The effect of two technologies on College Algebra students' understanding of the concept of function

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The effect of two technologies on College Algebra students' understanding of the concept of function
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by Gregory Kent Harrell.
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THE EFFECT OF TWO TECHNOLOGIES ON COLLEGE ALGEBRA STUDENTS'
UNDERSTANDING OF THE CONCEPT OF FUNCTION

















GREGORY KENT HARRELL


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


2001




























Copyright 2001

by

Gregory Kent Harrell



























This dissertation is dedicated to those who have demonstrated
unconditional love and support for me:
my parents, Franklin and Mavis Harrell, who taught me to value education,
my wife, Sherry Harrell, who is my lifelong friend and companion,
my children, Sean and Brandon Harrell, who are a constant source of pride and joy,
and my Creator, God, who has made it all possible.















ACKNOWLEDGMENTS

I am very grateful not only for the people who have been directly involved in

providing support in the completion of this dissertation, but also for the many people who

have invested their time and efforts in me throughout my lifetime. I am especially

thankful for my doctoral committee members. Each of them has shaped me as a

mathematics educator and as a person. Dr. Thomasenia Lott Adams, my committee chair

and mentor, has been an excellent model of both a mathematics educator and a person.

Her guidance helped my work progress smoothly, yet she allowed me the freedom to

think for myself and learn on my own. Dr. David Miller provided me with a deeper

understanding of statistics and broadened my knowledge of measurement, reliability, and

validity. Dr. Colleen Swain shaped my understanding of how technology can be

effectively used in the curriculum. Dr. Dale Campbell helped me better understand the

environment in which I have worked for the past ten years, the college and university

environment. He provided me with insight for developing and guiding change in the

college curriculum.

I greatly appreciate the time and efforts of the faculty members and students who

participated in this research study. Without them, none of this would have been possible.

I am indebted to Dr. Ben Nelms, who as graduate coordinator guided me to the

University of Florida. Without the financial support of the University of Florida and the

College of Education, this would not have been possible.










For my parents, wife, and children, I am forever grateful. My parents instilled in

me the strong work ethic needed to complete this dissertation. My wife, Sherry, provided

constant support and ran our household almost single-handedly. My children, Sean and

Brandon, showed me unwavering love and devotion, even when I had to spend time away

from them. Time with my family provided relief from my work and brought balance to

my life.
















TABLE OF CONTENTS

page

ACKN O W LED GM EN TS ................................................................................................. iv

LIST OF TABLES ............................................................................................................. ix

LIST OF FIGURES ........................................................................................................... xi

ABSTRA CT...................................................................................................................... xii

CHAPTERS

1 DESCRIPTION OF THE STU DY .................................................................................. 1

Introduction..................................................................................................................... 1
Learner A abilities Related to the Function Concept..................................................... 3
Technology ................................................................................................................. 5
Statem ent of the Problem ................................................................................................ 9
Justification of the Study .............................................................................................. 12
Student D iffi culties................................................................................................... 12
D different Population.................................................................................................. 12
Lack of Research Concerning Dom ain and Range................................................... 13
A Technological Research Environm ent.................................................................. 13
Theoretical Fram ework................................................................................................. 15
Concept ..................................................................................................................... 15
Schem a..................................................................................................................... 16
Conceptual Change................................................................................................... 16
Logico-Mathematical Experiences and Reflective Abstraction............................... 19
Concept Im age .......................................................................................................... 21
The Function Concept............................................................................................... 22
H historical Developm ent of a Concept....................................................................... 23
Definition of Term s....................................................................................................... 24
Significance of the Study.............................................................................................. 25
Organization of the Study ............................................................................................. 27

2 REV IEW OF THE LITERA TURE ............................................................................... 28

Historical D evelopm ent of the Concept of Function.................................................... 28
Early Developm ents.................................................................................................. 28
Eighteenth Century Developm ents........................................................................... 30









N ineteenth Century Developm ents........................................................................... 32
Twentieth Century Developm ents ............................................................................ 33
Sum m ary of the Historical Developm ent................................................................. 36
History of Function in M them atics Education............................................................ 36
Standards Related to the Function Concept.................................................................. 41
M multiple Representations .............................................................................................. 43
Difficulties and M isconceptions ................................................................................... 52
Identify and Define ................................................................................................... 52
Dom ain and Range.................................................................................................... 56
Representations and Translations .............................................................................. 58
Reify.......................................................................................................................... 60
M odel ........................................................................................................................ 61
Sum m ary of Difficulties and M isconceptions.......................................................... 61
Technology Standards................................................................................................... 62
Graphing Calculators and the Function Concept.......................................................... 63
Com puters and the Function Concept........................................................................... 79
Com prison of Com puters and Graphing Calculators.................................................. 87
Summ ary ....................................................................................................................... 91

3 RESEARCH DESIGN AND METHODOLOGY ......................................................... 95

Research Objective ....................................................................................................... 95
Instrum ents.................................................................................................................... 96
Dom ain/Range/Identify/Define/Translate (DRIDT)................................................. 97
M odel/Reify (M R) .................................................................................................... 98
Galbraith-Haines Technology-Mathematics Interaction Surveys............................. 98
Hannafin-Scott Preferred Amount of Instruction Survey......................................... 99
Pilot Study................................................................................................................... 100
Population and Sam ple ............................................................................................... 101
Instructors ................................................................................................................... 105
Instructional M materials ................................................................................................ 107
TGC Treatm ent....................................................................................................... 107
Activities related to identifying functions and non-functions............................. 108
Activities related to dom ain, range, and translate............................................... 109
Activities related to reification............................................................................ 111
CGC Treatm ent....................................................................................................... 112
Activities related to identifying functions and non-functions............................. 113
Activities related to dom ain, range, and translate............................................... 113
Activities related to reification............................................................................ 113
Design of the Study..................................................................................................... 114
Procedures................................................................................................................... 114

4 RESULTS .................................................................................................................... 117

Analysis for Domain/Range/Identify/Define/Translate Instrument............................ 117
Analysis for M odel/Reify Instrum ent......................................................................... 120
Analysis for Technology-Mathematics Interaction Instrument.................................. 122










Exploratory Analysis .................................................................................................. 123
Domain and Range.................................................................................................. 123
Concept Im age Identify and Define..................................................................... 127
Reify........................................................................................................................ 129
Classroom Observations ............................................................................................. 130

5 CONCLUSION ............................................................................................................ 132

Summary..................................................................................................................... 132
Discussion................................................................................................................... 136
Limitations of the Study.............................................................................................. 141
Implications................................................................................................................. 143
Implications for M mathematics Curricula................................................................. 143
Implications for M mathematics Instruction............................................................... 144
Recommendations....................................................................................................... 146

APPENDICES

A DOMAIN/RANGE/IDENTIFY/DEFINE/TRANSLATE INSTRUMENT............... 149

B M ODEL/REIFY INSTRUM ENT ............................................................................... 154

C MATHEMATICS-COMPUTING ATTITUDE SCALES-COMPUTER.................. 156

D MATHEMATICS-COMPUTING ATTITUDE SCALES-CALCULATOR...........163

E STUDENT QUESTIONNAIRE.................................................................................. 170

F COLLEGE ALGEBRA TOPICAL OUTLINE........................................................... 173

G TEXAS INSTRUMENTS TI-83 GRAPHING CALCULATOR FEATURES.......... 174

LIST OF REFERENCES................................................................................................. 176

BIOGRAPHICAL SKETCH ........................................................................................... 186
















LIST OF TABLES


Table Page

3-1. Student Body Profile Fall Semester 1998................................................................... 101

3-2. Frequency and Percentage of Sex by Group .............................................................. 103

3-3. Frequency and Percentage of Race/Ethnicity by Group............................................. 104

3-4. Average Age in Years by Group ................................................................................ 104

3-5. Prior Graphing Calculator Use by Group................................................................... 105

3-6. College Teaching Experience by Group..................................................................... 106

3-7. A Tabular Relationship that is not a Function............................................................ 109

3-8. Administration of Instruments and Treatment............................................................ 116

4-1. DRIDT Pretest and Posttest Descriptive Statistics..................................................... 117

4-2. DRIDT Analysis of Covariance ................................................................................. 119

4-3. Adjusted Posttest Means for the DRIDT Instrument.................................................. 119

4-4. MR Pretest and Posttest Descriptive Statistics........................................................... 120

4-5. MR Analysis of Covariance ....................................................................................... 121

4-6. Technology-Mathematics Interaction Descriptive Statistics...................................... 123

4-7. Domain/Range Descriptive Statistics......................................................................... 124

4-8. Algebraic Domain/Range Descriptive Statistics ........................................................ 124

4-9. Graphical Domain/Range Descriptive Statistics ........................................................ 125

4-10. Domain/Range Components Analysis of Covariance ............................................... 126

4-11. Attention to Domain during a Translation Activity................................................... 126









4-12. Identify / Define Descriptive Statistics...................................................................... 127

4-13. Categorization of Definitions .................................................................................... 129

4-14. Reify Descriptive Statistics ....................................................................................... 129















LIST OF FIGURES



Figure Page

1. Standard V iew offx +3 .....................................................................................110

2. Standard View of f(x)=(x-1)3..................................................................................... 110

3. Standard V iew of f(x) = x4- 30x2 .............................................................................. 110

4. M oving an Object Up 2 Units.................................................................................... 111
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE EFFECT OF TWO TECHNOLOGIES ON COLLEGE ALGEBRA STUDENTS'
UNDERSTANDING OF THE CONCEPT OF FUNCTION

By

Gregory Kent Harrell

December 2001


Chairman: Dr. Thomasenia Lott Adams
Major Department: School of Teaching and Learning

This research project examined the effect of two different technological curricula

on students' understanding of the concept of function. Treatment Group I used a text-

based, graphing calculator curriculum. Treatment Group II used a computer-based,

graphing calculator curriculum. Treatment Group I participated in graphing calculator

activities related to the domain and range of functions that were designed to promote

conceptual change via cognitive conflict. Because Treatment Group I instruction was

predominately lecture and Treatment Group II used self-paced computer software, the

research also explored the effect of students' preferred amount of instruction on their

understanding of the concept of function.

The research sample included 281 students in 10 college algebra classes with five

classes per treatment group. Function understanding was measured in terms of students'

ability to apply domain/range concepts and their ability to identify, define, translate,

model, and reify functions. Based on posttest group means, neither students' preferred









amount of instruction nor the interaction of students' preferred amount of instruction with

the technological curriculum had a significant effect on their understanding of the

concept of function.

The text, graphing calculator group demonstrated a significantly stronger

understanding of the function concept than the computer, graphing calculator group in

terms of their application of domain/range concepts and their ability to identify, define,

and translate functions. The abilities to apply the domain/range concepts and translate

between algebraic and graphical representations were the major sources of their better

understanding.

An analysis of the posttests for both treatment groups revealed difficulties.

Students had difficulty (a) considering the domain when translating between algebraic

and graphical representations, (b) determining the range of a function in its algebraic and

graphical representations, and (c) identifying whether or not algebraic formulas and

written descriptions of relationships represent functions.

Students demonstrated a limited concept image of function. The majority of

students' concept image of function did not include the definition of function. Students'

strongest concept image of function was that a function is an equation. When

considering a graphical representation, students demonstrated a concept image of

function in terms of the vertical line test and continuity.














CHAPTER 1
DESCRIPTION OF THE STUDY


Introduction

The concept of function is an essential part of the algebra curriculum and is one of

the most important concepts in all of mathematics (O'Callaghan, 1998). According to

Eisenberg and Dreyfus (1994), "having a sense for number and having a sense for

functions are among the most important facets of mathematical thinking" (p. 45). The

National Council of Teachers of Mathematics (NCTM) suggested that the concept of

function is a "foundational idea" that "should have a prominent place in the mathematics

curriculum because [it enables] students to understand other mathematical ideas and

connect ideas across different areas of mathematics" (NCTM, 2000, p. 15). One of the

main reasons that students have difficulty with calculus is due to a weak conceptual

understanding of the function concept (Harel & Trgalovi, 1996; Orton, 1983; Pinzka,

1999; Selden et al., 1994).

In Crossroads in Mathematics: Standards for Introductory College Mathematics

before Calculus, the American Mathematical Association of Two-Year Colleges

(AMATYC) stresses the importance of the function concept in mathematics education by

including a standard focused on functions. The College Standards content standard C-4

states that "students will demonstrate understanding of the concept of function by several

means (verbally, numerically, graphically, and symbolically) and incorporate it as a

central theme into their use of mathematics" (AMATYC, 1995, p. 13).









The concept of function as it is known today has explicitly emerged and evolved

over the past 300 years (Kleiner, 1989). The calculus of Newton and Liebniz at the turn

of the 18th century included the view of a function as a geometric curve (graph), while

later in the 18th century Euler viewed a function as an algebraic expression. The

definition of function plays a vital role in the mathematical community (Tall & Vinner,

1981; Vinner, 1976). This research study uses the Dirichlet-Bourbaki definition of

function: function is a relationship between two sets A and B that assigns to each

element x in set A, called the domain, exactly one element y in set B, called the range

(Barros-Neto, 1988; Larson & Hostetler, 1997).

The domain and range components of function play a central role in the definition

and concept of function (Adams, 1997; Hamley, 1934; Markovits et al., 1988). This

research study focuses primarily on these components of the function concept. The

domain contains the elements on which the function operates. The range contains the

elements that result from the function's operation. To represent a function graphically,

whether on paper or via technological graphing utilities, one must have an understanding

of the domain and range. The horizontal axis, called the x-axis, of the two-dimensional

graph reflects the elements of the domain. The vertical axis, called the y-axis, of the

graph reflects the elements of the range.

Students must understand the domain to obtain graphs of functions (Caldwell,

1995; Markovits et al., 1988). In order to obtain the x-axis elements of a graph, students

must properly choose elements on which the function operates (domain). Students then

use the function to assign each of the chosen domain elements an element in the range to

obtain y-axis elements of the graph. When graphing on paper, the learner performs this









function assignment one-by-one as each individual domain element yields one range

element. Individual points are then plotted on paper. When the learner uses a

technological graphing utility, however, the graphing utility performs all of the function

assignments and displays the entire graph (all plotted points) at once. In order for the

graph to properly display on the graphing utility, the learner must first hypothesize about

the range, the results of the function's operation.

Students who lack an understanding of the domain demonstrate limited

knowledge of the function concept because they do not understand the function well

enough to understand the elements on which it operates (Slavit, 1994; Tuska, 1992).

Learner Abilities Related to the Function Concept

The Dirichlet-Bourbaki definition of function is very abstract and allows a

function to be represented in a variety of ways. This research study will focus on the

algebraic and graphical representations of functions. In addition, this research will

include the following four learner abilities related to function representations that are

essential for students to understand the concept of function: (a) identification, (b)

translation, (c) construction of a model, and (d) reification. First, a student's ability to

identify an algebraic or graphical representation as a function or non-function indicates

his/her mental image of what a function represents. If a student has a limited perspective

of what represents functions, this limitation can hinder problem-solving abilities as well

as teacher-student communication (Vinner & Dreyfus, 1989). When considering a

graphical representation, one typically uses the vertical line test to determine whether or

not a graph represents a function. If any vertical line hits (intersects) the graph more than

once, then the graph does not represent a function.









Secondly, a student's ability to translate from one representation of a function

(such as a graph) to another representation of the function (such as an algebraic

expression) and to develop strong connections between these different representations is

important (AMATYC, 1995; Eisenberg, 1992; Kaput, 1989; Moschkovich et al., 1993;

NCTM, 2000). If a student can only problem-solve in one representation, then his/her

problem-solving abilities are limited.

Thirdly, a student's ability to construct a model refers to his/her ability to

represent a real-life situation using a function representation (O'Callaghan, 1998).

Typically, students must translate a verbal or written description of a problem situation

into a graph or algebraic expression. Kaput (1989) considered translations between

mathematical representations and non-mathematical representations as one of four

sources of meaning in mathematics. Connecting the mathematical system to the non-

mathematical system leads to deeper understanding of the concept (NCTM, 2000). In

addition, the need for mathematical concepts in society is one of the main reasons for

providing mathematics instruction. A student who cannot place the concept of function

beyond the mathematics classroom demonstrates a limited understanding of the concept

and will have limited problem-solving abilities.

Fourthly, a student's ability to reify is important (Kaput, 1989; O'Callaghan,

1998; Sfard, 1991). Reification refers to mentally seeing what was formerly viewed as a

process or procedure as an object that can have operations performed upon it

(O'Callaghan, 1998; Sfard, 1991). New mathematical objects are the outcomes of

reification (Sfard & Linchevski, 1994). Prior to reification, one views a function simply

as a process that is performed on mathematical objects, such as numbers. An example of









the process view of function is evaluating the function y = 4x2 for the number x = 3 to get

the number y = 4(3)(3) = 36 (Sfard, 1991; Sfard & Linchevski, 1994). When one views

the function represented by y = 4x2 as an object in its own right, then one can perform

processes on the function itself. These processes include, for example, composition of

functions in algebra and differentiation and integration of functions in calculus.

Technology

The function concept is taught at different levels of understanding from

kindergarten to graduate level courses in mathematics. This research study focuses on the

function concept within college algebra courses that are taught with technology.

The advancement of technology and the reduction in cost of computing power are

major forces in society that are shaping the future of mathematics education (AMATYC,

1995; Held, 1997; NCTM, 2000; Ornstein & Hunkins, 1998). Technological tools are

influencing both what is taught and how teaching occurs. The NCTM Standards (2000)

includes technology as one of six principles for school mathematics and supports the use

of technology as "essential in teaching and learning mathematics" (NCTM, 2000, p. 11).

College algebra is an introductory college mathematics course that provides the

necessary algebraic skills and understanding of functions that are necessary for future

success in subsequent college mathematics courses. Students learn to treat functions as

objects and to formally reason about operations on sets of functions in college algebra

and precalculus courses prior to calculus (Thompson, 1994). According to the College

Standards, "Today, introductory college mathematics plays a critical role in so many

professions that improving instruction at this level is essential for our nation's vitality"

(AMATYC, 1995, p. 69).









Some institutions are trying instructional methods that are alternatives to the

traditional textbook approach in college algebra. Technology has provided both an

impetus and a vehicle for instructional change. The College Standards states that

computers, graphing calculators, and software "should be fully utilized in college

classrooms" (AMATYC, 1995, p. 55) in order to enhance learning. The availability and

use of computers and graphing calculators now play a major role in the teaching and

learning of college algebra.

The function graphing utility is one of the most important tools so far in

reformulating algebra. Whether the graphing tool is on a mainframe computer, personal

computer, or graphing calculator, its use influences the algebra curriculum (a) by

allowing quick visualization of relationships, (b) by allowing for the solution of equations

and inequalities that are not possible through algebraic manipulation, (c) by allowing

numerical and graphical solutions to support algebraic solutions, (d) by promoting

exploration and understanding in terms of how a change in one representation affects

another representation, and (e) by promoting modeling of realistic situations (Heid,

1995).

According to Tall (1996), understanding begins with actions performed by the

learner. A graphing utility can promote this active learning on two levels. First, the

graphing utility provides a tool with which a learner can obtain graphs without relying on

a teacher. Secondly, in order to obtain a graph of a function, the learner must go through

an active process. The learner must first put the algebraic expression in a form that is

acceptable to the graphing utility, typically "y =" form (Ruthven, 1996). The learner

must then correctly enter the algebraic representation of the function into the graphing









utility. Then, the learner must choose and enter the domain and range for the two-

dimensional graph or accept the default values established by the program. This can

yield misleading graphical representations of the algebraic expression. According to

Ruthven (1996), the learner will "develop strategies to recognize when this has happened

and to guide the selection of an appropriate change of [domain and] range" (p. 452).

Once a graph is obtained, the zoom feature of a graphing utility allows the learner to

magnify, or zoom in on, particular portions of the graph. When the learner uses the zoom

feature, he/she is modifying the previously-entered domain and range values. By

combining the use of the zoom feature with the trace feature, which allows the learner to

determine a particular point on the graph, the learner can determine key features of the

graph such as x-intercepts, where the graph of the function crosses the x-axis.

Through the active process of finding an appropriate domain and range for

viewing the graph and using zoom and trace features for exploring the graph, students are

expected to obtain a better understanding of domain and range (Ruthven, 1996). The

availability of technology and the importance of multiple representations have also

elevated the importance of domain and range in the mathematics curriculum (Demana &

Waits, 1990). A student's lack of understanding of domain and range can lead to

misconceptions concerning the graph (Balacheff& Kaput, 1996; Ruthven, 1996). For

example, the parabolic function f(x) = x2-15x+30 looks like a steep diagonal line if the

student chooses the "standard" graphing calculator window -10 < x < 10 and -10 < y < 10

for graphing.

Students can display and explore multiple representations of functions on

graphing utilities. They can also translate one representation (table, graph, algebraic









expression) of a function to another on the graphing utility. Although students can

perform translations by hand using paper and pencil, this process is very time consuming.

When students have the ability to quickly and efficiently translate between

representations, they may better understand how to translate representations (Norris,

1994; Slavit, 1994) and be able to build stronger connections between representations

(Chandler, 1992; Ruthven, 1990).

In addition, graphing utilities may promote the object view of functions

reificationn) by providing graphs that can be viewed and manipulated (Balacheff&

Kaput, 1996; Slavit, 1994). For example, one can enter an algebraic expression such as y

= x2 in a graphing utility to obtain the graph. Then, by changing x2 to x2 + 1, the learner

can see that the original graph was moved up one unit. Graphing utilities are able to

provide objects (the original graph) on which to operate as well as provide the results (the

new graph) of those operations quickly and efficiently so that the learner can focus

his/her attention on what happened to the entire graph, thereby promoting the object view

of function reificationn).

Students can use both computer software and graphing calculators as graphing

utilities. The technology must be readily available to students for its use to be appropriate

(Ruthven, 1996). A graphing calculator is portable and inexpensive, which makes it

convenient for students to use (Wilson & Krapfl, 1994).

Computer software allows for audio/visual presentation, interactive feedback and

response, and dynamic/interactive manipulation of functions. An instructor can

implement an entire course on the computer with computer software. With computer

software, students can progress through the material at their own pace, slowing down for









difficult material and speeding up for easier material. Research by Tennyson et al.

suggests that college students benefit when allowed to control elements of their

instruction (as cited in Hannafin & Sullivan, 1996). In addition, Hannafin and Scott

(1998) have found that the amount of instruction which students prefer when learning

something new in a computer software environment affects their conceptual

understanding. By structuring the course so students use the software during class

sessions, the teacher's role changes from the traditional college lecturer. He or she can

now facilitate learning on an individual basis during class by assessing an individual

student's current knowledge via verbal interviews; asking guiding questions; encouraging

reflection upon what is being learned, and why/how it is being learned; helping students

connect new material with what they already know; and urging students to communicate

their knowledge in a variety of ways, including verbally (Stark & Lattuca, 1997).


Statement of the Problem

College students have great difficulty understanding the concept of function

(Eisenberg, 1992). The lack of understanding of domain and range plays a major role in

the weak understanding of the function concept. In fact, Markovits et al. (1988) found

that algebra students tend to ignore the domain and range of a function. Students have

also been found to have difficulty translating between representations of functions

(Eisenberg, 1992; Markovits et al., 1988). Their lack of understanding of domain and

range has been found to contribute to their problems with translation (Adams, 1997;

Caldwell, 1995).

Students have difficulty understanding what constitutes a function and what does

not (Vinner, 1983). In addition, they do not tend to use the mathematical definition of









function when problem-solving (Adams, 1997; Vinner, 1983). Students have trouble

finding a mathematical model (representation) of real-life situations (Carlson, 1998).

Furthermore, students struggle with the ability to reify and view a function as a

mathematical object (Breidenbach et al., 1992; O'Callaghan, 1998; Sfard, 1992; Sfard &

Linchevski, 1994).

Most research on technology compares a "traditional" group of students who does

not use technology with an experimental group who does use technology (Dunham &

Dick, 1994). The traditional, non-technology group represents the current method of

instruction. The expanding role of technology in the college mathematics curriculum,

however, is quickly changing the notion of "traditional." The traditional classroom is

quickly becoming a classroom with technology.

The purpose of this study is to examine the effects of two different technological

environments that use graphing calculators on students' understanding of the function

concept in college algebra as represented by their understanding of domain and range as

well as their ability to identify, define, translate, model, and reify functions. One

environment includes a text-based course with the use of graphing calculators integrated

throughout the course while the other environment includes delivery of instructional

material via computer software that also uses graphing calculators.

The purpose of this research is to answer the following questions:

1. How does differing technology use affect student understanding of the

concept of function?

2. How does the amount of instruction that a student prefers interact with the

different technologies used to affect understanding of the function concept?









3. Which technology, computers or graphing calculators, do students perceive as

being most beneficial for enhancing their mathematical learning?

In order to address the research questions, the following null hypotheses will be

tested:

1. A text-based, graphing calculator (TGC) curriculum and a computer-based,

graphing calculator (CGC) curriculum will have the same effect on students'

understanding of the concept of function in terms of the following function

components: (a) application of domain and range concepts, (b) identification

of functions and non-functions, (c) definition of function, (d) translation of

functions, (e) modeling, and (f) reification.

2. The students' preferred amount of instruction will not affect students'

understanding of the concept of function in terms of the following function

components: (a) application of domain and range concepts, (b) identification

of functions and non-functions, (c) definition of function, (d) translation of

functions, (e) modeling, and (f) reification.

3. The students' preferred amount of instruction will not interact with the TGC

or CGC curricula to affect students' understanding of the concept of function

in terms of the following function components: (a) application of domain and

range concepts, (b) identification of functions and non-functions, (c)

definition of function, (d) translation of functions, (e) modeling, and (f)

reification.









4. The graphing calculator technology used in the TGC curriculum will be

viewed by students as having the same effect on their mathematics learning as

the computer technology used in the CGC curriculum.


Justification of the Study

Student Difficulties

Numerous studies have denoted student difficulties with the concept of function.

Students have a limited mental image of what constitutes a function (Carlson, 1998;

Harel & Trgalovd, 1996; Tall, 1996; Vinner & Dreyfus, 1989). They ignore and show a

lack of understanding of domain and range (Adams, 1997; Caldwell, 1995; Markovits et

al., 1988). Also, they have difficulty reifying, or viewing a function as a mathematical

object (Breidenbach et al., 1992; Sfard, 1992; Sfard & Linchevski, 1994).

Graphing calculators and computer software have shown much promise in

increasing student understanding of the function concept (Adams, 1997; Dunham &

Dick, 1994; Heid, 1995; Hollar & Norwood, 1999; O'Callaghan, 1998; Wilson & Krapfl,

1994). However, the use of computer software and graphing calculators to represent

functions has limitations that can cause difficulties (Balacheff & Kaput, 1996; Ruthven,

1996; Tall, 1996). More research is needed in order to better understand student

difficulties in a technological environment.

Different Population

The current body of knowledge on the concept of function and the difficulties that

students have with the concept of function in college algebra is deficient. Many of the

studies on the function concept have been done outside of the United States (Ruthven,

1990; Schwarz & Hershkowitz, 1999; Vinner, 1983; Vinner & Dreyfus, 1989) or on









middle grades or early secondary students (Chandler, 1992; Leinhardt et al. 1990;

Markovits et al., 1988; Olsen, 1995; Slavit, 1994; Thompson & Senk, 2001). These

studies do not generalize to the college classroom in the United States. Many studies

done within the U. S. college classroom have concerned college calculus students (Hart,

1991; Keller & Hirsch, 1998; Pinzka, 1999; Porzio, 1995; Tall & Vinner, 1981), college

precalculus students (Norris, 1994; Quesada & Maxwell, 1994; Rich, 1990; Slavit, 1998),

or developmental college algebra students (DeMarois, 1997). Students in these courses

place below or above college algebra students in their mathematical understanding and,

therefore, constitute a different population from students in the college algebra

classroom. Therefore, these studies do not generalize well to the college algebra

classroom either.

Lack of Research Concerning Domain and Range

The concept of function is a very complex concept with many constructs within it.

The domain and range constructs of function are very important concepts, yet very few

research studies include the domain and range constructs of the function concept (Adams,

1997; Caldwell, 1995; Markovits et al., 1988; Sfard, 1992; Slavit, 1994; Tuska, 1992).

Even fewer studies include domain and range at the college algebra level (Adams, 1997;

Caldwell, 1995). More research is needed concerning the domain and range concepts of

function, particularly at the college algebra level.

A Technological Research Environment

The technological research environment may affect the understanding of the

overall function concept. In addition, it may affect different aspects of the function

concept in different ways (Adams, 1997; O'Callaghan, 1998). Existing studies on the

function concept include a research design that compares a graphing calculator group









with a no-technology group or a computer group with a no-technology group (Dunham &

Dick, 1994).

The latest research on the function concept (Adams, 1997; Hollar & Norwood,

1999; O'Callaghan, 1998) also uses a technology versus no-technology design. In

separate research studies at different colleges, O'Callaghan (1998) and Hollar and

Norwood (1999) studied algebra students in college using the same theoretical

framework and the same assessment instruments for conceptual understanding of the

function concept. The O'Callaghan (1998) study compared a computer-intensive-algebra

group with a no-technology group, while the Hollar and Norwood (1999) study compared

a graphing calculator group with a no-technology group. In both studies, the technology

group demonstrated a better overall conceptual understanding of the function concept

than the no-technology group. However, the computer group in the O'Callaghan (1998)

study did not outperform the no-technology group on the ability to reify functions, while

the graphing calculator group in the Hollar and Norwood (1999) study did outperform the

no-technology group on the ability to reify function. No college algebra studies on the

function concept exist that compare a computer group with a graphing calculator group.

Further research is needed concerning the effect of these different technological

environments on the understanding of the function concept, particularly on the ability to

reify. Student perceptions concerning which technology, computer or graphing

calculator, best enhances mathematical learning need to be explored.

Standards for curriculum and instruction require the use of technology, including

computers and graphing calculators, by all students when appropriate for enhancing

learning (AMATYC, 1995; NCTM, 1989; NCTM, 2000). College students who are









subjected to a no-technology group in a research study may see the restricted use of

technology as an "undeserving hardship" (Meel, 1998, p. 192). The question of interest

has moved beyond whether or not technology should be used, and toward which

technological environment best enhances learning.


Theoretical Framework

Early studies on the function concept used a Piagetian framework (Orton, 1971;

Thomas, 1975). Most current research on the function concept also uses a Piagetian

framework (Adams, 1997; Vidakovic, 1996) or a framework based on Piagetian ideas

(Breidenbach et al., 1992; Carlson, 1998; Hollar & Norwood, 1999, Moschkovich et al.,

1993; Sfard, 1991, 1992; Schwarz & Yerushalmy, 1992). This research study uses a

framework based on the ideas of Piaget as well.

Concept

"Abstracting is an activity by which we become aware of similarities among our

experiences. Classifying means collecting together our experiences on the basis of these

similarities. An abstraction is some kind of lasting mental change, the result of

abstracting, which enables us to recognize new experiences as having the similarities of

an already-formed class" (Skemp, 1987, p. 11). According to Skemp (1987), a concept is

an abstraction, or the defining property of a class. To form a concept, one needs a

number of experiences that have something in common. If concept A is an example of

concept B, then concept B is of a higher order than concept A. For Skemp (1987), the

first and second principles of learning mathematics are

1. Concepts of a higher order than those which people already have
cannot be communicated to them by a definition, but only by arranging
for them to encounter a suitable collection of examples.









2. Since in mathematics these examples are almost invariably other
concepts, it must first be ensured that these are already formed in the
mind of the learner. (p. 18)

Lower order concepts must be in place before students can abstract to a higher

level. For example, when learning the Dirichlet-Bourbaki definition of the function

concept, students must know the concepts of set, correspondence, and unique before

understanding the definition. While a definition cannot communicate a concept, a

definition can be "a way of adding precision to the boundaries of a concept, once formed,

and of stating explicitly its relation to other concepts" (Skemp, 1987, p. 14). In the early

stages of learning a new concept, Skemp (1987) recommended using examples that

clearly embody the concept and that have little distracting detail. Once students graph

the basic ideas of the concept, then teachers can increase distracting detail which will

reduce dependence on the teacher.

Schema

A schema is a conceptual structure that (a) is made up of individual, yet

interrelated, concepts and their properties, (b) integrates existing knowledge, (c) acts as a

tool for future learning, and (d) makes understanding possible. When connected strongly,

the interrelating of concepts may yield properties not seen in the individual concepts. A

schema determines how a person will react to events. Schema "reflect prior experiences

and comprise the totality of one's knowledge at any given time" (Schunk, 1996, p. 104).

Thoughts and actions are manifestations of schema.

Conceptual Change

For Piaget, the fundamental aspects of learning are assimilation and

accommodation (Copeland, 1970; Piaget & Garcia, 1989). Piaget (1969, 1989) used the

term assimilation to refer to the incorporation of new experiences with old ones.









Assimilation can be viewed as placing new experiences within existing, but growing,

schema (Skemp, 1987; Vidakovic, 1996). Assimilation occurs when new information is

consistent with what the student already knows. For example, students enter college

algebra with a view of domain and range as being set, typically between the values of-10

and 10 or -5 and 5. When a student graphs functions, assimilation occurs when the

student continually graphs on this "standard" domain/range.

If the learner encounters situations that do not fit within existing schema,

however, then the learner must make a change in the cognitive structure for learning to

take place (Skemp, 1987). Piaget used the term accommodation to refer to this

modification of existing schema in response to new experiences (Copeland, 1970; Piaget

& Garcia, 1989). In order for the learner to accommodate a new concept, the new

concept must bring about disequilibrium in the learner a state of mind that involves

conflicting concepts or schema, which is sometimes referred to as a cognitive conflict

(Sfard & Linchevski, 1994). The process of equilibration seeks to resolve this cognitive

conflict (Copeland, 1970; Piaget & Garcia, 1989; Vidakovic, 1996). Accommodation

then involves an exchange of one conception for another (Hewson, 1996). Facing

cognitive conflict triggers the reflection needed for accommodation (Sfard & Linchevski,

1994). For example, if teachers provide activities related to graphing functions with

technology in which the default or "standard" domain/range does not yield the expected

results, students will face a cognitive conflict which may result in accommodation that

expands their understanding of the function concept.

According to Skemp (1987), "a schema is of such value to an individual that the

resistance to changing it [accommodation] can be great" (p. 28). For Hewson (1996),









accommodation does not occur unless there is an accompanying change in the relative

status of different conceptions. A high status conception is a conception that is very

understandable, plausible, and useful. According to Hewson (1996),

Learning a new conception means that the status rises. If a learner sees
that a new conception conflicts with an existing conception (i.e., one that
already has high status for the learner), he or she cannot accept it until the
status of the existing conception is lowered. (p. 133)

The teacher must be aware of the importance of the status of students' views and

explicitly include status in classroom teaching. Reflecting about the content of the

conceptions themselves is an important aspect of conceptual change. If teachers elicit

different student views in the classroom, then the status of student conceptions can

change (Hewson, 1996).

While teachers can prepare activities to facilitate accommodation,

accommodation is difficult and cannot be assured. According to Skemp (1987),

"[accommodation] can be difficult, whereas assimilation of a new experience to an

existing schema gives a feeling of mastery and is usually enjoyed" (Skemp, 1987, p. 28).

Students do not learn or only temporarily learn new information that cannot be

assimilated, or fit into existing schema (Skemp, 1987). In order to facilitate conceptual

change, Hewson (1996) acknowledged that the following are important: (a) providing

appropriate contexts for classroom activities, (b) posing problems that have relevance and

meaning to the students, (c) finding ways to help students become dissatisfied with their

own ideas, and (d) introducing tasks in which students apply newly acquired ideas. Yet,

"while these features might be necessary [for conceptual change], they are certainly not

sufficient" (Hewson, 1996, p. 139).









In the early stages of learning, teachers must make certain that students develop

schematic learning with rich connections, not just symbol manipulation (Skemp, 1987).

They must know when students require straightforward assimilation for learning a new

concept and when students require accommodation. Skemp (1987) recommended that

teachers "lay a well-structured foundation of basic mathematical ideas on which the

learner can build" (p. 34).

Logico-Mathematical Experiences and Reflective Abstraction

For Piaget, learning can occur through two types of experiences: physical

experiences and logico-mathematical experiences. "Physical experience relates to

objects, with the acquisition of knowledge by abstraction starting from these objects.

Logico-mathematical experience has to do with the actions which the subject carries out

on the objects" (Piaget, 1966, p. 232). The acquisition of knowledge then results from an

abstraction that started with actions. The abstraction of physical experience is illustrated

by a child determining one pebble is heavier than another by weighing (acting on) the

pebbles (objects) to discover a property of the pebbles themselves. The abstraction of

logico-mathematical experience is illustrated by a child counting five pebbles left to

right, then right to left. If the child discovers there are five pebbles no matter which one

is counted first, then this is not a property of the pebbles (objects) themselves and

represents a logico-mathematical experience (Piaget, 1966). Whether learning through

physical or logico-mathematical experiences, Piaget viewed learning as beginning with

actions carried out on objects.

Piaget associated empirical abstraction with physical experiences and reflective

abstraction with logico-mathematical experiences. Empirical abstraction consists of

"deriving the common characteristics from a class of objects" (Piaget, 1966, p. 189).









Empirical abstraction involves deriving information "from the objects themselves"

(Piaget & Garcia, 1989, p. 2). Reflective abstraction applies to the learner's actions and

operations as well as to the schema that the learner constructs (Piaget & Garcia, 1989).

Reflective abstraction starts with "actions and operations" (Piaget, 1966, p. 188) and

proceeds to:

1. a projection onto a higher level (for example, of representation) of
what is derived from a lower level (for example, an action system),
and

2. a reflection, which reconstructs and reorganizes, within a larger
system, what is transferred by projection. (Paiget & Garcia, 1989, p.
2)

Reflective abstraction consists of "reconstructing an earlier structure but on a

higher plane, where it is integrated in a larger structure" (Piaget, 1966, p. 203). English

and Halford (1995) referred to Piaget's reflective abstraction as thinking about one's own

actions, and note that it is similar to the modem notion of metacognition.

Piaget carried out extensive studies on the development of children's thinking,

which he categorized into four stages: sensorimotor, preoperational, concrete operations,

and formal operations (Copeland, 1970). The final stage, the formal operations stage, is

the stage at which children and adults are capable of thinking logically and abstractly

(Copeland, 1970). The modem-day definition of function is dependent on Piaget's stage

of formal operational thought (Lovell, 1971). The structures that underlie the formal

operations stage develop through logico-mathematical experiences where the learner

engages in reflective abstraction (English & Halford, 1995). This research, therefore,

will be concerned with learning through logico-mathematical experiences.

What type of action through logico-mathematical experiences can teachers

provide students in order for reflective abstraction to occur? The function concept is









abstract and cannot be seen with the eye or manipulated with the hand. When a student

draws a graph on paper, the displayed graph is his/her end product and the graph is not

manipulated in any way. However, the use of technological tools allows the tabular,

graphical, and algebraic representations of a function to be manipulated. Technological

tools provide a dynamic media in which the states of a representational object can change

over time (Kaput, 1992). Traditional video is dynamic, but paper-pencil media is not.

According to Kaput (1992), "interactivity of the computer medium strongly distinguishes

computers both from static media [paper-pencil] and from traditional video media" (p.

526). In an interactive medium, the student not only can see the display of a

representational object change over time, but they can also take action on that object

themselves. Technological tools allow for a dynamic, interactive learning environment in

which students can perform actions on abstract mathematical objects, such as function

representations, and see the results of those actions.

Concept Image

Tall (1992) credits Vinner and Hershkowitz (1980) for introducing the terms

"concept image" and "concept definition." A concept definition refers to the formal

definition of a concept as agreed upon by the mathematical community. The Dirichlet-

Bourbaki definition of the function serves as the concept definition of the function. On

the other hand, the concept image "consists of all the cognitive structure in the

individual's mind that is associated with a given concept" (Tall & Vinner, 1981, p. 151).

Different individuals then have different concept images of a certain concept (Vinner,

1983). The concept image for an individual includes all of the mental pictures (such as

graphs, symbols, diagrams, formulas, etc.), associated properties, and associated

processes (Tall & Vinner, 1981; Vinner, 1983; Vinner & Dreyfus, 1989). Examples and









nonexamples of the concept shape the student's concept image (Skemp, 1987; Vinner &

Dreyfus, 1989), which is developed over the years (Tall & Vinner, 1981).

A gap between the concept image and the concept definition is called a

misconception. According to Vinner and Dreyfus (1989), even students who provide the

Dirichlet-Bourbaki definition when asked to define a function generally have an image of

a function as a formula only. This inconsistency of having two potentially conflicting

schema in one's cognitive structure is referred to as compartmentalization (Vinner &

Dreyfus, 1989). Another indication of compartmentalization is use of a schema that is

less relevant to a particular situation/problem instead of a more relevant schema.

Different situations can activate different parts of the concept image. The part of the

concept image that is activated at a given time is called the evoked concept image (Tall &

Vinner, 1981).

The Function Concept

According to Sfard (1991), the mathematical universe is populated with objects

that are only accessible mentally. The concept of function is an abstract concept. One

cannot hold or see the "function" object. One can only represent the abstract function

object in some way, such as with a graph, a table, or an algebraic expression (formula).

The ability to mentally "see" functions as objects is "an essential component of

mathematical ability" (Sfard, 1991, p. 3). The function concept can be seen as an abstract

object, a structure that can be manipulated whole, or one can see the function concept as

processes and actions (Sfard, 1991). "The ability of seeing a function both as a process

and as an object is indispensable for a deep understanding" (Sfard, 1991, p. 5). Different

representations of a function can be useful for enhancing the concept image of function.

For example, a graphical representation of the function y = 4x2 encourages an object view









of the function concept because the infinitely many points can be viewed as a whole.

Evaluating the function for x = 3 to get y = 4(3)(3) = 36 promotes the process view of

function (Sfard, 1991).

According to Sfard (1991), an action/process view of function will precede the

object view of function. Moving from the action/process view to the object view of

function is a lengthy process called reification. Viewing functions as new mathematical

objects is the result ofreification (Sfard & Linchevski, 1994). According to Sfard

(1991), reificationn increases problem-solving and learning abilities" (p. 29).

In order to reify, or to see a function as an object, "one must try to manipulate it

as a whole" (Sfard, 1991, p. 31). When a student graphs a function that is represented

algebraically, he/she may obtain a variety of graphs (objects) based on the domain and

range used for the function. With technological graphing utilities, one can quickly and

easily take action on the graph (object) by modifying the domain and range (Schwarz &

Dreyfus, 1995). This process of performing actions on the graph (object) allows the

learner to see the graph as an object, thereby promoting reification.

Historical Development of a Concept

According to Piaget and Garcia (1989), "a piece of knowledge cannot be

dissociated from its historical context and, consequently, the history of a concept gives

some indication as to its epistemic significance" (p. 7). Piaget's search for this source of

knowledge is referred to as genetic epistemology. By understanding the historical

development of the function concept, one can gain insight concerning the situations that

fostered the growth of the function concept and the difficulties associated with the growth

of the function concept. Viewing these situations and difficulties from a modem

perspective will inform instruction and research related to the concept (Dennis, 2000).









Definition of Terms

An algebraic representation refers to expressions whose quantities vary such as y

= x2 x + 2 and expressions whose quantities do not vary such as y = 3. Using Euler's

notation, these algebraic representations are written f(x) = x2 x + 2 and g(x) = 3, where

and g are the names of the functions. At times, the output value y is not explicitly stated,

but is implied by the context. For example, y = x2 x + 2 is sometimes written only

using the expression x2 x + 2. For the purpose of this study, the terms symbolic,

formula, algebraic expression, and analytic expression all refer to algebraic

representations.

The Cartesian coordinate system is oftentimes called the rectangular coordinate

system. It is a two-dimensional system made up of a horizontal line, called the x-axis,

and a vertical line, called the y-axis. The x-axis and y-axis are both number lines that

contain all real numbers. The x-axis and y-axis are perpendicular to each other and

intersect at the point where x = 0 and y = 0.

A graphical representation refers to a set of points on the Cartesian coordinate

system. The set of points may form a curve or line. For the purpose of this study, the

terms graph, graphic, Cartesian graph, curve, and geometric curve all refer to a

graphical representation.

A tabular representation refers to a table with multiple rows and two columns or

multiple columns and two rows. A table can contain any type of symbols, such as

numbers and text. For the purpose of this study, a table will contain numbers unless

noted otherwise. Therefore, the terms numeric and table will refer to a tabular

representation.









A correspondence is a pairing of items in one set with items in another set. For

example, the names of states can be paired with their population to form a

correspondence. A table is oftentimes used to clearly illustrate a correspondence.

A transformation is a modification within a representation. For example, adding

5x + 3x to yield 8x is a transformation within the algebraic representation. A translation

refers to movement between representations (Kaput, 1992). For example, obtaining the

graphical representation from the algebraic representation represents a translation. To

avoid confusion, the common use of the term translation in geometry to refer to

directional movement of geometric objects will be avoided. For the purpose of this

study, directional movement of a graph is a transformation, but not a translation.

Therefore, when discussing the horizontal and vertical movement of graphs, the term

graph shifting will be used.


Significance of the Study

Wilson and Krapfl's (1994) review of the literature reveals that the impact of

graphing calculators on early college student understanding of the function concept is not

yet certain. Later research has added much to the understanding of that impact, but many

questions still remain. This research study is designed to add to the body of research

related to the teaching and learning of the concept of function. Different teaching

methods will be explored in order to provide important information that can lead to better

learning in college algebra.

While domain and range are important components of the concept of function,

very little research is available concerning these components. The growth in the use of

graphing utilities has magnified the importance of domain and range. Students must









enter a graphing window (domain and range) in order to graph a function. Through the

process of obtaining an appropriate graph, students have opportunities for cognitive

conflict which can promote connections between algebraic, tabular, and graphical

representations of functions, thereby strengthening student understanding of domain and

range as well as their ability to translate between representations of functions.

Reification is an aspect of the function concept that is very difficult for students.

Graphing utilities allow for the manipulation of graphs. A student's ability to manipulate

these representations of function may promote what Piaget and Garcia (1989) refer to as

reflective abstraction and what Sfard (1991) refers to as reification.

Existing research is unclear concerning the effect of different learning

environments on the reification of the function concept. For Piaget and Garcia (1989) as

well as Sfard (1991), human knowledge starts with actions on objects that result in

entities that may be viewed as objects in their own right at some time. This reification

requires time and action on objects (functions). By comparing two different

technological environments that seek to promote this action, this research study will

obtain information that is not available in the existing research in relation to the concept

of function.

This research is based soundly in existing research, but will extend existing

research by including a sample of college algebra students within the same university and

the factor of technology use as it pertains to the concept of function, particularly in terms

of domain, range, and reification.









Organization of the Study

This chapter contains a description of the problem and its relevance in the field of

mathematics education. Chapter II contains a review of the relevant literature concerning

the concept of function, the use of graphing calculators in teaching the concept of

function, and the use of computers in teaching the concept of function. Chapter III

contains a description of the research design and methodology. Results of the analysis

and limitations of the study are reported in Chapter IV. A summary of the results,

implications, and recommendations for future results are presented in Chapter V.














CHAPTER 2
REVIEW OF THE LITERATURE


Historical Development of the Concept of Function

In modem-day mathematics, numerous terms are associated with the concept of

function: domain, range, graph, definition, analytic expression, relation, model,

correspondence, unique, dependence, mapping, independent variable, etc. Examining the

historical development of the concept of function provides insight into the meaning and

importance of these terms as well as how and why the terms are components of the

concept of function. In addition, examining the historical development of the function

concept:

1. provides insight into modem-day understandings of function, such as the

importance of the algebraic, graphical, and tabular representations of function;

2. identifies possible curriculum suggestions by identifying the order in which

aspects of the function concept were developed and extended;

3. identifies difficulties related to the function concept from a historical

perspective; and

4. demonstrates the central role of the function concept in many branches of

mathematics.

Early Developments

According to Kennedy and Ragan (1989), Eric Temple Bell in The Development

of Mathematics suggested that the Babylonians circa 2000 B.C. demonstrated the concept









of function with the use of tables like the one for n3 + n2 for n = 1, 2, ..., 30. This view

of function suggests a definition of function as a table or correspondence. According to

Boyer (1968), circa 1360 A.D., the French physicist and mathematician Nicole Oresme

drew a velocity-time graph for a body moving with uniform acceleration. He marked

points representing instants of time (longitudes) along a horizontal axis, then for each

instant of time drew a line segment (latitude) that represented the velocity. Oresme's

work provides early evidence of a graphical representation of a function. According to

Boyer, Oresme "seems to have grasped the essential principle that a function of one

unknown can be represented as a curve [graph], but he was unable to make any effective

use of this observation except in the case of the linear function" (p. 291). Oresme's work

also provides early evidence of a common modem-day misconception about functions -

that they must be linear (Markovits et al. 1988; Knuth, 2000). Known then as the latitude

of forms, Oresme's graphical representation of functions remained a popular topic from

his time until the time of Galileo (1564-1642).

While the concept of function has evolved for over 4000 years (Kennedy &

Ragan, 1989; Kleiner, 1989), the concept at it is known today explicitly emerged and

evolved in the last 300 years. The concept of function has evolved in connection with

problems in calculus and mathematical analysis (Kliener, 1989). The foundation for the

explicit form of function began with understandings of the function concept in the 17th

century with Descartes (Hamley, 1934; Kennedy & Ragan, 1989). The formal blending

of algebra and geometry, in what is known today as analytic geometry, by Descartes and

Fermat in the early part of the 17th century was a fundamental development for the

growth of the function concept (Kleiner, 1989). Descartes made analytic geometry









known to others when he published La G6ometrie in 1637 (Boyer, 1968). In La

G6ometrie, Descartes implicitly included two ideas of the function concept that signified

dependence. First, that unknown quantities can be expressed in terms of a single quantity

and second, that a curve (graph) pictures the dependence of one variable on another

variable (Hamley, 1934).

Eighteenth Century Developments

As the 18th century was ushered in, the function concept was made more explicit.

Beginning with the calculus of Newton and Liebniz and continuing to this day, the

evolution of the function concept has included focus on function as a geometric curve

(graph), as an algebraic expression, and as a correspondence between sets (Kleiner,

1989). A correspondence between sets is oftentimes organized in a table. Today, the

technology of graphing utilities provide an excellent method of displaying a function in

these three multiple representations tables, graphs, and algebraic expressions (Kaput,

1989; Moschkovich et al., 1993; Schwarz & Dreyfus, 1995). This research study focuses

on the graphical and algebraic representations of functions.

The calculus of Newton and Liebniz was a calculus that focused on graphs

(geometric curves). The German mathematician Liebniz is credited with introducing the

word "function" (Boyer, 1968; Hamley, 1934; Kleiner, 1989). According to Kleiner,

Liebniz introduced the word function in 1692 to designate a geometric object associated

with a curve. For example, Liebniz stated that "a tangent is a function of a curve"

(quoted in Hamley, 1934, p. 13).

In 1718, the Swiss mathematician Jean (English John; German Johann)

Bernoulli provided the first formal definition of function (Kleiner, 1989). His definition

focused on function as a quantity that depends on another quantity. Boyer (1968, p. 462)









and Kleiner (1989, p. 284) provide Bernoulli's definition: "One calls here Function of a

variable a quantity composed in any manner of this variable and of constants."

The Swiss mathematician Leonhard Euler introduced function notation f(x)

(Boyer, 1968; Hamley, 1934; Kennedy & Ragan, 1989; Kleiner, 1989). According to

Boyer, Euler's two-volume treatise of 1748 Introductio in Analysin Infinitorum "served as

a fountainhead for the burgeoning developments of mathematics throughout the second

half of the eighteenth century. From this time onward, the idea of function became

fundamental in (mathematical) analysis" (p. 485). Kleiner (1989) credited Euler's

Introduction as the first work that placed the concept of function in a central role. Euler

defined a function as an analytic expression (algebraic formula): "A function of a

variable quantity is an analytic expression composed in any manner from that variable

quantity and numbers or constant quantities" (Boyer, 1968, p. 485). According to

Kleiner (1989), Euler's Introductio has no pictures or drawings. Euler's algebraic

approach began an era that focused on function as an algebraic expression.

Attempts to solve the Vibrating String Problem led to controversy that centered

around the meaning of function. This controversy led to the concept of function being

further extended (Boyer, 1968; Kleiner, 1989). As described by Kleiner, the goal of the

Vibrating String Problem is to find the function that best describes the shape of an elastic

string at time t, where the string is fixed at ends 0 and 1, deformed in an initial shape, then

released to vibrate. The solutions offered for the problem by d'Alembert in 1747, Euler

in 1748, and Daniel Bernoulli in 1753 signified differing conceptions of function. For

d'Alembert, a function must be an analytic (algebraic) expression. From the physical

considerations of the initial shape of the string, Euler expanded his view of function









beyond his previous view as an analytic expression, and included curves (graphs) drawn

by free-hand as functions. Bernoulli, a physicist whose main interest was in solving the

physical problem, viewed an arbitrary function as an arbitrary shape of the vibrating

string (Kleiner, 1989). Bernoulli's solution was in terms of an infinite series of

trigonometric functions. His solution caused a conflict with an incorrect assumption of

18th century mathematics that if two analytic expressions agree on an interval, then they

agree everywhere. According to Kleiner (1989), this assumption "implicitly assumes that

the independent variable in an analytic expression ranges over the whole domain of real

numbers, without restriction" (p. 285). In addition to bringing the concept of domain to

the forefront, the Vibrating String Problem extended the concept of function to include

functions defined piecewise in different intervals by analytic expressions and to include

functions drawn free-hand, even if they cannot be specified by a combination of analytic

expressions (formulas).

Nineteenth Century Developments

In his work with heat conduction in 1822, Joseph Fourier advanced Bernoulli's

idea of infinite series of trigonometric functions by claiming that any arbitrary function

can be written in terms of an infinite series of trigonometric functions, now known as the

Fourier series (Boyer, 1968). In demonstrating an arbitrary function that could not be

written in terms of the Fourier series, Lejeune Dirichlet in 1829 advanced the concept of

function by providing "the first explicit example of a function that was not given by an

analytic expression and was not drawn by freehand" (Kleiner, 1989, p. 292). Dirichlet

broadened the definition of function as follows:

y is a function of a variable x, defined on the interval a < x < b, if to every
value of the variable x in this interval there corresponds a definite value of









the variable y. Also, it is irrelevant in what way this correspondence is
established. (Kleiner, 1989, p. 291)

In his definition, Dirichlet highlighted the concept of function as an arbitrary

pairing or correspondence, giving function a meaning separate from an algebraic

expression. He also furthered the idea of restricted domains by specifying the interval a

< x < b in his definition. In addition, Dirichlet furthered the concept of function by

making physical interpretation and graphical representations recognized aspects of

function (Hamley, 1934).

In 1854, the work of Riemann permanently placed discontinuous within the

concept of function, while in 1887 Dedekind defined a function as a "mapping" between

arbitrary sets. As the concept of function expanded, so too did the sets being mapped

"from" and "to." Therefore, Diriclet's correspondence between real numbers in 1829 had

grown to the mapping of functions to functions with Volterra by 1887 (Kleiner, 1989). In

college algebra today, the mapping of functions is seen when functions are combined to

form a third function, for example as in the composition of functions. Transformations of

graphs can also be viewed as a mapping of functions. This ability to view a function as

an object in its own right has proven difficult for students (O'Callaghan, 1998; Sfard,

1992). In addition, many current algebra students have the misconception that a

discontinuous graph does not represent a function because it is broken or unusual

(Carlson, 1998; Markovits et al., 1988).

Twentieth Century Developments

In 1934, Herbert Russell Hamley authored the Ninth Yearbook of the National

Council of Teachers of Mathematics in its entirety. In the Yearbook, Hamley described

the function concept and promoted its importance in mathematics education. His









inspection of the "modem textbooks" of his day provides valuable insight into the

evolution of the function concept. In the included definitions from textbooks spanning

the years 1919 to 1930, some definitions allow for multiple y values for each x value,

while others specify that only one y value is allowed for one x value. Hamley described a

function as "a correspondence between two ordered variable classes" (p. 6). After

synthesizing the definitions found in the textbooks, Hamley settled upon the following

definition of function:

Two variables y and x are in functional relation when there is a
determinate correspondence between the quantities xl 1, x2, x3, ... of the x
variable and the quantities yl 1, y2, y3, ... of they variable, the order of the
arrangement of the quantities of the two variables being alike. (Hamley,
1934, p. 20)

Hamley stressed the concept of dependence, and used much of our modem-day

terminology when writing about the function concept. For example, he described the

function concept in terms of a rule which, when applied to the "domain of the

independent variable" (p. 20), allows the corresponding dependent variables to be

specified (determined).

Hamley also stressed the relationship aspect of function and preferred to use the

term functional relation instead of function. In the early 20th century, the concept of

function included a strong focus on "functional thinking." Hamley sometimes referred to

functional thinking as relational thinking, because his view of function stressed the

relationship of the correspondence. Hamley summarized J. S. Georges' three abilities of

functional thinking as follows:

First, the ability to recognize mutual dependence between variables and
varying quantities; second, the ability to determine the nature of the
dependence or relationship between variable quantities; and third, the
ability to express and interpret quantitative relationships. (1934, p. 80)









When putting the concept of function in practice, Hamley tended to view function

in terms of physical representations, such as "the extension of a strained spring is a

function of the tension applied" (p. 21). From this applied perspective, functional

thinking stressed the ability to recognize relationships, interpret relationships, and model

relationships. These components of the function concept are currently stressed in the

mathematics curriculum (NCTM 1989; NCTM, 2000). This applied perspective may

imply that only one y value is assigned to each x value, but Hamley did not explicitly

state this idea, which is contained in today's modem definition of function.

Nicolas Bourbaki was the name used by a society of mathematicians who wrote

several volumes in a work titled l61nents de Mathdmatique (Boyer, 1968). Their goal

was to survey the important mathematics of their time. In 1939, the first volume in

l6ments contained the following definition of function:

Let E and F be two sets, which may or may not be distinct. A relation
between a variable element x of E and a variable element y of F is called a
functional relation in y if, for all x (symbol an element of) E, there exists a
unique y (symbol an element of) F which is in the given relation with x.
We give the name of function to the operation which in this way associates
with every element x (element of) E the element y (element of) F which is
in the given relation with x; y is said to be the value of the function at the
element x, and the function is said to be determined by the given
functional relation. (Kleiner, 1989, p. 299).

Bourbaki's definition of function as a relationship between sets stresses the idea of

uniqueness. With relationship and uniqueness included, Bourbaki's definition captures

the key ideas in the function definition as found in modem college algebra books.

Bourbaki also defined function as a subset of the Cartesian product E x F, where the sets

E and F are described above (Kleiner, 1989). This definition of function as a set of

ordered pairs is also commonly found in modem college algebra books (Kaufmann,

1994; Larson & Hostetler, 1997).









Summary of the Historical Development

The historical developments of the concept of function reveals that it is a complex

concept, which is comprised of many concepts such as domain, range, uniqueness,

variable, and discontinuous. To add to the complexities, the concept of function occurs

both in the "real-world" and in the purely mathematical realm. The function concept was

seen in "real-life" applications such as the heat conduction problem and the Vibrating

String problem. This focus on real-life contextual problems and modeling has a strong

presence in the mathematics curriculum today (NCTM, 2000). Also, the historical

development revealed that a function can be represented in many ways. These multiple

representations included tables of numbers, graphs, and algebraic expressions. The

historical development of the concept of function reveals a concept of breadth and depth

that can lead to difficulties and misconceptions.

This research study is well-justified by the historical development of the function

concept. This research study focuses on domain and range, graphical and algebraic

representations, and difficulties associated with understanding the function concept. The

ability to define function and identify functions is also justified by the historical

development. The ability to model, to work with real-life situations, is also justified by

the historical development. Lastly, the view of function as a process is evident in the

tables of the Babylonians, while Newton, Liebniz, and Volterra all demonstrated the

ability to view a function as an object.


History of Function in Mathematics Education

As early as 1893 Felix Klein, the mathematician and leader of the early 20th

century German mathematics reform movement, originated the idea that the concept of









function should be the central unifying theme of school mathematics (Hamley, 1934). At

a conference in 1904, Klein asserted "the function concept graphically presented should

form the central notion of mathematical teaching" (as quoted in Hamley, 1934, p. 52).

By 1914, calculus and analytic geometry was established in French secondary schools,

which thereby stressed the concept of function in those schools.

The reform movement in the United States is considered to have started with E.

H. Moore's presidential address to the American Mathematical Society in 1902. David E.

Smith and E. R. Hedrick were the first proponents of including the function concept in

American schools (Hamley, 1934). In 1911, Hedrick envisioned the function concept

firmly placed in school algebra: "The real subject matter of algebra consists of variable

quantities, the relations between variable quantities, and the acquisition of the ability to

control and interpret relations" (as quoted in Hamley, 1934, p. 77). In 1923, the National

Committee on Mathematical Requirements, a committee of the Mathematical Association

of America, published The Reorganization of Mathematics in Secondary Education

(Hamley, 1934). This report is a major milestone in the history of mathematics education

(Hamley, 1934). In the seventh chapter of the report, titled The Function Concept in

Secondary School Mathematics, the committee states that the idea of functional

relationship is "best adapted to unify the course" (as cited in Hamley, 1934, p. 78).

In 1923, The National Committee on Mathematical Requirements viewed the

fundamental importance of the functional relation as "the dependence of one variable on

another" (as cited in Hamley, 1934, p. 78). In addition, the committee emphasized

placing the relationship in context: "Indeed, the reason for insisting so strongly upon









attention to the idea of relationships between quantities is that such relationships do occur

in real life." (as cited in Hamley, 1934, p. 79).

The function concept made in-roads into the school classrooms of the United

States. By the late 1920s, articles authored by secondary school teachers appeared in

journals promoting teaching the concept of function in secondary education (Hamley,

1934). Teacher Eleanor Booher (1926) viewed the ability to recognize relationships as

the "very essence of intelligence" (as quoted in Hamley, 1934, p. 83) and stressed the

concept of function in her classroom.

The Dirichlet-Bourbaki definition of function is stated as follows: a function is a

relationship between two sets A and B that assigns to each element x in set A, called the

domain, exactly one element y in set B, called the range (Larson & Hostetler, 1997). It

has taken time for the Dirichlet-Bourbaki definition of function to become the standard

that is used today. In a study of eleven elementary algebra texts published before 1959

and nine published after 1959, Kennedy and Ragan (1989) did not find any elementary

algebra texts before 1959 that included both the relation between sets and the uniqueness

of y as specified by Bourbaki. Two common definitions allowed "one or more values of

y" (p. 312), while a third common definition contained the misconception that a function

must be an algebraic expression. This misconception dates back to Euler's limited view

of function as an analytic expression (algebraic formula) in his 1748 Introductio, and is

still a common misconception among students today (Carlson, 1998; Eisenberg, 1992;

Vinner, 1983; Vinner & Dreyfus, 1989). Similarly, none of the seven college algebra

texts published prior to 1959 contained the modem day Dirichlet-Bourbaki definition.

The Dirichlet-Bourbaki definition made great inroads in texts that were published after









1959. The Dirichlet-Bourbaki definition was included in six of nine elementary algebra

texts and four of eight college algebra texts. Today, the Dirichlet-Bourbaki definition of

function is used in "almost all" algebra text-books (Kieran, 1992, p. 408).

For the purposes of this study, a conceptually correct definition of function must

include two key ideas: (a) it must include a relationship between two sets (called the

domain and range), and (b) every element in one set must get assigned exactly one

element in the other set. This guiding statement, based on the Dirichlet-Bourbaki

definition of function, delineates the assessment method for student definitions of

function on the Domain/Range/Identify/Define/Translate instrument (Appendix A). The

Dirichlet-Bourbaki definition of function that is conceptually correct by today's standards

is stated as follows: A function f is a relationship between two sets A and B that assigns

to each element x in set A exactly one element y in set B. The set A is called the domain

of the function and the set B is called the range of the function.

The Dirichlet-Bourbaki definition is justified through the historical development

of the function concept, the literature concerning the function concept (Kieran, 1992;

Markovits et al., 1988; Vinner, 1983), and current practice in college algebra courses as

indicated by textbooks over the past 12 years (Barros-Neto, 1988; Kaufmann, 1994;

Larson & Hostetler, 1997).

From the historical development of the concept of function as well as its history in

mathematics education, one can see that each generation developed its own concept

image of the function concept. In addition, each generation was faced with conflicts of

understanding that led to the restructuring of the notion of function in the mathematics

community. This restructuring came about with great difficulty and required much time.









While accommodation and reification are typically used to refer to an individual's

cognition, these terms also apply to the historical development of the function concept

within the mathematics community. According to Sfard (1992), the historical

development of the concept of function can be seen as a 300 year struggle for reification.

The concept changed and expanded as individuals were faced with applied or pure

mathematical problems that forced them to re-conceive their notion of function. Euler's

re-conception of function to include a graph drawn by free-hand when faced with the

Vibrating String Problem provides one such example.

While the definition of function evolved to its present-day definition, a variety of

definitions were present at any given time. For example, in Hamley's (1934) inspection

of function definitions in early 20th century textbooks, some definitions allowed for more

than one y value to be assigned to the same x value, while other definitions stated that

only one y value was to be assigned. No matter which definition was used, however,

there was oftentimes a gap between the concept definition of function and a person's

mental image of function. Although, the terms concept definition, concept image, and

misconception were not used, these ideas were nonetheless present as early as 1870.

After examining the better mathematical analysis textbooks of his day, in 1870 Hankel

wrote:

One [text] defines function in the Eulerian manner; the other that y should
change with x according to a rule, without explaining this mysterious
concept; the third defines them as Dirichlet; the fourth does not define
them at all; but everyone draws from them conclusions that are not
contained therein. (as quoted in Kleiner, 1989, p. 293)

Similarly, although the representation of a function as an algebraic expression had

a long and established history at the time, in 1934 Hamley lamented









So many writers of school textbooks have fallen into the error of
supposing that the function concept was synonymous with the graphical
representation of functions. Few seem to have grasped the idea that the
function concept is a mode of thinking rather than a method of illustration.
(p. 79)

Dealing with misconceptions concerning the function concept has a long history for both

mathematicians and mathematics educators.


Standards Related to the Function Concept

The historical development of the function concept revealed its importance in

mathematical analysis and calculus as well as mathematics education. In addition, the

historical development of the function concept demonstrates the importance of various

components (domain, definition, modeling) and multiple representations (table, graphs,

algebraic expressions) of the function concept. Today, the concept of function is a

"central theoretical construct" in calculus courses (Tall, 1996, p. 320), "a pivotal concept

in higher mathematics education" (Harel & Trgalovd, 1996, p. 675), and "among the

most powerful and useful notions in all mathematics" (Romberg, Carpenter, & Fennema,

1993, p. 1). According to Eisenberg (1992), "the development of a sense for functions

should be one of the main goals of the school and collegiate curriculum" (p. 153).

Current national standards for mathematics education at both the school and

college level support these views that place a strong emphasis on the function concept.

The NCTM (1989) Curriculum and Evaluation Standards for School Mathematics

recognize the function concept as "an important unifying idea" (p. 154) in secondary

school mathematics. According to the NCTM Principles and Standards for School

Mathematics (2000), the concept of function is a "foundational idea" that "should have a

prominent place in the mathematics curriculum because [it enables] students to









understand other mathematical ideas and connect ideas across different areas of

mathematics" (p. 15). Within the Algebra Standard of the NCTM Standards (2000), K-

12 students are expected to "understand patterns, relations, and functions" (p. 37).

Secondary school students are expected to "convert flexibly among, and use various

representations" (p. 296) of functions. As students work with "multiple representations of

functions including numeric [tables], graphic, and symbolic [algebraic] they will

develop a more comprehensive understanding of functions" (p. 38). Within the

Representation Standard, the NCTM Standards (2000) state that students should be able

to "select, apply, and translate among mathematical representations to solve problems;

and use representations to model and interpret physical, social, and mathematical

phenomena" (p. 67).

In Crossroads in Mathematics: Standards for Introductory College Mathematics

before Calculus, the American Mathematical Association of Two-Year Colleges

(AMATYC, 1995) also stresses the importance of the function concept in mathematics

education. As the College Standards content standard C-4 states "students will

demonstrate understanding of the concept of function by several means (verbally,

numerically, graphically, and symbolically) and incorporate it as a central theme into

their use of mathematics" (AMATYC, 1995, p. 13). The College Standards further states

that students "will formulate such (functional) relationships when presented in data sets,

and transform functional information from one representation to another. Suggested

topics include generalization about families of functions, use of functions to model

realistic problems, and the behavior of functions" (AMATYC, p. 13).









Content standard C-2 of the College Standards states that "students will translate

problem situations into their symbolic representations and use those representations to

solve problems" (AMATYC, 1995, p. 13). The College Standards emphasize the use of

a "combination of appropriate algebraic, graphical, and numerical methods to form

conjectures about problems" (p. 13). Suggested topics include the translation of realistic

problems into mathematical statements (modeling) as well as the solutions of equations

by graphical, algebraic, and numerical techniques.

The historical development of the function concept, the history of the function

concept in mathematics education, and the mathematics education standards of today

demonstrate that the idea of multiple representations is important. In addition, the

historical development of functions clearly shows that the concept of function is

complex, is understood with great difficulty, and is prone to misconceptions. The

following two sections look at multiple representations and student difficulties within the

context of today's literature on the teaching and learning of the function concept.


Multiple Representations

Student understanding of the multiple representations of functions and the ability

to translate/connect from one representation to another is an important aspect of

understanding the concept of function (AMATYC, 1995; NCTM, 2000). A function can

be represented in a variety of ways. The historical development of the function concept

includes: numeric/tables, graphs, algebraic expressions, verbal/written descriptions, and

the Dirichlet-Bourbaki definition. Current researchers echo these views. For example,

according to Tall (1996) the function concept manifests itself in five representations:

1. visuo-spatial, from observing and experiencing distance, velocity, etc.;









2. numeric, that can be manipulated and computed;

3. symbolic, using algebraic symbols and expressions;

4. graphic, using graphs; and

5. formal, using the Dirichlet-Bourbaki definition.

While the abstract Dirichlet-Bourbaki representation of function is oftentimes

viewed from a mathematical perspective as having high status, the "richer features of the

other [representation] systems can support the building and interrelating of cognitive

structures" (Kaput, 1989, p. 170). The numeric (table), symbolic (algebraic), and graphic

representations mediate between the formal/abstract representation and the "infinitely

varied features of the world they model" (Kaput, 1989, p. 170). Due to this facilitating

bridge as well as due to the role of technological advances, students can represent a

function in three main ways: (a) in a graph, (b) in an ordered pair table, and (c) as an

algebraic expression (Romberg et al., 1993).

The graphical, tabular, and algebraic representations of function occur throughout

undergraduate mathematics, including college algebra. These three representations are

also seen throughout calculus, with technology-oriented calculus reform somewhat

restricting the use of algebraic manipulations in favor of stressing the connections

between the algebraic, numeric, and graphical approaches (Harel & TrgalovA, 1996). In

solving differential equations in calculus, Artigue (1992) proposed approaches from these

three representations in order to get the exact solution from the algebraic representation,

an approximate solution from the numerical representation, and the qualitative solution

from the graphical representation.









Pinzka's (1999) study of college calculus students clearly demonstrated the

importance of multiple representations of the function concept in calculus. In particular,

she related students' understanding of the derivative concept to the students' (a)

geometric concept image of function, (b) ability to understand and interpret graphs of

functions, and (c) ability to make connections among the various representations of

functions.

Eisenberg (1992) stressed the importance of the graphical representation of

function because (a) the ability to solve problems visually represents a deeper

understanding than if one only has the ability to solve algebraically, and (b)

mathematicians use visual/graphical exploration. For Eisenberg (1992), "single-valued

real variable functions should be thought of as being inherently tied to a graphical

representation, and all elementary concepts concerning functions [should] be defined in a

visual format" (Eisenberg, 1992, p. 159).

According to Kaput (1989), the different representations of functions provide

different strengths. The graphical representation of a function allows us to "consolidate

a binary quantitative relationship into a single graphical entity a curve or a line" (p.

172) with which one can reason. A table of data displays data that is more quantitative in

nature where changes in data values can be explicitly read from the table. The algebraic

expression, such as y = 2x+3, explicitly provides the quantitative relationship between x

and y. This relationship is hidden in the graphical and tabular representations of the same

function. In terms of the domain of the function, the domain is implied in the algebraic

expression in the sense that the student must provide input values x, but the tabular and

graphical representations make the domain more explicit (Kaput, 1989). Knowing the








strengths of the various representations and when/how to use them is an important part of

understanding the function concept.

Moschkovich et al. (1993), Eisenberg (1992), and Kaput (1989) all stressed the

importance of seeing the connections between representations. Knowledge of the

different representations should not be compartmentalized. For example, "the zeros of a

function should be thought of as points where the graph crosses the x-axis" (Eisenberg,

1992, p. 159). For Eisenberg (1992), a student with good function sense should be able to

count the number of solutions to the equation sin(x) = x by graphically visualizing the

graphs of y = sin(x) and y = x.

For Kaput (1989), mathematical meaning resides in the connections of

representational systems. In particular, meaning comes from: (a) translations between

mathematical representation systems (graphs, tables, algebraic expressions), (b)

translations between mathematical representation systems and non-mathematical systems

(modeling an algebraic expression from a written description for example), (c)

transformations and operations within a mathematical representation system (simplify the

algebraic expression, shift the graph, etc.), and (d) through the reification of actions,

procedures, and concepts into objects that can serve as the basis for new actions,

procedures, and concepts at a higher level of organization (Piaget's reflective

abstraction).

Not only are multiple representations important, but multiple perspectives are also

important. An algebraic expression, for example, can be viewed in terms of a

process/procedure or an object/structure (Sfard, 1991). From the process perspective, a

function is viewed as linking x and y values. For each x value, the function has a









corresponding y value. From the object perspective, a function and its representations are

thought of as entities for example, algebraically as classes of functions or as graphs that

can be picked up whole and shifted, rotated, or reflected (Moschkovich et al., 1993).

For Moschkovich et al. (1993), competent understanding of the function concept

"consists of being able to move flexibly across representations tabularr, graphical, and

algebraic) and perspectives (process and object), where warranted: to be able to "see"

lines in the plane, in their algebraic form, or in tabular form, as objects when any of those

perspectives is useful, but also to switch to the process perspective (in which an x value

of the function produces a y value), where that perspective is appropriate" (p. 97).

Moschkovich et al. (1993) provide numerous examples of problems in which students

must connect between different representations (table, graph, equation) of a function

and/or different perspectives (process/object) in order to solve the problems.

Research indicates that developing this ability is difficult (Moschkovich et al.,

1993). According to Kieran (1992), both process (procedural) and object (structural)

conceptions of function are important, but the challenge is to develop the ability to move

back and forth.

Research indicates that activities that involve the use of multiple representations

of functions (tables, graphs, algebraic expressions) lead to a broader understanding of the

function concept (Confrey & Doerr, 1996; NCTM, 2000). With technological tools,

students can easily display representations of functions that were formerly only available

in the mind's eye (Heid, 1995), thereby possibly enhancing their understanding of the

function concept.









In a study consisting of 98 undergraduate students, Johari (1998) randomly

assigned students taking a computer literacy course into one of two treatment groups.

Both treatment groups used self-paced software that contained instruction to facilitate the

understanding of (input and output) variables as well as the construction of linear

functions in word problem contexts. The first treatment group's software contained a

table representation of functions. The software for the second treatment group was

identical to the first with the addition of a graphical representation of function. The post-

test measured the ability to construct functions as well as the understanding of variables.

The table-graph treatment group scored significantly higher on achievement post-tests

than the table-only treatment group. Johari's study indicates that when students are

provided activities involving multiple representations, they develop a deeper

understanding of the function concept.

Technological environments alter the static display character of some

representation systems and provide new forms of actions. For example, with some

graphing utility software the learner can directly manipulate a static graph. According to

Kaput (1989), most of the mathematics related to algebra was developed under the

constraint of static displays and difficult and time-consuming computations. The

availability of technology, however, has lifted these constraints. Technology use

supports (a) transformations within the algebraic representation system through

Computer Algebra Systems (CAS) that manipulate algebraic symbols, (b) dynamic

linkages between representation systems, (c) new actions within representation systems,

(d) intelligent tutoring within representation systems, and (e) the capturing and









generalizing of actions into repeatable, nameable, and inspectable procedures (Kaput,

1989).

Computers can support multiple linked representations. The correspondence

between representations is explicitly and immediately made. The software can link the

action components and the visual display of multiple representations, so students can act

on (manipulate) one representation, then see the effect of that action on both that

representation and other representations. For example, a student can simultaneously see

the effect of changing an algebraic expression through the displays of the modified

algebraic expression as well as the displays of the related graphical and tabular

representations. By having multiple representations available for manipulation (action)

as well as visual display on the computer, a student does not have to rely on a single

representation, with its inherent weaknesses (Kaput, 1989). The strengths of the various

representations are all available. "In this type of environment the computations required

to translate actions across representations are done by the computer, leaving the student

free to perform the actions and monitor their consequences across the representations"

(Kaput, 1989, p. 179). Most importantly, "the cognitive linking of representations creates

a whole that is more than the sum of its parts" (Kaput, 1989, p. 179).

The actions performed by a student in any single representation system to perform

a task vary. For example, to solve an equation using a table with technology, one must

generate multiple columns of the table using step-sizes for the variable column and

algebraic expressions for the other two columns. Yet, with a graphical approach to solve,

the graphs are generated from the algebraic expressions of the equation, then tracing can

yield an approximate solution. Kaput (1989) recommends the linkage of at least two









representation systems. For example, while the learner solves an equation algebraically,

the computer can display both the algebraic equation and the graphs of both sides of the

equation at each step. The transformations within the algebraic system will be

accompanied by the corresponding transformation of the graphs. If the x-coordinate of

the intersection changes, then an incorrect algebraic transformation was performed

(Kaput, 1989). Using technology, students can see and evaluate the results of actions

taken without the cognitive difficulties and large amounts of time that is required using

paper-and-pencil methods.

With technology, multiple graphical displays are available in a short amount of

time, and "students are free to move and manipulate graphical objects just as we have

always been free to manipulate algebraic objects" (Kaput, 1989, p. 185). Through point-

and-click mouse motions, graphical objects can be transformed through reflections and

vertical/horizontal shifts.

The strength of a computer-based learning environments is (a) they provide

students with the ability to visually represent graphs, algebraic expressions, and tables in

much the same way they are traditionally represented; (b) they support student interaction

with mathematical objects; and (c) they support dynamic linkages between the

representational systems (Kaput, 1989).

The use of technological graphing tools elevates the importance of the

domain/range component of the function concept. According to Ruthven (1996),

This new emphasis on visualizing through graphs has important curricular
implications: in particular, it increases the importance of developing
understanding of the scaling of axes [domain/range] and the
transformation of graphs, as well as the relationships between symbolic
[algebraic] and graphic representations. (p. 459)









According to Schultz and Waters (2000), students can find approximate solutions

to equations and systems of equations using the zoom/trace features of technological

tools on the graphical representation of the accompanying functions. This capability

provided by technological tools, however, requires that students obtain the additional

ability to estimate domain and range (Schultz & Waters, 2000).

To summarize, the ability to work within a variety of representations, to translate

between those representations, and to have a strong connection between representations

when problem-solving is an important aspect of understanding the concept of function.

According to Schultz and Waters (2000), in order to increase problem-solving abilities

and facilitate the understanding of concepts, students need to be familiar with the various

representations and have the opportunity to choose and create suitable representations

(Schultz & Waters, 2000). In addition, having the ability to view a representation as a

process or an object, when appropriate, is also important. Lastly, technological tools

allow students to display multiple representations of functions that may facilitate

conceptual understanding of the function concept. With these tools comes an increased

importance on domain and range. There are, however, many questions that remain.

According to Keller and Hirsch (1998), "while the need for students to use and reason on

multiple representations is widely accepted, the research-based knowledge on how to best

accomplish this goal is just beginning to emerge" (p. 1). This research study focuses on

the algebraic and graphical representations of function as well as the domain and range of

functions.









Difficulties and Misconceptions

The concept of function involves many concepts (domain, range, relationship,

unique, etc.) and many abilities (identify, define, translate, model, reify, etc.). According

to Eisenberg (1992), even college students who have taken a number of mathematics

courses do not have much understanding of the concept of function. Research indicates

that a major cause of this lack of understanding is the complexity of the concept (Carlson,

1998; Markovits et al., 1988; Vinner, 1976). "Gaining an understanding of the many

components of the function concept is complex. It requires acquisition of a language for

talking about its many features and the ability to translate that language into several

different representations" (Carlson, 1998, p. 137).

Identify and Define

The complexity of the function concept is aggravated by the differing views of

definitions by the mathematical community (teachers) and people in general (students).

According to Vinner (1976), the structure of mathematics is formalist. Undefined

primitive terms are used to define nonprimitive terms. The sentence that provides the

meaning of nonprimitive terms, such as function, is called a definition. Definitions then

become a major part of the mathematical structure and play a major role in proving

theorems.

While high school and college mathematics teachers have a formalist view of the

structure of mathematics, at least 92% of college students do not (Vinner, 1976). Instead,

students view mathematical definitions as lexical definitions. In a lexical definition, the

meaning of a word is explained by other words. When a concept has several different,

but closely related meanings, people oftentimes assign it the first meaning they

saw/understood (Vinner, 1976). For the vast majority of students, the function definition








can be viewed as a lexical definition. It is composed of other words (domain, range,

rule/correspondence, unique), and oftentimes students think of a function as being an

algebraic expression, a graph, a linear function, etc., depending on how they were first

exposed to the function concept (Tall & Vinner, 1981).

Viewing the definition of function as a lexical definition can lead to a limited

concept image of function. In a study of ninth and tenth graders, Markovits et al. (1988)

identified consistent misconceptions among the students. Markovits et al. (1988) found

the following misconceptions in students' concept images: (a) every function is a linear

function, (b) discontinuous functions (graphical representation) are not viewed as

functions, and (c) piecewise-defined functions (algebraic representations) are not viewed

as functions.

In a study of student understanding of the concept of function among 65 tenth and

eleventh graders in Israeli high schools, Vinner (1983) focused on the ability to identify

and construct functions. Vinner found that students use concept images, not the concept

definition, when faced with a task. The (formal) definition of the concept remains unused

or is forgotten. Teachers assume that students will use the concept definition when faced

with a task, so there is no need for numerous examples, but this is a false assumption

(Vinner, 1983). Vinner found that students tended to view a function as an algebraic

formula or as involving manipulations of doing something with numbers (action/process

view). According to Vinner (1983), activities need to provide students with examples

that help form the desired concept image throughout the entire period of learning, not just

at the beginning (of the chapter). Vinner (1983) further promotes finding interesting

examples in the right context.









Vinner and Dreyfus (1989) studied 271 first-year college students at the

beginning of their calculus course and 36 junior high school teachers in Israel. A 50-item

questionnaire consisting of identification and construction items to assess function

understanding was administered and analyzed. Upon comparing students' images of the

function concept with their (formal) definition, the researchers found that many of the

definitions and images were very primitive among all participants except the mathematics

majors and teachers. They also frequently found discrepancies between the concept

image and definition for participants who gave the Dirichlet-Bourbaki definition. Student

definitions of functions were placed into one of six categories:

1. Correspondence: A function is any correspondence between two sets
that assigns to every element in the first set exactly one element in the
second set (Dirichlet-Bourbaki definition).

2. Dependence Relation: A function is a dependence relation between
two variables (y depends on x).

3. A function is a rule that has some kind of regularity.

4. Operation: A function is an operation or a manipulation (one acts on a
given number, generally using algebraic operations, in order to gets its
image).

5. Formula: A function is a formula (algebraic expression) or equation.

6. Representation: A function is identified with one of its graphical or
symbolic representations. (Vinner & Dreyfus, 1989, pp. 359-360)

In the Vinner and Dreyfus (1989) study, only 27% of the 307 sampled students

gave the Dirichlet-Bourbaki correspondence definition.

Leinhardt et al. (1990) identified function concept difficulties for first-year

(middle school or high school) algebra students. They found that students desire

"regularity." In particular, students (a) only consider "regular" graphs as graphs of

functions, (b) default to properties of linear functions when problem-solving, and (c) have








a tendency to connect points when graphing because it looks better (Leinhardt et al.,

1990).

In a review of student misconceptions concerning the function concept, Tall

(1996) identified the following common misconceptions that students have concerning

the function concept: (a) graphs that look familiar, such as the unit circle, are functions;

(b) a function is a formula (algebraic expression); (c) if y was a function of x, then it must

include x in the formula; (d) the graph of a function must have a recognizable shape (line,

parabola, etc.); and (e) the graph of a function must have certain continuous properties.

In a review of research on the function concept, Harel and Trgalovi (1996)

echoed many of these misconceptions. In particular, Harel and Trgalovai (1996)

identified the following misconceptions among students: (a) a function is a "regular"

graph, (b) the graphical representation must be continuous, (c) a function is a formula

(algebraic expression), (d) a function involves manipulations such as inputting x to get y,

and (e) a function is a formula with x in it. Interestingly, students did not recognize the

algebraic expression y = 4 as a function, but the graphical representation of y =4 was

recognized as a function.

In a study of college algebra students, Adams (1997) also found that students had

difficulty with the Dirichlet-Bourbaki definition of function. Upon analyzing the

definitions of functions provided by students, Adams found that 73% of students who

provided a definition gave an ordered pair representation. The second most common

definition was a graphical representation. The students' concept image of function was

dominated by the vertical line test. Lastly, 80% of students who provided an acceptable









definition of function either did not use or inaccurately used the definition when

responding to other items.

Carlson (1998) studied function concept understanding among college algebra,

second-semester calculus, and first-year mathematics graduate students. She

administered a function assessment and conducted interviews of five students from each

group who had just made a grade of "A" in their mathematics course. She found that

college algebra students had a narrow view of the function concept. In particular, college

algebra students believed all functions can be defined by a single formula (algebraic

expression) and thought functions must be continuous (Carlson, 1998).

A synthesis of research by Schwarz and Hershkowitz (1999) also indicates that

students have a limited view of the function concept as linear. Examples of functions and

their attributes are judged in terms of the linear function and its attributes (the graph is a

straight line, the line is determined by two points, the rate of change is constant, values

can be obtained by interpolation and extrapolation) instead of the mathematical definition

of function.

Domain and Range

Markovits et al. (1988), Sfard (1992), and Adams (1997) found that students have

difficulty with domain and range. In a study of ninth and tenth graders, Markovits et al.

(1988) found that students ignore the domain and range of the function. For example,

when asked to draw the graph ofh(x) = 3 for domain {natural numbers} and range

{natural numbers}, students ignored the domain and range given, drawing the horizontal

line with domain {real numbers} instead. In order to "convince students that the function

is influenced not only by the rule of correspondence but also by the domain" (p. 52),









Markovits et al. (1988) suggested providing examples and exercises for graphing where

the formula stays the same, but the domain and range change.

Markovits et al. (1988) also found that students work with functions better in the

graphical representation than in the algebraic representation (particularly for domain,

range, and rule of correspondence), yet they state the algebraic representation is almost

always taught in the curriculum prior to the graphical representation. They suggested that

more activities should be done in graphical form during the early development of

function concepts.

Adams (1997) found that students have difficulty:

1. Finding the domain and range of functions given both algebraically and

graphically,

2. Identifying functions that satisfy given domain and range restrictions,

3. Choosing appropriate domain and range restrictions with proper scales to

provide complete graphical representations of functions, and

4. Recognizing the effect that a domain restriction and axes scaling has on the

graphical representation of a function.

Adams (1997) promoted activities that emphasize domain and range, the

graphical representation of function, and multiple representations in the algebra

curriculum.

In a study that included four sections of college algebra students, Caldwell (1995)

found that students who had access to graphing utilities were able to graphically find the

domain of functions. These results support Kaput's (1989) view that the graphical








representation of function makes the domain more explicit than the algebraic

representation.

Representations and Translations

Markovits et al. (1988) found that students have difficulty finding x-coordinates

(preimages) and y-coordinates (images) on a given graph. According to Ruthven (1996),

students find the following areas of graphical representation difficult: the idea of ordered

pairs of projections, of graphs as collections of points, and of the interval characteristics

of graphs. Furthermore, the absence of numeric values on the axes displayed on the

screen might further aggravate an area of student difficulty: understanding how the

scaling of axes (domain/range) interacts with the visual appearance of a graph (Ruthven,

1996).

According to Leinhardt et al. (1990), students have difficulty (a) connecting

information from different settings compartmentalizationn), (b) interpreting graphs, (c)

translating between algebraic and graphical representations, and (d) translating from a

table of values to an algebraic expression. In addition, students have a pointwise focus

when working with the graphical representation of function. This pointwise focus causes

students to: (1) not use the pattern of the graph to get an equation, and (2) emphasize

single points at the expense of intervals and slope (Leinhardt et al., 1990).

In a review of research studies, Eisenberg (1992) found that students have a

strong tendency to think of functions algebraically rather than graphically, even if they

are explicitly guided to graphical methods. He states that this tendency is due to the

student belief that mathematical communication means analytic communication, and due

to students' weakness in the skills of graph interpretation and graph creation that are

needed to use graphs in problem-solving. This is due to the fact that students are not









explicitly taught graphical problem-solving skills, but instead are expected to get it on

their own (Eisenberg, 1992).

Carlson (1998) found that college algebra students were unable to interpret

information in a graphical model. For example, the graph of car speed as a function of

time was interpreted as the path of the car. She also found that students had a pointwise

view of functions. While they could interpret points on a graph, they had difficulty

interpreting graphical function information over intervals.

According to Adams (1997), students have difficulty translating from the

algebraic to the graphical representation of functions. Markovits et al. (1988) found that

students have more difficulty translating from graph to algebra than from algebra to

graph. In a study of 284 college preparatory high school students, Knuth (2000) found

that students are routinely given equation-to-graph translations, but they have difficulty

with graph-to-equation translations. In Knuth's study, problems were designed to

encourage a graphical solution, but students overwhelmingly turned to algebraic methods

with apparent lack of awareness of simpler graphical solution techniques. Many students

seemed to think the graph was unnecessary or only used to support algebraic solution

methods.

Eisenberg (1992) found that students have difficulty with multiple representations

of the function concept, particularly when moving from a graphical framework to an

algebraic one. In their review, Harel and Trgalovi (1996) noted that students have

difficulty connecting the different representations of function (graphical, tabular, and

algebraic).








Reify

Students tend to view a function as an action, process, or procedure (Carlson,

1998; Markovits, 1988; Sfard, 1992; Slavit, 1994; Vinner, 1983; Vinner & Dreyfus,

1989). Reification, or beginning to view a function as an object, is very difficult for

students. According to Sfard (1992), the vast majority of students view a function as a

computational process even after instruction centered on the object view of function. In a

research study among college students, O'Callaghan (1998) concluded "the general

indications here were that this level of abstraction was beyond the reach of both [control

and treatment] groups" (p. 36) Similarly, Hollar and Norwood (1999) found that students

in both the traditional and graphing calculator groups showed great difficulty on the reify

component of function understanding.

According to Sfard and Linchevski (1994), students confuse the concept of

function with its representations, thereby obtaining a concept image of function as a

graph or an algebraic expression. By settling on a concept image based on

representations, students cannot piece together the view of function as a whole/object

(Sfard & Linchevski, 1994).

According to Sfard (1991), the formation of an object view of function is a

lengthy and difficult process, "because to see something familiar in a totally new way is

never easy to achieve" (p. 30). In a study of college students, Carlson (1998) found that

an individual's view of the function concept "evolves over a period of many years and

requires an effort of sense making to understand and orchestrate individual function

components to work in concert" (p. 115). However, "the rapid pace at which new

information is presented eliminates needed time for reflection and appears to encourage

students to settle for superficial understanding" (p. 140). According to Carlson (1998),








providing students with engaging activities and time for reflection may promote student

understanding of the function concept.

Model

In the Vinner and Dreyfus (1989) study, the majority of students could not

construct a function representation from a verbal description of a function (translate

verbal to algebraic formula or verbal to graph). Furthermore, Carlson (1998) found that

college algebra students could not represent "real world" relationships using algebraic or

graphical function representations.

Summary of Difficulties and Misconceptions

Students have numerous difficulties and misconceptions related to the function

concept. The function concept is a complex concept that requires time to reflect in order

to understand (Carlson, 1998). Research indicates that many students don't know the

formal definition of function, and when they do know the formal definition they tend not

to use it when problem-solving (Adams, 1997; Schwarz & Hershkowitz, 1999; Vinner,

1983; Vinner & Dreyfus, 1989). Common misconceptions as indicated by the review of

the literature include:

1. A function must be linear (Knuth, 2000; Markovits et al., 1988; Schwarz &

Hershkowitz, 1999; Tall & Vinner, 1981).

2. A function must be continuous (Carlson, 1998; Harel & Trgalovi, 1996;

Markovits et al., 1988; Tall, 1996).

3. A function must be an algebraic formula (Carlson, 1998; Eisenberg, 1992;

Sfard, 1992; Vinner, 1983; Vinner & Dreyfus, 1989).

4. A function must involve manipulations, actions, or processes (Carlson, 1998;

Sfard, 1992; Vinner, 1983; Vinner & Dreyfus, 1989).








5. A function must be a formula with x in it (Harel & Trgalovi, 1996; Tall,

1996).

Students have great difficulty with the domain and range components of the

function concept (Adams, 1997; Markovits et al., 1988). Students are better able to

determine the domain when working in the graphical representation than when working

in the algebraic representation (Caldwell, 1995; Kaput, 1989; Markovits et al., 1988). The

use of graphing utilities, therefore, may enhance understanding of the concept of

function.

The ability to reify the function concept is also very difficult. Reification

(reflective abstraction) requires time (Piaget & Garcia, 1989; Sfard, 1991) and may be

promoted by the use of computers (Sfard & Linchevski, 1994).

The availability of technological tools may facilitate student understanding of the

function concept. The next sections include a review of the literature related to the use of

technology to enhance student understanding of the function concept.


Technology Standards

Published in 1995 by the American Mathematical Association of Two-Year

Colleges (AMATYC), Crossroads in Mathematics: Standards for Introductory College

Mathematics before Calculus provides the most detailed principles and standards for

content and pedagogy in college mathematics before calculus. These College Standards

include the use of technology as one of seven basic principles that form the foundations

of the standards. The use of technology is promoted in all three of the categories of

standards: standards for intellectual development, standards for content, and standards for

pedagogy. Standard 1-6 of the intellectual development standards states that "students








will use appropriate technology to enhance their mathematical thinking and

understanding and to solve mathematical problems and judge the reasonableness of their

results" (AMATYC, 1995, p. 11).

According to the College Standards, technology "should be used to enhance the

study of mathematics but should not become the main focus of instruction. The amount

of time students spend learning how to use computers and calculators effectively must be

compatible with the expected gain in learning mathematics" (AMATYC, 1995, p. 12) in

the standard 1-6). Further, Standard 1-6 states that graphing calculators should be "among

the technology staples to be used by students" (p. 12).


Graphing Calculators and the Function Concept

According to Dunham and Dick (1994), the graphing calculator provides a tool

that allows multiple representations, including numeric and graphical, to be a central part

of the mathematics curriculum. In addition, they make mathematical modeling with real

data possible. According to Wilson and Krapfl (1994), graphing calculators

quickly and easily display the graphical representation of a large number of
algebraic expressions;

provide the ability to adjust the scale of axes (graphing window) and to trace
points to analyze function graphs;

provide the capability for students to build conceptual links between the
algebraic, graphic, and tabular representations of functions;

allow students to solve more complex real-world problems as well as
problems that cannot be solve algebraically; and

are portable and relatively inexpensive (in comparison to a computer).

According to Slavit (1994), the key benefits of a graphing calculator include the

production of a graphical representation and the capability to solve problems numerically.








Graphing calculators are also portable, relatively inexpensive, and provide the ability to

analyze a function using multiple representations.

In their review of the literature on the impact of graphing calculators on

understanding the function concept in secondary and early college students, Wilson and

Krapfl (1994) also point out the following potential problems with graphing calculator

use:

The graphing calculator can be confusing for students, even after instruction.

When faced with multiple representations, students sometimes make incorrect
connections and draw incorrect conclusions.

Sometimes graphs are not presented accurately on the calculator.

Over-reliance on the graphing calculator may impede understanding by
shifting authority from the teacher and textbook to the calculator (instead of to
the student).

Furthermore, Wilson and Krapfl (1994) found that most early studies on graphing

calculators compared achievement and/or attitude as measured by calculator and non-

calculator users. They found many claims that graphing calculators can help students

"develop a deeper understanding of and appreciation for functions" (p. 255). For

example, graphing calculators allow students "to view the three most common function

representations (table, graph, and formula) and build conceptual links among these

representations" (p. 254). In 1994, research concerning these claims related to the

function concept was just beginning to emerge. Wilson and Krapfl identified studies by

Ruthven (1990), Dunham (1990), and Rich (1990). In a one-year study in England of 47

secondary pre-calculus students who used graphing calculators and 40 students who did

not, Ruthven (as cited in Wilson & Krapfl, 1994) found that students with access to

graphing calculators make stronger links between graphic and algebraic representations








of functions. When asked to translate a graph to an algebraic expression, the graphing

calculator group outperformed the control group. Ruthven also found that differences in

pretest and posttest between males and females were no longer present on the posttest for

the treatment group.

In a one-year study of two pre-calculus classes who used graphing calculators and

three who did not, Rich (1990) found no significant difference in overall achievement for

calculator and non-calculator groups. In addition, she found that students in the graphing

calculator group (a) better understood the global features of graphs (domain, asymptotic

behavior, end behavior), (b) learned that algebra problems can be solved graphically, and

(c) better understood the connections between the algebraic representation and graphical

representation of functions. Based on their review of the literature, Wilson and Krapfl

(1994) concluded that the impact of graphing calculators on secondary and early college

student understanding of the function concept is not yet certain. Since their review, more

research has been conducted.

Quesada and Maxwell (1994) studied 710 college precalculus students over three

semesters. The experimental group used graphing calculators with a text written for

graphing calculator use. The traditional control group used a "regular" text and scientific

calculators. There is no further description of the "traditional" class, but the experimental

class graphically solved equations and inequalities, worked real-life applications for each

topic, discussed the domain of the problem situation, and found both "exact" and

"approximate" solutions. The approximate solution was not accepted if the exact

solution was requested on an exam. The graphing calculator group scored significantly

higher in all categories of the final exam, including functions, graphs, word problems,








and equations. The control group, however, performed better on the multiple-choice

items.

In a year-long case study at a private high school, Slavit (1994) studied 18 Honors

Algebra students in one class for the entire year and 18 more Honors Algebra students in

another class during the second school semester only. The students were required to

purchase Texas Instruments TI-81 graphing calculators for the class. According the

Slavit (1994), the graphing calculators were "used extensively in the instruction and

course assignments" (p. 8). Data concerning conceptual understanding was gathered for

all students in the class via written assessments. In addition, three students who were

selected by the teacher and the researcher agreed to participate in interviews throughout

the school year.

Early in the year, student's initially viewed a function in terms of specific

procedures that produce an output, typically from an algebraic expression. This seems to

support Sfard's (1991) view of the development of the function concept that a process

view of function is necessary prior to development of an object view of function.

Throughout the year, students defined a function as a relationship between sets, but they

referred to specific outputs (procedural/process terminology) when asked to discuss what

it meant or when faced with a problem-solving situation. "All but one example of

function given by the students throughout the entire year was in symbolic [algebraic]

form, and most were examples of linear expressions" (Slavit, 1994, p. 14). As student's

conception of function grew during the year, Slavit found the following positive effects

of the graphing calculator on student understanding of the function concept: (a) Students

showed a multi-representational concept image when problem solving, with some








students relying heavily on the graphical representation; and (b) the graphical images of

the graphing calculator strengthened the students' object view (Sfard, 1991) of functions.

Specifically, 67% of students used the graphing calculator to find the zeroes off(x) = x2-

4x-32, and 41% used the graphing calculator to first find the zeroes of the function f(x) =

x3+3x2-4x- 12 when asked to factor.

After instruction, the students were better able to translate between graphical,

algebraic, and numeric/tabular representations of functions. Translation strategies were

most advanced (i.e. more likely to include global properties of a graph such as overall

growth instead of just local properties such as individual plotted points) for translations

involving graphs, providing evidence that the graphical representation facilitates an

object view of function.

In addition, Slavit (1994) identified misconceptions that he credited to graphing

calculator use. Students did not view equations that "could not be solved for y" (p. 36) as

functions, nor did they view functions with unusual domain restrictions as functions.

Discontinuous graphs were identified as non-functions because they did not "look" like a

function. Students did not refer to their own stated definition of function as exactly one

output for each input when determining whether these graphs were functions, indicating a

strong graphical concept image. The misconception that some graphs could not be

functions because one could not find an equation that went with the graph was also

present.

Domain and range problems were also evident. When given symbolic functions

to graph whose key features were not included in the standard [- 10,10,- 10,10] graphing

window, most students knew to extend the range of y-coordinates in order to make the








graph continuous (although this may be related to the function as continuous

misconception), but two-thirds of the students did not investigate the behavior of the

graph by expanding the window in terms of the x-coordinates (domain).

Caldwell (1995) studied the effect of graphing calculator use on college algebra

students understanding of function in a two-year college. The study included four

sections with two instructors each teaching one control section and one treatment section.

The control section instructors and students used scientific calculators while the treatment

sections instructors used a Texas Instruments TI-81 overhead display and students were

furnished TI-81 graphing calculators to use during the entire semester. Students

registered for a class at a specific time, then they were randomly placed in a treatment or

control section (the treatment/control pairs met at the same time). The treatment class

was provided with instruction on how to use the graphing calculator. The graphing

calculator was used for performing calculations, graphing functions and relations, solving

equations and inequalities, and solving systems of equations and inequalities. The

control sections covered the same topics, but used graph paper and traditional paper and

pencil techniques. Caldwell post-tested for conceptual understanding (function, domain,

range, symmetry, increasing/decreasing function, inverse, translation, and intercept),

procedural understanding (finding the slope of a line, domain, range, composition,

intervals of increase/decrease, solving equations, matching function and graph), and

attitude toward mathematics. The graphing calculator treatment group scored

significantly higher on the procedural understanding assessment. There was no

significant difference in the conceptual understanding, nor in attitude toward

mathematics.








In addition, Caldwell (1995) made the following observations during the study:

To produce useful graphs of functions, students need algebraic estimation
skills in order to determine a reasonable domain, range, and scale for the axes.

Students in the treatment group were able to graphically find the domain of
(rational) functions.

Many students were frustrated by the hands-on learning activities. Most were
used to the traditional lecture format and found it difficult to form and verify
conjectures.

It was less time-consuming to teach using traditional lectures than with hands-
on learning activities, yet the hands-on activities helped students become self-
directed learners.

Use of the graphing calculator saved time when producing graphs and allowed
for explorations not otherwise possible.

The graphing calculator allowed a needed change in focus from producing
graphs to interpreting graphs.

Caldwell's results call for more research on the impact of graphing calculators on

student understanding of the function concept. His study also stressed the importance of

domain, range, and translations between representations, but these components of the

function concept are imbedded in assessment instruments that include other components.

The effect of the graphing calculator on these components remains largely unknown.

In a study of 92 students enrolled in pilot sections of developmental algebra in

four community colleges, DeMarois (1997) sought to determine the effect of a beginning

algebra course that focuses on functions and integrates technology "as a tool to explore

mathematics" (p. 1). DeMarois does not explain in any detail how the technology was

used, but it appears that graphing calculators were used in the study. All students in the

study were enrolled in the technology sections. All students took written function

surveys at the beginning and end of the course to assess their understanding of the








function concept. In addition, DeMarois conducted interviews of some students. A

framework that categorizes function understanding in terms of breadth and depth was

used. Breadth consists of the various representations, such as verbal, numeric, algebraic,

graphic, etc. Depth refers to the layers of understanding: pre-procedure, procedure,

process, concept, and procept. The procept layer is considered to have the most depth. At

this level of understanding, students can view a function representation (graph, symbol,

table) as either a process or an object, depending on what is required of the problem

situation.

After analyzing both the quantitative function assessments and the qualitative

interviews, DeMarois (1997) concluded:

The function concept is accessible to the developmental algebra student.
When asked "what is a function?" 43% of the students went from a blank or
pre-procedure level to a process level of understanding during the semester.

Function machines serve as a reasonable starting point for introducing the
function concept.

In addition, DeMarois found the following difficulties:

Students remained weak on the graphical representation of functions.

Function notation was interpreted inconsistently, even by the most capable
students.

Constant functions caused confusion and were interpreted inconsistently
across representations.

The requirement for exactly one output given an input was applied
inconsistently.

Connecting representations was difficult.

Use of prototypes was more common with the algebraic representation than
the graphical representation.

DeMarois suggested the following curriculum changes:








The graphical representation of function as well as translating between
representations need more attention in the curriculum.

Function as an object needs to be discussed with students.

A focus on the best uses of each representation needs to be included.

Interferences caused by use of the graphing calculator need to be addressed.

Adams (1997) conducted a study concerning the concept of function among

college algebra students in a community college. There were 26 students in the graphing

calculator group and 39 students in the control group. Both groups used the same text

and followed the same departmental syllabus. The treatment lasted for three weeks,

during which time the students studied: introduction to functions, linear functions,

quadratic functions, algebra of functional equations, parabolas, and applications of

parabolas. The control group did not use graphing calculators. The graphing calculator

group only used the graphing calculator for in-class assignments during the study. They

used the calculators to graph functions and explore problem-situations described in the

text. On posttest, the graphing calculator group had higher achievement regarding the

application of the concepts of domain and range as well as the selection of appropriate

dimensions for viewing and graphing functions (instrument reliability coefficient was

0.84). The graphing calculator was found to have no effect on the concept of function in

terms of identification, construction, and definition.

Slavit (1998) studied the effect of graphing calculators in a college precalculus

classroom in the United States with thirteen students. By the end of the semester, six

students were still in the class. The researcher observed the classroom twice a week and

conducted a series of interviews with three students during the semester. The graphing

calculator was a vital part of course instruction. The instructor taught algebra using








multiple representations and applied graphical and algebraic methods in problem-solving

situations. Despite the focus on graphical representations and problem-solving strategies,

students continued to think in terms of algebraic representations and, when given a

choice, used algebraic methods instead of graphical methods when problem-solving.

According to Slavit (1998), the data provided the following reasons for the strength of

this algebraic concept of function:

1. The effect of past instruction which focused on symbolic [algebraic]
manipulation,

2. The presence of symbolically-based problems in the homework and
tests which were often not directly connected to the graphic-oriented
activities in the classroom, and

3. A general emphasis on procedures over concepts. (p. 370)

Most students came into the class with a strong algebraic view of function. The

heavy emphasis on graphical representations and de-emphasis of algebraic

representations during instruction caused these students to place this new knowledge in

isolation. They did not view the function concept from a multi-representational

perspective, and did not make connections between procedures used to solve tasks in

different representations. They tended to compartmentalize graphical and algebraic

methods, which prevented an object view of function. As a result of the study, Slavit

(1998) recommended that instruction and assessment provide balance between the

representations. Additionally, connections between the representations must be made

beyond the procedure/process level in order to promote an object view of function.

Keller and Hirsch (1998) conducted a study on one university calculus class (n =

39) that required a graphing calculator for all students, and one that did not (n = 40) at the

same university. Both classes met at the same time, and used the same text. The








graphing calculator class was enhanced with graphing calculator activities. Both groups

preferred to use an algebraic representation on tasks that were purely mathematical (non-

contextual). On the contextual problems, both groups of students preferred a tabular

representation at the start of the course, but they preferred a graphical representation at

the end of the course. This compartmentalization of "use a graph for contextual problems

and use an equation for non-contextual problems" was not as strong, however, for the

graphing calculator group, indicating the graphing calculator group was more flexible in

their use of multiple representations.

Beckmann et al. (1999) provide suggestions for assessing student understanding

of functions in a graphing calculator environment. In agreement with the NCTM (1995)

Assessment Standards for School Mathematics, Beckmann et al. (1999) stated that

technology should be an integral part of assessment if it is an integral part of instruction.

Furthermore, Beckmann et al. (1999) state that assessment should include a balance

between (a) items that require calculator use for their solutions, (b) items that can be

solved with or without a graphing calculator, and (c) items that provide no advantage or a

disadvantage for graphing calculator use. In order to achieve this balance, they suggest

(a) requiring students to explain their reasoning, (b) having students analyze graphs and

tables, and (c) using real contexts.

Hollar and Norwood (1999) studied the effects of a Texas Instruments TI-82

graphing calculator approach to teaching intermediate algebra in a university

environment. There were 46 students (two classes) in the treatment group and 44 (two

classes) in the control group. The treatment group used TI-82 graphing calculators in

class, for homework, and on in-class exams. The control group had no known access to








graphing calculators. The treatment group used a textbook that included a balance of

graphing calculator and traditional algebra work with exploration and discovery

examples. The control group text covered the same topics, but emphasized algebraic

manipulations. Neither group had access to calculators for the function test or the

traditional departmental final examination, which was composed mainly of algebraic

manipulations. The function test assessed student understanding of the function concept

in terms of their ability to model, translate, interpret, and reify. Hollar and Norwood

found that (a) the calculator group performed significantly better on all aspects of

understanding the concept of function (model, interpret, translate, reify) than the control

group, (b) there was no significant difference between the graphing approach and

traditional algebra approach groups in their posttest attitude toward mathematics, and (c)

there was no significance difference on the final examination of traditional algebra skills

between the groups.

Although students in the treatment group demonstrated a significantly stronger

ability to reify than the control group, Hollar and Norwood (1999) found that students in

both the traditional and graphing calculator groups showed great difficulty on the reify

component of function understanding. Hollar and Norwood identified the following

benefits of the graphing calculator curriculum: (a) it includes problems related to

modeling real-world applications that would be too time consuming or too complex to do

without a graphing calculator; and (b) students can quickly create the graphical,

algebraic, and tabular representations of function and easily move between these different

representations using the calculator.








Thompson and Senk (2001) studied eight second-year algebra classes at four

different high schools. There were two classes from each high school in the study. One

class at each school used the existing textbook, curriculum, and calculator usage. The

second class used the University of Chicago School Mathematics Project (UCSMP)

curriculum. The UCSMP curriculum includes problem solving, real-life applications,

and continuous review. In addition, graphing calculators are used by all students and are

an integral part of the curriculum. Activities are designed to encourage students to make

connections between algebraic, numeric, and graphic approaches to problem solving.

The existing (control) books did not assume calculators would be used, but contained

some optional activities with scientific calculators. The teachers in all sections but one,

however, did use graphing calculators. As a whole, the UCSMP group outperformed the

control group on multi-step problems, problems involving applications, and problems

involving graphical representations. There was no significant difference between the

groups on items testing algebraic skills. In one school, the control group used scientific

calculators, not graphing calculators. When comparing the treatment (graphing

calculator) and control (no graphing calculator) groups at this particular school, there was

no significant difference between algebraic skill items. However, the UCSMP (graphing

calculator) students performed better on graphical representation and application items.

Because published articles concerning the effect of graphing calculator use on

secondary and college students' understanding of functions are not numerous, one must

turn to dissertation abstracts for more information. While the following dissertations

written since 1990 provide additional insight, the abstracts available do not provide much

detail concerning the studies.









In a study of calculus students, a traditional group was compared to an

experimental group by Hart (1991). The experimental group used graphing calculators

with a curriculum that emphasized algebraic, numeric, and graphic representations. The

experimental group showed greater conceptual understanding of numeric and graphic

representations and exhibited stronger connections among the three representations. In

addition, there was evidence of more compartmentalization among the traditional

students.

In a study of 1000 precalculus students in a graphing calculator environment,

Tuska (1992) analyzed difficulties that students had on a multiple choice midterm. Tuska

found the following misconceptions:

1. The domain of a function cannot skip intervals.

2. The domain is a subset of the range.

3. The graph of a function on a large viewing window is always
enough to determine the end behavior. (p. 2725)

Upon providing intervention using examples and nonexamples, Tuska (1992)

recommends using a larger variety of examples, placing more emphasis on verbal

representations, and emphasizing the power of multiple representations.

Chandler (1992) studied high school precalculus students. Five classes used a

graphing calculator and four classes did not (n = 173). The calculator treatment lasted for

two weeks while both groups studied transformations of trigonometric functions. The

calculator group scored significantly higher on post-test achievement than the control

group. Chandler (1992) concluded there is a positive increase in understanding when

students are able to use a graphing calculator to visualize. Furthermore, students








demonstrated a better understanding of the relationship between the algebraic and

graphical representations of function.

Norris (1994) studied three university precalculus classes which required

graphing calculators and a control group of four classes that did not require graphing

calculators for a total sample of 304 students. Norris assessed algebraic skills, function

concept knowledge, and attitude toward mathematics. He found no significant difference

in algebraic skills between the two groups, indicating that use of graphing calculators

does not negatively impact algebraic skills. The graphing calculator group scored

significantly higher on the posttest of basic function concepts and graphing. In addition

their mean performance improvement from pretest to posttest was also significantly

higher than the control group. There was no significant difference in posttest attitudes

and in mean improvement of attitudes from pretest to posttest.

The review of the literature concerning the effect of the graphing calculator on

conceptual understanding of the function concept reveals that graphing calculator groups

may develop a stronger concept image of function in terms of multiple representations

than do "traditional" groups. In particular, the literature review reveals the following:

1. Treatment groups who used graphing calculators demonstrated a better overall

understanding of function that those who did not use graphing calculators.

Studies by Hollar and Norwood (1999), Norris (1994), and Quesada and

Maxwell (1994) support this conclusion. Nonetheless, a few studies have

found no significant difference or a negative influence (Dunham & Dick,

1994).








2. Use of graphing calculators can effect different aspects of function

understanding in different ways. While Caldwell (1995) found the treatment

group to have better procedural understanding, there was no significant

difference in conceptual understanding. Similarly, the graphing calculator

group in Adams' (1997) study showed a better understanding of domain,

range, and scale, yet there was no significant difference in their ability to

identify, define, and construct functions.

3. Graphing calculators may help students reify to obtain an object conception of

function (Hollar & Norwood, 1999; Slavit, 1994).

4. As a whole, graphing calculator groups develop stronger connections between

the graphical, algebraic, and tabular (numeric) representations of function

(Chandler, 1992; Hart, 1991; Rich, 1990; Ruthven, 1990; Thompson & Senk,

2001).

5. Graphing calculator groups show less evidence of compartmentalization (of

the representations) than do traditional groups (Hart, 1991; Keller & Hirsch,

1998).

6. Graphing calculator groups can better translate from one representation to

another (Norris, 1994; Ruthven, 1990; Slavit, 1994).

7. Graphing calculator groups better understand global features of graphs (Rich,

1990; Slavit, 1994).

8. Graphing calculator use does not harm traditional algebraic skills as

demonstrated by traditional final exams in algebra (Hollar & Norwood, 1999;

Norris, 1994).








9. Graphing calculator use does not affect attitude towards mathematics

(Caldwell, 1995; Hollar & Norwood, 1999; Norris, 1994).


Computers and the Function Concept

The computer-based software Grapher was developed at the University of

California-Berkeley to enhance students' concept image of function (Moschkovich et al.,

1993). Through detailed video-taped studies of how an individual student used the

software and subsequently worked problems, Moschkovich et al. (1993) suggest that by

allowing students the opportunity to actively manipulate graphs using software, the object

view of function can be enhanced. Further, they believe that the process to object

framework proposed by Sfard (1991) is not necessarily a hierarchy that must follow in

the process first, then object second sequence. According to Moschkovich et al. (1993),

the use of technology provides opportunities for facilitating the object perspective of

function that were not previously available to students. Moschkovich et al. (1993) also

stress that action on the screen is not sufficient for learning, but that the student must be

the impetus of that action via manipulations. For complex mathematical concepts,

learning takes time and experience.

Cuoco (1994) found that students using Logo to study the function concept could

think of a function as an object. Li and Tall (1993) and Breidenbach et al. (1992) drew

similar conclusions using structured BASIC and ISETL (Interactive SET Language),

respectively. ISETL allows the name of a function to be used as an input for another

function, thereby enhancing the object view of function.

Olsen (1995) studied the effect of the multi-representational software Function

Explorer on 74 eighth-graders understanding of the concept of function. Function








Explorer is an interactive computer-based learning environment which provides dynamic,

linked representations of functions. The software has three representations: a table,

parallel number lines, and perpendicular number lines. Students input the independent

variable into any of the three representations by using a mouse click for the parallel and

perpendicular number line representations or by keying in the value for the tabular

representation. Upon input, the output value is displayed in all three representations.

The students in the study were pre-tested, then post-tested after six days of solving

worksheet problems using the software. The students showed significant improvement

on pointwise and global interpretation of graphs. In addition, students preferred the

parallel number lines representation. Olsen (1995) found that many students could answer

questions using the software, but could not answer similar questions reading a static

Cartesian graph. He states that students who are not yet able to interpret Cartesian

graphs can still gain function understanding using the software.

Schwarz and Dreyfus (1995) studied the effect of the Triple Representation

Model (TRM) computer software on student understanding of the function concept. The

study lasted for twelve weeks and included three experimental (TRM) and three control

classes in Israel. Schwarz and Dreyfus describe the TRM software as dynamic and

interactive, similar to Function Analyzer and Function Probe software. In particular, the

function representations (graph, table, algebraic expression) are linked together so that

when a student manipulates one representation, it affects the other representations. In

Schwarz and Dreyfus' view, graphs of functions are representatives of the functions.

Based on scaling (domain/range), a variety of representatives can be obtained from a

single algebraic expression. The TRM software served as a source of representatives of








functions, made it easy to move between representatives, and was designed to stress the

parallels between representations (table, graph, algebraic expression). The TRM students

were given open-ended problem-solving activities that required them to construct, use,

and compare different representatives. In addition, they were given activities in which

several representatives were in conflict. Schwarz and Dreyfus (1995) found that the

TRM students were stronger than the control group at (a) recognizing properties of

functions (linear, maximum, etc.) while performing actions on their representatives, and

(b) understanding the conservation of manipulated objects (i.e. properties of functions are

conserved under actions on representatives).

Research by Confrey and Doerr (1996) indicates that the use of multiple

representations of functions (tables, graphs, algebraic expressions) leads to a broader

understanding of functions. By using the software Function Probe in conjunction with

contextual problems (applications), students perform transformations on the various

representations and explore their effect on the other representations. The software is

flexible to allow for tabular and graphical transformations to be accomplished

independently of any algebraic formula. The computer mouse is used to transform

graphs by shifting, stretching, and reflecting. This approach treats the graph as a single

object to be transformed. "The importance of contextual problems for multiple

representations is that students are encouraged to seek out how the actions, operations,

and roles are made visible (more or less) in the different representations" (Confrey &

Doerr, 1996, p. 166).

O'Callaghan (1998) studied the effects of a Computer-Intensive Algebra (CIA)

curriculum and a traditional algebra curriculum on college algebra students'









understanding of the concept of function as well as student attitudes toward mathematics.

The study included one experimental CIA class and two traditional algebra classes. The

traditional algebra classes are not described. According to O'Callaghan, the CIA

curriculum was developed in the 1980s under the direction of James Fey and Kathleen

Heid. The CIA curriculum represents a non-traditional approach to algebra as a

"function-oriented curriculum that is characterized by: (a) a problem-solving approach,

(b) an emphasis on conceptual knowledge, and (c) the extensive use of technology" (p.

21). Students experience activities in the curriculum that require them to solve problems

and to describe their method of solution within concrete (applied) situations. In addition,

they have access to a Computer Algebra System (symbol manipulation tool). The teacher

of the experimental CIA class served as a guide, provided motivation and feedback, and

encouraged student reflection on activities. Functions are described in the CIA class as

relations among variables.

The concept of function was assessed in terms of four components: modeling,

interpreting, translating, and reifying which were based on Kaput's (1989) four sources

of meaning in mathematics. The CIA students demonstrated a significantly better overall

understanding of functions than did the traditional algebra groups. This better

understanding included the individual components of modeling, interpreting, and

translating. There was no significant difference between the groups in their ability to

reify the concept of function. This led O'Callaghan (1998) to conclude "the general

indications here were that this level of abstraction was beyond the reach of both groups"

(p. 36).









CIA students also showed significant improvement in their attitude toward

mathematics, whereas the traditional algebra group did not show significant

improvement. However, there was no significant difference on post-test between the CIA

group and the traditional algebra group in their attitude toward mathematics. Lastly,

results on the common departmental final examination yielded mixed results. When

analyzed using analysis of variance (ANOVA), the CIA students scored significantly

lower on this examination of "operations and procedures emphasized by the traditional

curriculum" (p. 34). On the other hand, when using adjusted post-test means (ANCOVA)

based on the Math ACT scores as a covariate, there was not a significant difference in

these traditional skills.

O'Callaghan (1998) identified four factors that contributed to the better

understanding of the function concept for the CIA group:

1) the early introduction of functions, 2) the definition of function as a relation
among variables, 3) the use of concrete situations to be modeled and explored,
and 4) the expression of functions in different representation systems. (p. 37)

O'Callaghan called for more research on the function concept, particularly the

reify component. According to O'Callaghan, "a more complete and refined

understanding of this and other aspects of the function concept and its acquisition is key

to designing ways to help students develop powerful conceptions about this most

important mathematical entity" (1998, p. 38).

Schwarz and Hershkowitz (1999) studied two groups of students in ninth grade in

Israel who participated in a one-year course on functions. The first class (n = 32) used

multi-representational software or a graphing calculator. Schwarz and Hershkowitz

(1999) also used the term computer tools to describe the treatment, making it difficult to

discern the role of the graphing calculator and computer software. However, through the








discussion the researchers make it apparent that the students used multi-representational

software. The control group (n = 71) consisted of two classes. In the study, an eight-item

questionnaire was administered as a posttest that the researchers divided into the

following three components of function understanding: (a) prototypicality, (b) part-

whole reasoning, and (c) understanding attributes. Items were analyzed quantitatively

and qualitatively.

Quantitative analysis of the results revealed that the technology group scored

significantly better on the prototypicality component. This indicated that the concept

image of the technology group was broader than the traditional (control) group. Through

further qualitative analysis, the researchers found that the traditional group was more

restricted to their view of functions as linear or quadratic prototypes (thinking, for

example, that graphs of functions are only lines or parabolas) than was the technology

group.

In terms of part-whole reasoning, there was no difference between groups when

asked to construct a single graph (whole) from three partial graph representatives (parts)

displayed on different graphing windows, but the technology group was better when

asked to determine whether a given partial graph was part of a given whole graph. An

understanding of domain, range, and scale was instrumental in the successful completion

of the part-whole items. The traditional group tended to view graphical representations

locally (pointwise), while the technology group was more likely to view the

representations globally (in terms of manipulations on the graph, overall shape of the

graph, and intervals of increase and decrease).









They did not find a significant difference between the technology group and

traditional group in terms of the attribute understanding component of function. The

items for this component required students to translate between the following

representations: (a) an algebraic expression to a partial graph, (b) ordered pairs to a

graph, and (c) a graphical representation to an algebraic equation where the graph is a

line with no given domain/range/scale. Through analysis of students' justification of

their answers, Schwarz and Hershkowitz (1999) found that the technology group

provided more justification, more complex justifications, and were better able to analyze

properties of graphs without mapping to possible prototypes. Therefore, their translations

between representations indicates richer concept images of the function concept.

The researchers attribute differences between the groups to the multi-

representational software, the types of activities, and classroom practice. The traditional

group (a) was exposed to a variety of functions beyond linear and quadratic, (b) worked

with functions in parallel in several representations, (c) was exposed to the formal

definition of function early, and (d) periodically worked on explorations as guided by the

text or teacher.

The technology group (a) was encouraged to make their own decisions

concerning selection of the representation desired as well as when and how to link

representations, (b) worked collaboratively in small groups on investigative activities, (c)

wrote group and individual reports to compare and critique solution processes, and (d)

was exposed to a classroom environment in which the teacher was a facilitator and

model.









Schwarz and Hershkowitz (1999) stressed the importance of the multi-

representational software availability of (a) the zoom, scale, and scroll (trace)

manipulations of graphs and numerical data, and (b) the ability to transform algebraic

expressions to manipulative graphs and tables. They credited these features for the

enhanced function understanding of the technology group for the following particular

areas:

invoked more examples and linked them to transformations than did the
traditional group,

recognized partial representations as different "windows" representing the
same function,

more often used global and complex justifications when working with
graphical representations.

Lastly, the classroom environment emphasized reflection and use of multiple

software representations, thereby starting "a process of internalization that enabled

students, using the computer tools, to carry out mental actions abstracted from their

physical actions on functions' representatives" (Schwarz & Hershkowitz, 1999, p. 387).

In a review of research in relation to computer technology and the function

concept, Smith (1997) concluded that the benefits of the computer are not yet convincing.

This review yielded similar, although somewhat more positive, results. Several studies

indicated that the use of computer software results in a better understanding of the

function concept than does traditional instruction (Confrey & Doerr, 1996; Moschkovich

et al., 1993; O'Callaghan, 1998; Olsen, 1995; Schwarz & Dreyfus, 1995; Schwarz &

Hershkowitz, 1999). Computer software groups better demonstrated a global perspective

of graphs of functions (Olsen, 1995; Schwarz & Hershkowitz, 1999). In addition,

computer software facilitated the object view of function in studies conducted by Cuoco









(1994), Li and Tall (1993), and Moschkovich et al. (1993). However, in a study by

O'Callaghan (1998), the computer group did not reify (view a function as an object)

better than the no-technology control group. Similarly, the effect of computer software

use on students' ability to translate representations is not clear. The computer group in

O'Callaghan's (1998) study translated significantly better than the control group, yet

there was no significant difference in the Schwarz and Hershkowitz (1999) study. These

studies indicate that computer software has great potential, but the effect of using

computer software to facilitate understanding of the function concept is still not certain.

A clear trend does, however, emerge concerning the use of computer software in

current research on the function concept. The software used is dynamic, interactive, and

links multiple representations of functions together (Confrey & Doerr, 1996; Olsen,

1995; Schwarz & Dreyfus, 1995). In addition, the software is used in a problem-solving

environment (Confrey & Doerr, 1996; O'Callaghan, 1998; Schwarz & Dreyfus, 1995;

Schwarz & Hershkowitz, 1999) using contextual/applied problems (Confrey & Doerr,

1996; O'Callaghan, 1998).


Comparison of Computers and Graphing Calculators

Porzio (1995) studied college calculus students' ability to use multiple

representations and connect multiple representations when solving calculus problems.

Three calculus classes at the same U. S. university were studied for one quarter. The first

class (n = 40) used a traditional approach to calculus that emphasized algebraic

manipulations to introduce concepts and solve problems. The second class (graphing

calculator) was similar in content but stressed algebraic and graphical representations and

used graphing calculators. The third group (computer) used the electronic calculus