates provided by the researchers for the questionnaire. In analyzing the
questions the researchers consulted a panel of mathematical experts and used
trained assistants to help organize the data. The subjects, students of various
academic majors, were divided into four groups depending on the level of
mathematics course needed for their academic major. Given this basis for
division, similar groups of students at other universities could have been
divided differently based on those universities' requirements.
The first four questions were identification questions, the fifth and sixth
questions were construction problems, and and the seventh question was a
request for a definition of function. Students were asked to explain all answers.
After randomly choosing and analyzing 50 questionnaires of correct definitions,
they were able to determine that the students' definitions could be sorted into
six categories, which unmistakenly were parallel to some of the definitions
found in the historical development of the concept of function. For example,
one of the categories was function as defined as an algebraic expression or
equation, which parallels Bernoulli's and Euler's conceptions of the definition.
Of the various results, they found that the students supplying a Dirichlet-
Bourbaki definition increased with mathematics course level and these students
gave more and better justifications for their responses to identification and
construction exercises regarding the concept of function.
Papakonstantinou (1993) administered a six item questionnaire to 552
geometry, algebra, precalculus, and calculus high school students. The
students were enrolled in regular and honors classes in two urban schools.
The purpose of the study was to examine the students' knowledge of the
concept of function and the relationship between this knowledge and the
students' ability to provide mathematical and non-mathematical examples of the
concept of function. The researcher found that the students had difficulty
defining the concept, providing sufficient examples, and providing justifications
for their responses. There was a relationship between the students' ability to
define the concept and their ability to provide examples of the concept.
Moreover, she concluded that the students' understanding of the concept was
stronger from a visual or graphical representation approach that from any other
Dreyfus (1990) reported two other "interrelated" difficulties which were
difficulty in visualizing properties of functions graphically and difficulty in
allowing functions to be conceived as mathematical objects. Both observations
were also reported Ayers, Davis, Dubinsky, and Lewin (1988) and Schwarz,
Dreyfus, and Bruckheimer (1990). This is not a new phenomenon. Van Engen
(1953) suggested that this deficiency in acquisition of a concept was possibly
due to the fact that there was a lack of visual aids that provided students with an
opportunity to see the concept of function in action. The employment of
graphing calculators would ease this problem for classroom instruction today.
Markovits, Eylon, and Bruckheimer (1988) investigated the concept of
function of a sample of ninth and tenth grade students. No other information
was given regarding the sample. The researchers were able to determine, by
way of a questionnaire, at least seven difficulties the students had with the
concept. There was not an indication of the statistical strength of the
questionnaire or of the procedures for administration. The framework adopted
by the researchers included two basic facts: The first was the adoption of the
Dirichlet-Bourbaki definition of function and the second was the realization that
multiple representations of the concept of function exist.
The difficulties uncovered by Markovits, Eylon, and Bruckheimer (1988)
which have not been previously addressed will now be discussed. These
researchers found that many students were deficient in understanding
mathematical vocabulary. Vinner and Dreyfus (1989) also reported that many
students' difficulties were due to the fact that they did not have ample mastery of
terms which related to the concept of function, such as image and preimage.
Confrey (1981 a) reported that the terms which accompany a concept are
important in the sense that they help communicate the characteristics of the
concept. Dunham and Osborne (1991), Dreyfus and Eisenberg (1982), and
Rosnick and Clement (1980) were able to add students' lack of understanding
of variables and functional notation.
Markovits, Eylon, and Bruckheimer (1988) also observed that students
often dismissed related concepts of the concept of function, such as domain
and range. The researchers suggested that such related concepts should be
fully understood in all representations of the concept of function or should be
deemphasized in instruction. Of course, the concepts of domain and range are
essential to understanding the concept of function, and emphasis on them
should not be decreased, thus leaving educators the responsibility of focusing
on the concepts in instruction. Orton (1971) found that students misunderstood
the concepts of domain and range as related to the graphical representation of
functions. Given the graph of a function, many of the children in his study were
not able to denote the domain and range of a function
Regarding students' choices of domain and range for graphing a function,
Laughbaum (1989) suggested that if students think that the graph of the
function stops with the "edges" of the medium on which the graph is viewed and
hence implying that the domain and range of the function end at those points,
then the mathematics educator should encourage the students to view of the
function using various domains and ranges. With access to a graphing
calculator, this becomes a small task to facilitate.
Other ideas related to the concept of function are variable and single-
valuedness (Reeves, 1969), images, preimages, zeros, and extremum (Dreyfus
& Eisenberg, 1982, 1984), inverse, many-to-one, and one-to-one (Orton, 1971).
Finally, the researchers discovered that students had difficulty with multi-step
problems and rational number operations. In these instances, we cannot
expect students to succeed with mathematics until they are able to work
comfortably with integers and rational numbers (Leitzel, 1984), and are fully
prepared to use skills to work problems with multi-steps.
Particularly for graphing functions, students have shown to have a lack of
awareness of the importance of the scales of the axes. Dunham and Osborne
(1991) reported that students often ignore the scales of the axes, and they have
the tendency to assume that the scales of both axes of the Cartesian coordinate
system are identical and marked in intervals of one. The researchers found that
when students were encouraged to focus on the scales of the axes, the
students showed improvement in working with points on the graph, scaling, and
using graphing space efficiently. They suggested that teachers should require
students to examine the effect on the graph of the function when the scales of
the axes are changed using graphing devices. Demana and Waits (1988)
found that lack of choosing appropriate scales of axes for graphing functions
could cause one to be a victim of the "pitfalls of graphing". They suggested that
inappropriate scales could cause one to draw incorrect conclusions from the
graph of a function. The researchers proposed that one solution to the problem
was to require that students graph the same function on different sets of axes to
review the effects that the scales may have on the production of the graph.
Yerushalmy (1991) reported on a study involving 35 secondary school
students using the Function Analyzer (Schwartz & Yerushalmy, 1988). Of
current interest are the responses of the students to tasks involving an
understanding of scale. One such tasks requested that the subjects match an
algebraic representation of function with its graphical representation. The task
involved four linear functions graphed on sets of axes with different scale units.
There were "two different pictures of the same function and two identical
pictures of different functions" (Yerushalmy, 1991, p. 48). The results of the
responses showed that the students were relatively successful with the two
identical pictures of different functions (91% and 94%) and were not misled by
the scale of the axes. However, they were less successful with the two different
pictures of the same function (61% and 70%).
The researchers also presented the students with the graph of a function
for which the axes were not labeled. The students were asked to make a
choice between matching the graph with an algebraic equation or denoting that
such an equation could not be determined from the information provided. A
majority of the students (28 out of 30) were correct in responding that the
equation could not be determined. Of the 28 students, 71% had an argument
involving the lack of labels and units on the axes and 29% suggested that they
needed the graph of another function to make comparisons or a different
Educators should find a better or best method for teaching the concept of
function (Cohen, 1983). Henderson (1967, 1970) outlined three ways of
teaching concepts in mathematics, including the concept of function. The three
ways were as follows: In the connotativee" mode, one would use the term
"function" to discuss the properties and characteristics of functions which would
cause one to apply the name to a given mathematical object. In the
denotativee" mode, one would use the term "function" to distinguish between
relations which were and were not functions and produce examples of
functions. "Implicatively", the definition of the concept of function would be
presented to the learner. In the case of the last method which is the most
passive of the three, Vinner and Dreyfus (1989) recommended that formal
definitions of concepts should only be employed to finalize the concepts
through instruction once it is clear that the students have acquired the concept.
In agreement, Tall (1986) stated that "formal definitions.., are totally
inadequate starting points for an unsophisticated learner lacking the cognitive
structure to make sense of them" (p. 23). Likewise, it was proposed by Confrey
(1981a) that "a precise definition fails to communicate the importance of the
concept of function within the conceptual framework of ... algebra" (p. 9).
Other researchers (e.g., Henderson, 1967, 1970; Skemp, 1987) have proposed
that the definition would help students acquire the concept. Wilson (1990)
suggested that definitions, along with examples and nonexamples of a concept
work together to help students acquire concepts.
Regardless of the instructional methods used, students must become
active in learning process (Kaput, 1979). In 1971, Collis reported that "the
attainment of concepts in mathematics is the result, in the main, of deliberate
formal teaching" (p. 12). While, Fischbein (1987), through his research on
students' intuition, found that formal teaching does not provide for all of the
students' academic needs. By accepting the notion that a formal presentation of
definitions and concepts alone does not promote conceptual understanding,
the present research study is conducted with an effort to actively involve the
learner in the instructional process. Studies aimed at increasing the
instructional component of the concept of function are rare compared to the
work being done on students' understanding of the concept and its components
(Stein, Baxter, & Leinhardt, 1990). Teachers should be prepared to organize
and present concepts in a manner which would afford the students greater
access to the concept (Leitzel, 1984). Cohen (1983) and Cornelius (1982)
suggested that concepts should be instructionally developed so that students
can understand and acquire them. One way to encourage this would be for
educators to present concepts in a way such that the new concepts are related
to concepts students have already acquired (Tirosh, 1990). By examining
cognitive conflict in students' mathematical knowledge, we may be able to
avoid compartmentalization (Steffe, 1990). Regarding the concept of function,
Confrey (1981a) proposed that the aim of mathematics education was to
attempt to encourage students to succeed at accommodation of the concept.
Given that "assimilation is typically the way that abstract formally defined
concepts are acquired" (Thomas, 1969, p. 52), the task of the current study was
to examine an attempt to assist students in avoiding compartmentalization
regarding the concept of function either by assimilation, and to a greater
Several researchers have attempted to design methods that will assist
teachers in helping students acquire the concept of function and alleviate
misconceptions. Many of these same researchers have studied students
misconceptions and inconsistencies regarding the concept of function.
Although "there is no proven optimal entry to functions and graphs" (Leinhardt,
Zaslavsky, & Stein, 1990, p. 6), the Function Block introduced by Dreyfus and
Eisenberg (1981, 1982, 1984) was one such attempt to provide dialogue
concerning the presentation of function in the mathematics curriculum. The
main theme of this research has been that students' intuition of the concept of
function should be considered, especially when the topic is first presented.
According to some researchers (e.g., Dreyfus & Eisenberg, 1982, 1983, 1984;
Vinner & Dreyfus, 1989), students' intuition should be assessed before the topic
of function is encountered, because intuition exists before the onset of formal
instruction (Leinhardt, Zaslavsky, & Stein, 1990). Fischbein (1987) advocated
that one's intuition goes beyond the obvious facts one is given. Confrey
(1981a) presented one characterization of a concept as its acquisition which
can allow the learner to go beyond the given information. By examining
students' intuition, the researcher can determine whether or not the learner can
use the concept for generalization. Dreyfus and Eisenberg (1981, 1982, 1983)
suggested that the concept of function be introduced into the instruction by
three interrelated properties of function. The three dimensional block model
displays the function concept on three axes. The x-axis represents the various
settings (representations) in which functions can be presented, which allows for
lateral transfer from one setting to another. The y-axis represents concepts
related to the concept of function, which involves new learning, and the z-axis
represents the level of abstraction or generalization, such as problem type and
number of variables, as a representation of vertical transfer. The researchers
did not address the definition of the concept of function as an element of the
Function Block (Kolb, 1985), nor was the idea that a function represents a
quantitative relationship or functionality emphasized in the model.
According to the designers of the model, the Function Block can be used to
encourage decisions regarding the sequencing and presentation of the concept
in instruction and curriculum by choosing various aspects of the different axes
to emphasize. One has to be careful not to overwhelm the students by mis-
sequencing (Tirosh, 1990) the concept, if the Function Block is used for
Dreyfus and Eisenberg (1981, 1982) administered a questionnaire to 443
Israeli students of grades six through nine in 24 classes. The questionnaire
was designed to examine differences, if they existed, between students'
intuition of the concept of function regarding the independent variables of
ability/social level, grade, sex, and setting of the concept of function. The
ability/social level variable was questionable. The classification was based on
social level as being dominant. Hence a student with high ability but low social
economic status would be classified as "low". Garofalo (1983) pointed this out
as one of the weaknesses of the variable. He also noted students then are not
compared as a group by using any standardized instrument.
The representations they used in the questionnaire were arrow diagram,
graph, and table. Each representation involved a function given concretely and
a function given abstractly. Concepts related to the concept of function and
addressed in the instrument were image, preimage, growth, extrema, and
slope. Reliability coefficients were estimated by Kuder Richardson formula 20
(KR-20) and reported as .91 for the full test and .86 and .81 respectively, for the
subtests. The researchers used a four-way analysis of variance for the 1982
report of the study, but the model does not match their hypotheses. Huber
(1983) noted that interactions were included in the model, but were not
included in their hypotheses. He went on to highlight the fact that there were
significant interactions, but the researchers only discussed the main effects.
They found that ability/social level and grade each had a strong effect on
the total mean scores. The diagram representation caused students the most
difficulty, and it was concluded that students with high ability/social level were
most likely to perform better in the graphical representation than in the table
representation. Students with low ability/social level performed better in the
table representation than in the graph representation. In a study of middle and
high school students, Wagner (1981) found that the sample found it easier to
interpret functions presented in chart form, table, or ordered pair. The
implications of these findings propose to the educator the task of determining
which representations are most appropriate for the population of students in
In a study reported in 1984, Dreyfus and Eisenberg examined a sample of
seventh and eighth grade Israeli junior high school students regarding the
usefulness of the Function Block. Each third of the sample was given an arrow
diagram test booklet, a graph test booklet, and a table test booklet, respectively.
Each booklet had a concrete situation and an abstract situation. The decision
to include both a concrete situation and an abstract situation was based on the
z-axis of the Function Block, however, Even (1989) suggested that this axis was
not precisely clarified and may itself have axes or branches of its own.
Computers and Calculators
Several researchers (e.g., Ayers, Davis, Dubinsky, & Lewin, 1988;
Confrey, 1992; Schwarz & Bruckheimer, 1990; Schwarz, Dreyfus, &
Bruckheimer, 1990) have designed computer programs to deal with the concept
of function in instruction. A Triple Representation Model (TRM) for the concept
of function was developed in Israel (Dick, 1992) and introduced into
mathematics education by Schwarz and Bruckheimer (1990) and Schwarz,
Dreyfus, and Bruckheimer (1990). This model for instruction of the concept of
function was designed to address students' misconceptions about and
difficulties with the concept of function. There are several main characteristics
of the TRM. The model can be used to provide the learner with a path between
the algebraic, graphical, and tabular representation of function. Instruction
which relates these three representations of functions assist students in
developing a deeper understanding of functions, and may help
"decompartmentalize" students' concept of function (Dick, 1992).
The model is useful for providing for transfer between the three
representations. Smith's (1972) research report on transfer between
representations of function with grade nine students provided evidence that
when students are not taught with the representation which they are asked to
transfer to, then the transfer may not be completely successful. The sample of
this study consisted of students participating in the Secondary School
Mathematics Curriculum Improvement Study (SSMCIS). Thomas (1969, 1971)
reported on a study of 201 seventh and eighth grade students participating in
the same project and using an experimental text. At that time, the mapping
approach to function was emphasized, with some use of the ordered pair
approach. Of the students who were competent in working with function given
by arrow diagram and algebraic rule, only 8%/o of them were successful with
the graph and ordered pair representation. As found by Even (1989,1990),
because a student understands the concept in one representation, does not
guarantee that this understanding will transfer to another, newly introduced
The transfer between the representations is completely automatic. The
learner has the option of using the computer to operate within the transfer of the
concept of function from one representation to another. However, the learner
may still be hindered by the inability to conceptually apply transfer to the
concept. For example, Yerushalmy (1991), in a study employing the Function
Analyzer (Schwartz & Yerushalmy, 1988) a computer program involving three
representations of functions, found "that learning within the linked multiple
representation environment does not necessarily motivate a linked
performance. Although the students presented a wide repertoire of visual and
other arguments, they did not tend to use these arguments for cross-checking
purposes" (p. 54).
Work within any representation is operational. This provides for activity in
each representation, which will give the student the opportunity to view each
representation in action. In addition, the computer environment is the focus of
The curriculum for function suggested by the researchers will help students
to understand the concept of function through experiences with the three
representations, transfer between them, and problem solving. The researchers'
observations of students using the TRM have shown significant success with
the activities in the TRM. The researchers allowed an introductory unit on
functions to be taught to ninth grade students for ten weeks for a total of 20
hours of instruction. Students were paired together at each computer with an
enlarged screen for class demonstrations and discussions. They suggested
that the model should not be used at large until more research is conducted
regarding the management of computers as tools of instruction. Although
research on teaching and learning algebra with computers is not new (Kieran,
1990), caution of the use of computers is still recommended by IGPME (Nesher
& Kilpatrick, 1990).
Several of the previous researchers have provided answers to questions
posed by Dufour-Janvier, Bednarz, and Belanger (1987) regarding the use of
multiple representations of mathematical concepts:
1. Which representations should be retained? This question is attended to
whenever researchers draw conclusions concerning the usefulness of certain
representations and difficulties that students may have with them. One can
safely assume that the representations of the concept which are evident to us at
this time must be used appropriately. Perhaps all representations (e.g., graph,
arrow diagram, equation, table, sets of ordered pairs) should be retained with
the condition that they are called upon at appropriate times during instruction.
In Confrey's (1992) "Function Probe" software tool, "the function concept is
viewed as evolving from an interweaving of a variety of representations, each of
which provides a different way of constructing the idea and experimenting with
it. Contrasts and commonalities among the representations provide the basis
for understanding the concept" (p. 169).
2. Are there representations that are more appropriate than others for
developing a concept? Thorpe (1989) suggested that "the definition of a
function as a set of ordered pairs is not only too abstract for an initial
introduction; it is inconsistent with the way functions are viewed and used by
professionals" (p. 13). However, discussion of the concept using the ordered
pair representation could be beneficial for characterizing one-to-one, many-to-
one functions and inverses of functions. The graphical representation has been
advocated as one of the most useful settings in which students may develop the
concept of function. Orton (1971) analyzed interviews with 72 students (ages
12-17) who were either average or above-average mathematically, and he
concluded that in regards to the ordered pair representation of function, "some
children who could recognize functions were confused when they were asked
to explain the difference between a relation and a function, and to relate their
responses to sets of ordered pairs if they could" (p. 46).
3. How should the representations be used? In which context? The
researchers employing the Function Block addressed the issue of presenting
functions in a concrete and abstract environment and employing various
representations. The context in which the representations are used is ultimately
decided upon by the educator. Proper judgment must be used in determining
whether abstract and/or concrete situations are best for approaching and
discussing the topic of function.
4. What are the difficulties and the children's conceptions that need to be
taken into account when a representation is used? The preceding portion of
this review has been evidence that students have exhibited various difficulties
concerning the concept of function.
5. Are there representations that are more appropriate to the level of
development of the child and to where he is in regard to the learning of
mathematics? Based on some of the previous research results, students of
different mathematical ability may prefer different representations of the concept
of function. The learner's general mathematical abilities might be factors in
determining how well the learner can deal with such things as the algebra of
Friedlander, Rosen, and Bruckheimer (1982) described a parallel
coordinate axes much like Arcavi and Nachmias (1990) and Nachmias and
Arcavi (1990) who designed a computer environment, called the Parallels Axes
Representation, as an attempt to help students in working with the symbolic
and graphical representation of function. The model consisted of two vertical,
parallel number lines: the left for the domain and the right for the range.
Mapping lines represent the correspondence between the two, and can be
used to determine characteristics of the function they represent. For example
"slope is reflected as the inclination of the line with respect to the left axes"
(Friedlander, Rosen, & Bruckheimer, 1982, p. 81). This is not meant to replace
the Cartesian or rectangular coordinate system, but the researchers suggested
that students need to see functions represented by different systems.
The microcomputer has proven to be beneficial when used to assist
learners with their acquisition the concept of function (Waits & Demana, 1988).
In recognizing that even college students have difficulty with the concept of
function, Ayers, Davis, Dubinsky, and Lewin (1988) employed the computer as
a method for instruction. The goal was to give students an environment in
which they could experience multiple examples of function. They found that the
students in the computer group were more effective in dealing with concepts
and composition of function than students in the paper and pencil group. The
researchers did not employ any qualitative measures to further examine the
differences between the two groups. Fischbein (1990) has proposed that
research techniques of "pure statistics" should be accompanied by such
methods as observations and interviews, in order to get a more complete
picture of the situation. Another limitation of this study involved a lack of
generalizability to a target population. Their sample was unique in that every
student enrolled at the university was required to own a computer, and the
students were already divided into groups by the university registrar's office.
Statistics were not provided for the pretest nor for the posttest, which were
designed for the study.
Tall and Thomas (1988) reported on employing computers and the Socratic
Method of instruction to examine students' concepts in algebra. Their "generic
organizers" (Tall, 1986, 1990) were microworlds (computer environments)
designed to allow students to work with examples and concepts or systems of
concepts. They conducted three experiments to determine if the combination
worked. Tall and Thomas (1986) used the topic of algebraic variable and
found that the experimental group performed conceptually better than the
control group. They administered another posttest after one year and found that
the students using computers still faired better conceptually than those who had
not. Research involving whole class treatment is representative of what is
needed in mathematics education, because whole class instruction is a reality
in education and quality of education is more important than quantity in
education (Brophy, 1986).
In regard to conceptual learning in mathematics, researchers have begun
to experiment with computer algebra systems to determine their value
regarding the enhancement of concepts in algebra and calculus. Palmiter
(1991) employed the computer algebra system MACSYMA in a study to
examine conceptual and computational differences between students
using the system and students without access to the system. Her study
included 40 students in the experimental group and 41 students in the control
group. The students who were enrolled in a calculus course at a university,
were randomly assigned to the two groups. Both groups studied the same
topics, which included the fundamental theorem of calculus, inverse of
functions, and integration during the course of the experiment.
The study lasted 5 weeks for the MACSYMA group and 10 weeks for the
control group. At the end of the sessions, a conceptual and computational
exam were administered to both groups. The instruments were designed by the
course lecturers. There was no indication in the research report of the statistical
strength of the instruments. However, the lecturers agreed that the content of
the exams reflected the content and difficulty of the course material. The
researcher reported means and standard deviations as an indication of the
differences between the two groups. Only the conceptual exam results will be
an issue in this review.
An example of a problem on the conceptual exam was an analysis of a
graph of a function for a given interval. It was determined that the students
using the computer algebra system scored significantly higher on the
conceptual exam than the students in the control group using paper and pencil
only. Palmiter (1991) admitted to several weaknesses in this conclusion. The
students using the computer algebra system were fully aware that they were
participating in an experiment, and thus this knowledge may have led to a
"Hawthorne Effect". Secondly, the results could be reflective of the individual
instructor and not the treatment itself. Lastly, the experimental group's sessions
only lasted five weeks, while the control group sessions lasted ten weeks. In
this time, the control group had to deal with computational and conceptual
material, while the experimental group used the computer for computational
work. Hence the control group had to work manually on more material and for a
greater period of time. An examination of the students using the computer
algebra system may be more beneficial than examining a comparison
The Function Analyzer (Schwartz & Yerushalmy, 1988) was the
technological tool used by Yerushalmy (1991) to examine its effects on the
learning of function and graphs for a sample of 35 students in the eighth grade.
The main characteristic of the computer program is the manipulation of
functions in three representations: graphical, symbolic, and numerical. From
observations and examinations of five tasks over the 3-month period of the
study, the researcher found that the benefit of visual representation was
hindered by students' lack of "formal algebraic knowledge". In classifying
functions, "students were [more] concerned with [other] factors in the
description of the function, such as symmetry, inflection, location on the grid, or
visual discontinuity" (Yerushalmy, 1991, p. 50). This suggested that stressing
the visual alone by providing a computer screen is not sufficient. The learner
will still be inhibited if an understanding of the concept of function is not
Lynch, Fischer, and Green (1989) reported on a project concerning the use
of computers in elementary algebra that would enhance students' concepts in
algebra. The concept of function was one focus of the project. The
representations available by computer program were tables, graphs, and
equations. Homework, classwork, and exams were completed with emphasis
on the computer and on the algebraic concepts introduced by the activities.
The project directors emphasized the need to reorganize if technology will be a
part of the educational process. Students became more involved in the
learning process and used the tools available to explore and conjecture. Like
the computer, the graphing calculator is powerful enough to be used to
enhance algebraic concepts, such as the concept of function.
Roberts (1980) reviewed the impact of calculator usage provided by 34
studies of mathematics achievement and attitude. Most researchers have been
only concerned with these two variables (Bell, Costello, & Kuchemann, 1983).
The levels of the studies were elementary, secondary, and college. There were
ten articles and 24 dissertations included in the research study. The research
design for most of the studies were pretest-treatment-posttest, with the
experimental group receiving instruction with a calculator (treatment), and a
control group receiving instruction without a calculator.
The two areas of interest were the review for the college level studies and
report of conceptual effects of calculator usage. Of the ten studies reported for
the college level, only three had any concern with the conceptual effect of
calculator usage, and of these, only two showed support for concept
development. In the review of the ten studies, Roberts (1980) did not mention
which of the ten studies involved concept development. The conclusion is
weakened by the omittance of this information.
In the review by effect, Roberts (1980) stated that "few studies made any
real attempt to carefully integrate calculator use into the curriculum that would
illustrate how calculators can facilitate concept learning" (p. 84). This indicates
the need for more researchers to conduct studies which represent an emphasis
on calculators for concept development. The researcher noted several
weaknesses with the studies reviewed. First, he noted that because of the
difficulty of assigning students randomly to treatment and control groups,
researchers assigned groups at random, but still performed statistical analysis
on the individual student, rather than the class as the unit of study and used the
ANCOVA to correct for the use of nonrandom groups. In addition, most of the
studies were conducted with the control and treatment groups in the same
schools and with the same grade levels, thus there was a greater opportunity
for exchange of information during the study. Roberts also criticized the
researchers for using the same teachers for both control and treatment groups
and for not allowing the use of calculators on posttests.
In the most complete study conducted regarding the use of calculators
(Dick, 1988), Hembree and Dessart (1986, 1992) conducted a meta-analysis.
The method of meta-analysis was developed by Gene Glass, Barry McGraw,
and Mary Smith (Borg & Gall, 1989), and is used to convert the statistical
findings from individual studies to an effect size. "The mean of the effect sizes
for all studies included in the research review is the calculated estimate of the
typical effect of the phenomenon under study" ( Borg & Gall, 1989, p. 173), in
this case, calculator effects for students in grades K-12. If other researchers
used the same studies and statistics, the results should be the same.
(Hittleman & Simon, 1992). A total of 79 studies (not referenced in the research
report) were used which were composed of journal articles, published and
unpublished reports, and dissertations. Because one of the weaknesses of
meta-analysis is the inclusion of poorly conducted studies (Borg & Gall, 1989),
the researchers indicated that they took extra precautions, such as contacting
researchers, to clarify content in questionable articles. They did not indicate in
this particular report the characteristics of the weak or strong studies. The
studies differed in such aspects as sample size, grade level, instrumentation,
and length of treatment. In each study, there were a treatment group using
calculators and a control group compared by average scores to measure effects
of calculators in instruction; in most cases the students were not allowed to use
the device for testing. Branca, Breedlove, and King (1992) and Wheatley and
Shumway (1992) found that when students are allowed access to calculators
for evaluation then their teachers can concentrate on evaluating students'
conceptual growth, rather than their paper and pencil performance.
There were 17 independent variables by which the 79 studies were
classified. The dependent variables were achievement and attitude. Concept
development with the calculator was evident in 13 studies, but "regarding the
achievement of concepts, a nonsignificant effect was found across all grade
and ability levels" (Hembree & Dessart, 1986, p. 94).
Koop (1982) explored the effect of the calculator at the community college
level, for the dependent variables of attitude, course completion, retention,
achievement, and effects on different populations of students. Practically no
research has been conducted at this level concerning concept development.
The 150 subjects were randomly assigned to classes. There were three
classes with calculators and three classes without calculators. They could use
calculators for all classwork, homework, and exams. There were three
instructors who had one of each type of class. The independent variables were
treatment, sex, instructor, ethnicity, and age.
An analysis of variance was performed first with pretest data to determine
whether or not this covariate could be relied on more to equate the groups or to
increase the power of the analyses. When the researcher determined that the
pretest could be used to increase the power, he conducted an analysis of
covariance, using the pretest and a posttest. Of interest, was the result that the
older students (those over 29 years of age) confounded the statistics for the
main effect of the calculator. When this group of students were excluded from
the analysis, the effect was much more profound. The researcher did not
suggest that this should be done, but only wanted to point out the situation.
Although ethnicity was listed as a independent variable, it was not discussed in
the research report.
The scarceness of research projects examining the use of calculators for
teaching and learning mathematics emphasizes the need for research aimed at
using the graphing calculator in instruction to alleviate misconceptions in
students' knowledge (Tirosh, 1990), and encourage conceptual change.
Students require an atmosphere for learning that encourages trial and error
(Mathematical Sciences Education Board (MSEB) & National Research Council
(NRC), 1990); exploration, and conjecturing (Burrill, 1992; Demana & Waits,
1989; NCTM, 1990; Ruthven, 1992; Vonder Embse, 1990). The graphing
calculator makes teaching and learning functions an active rather than a
The graphing calculator is very similar to the microcomputer (Willoughby,
1990) and in fact, has been described as a hand-held computer (Cooney,
1989; Dick, 1992; Shumway, 1989), a mini-computer (Debower & Debower,
1990), a programmable computer (Trotter, 1991), and a pocket computer
(Demana, Dick, Harvey, Kenelly, Musser, & Waits, 1990; Demana & Waits,
1988, 1989, 1990). There is still dialogue regarding the use of calculators in
the undergraduate mathematics curriculum (Dion, 1990), although they can be
used to have a great impact on the learning and teaching of mathematics
(Leitzel, 1989). Because graphing calculators are programmable, they are
equally capable of graphing as the computer (Demana & Waits, 1990). On a
more practical level, graphing calculators are less expensive than computers
(Demana & Waits, 1990), and because of their size, they can be physically
handled by the students as a hand-held tool for learning. The graphing
calculator is proving to be most beneficial for graphing and analyzing functions
(Barrett & Goebel, 1991; Hector, 1992) and providing students the opportunity to
enhance their understanding and intuition regarding the concept of function
(Demana & Waits, 1991). It has been proposed that the graphing calculator will
be the technological tool that will have an "immediate" effect on the hig school
mathematics curriculum (Demana & Waits, 1990), and according to Leitzel
(1989) this effect will be greater than that produced by the use of
NCTM (1980, 1989) and MSEB and NRC (1990) suggested the availability
and use of calculators in mathematics at all levels of instruction, from the
elementary level (Harvey, 1991) to the community college level (Koop, 1982).
The availability of the graphing calculator will increase its impact on learning
and teaching mathematics far beyond the computer, that is, until perhaps
computers are more accessible than graphing calculators (Shumway, 1990)
The 1992 NCTM Yearbook, Calculators in Mathematics Education (NCTM,
1992) was compiled of articles written exclusively about the use of calculators
in mathematics education. In his chapter of the Yearbook, Hector (1992)
suggested that graphing calculators affect the way functions are taught in the
mathematics curriculum. NCTM (1989) proposed that students should not
graph functions by first constructing a table of ordered pairs by hand, one
reason being that this action emphasizes a point-wise characteristic of the
graph of functions (Even, 1990). The graphing calculator can not only be used
to decrease the time needed for producing graphs of functions, but it can be
employed to refocus attention to concepts and not skills of graphing by hand
(Hirsch, Weinhold, & Nichols, 1991). Without graphing calculators students do
not have an alternative (Hector, 1992), and teachers may be left without a way
of introducing and focusing on the graphical representations of certain
functions, if chalk and blackboard are the only resources (Morris, 1982).
Graphing calculators are programmed to allow analysis of functions.
Students can analyze the graph section by section by applying the zoom-in
feature to view the graph more closely or the zoom-out feature to view a
broader portion of the graph (Dion, 1990; Hector, 1992; Shumway, 1989). With
the trace feature, the learner can analyze more closely point on the graph
(Burrill, 1992). In this sense, the function becomes a mathematical object to be
visualized and studied. As a visual aid, the graphing calculator can help clarify
meaning of concepts, as visual aids are capable of doing (Van Engen, 1953).
By examining multiple examples of functions, students are able to compare
functions (Dion, 1990), and gain a better understanding of the concept (Barrett
& Goebel, 1991; Demana & Waits, 1988).
For a sample of 67 college calculus students, Vinner (1989) applied a
treatment of visualizing algebraic notions in the course and providing
opportunities for students to choose between algebraic and visual proofs. He
administered several questionnaires from which he was able to determine that
the students were more likely to choose algebraic over visual. However,
"reliable access to graphic calculators is likely to encourage both students and
teachers to make more use of graphic approaches to solving problems"
(Ruthven, 1990, p. 447). Like the computer the graphing calculator can be used
to provide visual images (Bloom, Comber, & Cross, 1987).
Ruthven (1990) compared secondary school mathematics students who
were classified into two groups: those using the graphing calculator as a tool in
the classroom and those not having regular access to graphing calculators, on
a questionnaire and 12 graphing items. The independent variables were
treatment, grade, school, and sex. The researcher did not give a reason for
including the variable of sex, nor was a reason given for including an
interaction of sex and treatment in the statistical model.
The students were from classrooms in England schools which were
participating in the Graphics Calculators in Mathematics Project, in which six
teachers had at least one class with permanent access to graphing calculators.
The researcher used four schools with parallel classes. The statistical strength
of the questionnaire was not reported. He did not report on the entire test of
graphing items, but only on a subtest of symbolization items. He found that the
graphing calculators encouraged relationships between the symbolic and
graphic representation of function. Describing the graphing calculator as a
"cognitive tool", he was convinced that it aids conceptual understanding, as are
many other researchers (e.g., Branca, Breedlove, & King, 1992; Dance, Jeffers,
Nelson, & Reinthaler, 1992; Rubenstein, 1992; Wheatley & Shumway, 1992).
In a study designed to examine the impact of graphing calculators on high
school students' acquisition of function concepts and instructional processes,
Rich (1990) used the t-test to compare the means of an achievement test and
an attitude survey of treatment and control groups. The treatment groups
consisted of in-tact classes with access to graphing calculators and a
precalculus textbook which was written with an emphasis on a graphical and
technological approach to function. The control group consisted of in-tact
classes with access to scientific calculators without graphing capabilities and a
traditional precalculus textbook. One of the main differences between the two
textbooks was the emphasis on graphing and the use of graphing calculators.
The researcher did not explore the effect that the use of different textbooks
might have had on the results of the analyses.
In regards to achievement, the treatment group had higher mean than the
control group, but the t-test analysis did not indicate any significant difference
between the two means. Further analysis of a formal interview, which dealt
with misconceptions regarding the concept of function, was conducted with
high, average, and low ability students. The interviews revealed the following
when comparing the treatment and control groups.
1. The treatment groups were better able to graph functions.
2. The students in the treatment groups were better able to work with
functions algebraically and graphically.
3. The treatment groups showed a better understanding of the
relationship between algebraic equations and graphs of functions.
4. The treatment groups understood the concept of inverse of a
function and absolute value functions better.
5. Students in the treatment groups were better prepared to respond
6. The treatment groups had a better understanding of functional
7. Both groups were capable of dealing with variables and functional
8. The treatment groups had more insight on initial images of graphs.
Conceptual Change Theory in Mathematics Education
Because accommodation is difficult to accomplish and is a more radical
action than assimilation, Strike and Posner (1985) and Posner, Strike, Hewson,
and Gertzog (1982), found that either compartmentalization will occur or an
attempt at assimilation will be made. Schwarz, Dreyfus, and Bruckheimer
(1990) described compartmentalization as the act of restricting one's
knowledge to a particular context. By this description, the researchers implied
that students either willfully or unconsciously employ certain pieces of
information depending on the situation they are involved in.
Students' misconceptions are real concerns for educators, and they need
to be aware of them (Wagner, 1981). Before curricula and instruction can be
altered to deal with students' misconceptions, attention must be give to
examining the nature of particular misconceptions (Nussbaum & Novick, 1981,
1982). Much of this work is taking place in science education (Dreyfus,
Jungwirth, & Eliovitch, 1990; Nussbaum & Novick, 1981, 1982; Pines & West,
1986). Brophy (1986) suggested that by developing curriculum and instruction
that addresses students' misconceptions, we may be able to significantly
reduce these misconceptions. As the theory of conceptual change has been
applied in mathematics education via the introduction of cognitive dissonance,
cognitive conflict, conceptual conflict, and the use of the conflict teaching
approach, encouraging assimilation and accommodation, it has become
evident through limitations, warnings and suggestions reported in several
studies (e.g., Movshovitz-Hadar & Hadass, 1990; Tirosh, 1990; Tirosh &
Graeber, 1990a; Williams, 1991; Wilson, 1990) that some modification to the
intervention of cognitive conflict will be necessary.
The application of the theory of conceptual change, although not widely
used in mathematics education, has provided for success in the teaching of
science concepts. Stavy and Berkovitz (1980) used a "conflict-producing
technique" to increase science students' development of the concept of
temperature. Upon discovering that the children had conflicting representations
for temperature, the researchers intervened with worksheets designed to
induce cognitive conflict through activities. For a sample of 77 students, the
researchers administered protests and post-tests to two treatment groups and
one control group. One treatment group received a revised curriculum and
whole group instruction to induce cognitive conflict; the other treatment group
received individual instruction to induce cognitive conflict, and the control group
received neither. The cognitive conflict was induced by worksheet activities.
Results supported the conclusion that the conflict inducement produced positive
results regarding the students' acquisition and understanding of the concept of
temperature. In applying the theory of conceptual change in science education,
Stavy and Berkovitz (1980) have recommended that curricula and instructional
methods be designed to induce cognitive conflict, with the goal of assisting
students in acquiring concepts. Hand and Treagust (1991) provided a model
for designing curriculum materials to encourage cognitive conflict. Their model
consisting of 15 lessons involved seven worksheets which addressed students'
misconceptions through practical activities and challenging questions. The
misconceptions identified in interviews were the basis for the worksheets.
Sierpinska (1987) and Williams (1991) examined students' concept of
mathematical limit. In his study examining college calculus students' concept of
limit, Williams (1991) applied treatment in which he attempted to encourage
conceptual change in the students by inducing cognitive conflict through a
conflict teaching approach. The researcher's aim was to present
characterizations of limit which would be in conflict with the students' concepts
of limit. A survey was administered to 341 university calculus students. The
objective of the survey was to determine the students' initial concept of limit.
Eventually, 10 students were chosen to participate in the treatment portion of
The 10 students met as a group with the researcher on five occasions
during a 7-week period, during which time clarification of a definition for limit
was performed, three characterizations of limit were presented as inducers of
conflict, and students were asked to explain whether or not their view of limit
was altered by the previous information. Williams (1991) concluded that the
students were unwilling to integrate a formal view and formal definition into their
knowledge structure. Although the researcher acknowledged failure in the
treatment based on his goal of having the students alter their concept of limit,
he concluded that a closer examination of the students' knowledge of function
was needed to determine the underlying assumptions which caused them to
have misconceptions about the concept of limit. The researcher did not suggest
that technological advances may have been beneficial in the study. It is
possible that the students were inhibited by the static representation of
functions that the researcher made available. The graphing calculator would
have been an excellent tool for the students to use to examine the functions as
dynamic models of mathematics. With graphing calculators, the subjects would
have had the opportunity to examine a great variety of functions. The lack of
employment of any technological tool in the study perhaps was a major
weakness considering the topic which was of focus. Similarly, Sierpinska
(1987) found that the induced conflict did not encourage major changes in the
students' concept of limit. However the researcher was encouraged that
discussion of the "mental conflict" would produce some change in the future.
In Festinger's (1957) original thesis, he proposed that the goal of inducing
cognitive dissonance was to provide motivation for a student to attempt to
reduce the dissonance or conflict; this goal was attained in Williams' (1991)
and Sierpinska's (1987) study, because they were able to create situations
where the students could examine their concepts and question the validity of
their concepts. These studies represent one of the obstacles which science
educators have found in attempting to induce cognitive conflict: the learner may
not encounter conflict which insures learning (Dreyfus, Jungwirth, & Eliovitch,
1990; Driver & Erickson, 1983). For example, Rosnick and Clement (1980)
obtained similar results from a study involving the testing and interviewing of
university students enrolled in precalculus and calculus courses. The
researchers asked students to translate information given in a table, graph, or
picture into an algebraic equation. They found that the students had serious
misconceptions regarding variable and equation. Through tutoring sessions,
they found that "although students's behavior for the most part was
changed,.. their conceptual understanding of equation and variable
remained, for the most part, unchanged" (p.18).
Tirosh and Graeber (1990a) employed the conflict teaching approach in
an attempt to examine preservice teachers' concept of division. Treatment
consisted of individual interviews with the 21 subjects. The researchers
conducted the interviews with the goal of assisting the students in becoming
aware of and reflecting upon their misconceptions and inconsistencies. One
type of conflict induced involved encouraging the subjects to review their
conceptions regarding division and their calculations of division. The subjects
in the study exhibited improvement in translating information given in word
problems and in translating written statements of division into mathematical
language. The results provided evidence that if the approach is used properly, it
is possible to have positive outcomes. This approach does have its limitations.
The student may not even recognize that he has a conflict (Wicklund & Brehm,
1976; Wilson, 1990), and if so, may not respond positively. In addition, the
learner's self-concept as a learner of mathematics may be negatively affected.
Cognitive conflict was addressed by Wilson (1990) in a study designed to
examine students' discrepancies between definitions and examples. Tirosh,
Graeber, and Wilson (1990) proposed two types of inconsistencies. The first
type of inconsistency involved discrepancies between students' mathematical
constructs and "conventional" mathematical constructs. An example of this
inconsistency regarding the concept of function would be a discrepancy
between the learner's definition of the concept and the definition which is
proposed by the mathematics community. The second type was an
inconsistency within the students' repertoire of mathematical constructs.
Wilson's (1990) study concentrated on the second type of inconsistency. She
warned that although students may acknowledge inconsistencies, they may still
resist conceptual change. (This resistance is a major component of Festinger's
theory (Festinger, 1957; Wicklund & Brehm, 1976)). However, she encouraged
the use of inconsistencies in instruction, which stresses definitions, examples,
and "logical reasoning".
The academic level of students has been a major factor in several studies
regarding inconsistencies. Vinner and Dreyfus (1989) found that subjects from
higher levels (academically) were able to provide more and better justifications
for their responses than their lower level counterparts. Differences between low
and high ability students also surfaced in Markovits, Eylon, and Bruckheimer's
(1988) study of students' concept of function. They found that students of lower
ability had more success with mathematical activities which were in "story" form
than with mathematical activities which were essentially algebraic in nature.
The highest ability students did not have this difficulty.
Regarding the issue of scale of axes for graphing, Goldenberg and Kliman
(1988) conducted interviews with high school juniors and seniors at the
precalculus level and eighth graders enrolled in first year algebra. All of the
students were characterized "bright and articulate" and chosen for this reason
to insure that the researchers would obtain usable data. The purpose of the
study was to accumulate, through video and personal observation, a set of
metaphors that the students would use for examining graphs regarding the
scales of the axes. For example, one such metaphor involved changing scale
as a way of "magnifying" the graph. More important than the metaphors which
the researchers collected, was the method which they used to encourage
students to react in the way that the researchers wanted them to react.
To insure themselves that the students would be very responsive, the
researchers selected problems which they predicted would cause conflict and
discrepancies in the students' knowledge of graphing. For instance, the
researchers asked the students to provide information regarding the graphs of
parabolas, and the responses from the students were typical of what students
"expect" the graphs of quadratic equations to look like: a curve with symmetry
with respect to the line through its vertex. However, the researchers presented
the students with a quadratic equation and the corresponding graph of the
parabola on a set of axes scaled in such a way as to cause the graph to
resemble a vertical line. This action caused conflict within the students'
conceptual structure of quadratic equations and graphs of parabolas. The
students were encouraged, during the interview, to examine their conceptions
and make suggestions as to why such an event would occur. Some students,
having determined that the choice of scale of the axes affected the graph of the
function, still did not believe that the graph was valid. However, encouraging
the students to examine their conceptions and to consider that perhaps their
conceptions regarding the graphing of quadratic functions and functions in
general were inadequate provided an environment for active learning.
Numerous misconceptions regarding the concept of function were found to
exist with learners by examining their intuitions about the concept.
Acknowledgement of this was profound; what was missing were examples of
curricula and instructional methods to help alleviate these inconsistencies in
students' mathematical knowledge. Yet this assistance would have to be such
that it is accessible to students and teachers. At least three models were
suggested: the Function Block, the Triple Representation Model, and the
Parallel Axes Representation model. They are all to be commended, but they
are not yet well known or even accessible and have not been proven to be
useful on a global scale.
It is apparent from the review that little attention has been given to domain
and range, two very important subconcepts of function. In order for each and
every function to be defined, it must have a defined domain. Changing the
domain of the function, changes the function itself and may do so to the degree
that the mathematical object is no longer a function. These two concepts make
up the foundation of the concept of function. It is proposed that by building up
students' understanding of these two concepts, educators will be closer to
insuring that students are on an upward path to acquiring the concept of
Besides the difficulty students showed in denoting the domain and range of
functions as stated in the literature, at least two of the other misconceptions
could be addressed by focusing on the subconcepts of domain and range. The
first was the students' inability to make a connection between the graphical and
algebraic representation of the same function. If the students' were aware of
the role of the domain in defining functions, the pairing of the different
representations of the concept might be less difficult for students. For example,
a function that is defined only in the domain of values greater than or equal to
zero would be expected to have a graph that appears to the right of the y-axis.
However, the learner would have to be aware of the importance of knowing the
domain of a function.
Secondly, some misconceptions focused on the fact that students expected
graphs of functions to be acceptable, precise, and predictable. Having
knowledge that the domain has a direct effect on the graph of the function
would alert students to the fact that graphs of functions are sometimes "out of
the ordinary", but with value placed on the role of the domain, "out of the
ordinary" can become ordinary. The idea of an "acceptable" graph is also
related to students lack of consideration of the scale of axes on which the graph
is produced. The omittance of such an important detail causes students to hold
on to their conceptions regarding what certain graphs should look like. When
students ignore the domain, range, and/or scales of axes for the graph of a
function, they may be overlooking important details of the function represented
by the graph. The graph of the function represents the functional relationship
between the domain and range of the function, without sufficient
encouragement to consider these three main aspects of the graph of a function,
students are limited in their understanding of the concept of function.
Furthermore, to assure that students will be appropriately guided in "reading"
graphs correctly, the effects that the domain, range, and scales of axes has on
the graph should be examined.
By adopting the theory of conceptual change, we can make these
misconceptions beneficial in the learning process by inducing cognitive conflict
through activities which will encourage students to deal with their
misconceptions. Activities allowing for an examinations of misconceptions will
assist students in dealing with their misconceptions. Secondly, graphing
calculators are viable educational tools and are as powerful as the
microcomputer. The graphing calculator would seem to be the most
appropriate tool for dealing with misconceptions surrounding the concept of
function. Yet these devices, although they are quite accessible, are rarely
employed as a tool for conceptual change. Use of the graphing calculator
forces one to examine appropriate domains, ranges, and scales of axes for
graphing functions. In fact, the graphing calculator is perhaps the most feasible
medium for conducting such activities. One should consider that one reason
why the topic has been omitted in traditional algebra courses because an
efficient medium for examining appropriate domains, ranges, and scales of
axes for graphing functions has not been readily accessible. Yet the use of the
graphing calculator for this purpose has not been exploited in the teaching and
learning of algebra.
Several trends surfaced from an analysis of the research reported. For the
most part, the subjects participating in the various studies were not chosen
randomly, but instead were volunteer or opportunity samples. Many of the
samples were classes of students participating in special education projects,
rather than "normal" classroom settings. This condition makes it almost
impossible to generalize the research findings to a target population, given the
special situations under which the studies were conducted. However, the
constraints placed on researchers using school samples are often unavoidable.
When students are assigned to classes by the school or students in colleges
and universities register for classes by choice, it is difficult to have a random
sample. In some cases, it may be possible to choose groups randomly to be
assigned to treatment and control but few researchers denoted having this
particular option. These studies cannot be dismissed without consideration for
the information that the researcher has provided for the mathematical
community. It is still important to consider the individual research results of
studies that are limited in their generalizability. Such information can serve as a
foundation for future research findings.
In most studies, questionnaires were administered to obtain information
rather than to make comparisons between groups or test performance. Several
of the researchers did not acknowledge the need to collect and/or report
reliability and validity evidence for their instruments. In many cases, the
instruments were used to make inferences regarding students' intuition,
misconceptions, and inconsistencies regarding the concept of function, yet the
reader could not be assured that the instruments were in their best condition for
completing the given tasks. Likewise, reliability estimates were not given for
many of the measurements, thus there is no indication of the replicability of the
study with the same or a comparable sample using the same instrument. In
cases where the instrument was used for comparison, lack of adequate
protesting measures and use of nonrandom sampling techniques may have
caused the power of statistical treatment to be weakened.
RESEARCH DESIGN AND METHODOLOGY
The purpose of this study was to investigate community college algebra
students' concept of function regarding their understanding of the concepts of
domain and range, their selection of appropriate domain, range, and scale for
the axes for graphing polynomial functions, and their identification, construction,
and definition of function. Of interest were the effects of employing the graphing
calculator and a conceptual change assignment on the above factors. The
study was designed to test the following null hypotheses:
1. Students' use of the graphing calculator during instruction will not
affect their concept of function regarding their (a) application of
the concepts of domain and range, (b) selection of appropriate
domain, range, and scale for the axes for graphing functions, and
(c) identification, construction, and definition of function.
2. Students' participation in a conceptual change assignment will not
affect their concept of function regarding their (a) application of
the concepts of domain and range, (b) selection of appropriate
domain, range, and scale for the axes for graphing functions, and
(c) identification, construction, and definition of function.
3. Students' use of the graphing calculator during instruction and the
students' participation in a conceptual change assignment will not
interact to affect their concept of function regarding their
(a) application of the concepts of domain and range (b) selection
of appropriate domain, range, and scale for the axes for graphing
functions, and (c) identification, construction, and definition
Conceptual Change Assignment
The Conceptual Change Assignment (see Appendix A) was designed by
the researcher. The assignment was modeled after the accommodation model
given in Nussbaum and Novick (1981, 1982). The assignment was
implemented immediately before the instructor formally presented the topic of
function and the concepts of domain and range. Part I of the assignment was
designed to stimulate a learning environment which would encourage the
learners to examine their conceptions before being presented with formal
instruction regarding the concepts of domain, range, and scales of axes for
graphing functions. By the time the students reached the topic of function
including the concepts of domain and range, they had covered sections
regarding graphing linear and quadratic equations. Part I of the assignment
represented an opportunity for students to review these exercises, although
they were unaware that they were formally dealing with the concepts of domain
and range and scaling axes for graphing functions.
Part II of the assignment was designed to create conflict between the
students' conceptions regarding the concepts of domain, range, and selection
of domains, ranges, and scales of axes for graphing functions. The exercises in
Part I were typical exercises in algebra regarding the domain and range of
linear, quadratic, and cubic functions and typical scales of axes for graphing
functions. However, the exercises in Part II were designed to encourage conflict
between what the learner already knew and what was being presented as
mathematical possibilities. The inducement of conflict was planned to arise
when the learner realized that the domains, ranges, and scales of axes did not
have to be typical, that the domains, ranges, and scales of axes used for the
equations in Part I did not work for equations in Part II, and that the domain and
range of the function dictated the design of the Cartesian coordinate system.
Part Ill of the assignment was designed to encourage and support the
learner's accommodation of the concepts of domain, range, and scales of axes
for graphing functions. To support the fact that domains and ranges did not
have to be typical, the learner was encouraged to view functions with different
domains and ranges. To support the learner's accommodation of the fact that
the scales of axes affect how the graph will look, the learner was encouraged to
view functions with different scales of axes.
Domain / Range / Scale Instrument
The Domain / Range / Scale Instrument (see Appendix B) was designed by
the researcher and administered as a pretest and posttest. The objectives of
the instrument and the relationship between the objectives and the hypotheses
of the study are given in Table 1.
Identification / Construction / Definition Instrument
The instrument denoted by the researcher as the Identification /
Construction / Definition Instrument was adapted from Vinner and Dreyfus
(1989) (see Appendix C). The objective of administering the instrument was to
determine students' concept image and concept definition of the concept of
function. One purpose of the instrument was to examine the students' image of
the concept and the students' definition of the concept. The instrument
consisted of seven items. Items one through four were identification items;
items five and six were construction items. These first six items addressed the
Objectives of Domain / Range / Scale Instrument
Items Hypothesis Objective: To test the student's ability to...
1 -4 (1-3)a denote the domain and range of functions
5 7 (1-3)a denote the domain and range of functions
8 10 (1-3)a identify specified function values for functions
11 -13 (1-3)a identify specified function values for functions
14-15 (1-3)b graph functions given algebraically with
16 (1-3)b graph functions given algebraically without
17* (1-3)b identify functions which satisfy specified
domain and range restrictions.
18 22* (1-3)a,b distinguish between the properties of the
function and the properties of its graph.
23 26 (1-3) a,b choose appropriate domain and range
restrictions and reasonable scales to provide
complete graphs of functions.
27 (1-3)a,b recognize the effect that a domain restriction
and scale of the axes may have on the graph
of a function.
Note: (1-3)a,b refers to hypotheses la, 1b, 2a, 2b, 3a, and 3b.
*This item appears in Yerushalmy (1991).
*These items appear in Markovits, Eylon, & Bruckheimer (1988).
students' image of the concept. The items required identification of functions
and nonfunctions and the creation of a functional equation based on a written
description of a functional relationship. The last item was a request to the
students to provide a definition of the concept. All of these items addressed
hypotheses 1c, 2c, and 3c. The students were asked to explain all of their
The researcher conducted a pilot study during the Fall 1992 term using two
community college algebra classes in a north-central Florida community
college. The classes participated in activities involving graphing calculators and
a conceptual change assignment regarding the concept of function.
After the implementation of the conceptual change assignment in one
class, several changes were made regarding the content of the assignment
before it was administered to the other class. First, the length of the
assignment was decreased to allow time for whole group discussion at the end
of the assignment. To facilitate this, two graphing activities which were
repetitive were eliminated. However, an additional graphing activity was added
to increase the diversity of the types of equations used in the assignment.
Furthermore, the researcher observed that the success of the assignment
for students using the graphing calculator was enhanced by the students'
previous experience with the graphing calculator.' Because the students were
already familiar with the operation of graphing calculators, they were able to
concentrate on the assignment and not on the operation of the calculator.
There were only a few questions asked regarding the operation of the
calculator. All other questions and comments posed by the students were
concerned with the activity. Observation of the students' responses to the
assignment revealed that they were able to relate their algebraic responses to
questions regarding the domain and range of functions to the graph of the
functions. Furthermore, the induction of conflict was successful given that the
students reached a point of conflict during the activity and indicated that certain
events did not correspond with what they "knew" to be true. After examining
the situation, the students began to seek solutions for the problem. At the
completion of the assignment, the students were able to discuss their written
and graphical responses with knowledge of what they had learned by using the
graphing calculator to complete the assignment. Prior use of the graphing
calculator did not negatively effect the objective of the assignment.
Two instruments were administered to the two classes of students to
examine the students' concept regarding their understanding of domain and
range, their selection of appropriate domain, range and scale for the axes for
graphing functions, and their identification, construction, and definition of
function. To make changes to the instruments and gather evidence of validity,
the instruments were not submitted to the classes simultaneously. The length of
the Domain / Range / Scale Instrument was decreased because of time
constraints of the class periods. To facilitate this, several exercises from the
various objectives of the instrument were eliminated. Several questions on the
Identification / Construction / Definition Instrument were determined to be too
difficult for the students. They were replaced with comparable, but less difficult
exercises. The objective of the exercises remained the same.
To insure that the content of the instruments reflected content domain of
college algebra regarding the concept of function, the content validity of
instruments was provided by using the literature as a guide, adoption of several
items from other instruments which were related to the current topic and
an analysis of the content of the instruments by a mathematician and a
mathematics educator. The researcher submitted the instruments to the two
experts for their review and later met with them to discuss the content and
objectives of the instruments. The two experts provided their professional
assessment of the instruments which the researcher incorporated into the
design of the instruments.
Reliability estimates for the final version of the Domain / Range/ Scale
Instrument and the final version of the Identification / Construction / Definition
Instrument were estimated by Kuder Richardson (KR-20) and were given as .84
and .77, respectively. The difficulty and discrimination indices for the individual
items of the two research instruments are reported in Tables 2 and 3. The
difficulty index for each item was given as the proportion of students who
answered the item correctly. The discrimination index for each item was
calculated by subtracting the proportion of students in the lower half of the
group who answered the item correctly from the proportion of students in the
upper half of the group who answered the item correctly.
Research Population and Sample
The population for this study consisted of students enrolled in college
algebra at public community colleges. Community colleges, although rightfully
deserving of a place in America's educational system, are often omitted from
the network of educational research, at least more so than elementary, middle,
and high schools, 4-year colleges and universities. Wattenbarger (1989)
reported that there were 1,300 community colleges in the United States and
that 38% of all college and university students seeking bachelors degrees were
graduates of a community college. Furthermore, 47% of all undergraduate
Difficulty and Discrimination Indices
Domain / Range / Scale Instrument
Item Difficulty Discrimination
1 .231 .190
2 .154 .024
3 .615 .714
4 .154 .024
5 .154 .333
6 .308 .357
7 .462 .690
8 .692 .262
9 .769 .429
10 .846 .286
11 .308 .047
12 .154 .333
13 .462 .381
14 .231 .500
15 .308 .357
16 .462 .071
17 .538 .547
18 .154 .333
19 .615 .404
20 .926 .143
21 .615 .096
22 .385 .524
23 .077 .167
24 .154 .024
25 .154 .024
26 .308 .357
27 .385 .833
Difficulty and Discrimination Indices
Identification / Construction / Definition Instrument
Item 1 2 3 4 5 6 7
Difficulty .615 .538 .615 .615 .615 .154 .385
Discrimination .675 .875 .675 .350 .350 .250 .625
minority students are enrolled in community colleges (Koltai & Wilding, 1991).
These statistics can be used to support the need to include the community
college system in the mathematics education research base of undergraduate
institutions. If this is not done, we as educators are taking the risk of omitting
many learners from our research sample. These facts provide reason for an
increased effort to include community colleges in more educational research.
More specifically, as society requires more persons who can work
efficiently in a quantitative and technological environment, it will be necessary
to focus attention on curricula and instruction which provides students the
opportunity to develop a strong concept of function. One recommendation
made in the report, "The Status of Science, Engineering, and Mathematics
Education in Community, Technical, and Junior Colleges" is that community
college faculty should attempt to maintain learning environments which
enhances students' learning of science, engineering, and mathematics (Koltai
& Wilding, 1991). The graphing calculator actually could be used to play a part
in all three disciplines, but in particular, this recommendation supports the need
to include the use of technology in the learning and teaching of algebra. In this
report, the authors also reminded the reader that the community college is an
active educational institution involved in the advancement of science and
The research sample consisted of 128 college algebra students enrolled
in a north-central Florida community college. There were eight classes of
students: six treatment classes and two control classes. This community
college was selected because the researcher had previously established a
professional relationship with administrators and instructors. The researcher
had also taught algebra and general mathematics courses at the college, and
thus was familiar with the atmosphere of the learning environment,
departmental policies, and general characteristics of instructors and students.
All classes were intact, thus random assignment of students to the classes
was not possible. The students in the classes represented characteristics of
the population of students enrolled in this particular community college and
community colleges in general. The researcher chose to use day classes
because the night classes are typically smaller classes and consist of
students who are employed full-time during the day. Otherwise, there was no
indication that the students in the sample were significantly different from all
other students enrolled in college algebra. To aid with the description of the
demographics and other descriptors of the students within the sample, each
student in the study was requested to complete an information sheet (see
Appendix D). The chi-square and one-way analysis of variance (ANOVA)
procedures were employed to determine if there were significant differences
among the groups in the sample regarding gender, age, race, enrollment status
number of secondary and postsecondary mathematics courses, and grade
There were more females participating in the study, however the
arrangement of the students by gender in the four groups was nonsignificant
(see Table 4).
Frequency and Percentage of Gender by Group
Group n Males Females
Calculator & Assignment 30 10 (33%) 20 (67%)
Calculator 26 11 (42%) 15 (58%)
Assignment 33 11 (33%) 22 (67%)
Control 39 19 (49%) 20 (51%)
Total 128 51 (40%) 77 (60%)
Note: The chi-square value of 2.46 for 3 degrees of freedom with p = .482
The average ages for the four groups in the study are presented in Table
5. The results from the ANOVA procedure revealed that age is significant for
both the calculator factor and the assignment factor (see Table 6). The students
who used calculators during the study were significantly younger than students
who did not use calculators during the study. Furthermore, students who
participated in the conceptual change assignment were significantly younger
than students who did not participate in the conceptual change assignment.
The categorization of the students by race is presented in Table 7.
Because of the number of categories with less than five students, employing the
chi-square procedure was not appropriate. However, the information in the
table revealed that a significant number of the students participating in the study
were Caucasian/White for all four groups.
Average Age by Group
Group n Average Age in Years
Calculator & Assignment 30 19
Calculator 26 21
Assignment 33 20
Control 39 25
Total 128 21
Analysis of Variance: Age
Source DF Type III Sum of Squares F p
Calculator 1 179.49 6.94* .0095
Assignment 1 259.76 10.04* .0019
Calculator Assignment 1 87.32 3.37 .0686
Model 3 607.86
Error 124 3209.01
Note: *significant for p = .05
Frequency and Percentage of Race by Group
Race I II III IV Total
African American 1(3%) 0 4(12%) 3 (7%) 8
Asian/Pacific Islander 1 (3%o) 1(4%) 2 (6%) 1(3%) 5
Caucasian/White 26 (88%) 23 (88%) 25(76%) 34 (87%) 108
Hispanic 1(3%0) 2 (8%) 1(3%) 1(3%) 5
American Indian 0 0 0 0 0
Other 1(3%) 0 1(3%) 0 2
Note: I--Calculator & Assignment Group(n=30) Ill--Assignment Group (n=33)
II--Calculator Group (n=26) IV--Control Group (n=39)
The enrollment status of the students by group is reported in Table 8.
According to the results from the chi-square procedure, there were no
significant differences between the groups regarding the number of students in
the groups who attended college full-time and the number of students who
attended college part-time.
The average grade point average of the students by group are reported in
Table 9. The averages are based on the grade point averages reported by the
students. Due to some circumstances, such as first-time enrollment, some
students did not have grade point averages at the time of the study. For the
reported grade point averages, the ANOVA results revealed that grade point
average was significant for the factor of calculator (see Table 10). Students
who used the calculator during the study had a significantly lower grade point
average than students who did not use calculators during the study.
Frequency and Percentage of Course Load by Group
Group n Full-Time Part-Time
Calculator & Assignment 30 28 (93%) 2 (7%)
Calculator 26 22 (85%) 4 (15%)
Assignment 33 31 (94%) 2 (6%)
Control Group 39 31 (79%) 8 (21%)
Total 128 112(88%) 16 (12%)
Note: The chi-square value of 4.67 for 3 degrees of freedom with p = .197
Average Grade Point Average by Group
Group n Grade Point Average
Calculator & Assignment 25 2.86
Calculator 20 2.85
Assignment 27 3.10
Control 37 3.12
Total 109 2.98
Note: Grade point average was based on a 4.0 scale.
The students were also asked to report the number of mathematics
courses taken in high school and in college. The results of the tabulation of this
information is given in Table 11. There were no significant differences between
the four groups regarding the number of mathematics courses taken by the
Analysis of Variance: Grade Point Average
DF Type III Sum of Squares F
Calculator 1 1.59 5.18* .0249
Assignment 1 .03 .10 .7535
Calculator Assignment 1 .12 .42 .5198
Model 3 1.88
Error 105 32.26
Note: *significant for p = .05
Fequency of Mathematics Courses Taken by Group
Group n High School College Total
Calculator & Assignment 30 98 64 162
Calculator 26 96 47 143
Assignment 33 108 67 175
Control 39 104 92 196
Total 128 406 270 676
Note: The chi-square value of 7.20 for 3 degrees of freedom with p = .066
All college algebra students in the study used the same college algebra
textbook: Algebra for College Students (Lial, Miller, & Hornsby, 1992), and all
instructors used the same topical outline according to departmental policy.
Before reaching the topic of function, the students studied linear and quadratic
equations, during which time, they were instructed in the processes of paper
and pencil graphing due to departmental policy. Sections of the text covered
for the unit on functions during the study are given in Table 12.
Text Sections Covered in Course During Study
Text Section Section Title
7.6 Introduction to Functions; Linear Functions
8.1 Algebra of Functions
8.2 Quadratic Functions; Parabolas
8.3 More About Parabolas and Their Applications
Each section listed objectives for the topic, followed by examples. Three
definitions for the concept of function presented in the text were as follows:
1. A function is a relation for which each value of the first component
of the ordered pair, there is one and only one value of the second
2. A function is a set of ordered pairs for which the first component is
3. A function is a rule or correspondence such that each domain
value is assigned to one and only one range value.