UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  Vendor Digitized Files  Internet Archive   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
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 singlehandedly. 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 LogicoMathematical 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 GalbraithHaines TechnologyMathematics Interaction Surveys............................. 98 HannafinScott 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 nonfunctions............................. 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 nonfunctions............................. 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 TechnologyMathematics 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 MATHEMATICSCOMPUTING ATTITUDE SCALESCOMPUTER.................. 156 D MATHEMATICSCOMPUTING ATTITUDE SCALESCALCULATOR...........163 E STUDENT QUESTIONNAIRE.................................................................................. 170 F COLLEGE ALGEBRA TOPICAL OUTLINE........................................................... 173 G TEXAS INSTRUMENTS TI83 GRAPHING CALCULATOR FEATURES.......... 174 LIST OF REFERENCES................................................................................................. 176 BIOGRAPHICAL SKETCH ........................................................................................... 186 LIST OF TABLES Table Page 31. Student Body Profile Fall Semester 1998................................................................... 101 32. Frequency and Percentage of Sex by Group .............................................................. 103 33. Frequency and Percentage of Race/Ethnicity by Group............................................. 104 34. Average Age in Years by Group ................................................................................ 104 35. Prior Graphing Calculator Use by Group................................................................... 105 36. College Teaching Experience by Group..................................................................... 106 37. A Tabular Relationship that is not a Function............................................................ 109 38. Administration of Instruments and Treatment............................................................ 116 41. DRIDT Pretest and Posttest Descriptive Statistics..................................................... 117 42. DRIDT Analysis of Covariance ................................................................................. 119 43. Adjusted Posttest Means for the DRIDT Instrument.................................................. 119 44. MR Pretest and Posttest Descriptive Statistics........................................................... 120 45. MR Analysis of Covariance ....................................................................................... 121 46. TechnologyMathematics Interaction Descriptive Statistics...................................... 123 47. Domain/Range Descriptive Statistics......................................................................... 124 48. Algebraic Domain/Range Descriptive Statistics ........................................................ 124 49. Graphical Domain/Range Descriptive Statistics ........................................................ 125 410. Domain/Range Components Analysis of Covariance ............................................... 126 411. Attention to Domain during a Translation Activity................................................... 126 412. Identify / Define Descriptive Statistics...................................................................... 127 413. Categorization of Definitions .................................................................................... 129 414. Reify Descriptive Statistics ....................................................................................... 129 LIST OF FIGURES Figure Page 1. Standard V iew offx +3 .....................................................................................110 2. Standard View of f(x)=(x1)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 computerbased, 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 selfpaced 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 TwoYear Colleges (AMATYC) stresses the importance of the function concept in mathematics education by including a standard focused on functions. The College Standards content standard C4 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 DirichletBourbaki 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 (BarrosNeto, 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 xaxis, of the twodimensional graph reflects the elements of the domain. The vertical axis, called the yaxis, 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 xaxis 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 yaxis elements of the graph. When graphing on paper, the learner performs this function assignment onebyone 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 DirichletBourbaki 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 nonfunction 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 problemsolving abilities as well as teacherstudent 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 problemsolve in one representation, then his/her problemsolving abilities are limited. Thirdly, a student's ability to construct a model refers to his/her ability to represent a reallife 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 nonmathematical 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 problemsolving 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 previouslyentered 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 xintercepts, where the graph of the function crosses the xaxis. 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) = x215x+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 problemsolving (Adams, 1997; Vinner, 1983). Students have trouble finding a mathematical model (representation) of reallife 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, nontechnology 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 textbased 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 textbased, graphing calculator (TGC) curriculum and a computerbased, 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 nonfunctions, (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 nonfunctions, (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 nonfunctions, (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 notechnology group or a computer group with a notechnology group (Dunham & Dick, 1994). The latest research on the function concept (Adams, 1997; Hollar & Norwood, 1999; O'Callaghan, 1998) also uses a technology versus notechnology 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 computerintensivealgebra group with a notechnology group, while the Hollar and Norwood (1999) study compared a graphing calculator group with a notechnology group. In both studies, the technology group demonstrated a better overall conceptual understanding of the function concept than the notechnology group. However, the computer group in the O'Callaghan (1998) study did not outperform the notechnology group on the ability to reify functions, while the graphing calculator group in the Hollar and Norwood (1999) study did outperform the notechnology 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 notechnology 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 alreadyformed 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 DirichletBourbaki 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 of10 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 wellstructured foundation of basic mathematical ideas on which the learner can build" (p. 34). LogicoMathematical Experiences and Reflective Abstraction For Piaget, learning can occur through two types of experiences: physical experiences and logicomathematical experiences. "Physical experience relates to objects, with the acquisition of knowledge by abstraction starting from these objects. Logicomathematical 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 logicomathematical 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 logicomathematical experience (Piaget, 1966). Whether learning through physical or logicomathematical experiences, Piaget viewed learning as beginning with actions carried out on objects. Piaget associated empirical abstraction with physical experiences and reflective abstraction with logicomathematical 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 modemday 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 logicomathematical experiences where the learner engages in reflective abstraction (English & Halford, 1995). This research, therefore, will be concerned with learning through logicomathematical experiences. What type of action through logicomathematical 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 paperpencil media is not. According to Kaput (1992), "interactivity of the computer medium strongly distinguishes computers both from static media [paperpencil] 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 DirichletBourbaki 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 problemsolving 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 twodimensional system made up of a horizontal line, called the xaxis, and a vertical line, called the yaxis. The xaxis and yaxis are both number lines that contain all real numbers. The xaxis and yaxis 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 modemday 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 modemday 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 velocitytime 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 modemday 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 (15641642). 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 twovolume 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 freehand 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 freehand, 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 modemday 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 "realworld" and in the purely mathematical realm. The function concept was seen in "reallife" applications such as the heat conduction problem and the Vibrating String problem. This focus on reallife 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 welljustified 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 reallife 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 inroads 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 DirichletBourbaki 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 DirichletBourbaki 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 DirichletBourbaki definition. The DirichletBourbaki definition made great inroads in texts that were published after 1959. The DirichletBourbaki definition was included in six of nine elementary algebra texts and four of eight college algebra texts. Today, the DirichletBourbaki definition of function is used in "almost all" algebra textbooks (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 DirichletBourbaki definition of function, delineates the assessment method for student definitions of function on the Domain/Range/Identify/Define/Translate instrument (Appendix A). The DirichletBourbaki 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 DirichletBourbaki 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 (BarrosNeto, 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 reconceive their notion of function. Euler's reconception of function to include a graph drawn by freehand when faced with the Vibrating String Problem provides one such example. While the definition of function evolved to its presentday 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 TwoYear Colleges (AMATYC, 1995) also stresses the importance of the function concept in mathematics education. As the College Standards content standard C4 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 C2 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 DirichletBourbaki definition. Current researchers echo these views. For example, according to Tall (1996) the function concept manifests itself in five representations: 1. visuospatial, 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 DirichletBourbaki definition. While the abstract DirichletBourbaki 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 technologyoriented 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), "singlevalued 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 xaxis" (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 nonmathematical 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 selfpaced 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 tablegraph treatment group scored significantly higher on achievement posttests than the tableonly 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 timeconsuming 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 stepsizes 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 xcoordinate 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 paperandpencil 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 andclick mouse motions, graphical objects can be transformed through reflections and vertical/horizontal shifts. The strength of a computerbased 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 problemsolving is an important aspect of understanding the concept of function. According to Schultz and Waters (2000), in order to increase problemsolving 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 researchbased 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) piecewisedefined 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 firstyear college students at the beginning of their calculus course and 36 junior high school teachers in Israel. A 50item 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 DirichletBourbaki 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 (DirichletBourbaki 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. 359360) In the Vinner and Dreyfus (1989) study, only 27% of the 307 sampled students gave the DirichletBourbaki correspondence definition. Leinhardt et al. (1990) identified function concept difficulties for firstyear (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 problemsolving, 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 DirichletBourbaki 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, secondsemester calculus, and firstyear 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 xcoordinates (preimages) and ycoordinates (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 problemsolving. This is due to the fact that students are not explicitly taught graphical problemsolving 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 equationtograph translations, but they have difficulty with graphtoequation 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 problemsolving (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 TwoYear 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 16 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 16). Further, Standard 16 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 realworld 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. Overreliance 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 oneyear study in England of 47 secondary precalculus 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 oneyear study of two precalculus classes who used graphing calculators and three who did not, Rich (1990) found no significant difference in overall achievement for calculator and noncalculator 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 reallife 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 multiplechoice items. In a yearlong 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 TI81 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 problemsolving 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 multirepresentational 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 4x32, and 41% used the graphing calculator to first find the zeroes of the function f(x) = x3+3x24x 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 nonfunctions 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 ycoordinates in order to make the graph continuous (although this may be related to the function as continuous misconception), but twothirds of the students did not investigate the behavior of the graph by expanding the window in terms of the xcoordinates (domain). Caldwell (1995) studied the effect of graphing calculator use on college algebra students understanding of function in a twoyear 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 TI81 overhead display and students were furnished TI81 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 posttested 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 handson learning activities. Most were used to the traditional lecture format and found it difficult to form and verify conjectures. It was less timeconsuming to teach using traditional lectures than with hands on learning activities, yet the handson 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: preprocedure, 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 preprocedure 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 inclass assignments during the study. They used the calculators to graph functions and explore problemsituations 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 problemsolving situations. Despite the focus on graphical representations and problemsolving strategies, students continued to think in terms of algebraic representations and, when given a choice, used algebraic methods instead of graphical methods when problemsolving. 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 symbolicallybased problems in the homework and tests which were often not directly connected to the graphicoriented 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 deemphasis of algebraic representations during instruction caused these students to place this new knowledge in isolation. They did not view the function concept from a multirepresentational 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 noncontextual 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 TI82 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 TI82 graphing calculators in class, for homework, and on inclass 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 realworld 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 secondyear 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, reallife 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 multistep 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 posttest 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 computerbased software Grapher was developed at the University of CaliforniaBerkeley to enhance students' concept image of function (Moschkovich et al., 1993). Through detailed videotaped 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 multirepresentational software Function Explorer on 74 eighthgraders understanding of the concept of function. Function Explorer is an interactive computerbased 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 pretested, then posttested 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 openended problemsolving 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 ComputerIntensive 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 nontraditional approach to algebra as a "functionoriented curriculum that is characterized by: (a) a problemsolving 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 posttest 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 posttest 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 oneyear course on functions. The first class (n = 32) used multirepresentational 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 multirepresentational software. The control group (n = 71) consisted of two classes. In the study, an eightitem 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 partwhole 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 partwhole 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 notechnology 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 problemsolving 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 