The impact of computing technology on the teaching and learning of mathematics at the secondary level: Implications for Standards 2000
M. Kathleen Heid
The Pennsylvania State University
Prepared for the NCTM Standards 2000 Technology Conference
June 5-6, 1998
ROUGH DRAFT - DO NOT CITE
Technology has been and continues to be one of the most important catalysts for change in todays mathematics education in North America. Students today have unprecedented access to a robust range of information and computing tools. The synergistic combination of computer algebra systems, dynamic geometry tools, interactive video, and the Internet has the potential to alter permanently students personal relationships to mathematics and mathematical thinking. When the 1989 Standards were being written, the future of technology was promising but its effects were unclear. As a consequence, although the leaders of the Standards committee were well aware of the potential impact of technology, 1989 Standards mentioned little about the potential for the mathematics curriculum to capitalize on the technological tools. As the Standards 2000 committee looks to the future, it cannot ignore the role of technology in the future mathematical lives of students.
One of the conditions that is different today than ten years ago is there has been much research on the impact of technology on the teaching and learning of mathematics. This paper will review that research relevant to the secondary mathematics (touching on middle school and focusing on the high school years) with respect to its potential for informing the role of technology in the Standards 2000 recommendations. This paper represents an effort to bring together relevant empirical research. It is organized under content and process categories since those categories seem to represent the ones in the 1989 Standards . The content categories are: Numbers and numerical reasoning, algebra and functions, mathematical modeling, and geometry. The category of probability and statistics is a notable omission, one that reflects what seemed to be a dearth of research in the teaching and learning of probability and statistics in a technological environment. The process categories are: Problem solving, reasoning, communication, connections, and multiple representations. The section on multiple representations is more illustrative than comprehensive.
This review was first undertaken two and one-half years ago when I embarked on the task of synthesizing the published mathematics education literature related to the impact of technology on the teaching and learning of mathematics at the secondary level. For the most part, I searched for published journal articles, and I did not conduct a comprehensive search of dissertations or of conference proceedings even though much of the current research in this area begins with one of these works. At the time of my first search, my review included over 400 articles or book chapters, including a number of theoretical works. In preparation for this paper I updated that search1, adding another 20 journal articles and a comprehensive set of dissertations on computer algebra systems in mathematics education and targeted by comments on the areas of content and process mentioned above.
A. Number and Operations
Calculators and computers have been available to students for nearly twenty years and they havebeen widely available in schools for over a decade. Many of the concerns initially registered about the potential detrimental effects of automated computation have been addressed, some in extensive meta-analyses of existing studies. The research consensus is that the use of calculators does not lead to an atrophy of basic skills (Hembree & Dessart, 1986, 1992). Hembree and Dessart examined studies in which the treatment group used calculators for computation or to help develop concepts and problem-solving strategies and in which the comparison group received instruction on the same mathematical topics but had no access to calculators during class time. The 1992 Hembree and Dessart study extended the meta-analysis reported in the 1986 study (thereby increasing the number of qualifying studies from 79 to 88), and corroborated or strengthened its results. The studies showed no effects from calculator use on tests of conceptual knowledge. There were, however, differences in effect sizes for measures of computational skills and problem solving (See Table I). For treatment groups which used calculators on tests, students of low, average and high abilities increased their performance significantly on measures of problem solving, and students at the low and average levels increased their performance significantly on tests of computational skills. For treatment groups which did not use calculators on tests, a positive effect size was noted for all levels of students on problem-solving measures and for all but fourth grade students on measures of computational skills. Not only does calculator use not harm students computational skills, but studies on the understandings of students who use calculators and computers as computational tools report that technologically equipped students made significant gains (or gains no different from students without technology) on tests of their estimation skills (Damarin, 1988), understanding of number concepts (e.g., factors, prime numbers, fractions) (Henderson et al., 1985), and mental arithmetic and calculation algorithms (Hedron, 1985).
Table I: Calculator effects on student achievement
Use of calculators on test
Student ability level
Average - Grade 4
Average - Grades other than 4
Adapted from Hembree and Dessart, 1992, pp. 24, 26. (+) and (-) denotes respectively higher and lower scores of calculator groups as compared to noncalculator groups.
Some would argue that the advent of widespread calculator use has ushered in an era in which numbers and computations have a reduced importance. Instead, a range of writers argue that calculators require a new knowledge of and give a new importance to numbers and numerical methods. Fey (1994) reminds us that the long list of computational skills that are traditionally part of the school mathematics curriculum are now easily replaceable "by a small number of widely applicable macroprocedures" (p. 25). He has also pointed out that "Numerical computation takes on new importance because of its potential for concretizing mathematical ideas that may have been obscured through abstract symbolic approaches" (Fey, 1989). Increasing access to technology increases the importance of numerical methods and gives students unprecedented access to large data sets. Number sense and knowledge of numbers are newly important in a technological world.
But technological empowerment does not come without a price. Along with the new role for numbers is a set of new skills and understandings that students will need in order to capitalize on technological approaches. Ruthven and Chaplin (1997) describe the United Kingdom Calculator-Aware Number (CAN) curriculum, an elementary school mathematics curriculum in which calculator was always available, traditional column methods of calculation were not taught, classroom activities were practical and investigational, explorations of "how numbers worked" were encouraged, mental calculation was emphasized, and students were encouraged to share their methods. In a microstudy of 56 students, some who were long-term participants in the CAN curriculum and some who were not, the researchers saw a calculator as a "viable alternative to poorly understood and frequently misexecuted written column methods" (p. 118) such as long division. The researchers discuss the cognitive conflicts that occur in students when faced with using a calculator. In contrast to a paper-and-pencil procedure in which students focus on the integer portion and the remainder, the calculator gives them an integer with a decimal part. They concluded: "Our observations demonstrate that the viability of this alternative is not guaranteed by the capacity to key the computation; rather it depends on the availability of appropriate mental schemes. The calculator does not simply curtail a non-calculator process; it changes the representation of division in a quite fundamental way." (p. 118)
B Patterns, Functions, and Algebra
Multi-representational technology has long raised questions about what students ought to learn about algebra and algebraic reasoning and in what order. Graphing tools provide compelling reason to make graphs and functions the organizing feature of school mathematics instruction. The question is what still needs to be taught when accessible technology can perform most of the symbolic manipulation on which students spend a vast majority of their instruction-related time. Courses and topics within courses can be resequenced. Concepts can precede skills. New topics can be taught. The following sections will discuss empirical research that ought to inform those decisions.
1. Effects on development of traditional algebraic symbolic manipulation and word problem skills
School algebra has long been defined as techniques for solving equations, producing equivalent algebraic expressions, and applying these techniques to a fixed set of word problem types. An initial question about the impact of technology might be how it affects the learning of these traditional routines. A variety of studies have looked at the role of computer programs in the enhancement of traditional algebra learning. For example, Nathan and Resnick (1993) describe an algebra tutor, ANIMATE, designed to improve students success on word problems. In a pretest/posttest control group design, ANIMATE training led to significant performance improvements on standard and novel algebra word problems.
In addition to the solution of traditional algebra word problems, central to traditional algebra has been the acquisition and refinement of by-hand symbolic manipulation skills. The effects of the use of a computer algebra system (CAS) on the development of algebraic symbolic manipulation skills has been a hotly contested issue. A collection of studies have been conducted on the relationship, in a CAS environment, between the acquisition of conceptual understanding and the development of related by-hand skills.
The role of technology in current calculus reform efforts varies (Johnson et al., 1996) from use of Mathematica (1990) notebooks (Davis et al., 1994), to incorporation of a programming language designed to parallel students' mathematical thinking (Dubinsky et al., 1995), to student use of CAS as a laboratory tool (Leinbach et al., 1991), to incorporation of symbolic manipulation calculators (Dick & Patton, 1994) and graphing calculators. In almost every case, technology is used to facilitate the focus of the course on the development of concepts and on understanding of the applications of calculus.
When CAS is used as a tool throughout calculus so that the course can focus on concepts and applications, a common result is that computer users do significantly better on conceptual tasks with little or no loss to by-hand symbolic manipulation skills (Heid, 1984, 1988; Palmiter, 1986; Schrock, 1989; Rosenberg, 1989; Park, 1993). When the CAS has been used to supplement calculus instruction as a lab or on homework assignments or in a smaller portion of the course and the research was designed to measure both conceptual understanding and procedural skills, CAS-using groups have done at least as well as their non-technology counterparts. Some of these studies showed higher scores for the CAS-using group on both skills and concepts measures (Campbell, 1994; Cooley, 1995), and others showed no significant differences on either concepts or skills measures (Hawker, 1986; Judson, 1988; Melin-Conejeros, 1992). CAS-oriented curricula may work differently for higher ability students. In one study of third semester honors calculus students, the traditional students outperformed the CAS-using group on conceptual items while neither group developed formalized understanding of underlying calculus concepts (Meel, 1995); in another study, Crocker (1991) noted that middle and low ability Calculus&Mathematica students, not the high ability students, were more likely to experiment and try different approaches.
A vast majority of studies using the CAS in the learning of calculus have shown some benefit for the CAS-using group. However, in a study in which five CAS labs were substituted for five lectures the traditional group outperformed the CAS-using group on the final examination (Padgett, 1994) results which may have been influenced by the short length of the intervention.
More recent studies are revealing qualitative differences between the CAS-using groups (often engaged in reformed calculus curricula) and the traditional groups which may explain the nature of the quantitative differences in studies like this. Parks (1995) noted that the CAS-using groups used a variety of problem-solving strategies. Roddick (1997) noted that Calculus&Mathematica students were more likely to approach problems in subsequent courses from a conceptual point of view whereas traditional students were more likely to approach problems procedurally. In an investigation of the effects of a reformed calculus course which focused on multiple representations, Hart (1991) found that CAS-using students showed greater facility with graphical and numerical representations and with the relationships among graphical, numerical, and symbolic representations, while traditional students showed more evidence of compartmentalizing their mathematical knowledge. Porzio (1994) showed that Calculus&Mathematica students were more facile with different representations than their counterparts who were using, at most, graphing calculators.
It is clear that the role of symbolic manipulation in school mathematics must be rethought as students learn to live and work in a technological world. Usiskins (1995) analysis of the relative roles in the UCSMP curriculum of particular paper-and-pencil and technology-based algorithms provides insight into the intricacies of the issue. As mathematics educators examine this role, they are often struck by ways in which by-hand symbolic manipulation has served them and their students in their respective mathematical careers. A study by Yerushalmy, for example, identified ways in which students built new understandings on old and how, in this case, a student developed his conceptual understanding of asymptotes on a symbolic manipulation, polynomial division. The student performed the symbolic manipulation () by hand, getting , recognized that the graph of f(x) = would get increasingly closer to the graph of f(x) = x - 1 as x got increasingly large. In other words, he reasoned that the graph of the function should have a slant asymptote. It seems that the important issue is that the student thought to perform a polynomial division and reason from it. While he could have used technology like the TI-92 to get an equivalent result (EXPAND () yields ), the question is whether a student who had learned his symbolic manipulation in the context of a CAS would have similarly powerful (or even more powerful) understandings. It seems that technology actually requires that students pay more attention to notation. Nickerson (1995) suggested that notation systems play an even greater role in computer-enhanced instruction because of the almost infinite flexibility of notation systems in a computer medium. As students use an increasing range of sophisticated technologies, their ability to adapt and understand different notations will become more important.
Prior to widespread access to computational tools, one of the most popular sequences within any given mathematical topic was for students to learn the procedures, then learn to apply them or to acquire an understanding of underlying concepts only after students had developed facility with related skills. Algebra students would first spend their algebra time mastering by-hand methods for solving equations and writing equivalent forms for algebraic expressions, and follow this with forays into applications which were intended to give them more practice with algebraic manipulations. Calculus students would focus their attention on the execution of a range of derivative formulas and integration techniques paying only passing attention to the theories that underpinned those routines.
With computers that could perform most of these routines, a concepts-and-applications-before-skills approach to algebra and to calculus was tested in a range of studies. In a study of calculus learning, Heid (1984, 1988) examined the effects of an applied calculus course during the first twelve weeks of which students focused on concepts and applications and used the CAS to execute routine symbolic manipulations. A composite picture gleaned from interviews, examinations, and classroom interactions suggested that students in the CAS group understood mathematical concepts more deeply and more robustly than their counterparts in a traditional applied calculus course. CAS students in this study spent the remaining three weeks on the development of traditional calculus skills and performed almost as well as the students in the traditional applied calculus course on the final exam of traditional calculus skills. Studies by Judson (1990) and Palmiter (1991) corroborated the conclusion that calculus courses that provide students access to computer algebra systems can be designed to focus attention on concepts and applications without concomitant loss of manipulative skills. The studies suggested that symbolic manipulation skills may be learned more quickly after students have developed conceptual understanding through the use of computing tools with the facility of a computer algebra system (Heid, 1988; Heid et al., 1988; Judson, 1990; Palmiter, 1991; Heid, 1992).
Similar results were obtained in introductory algebra courses. Heid and colleagues conducted a series of studies of the mathematical understandings of students who completed Computer-Intensive Algebra (Fey et al., 1995) courses. Computer-Intensive Algebra (CIA) focuses on the development of algebraic concepts like function, families, equivalence, and system, and gave students constant access to computer algebra systems. The CIA curriculum did not directly teach traditional by-hand algebraic manipulation skills. Students in the CIA classes consistently outperformed their counterparts in traditional algebra classes on measures of concepts, applications, and problem solving without significantly diminished skills (Heid, 1992; Heid et al., 1988; Matras, 1988; Oosterum, 1990; Sheets, 1993). These studies showed that beginning high school algebra students who studied the CIA curriculum for all but the last six to eight weeks of the school year, and who studied traditional skills for a maximum of six to eight weeks, had a much deeper conceptual understanding of fundamental algebraic ideas (such as function, variable, and mathematical modeling), and scored as well as their traditional course counterparts on final examinations of by-hand algebraic manipulative skills. O'Callaghan (1994) verified these results in a study of college students enrolled in a beginning algebra course who were using the CIA text.
5. Development of conceptual understanding
Research provides evidence in a variety of arenas that technology can be used to develop conceptual understanding, and as previously discussed, the understanding developed in these studies is typically accompanied by evidence that there was no concomitant detrimental effect on manipulative skills. For example, in a CAS-aided functions course (Hillel et al., 1992), when emphasis was redirected from practice of algebraic manipulation to reading graphs, working between and among representations, and interpreting solutions to equations, students in experimental classes performed better than their counterparts in traditional courses on conceptual questions, and at least as well on technical questions. Not every use of computer algebra systems resulted in advantages for the CAS group, however. Describing a one-week study of the learning of mathematics of 14 to 16-year olds, Thomas and Rickhuss (1992) reported that when the CAS muMath was used to hide the algebraic manipulation and to help instruction to concentrate on mathematical concepts, students did better on one, but not both, concepts, and CAS students took more time to complete the exercises. The short duration of the study may have significantly influenced the results, however, especially in light of the fact that students needed part of that time to become accustomed to this particular use of technology.
6. Development of understandings of specific algebraic concepts
New understandings of variables and functions and tools to support particular concept development
Particular types and uses of computing tools have the potential for enhancing student understanding in particular areas of algebra. The effects of graphing technology on conceptual understandings of graphs and functions are well-documented (Kieran, 1993; Leinhart, Zaslavsky, & Stein, 1990; Dunham, 1991; Dunham & Dick, 1994) and include: higher levels of graphical understanding (Browning, 1989); better work in interpreting graphs (Beckmann, 1989; Dugdale, 1986/87; Oosterum, 1990); better work in relating graphs to their symbolic representations (Dugdale, 1989; Rich, 1990; Ruthven, 1990; Shoaf-Grubbs, 1994); deeper understanding of functions (Beckmann, 1989; Rich, 1990); and better understanding of connections among a variety of representations (Beckmann, 1989; Browning, 1989; Hart, 1992). Yerushalmy (1997) examined videos and written summaries of the work of 30 precalculus high school students in eight 80-minute weekly sessions on horizontal, vertical, and slanted asymptotes. Based on class observations and student work with the following problem (p. 13),
Find a horizontal asymptote to
Find a function whose graph is between f(x) and its horizontal asymptote.
Yerushalmy found that graphing lots of examples with special purpose software helped students when they needed to think about function graphs without software.
With their inherently algebraic structure, it would seem that spreadsheets are a natural computer tool for the learning of algebraic ideas. The effects of spreadsheet use on students understandings of algebra, however, are among the most under-studied. In a study of ten to eleven year old Mexican and British students with no previous formal algebra instruction, Rojano and Sutherland (Rojano, 1996; Sutherland & Rojano, 1993) found that, after twelve hours of hands-on spreadsheet work aimed at enabling students to express the generality of symbolic relationships, students improved in their understanding of function and appeared to move from thinking with the specific to the general. The spreadsheet environment as used in this study seemed to move the students toward a "more algebraic" way of proceeding. Students seemed to develop an awareness of the role played by variation of the unknown and of the interdependent relationship among the unknowns, a more algebraic way of thinking. This link between algebraic thinking and spreadsheets, however, is not automatic. When students who have been exposed to formal algebra encounter spreadsheets, their knowledge does not automatically transfer. Based on research on fifteen-year olds learning to use spreadsheets during eleven two-hour sessions in an elective class, the researchers (Capponi & Balacheff, 1989) observed: "...despite the fact that use of a spreadsheet requires manipulation of formulas, there is not a mere transfer of the pupil's algebraic knowledge into the spreadsheet context" (p. 179).
A number of studies have targeted the use of special software to enhance students understandings of function. In Yerushalmys (1997c) study of students participating in a technology-intensive "Visual Mathematics" program, 12 pairs of algebra students were interviewed twice a year during each of grades 7 to 9. By the end of the problem-solving sessions with two "less able pairs" of students it was clear that they were both able to solve the problem successfully and discuss their solutions in a way which demonstrated a meaningful understanding of the situation. Yerushalmy observed that the use of technology in the teaching and learning of algebra from a technology perspective "requires and allows not just a change in the sequence of learning but rather a deep change in the scope of traditional actions and objects" (p. 3). Expanding the scope of traditional instruction were such actions as sketching from a story by observing rates of change, thinking continuously and directionally, and using graphs of functions as organizing features in conceptual models for equations. Other researchers have used specially designed algebraically oriented tools to teach the concept of function. For example, Dubinsky and his colleagues (Ayers et al., 1988) found that when college calculus students studied composition of functions using UNIX shells and scripts to induce reflective abstraction, they exhibited a better conceptual understanding of function and composition. Confrey built Function Probe to support "a covariation approach to functions, which gives equal weight to the relationships in the columns and the rows" (Noss, Healy, & Hoyles, 1997, p. 206). Enhancing the understanding of function is not the sole domain of technological tools typically associated with algebra and calculus. Hazzan and Goldenberg (1998) studied the work of three undergraduate mathematics majors on three previously prepared dynamic geometry constructions and noted that students recognized several new manifestations of features of function: the nature of the independent variable, constant function, undefined function, continuous function.
Just as technology can be used to enhance students understandings of function, it can also useful in developing students understandings of variable. Sutherland (1991) reports that the connection students make between the variable they use in Logo and the variable they use in an algebra setting depended very much on the nature of their Logo experience. For example, students without prior formal algebra experience who used "unclosed" algebraic expressions in Logo seemed better able to accept these ideas in algebra. Not all uses of technology provide advantages for the learning of variable. McCoy (1990) commented that results of previous studies were mixed as to whether computer programming improves understanding and skill in algebraic variables.
Families of functions
At times, computing software can provide students with representations that clarify their thinking about families of functions. A study conducted by Ruthven (1990) found that students with calculator experience outperform their peers both in recognizing the global shape of a graph, and in picking our local features that could be linked to its algebraic form. OKeefe (1992) observed eight college students using Dynagraph software to explore families of functions. The software featured dynamic parallel number lines which illustrated the mapping of the pre-image of a function to its image. Students frequently saw the domain value of zero as the starting point of the function. Students noted the y-intercepts more readily as they observed the motions of the Dynagraph than when asked to describe the same functions given in algebraic or graphical form. (p. 116) Students using Dynagraph quickly noted fixed points when they occurred, noting their location if the values were integers. They recognized and communicated about the meaning of optimum values for quadratic and sinusoidal functions. On the few functions having asymptotes, students readily identified asymptotic behavior, though describing the asymptote as a point instead of the line. Most students quickly identified critical values using the Dynagraph, while the same students were often unable to do so with symbolic or graphic representations.
C. Role of technology in mathematical modeling
An organizing theme for the 1989 Standards that has increased in importance today is that of helping students to learn to use mathematics to describe and understand the world around them. Because of the facility with which technology can generate and manipulate mathematical models, students can engage in authentic mathematical experiences, leaving the door open to reformulating major parts of the school mathematics experience as mathematical modeling experiences. Arguments for including mathematical modeling and problem solving in mathematics instruction are that it: develops general competencies; develops ability to see and judge uses of mathematics independently; prepares students to apply math to everyday life; provides a real picture of mathematics; helps in learning, keeping, choosing mathematical concept and strategies (Blum & Niss, 1991). Studies are beginning to surface that examine the modeling process and technological approaches to it.
1. Students can learn modeling skills through instruction
Srivastava (1983) examined the teaching of "modelling skills" prior to teaching a unit in Physics that used those skills. The "modelling skills" students retained and transferred to the unit in physics as the result of prior teaching were: 1) identifying the main problem, 2) classifying information into given and to find categories 3) classifying information into relevant and irrelevant categories, 4) expressing the problem in clear and concise terms, 5) generating variables, 6) selecting variables, 7) expressing relationships between the variables, 8) translating from English into mathematics, 9) building a mathematical model, 10) simplifying mathematical sentences, 11) solving linear equations/inequalities algebraically, 12) solving linear equations/inequalities graphically, 13) examining internal validity of the solution, 14) examining external validity of the solution, 15) translating from mathematics into English, and 16) interpreting the solution into the context of the real world situation. This list of modeling skills, however, does not address the intimate relationship between mathematics and context in the modeling process. Modeling skills, include, of course, the ability to interpret models proposed by others. OCallaghan (1998), in a study comparing a traditional and a CIA approach, a computer-intensive approach to introductory algebra, found that CIA students were better in terms of overall understanding of functions and at the components of modeling, interpreting and translating. OCallaghans study underlines the importance of certain approaches that allow students to better understand the modelling process and supports the need to connect mathematics to real world situations that are comprehensible to students if they are to increase their understanding of mathematics. The CIA approach provides a first step for students on the road to seeing mathematical connections like those to which Noss and Hoyles allude: "We see modelling not simply as a matter of decontextualization and translation but of a synergy between mathematical and situational meaning" (Noss & Hoyles, 1996, p. 4).
Tools for modeling
Technology affords students with tools for collecting data and for creating, critiquing and using mathematical models. Because data storage is often conducted with an approximate facsimile of the data, students must develop new understandings of the role of data in their reasoning about mathematical modeling. Yerushalmy (1990) has analyzed the role of information in Supposer-enhanced geometry and has pointed out three areas of potential difficulty or challenge for teachers and students: accuracy, large amounts of data, and representativeness of the data.
Prior to the release of the first NCTM Standards, Zollman pointed out the viability of videodisc technology as a means for quantitative analysis of real-world phenomena because of its capacity for assisting in collecting data, developing models, and comparing the predictions of models to actual experimental results. Most notable among studies which have tested various ways to incorporate applications and modeling into the school mathematics curriculum are studies of the use of technology-based laboratories (Tinker & Thornton, 1992; Nobel & Nemirovsky, 1995; Boyd & Rubin, 1996). Microcomputer-based laboratories (MBLs) and calculator-based laboratories (CBLs) are probes that allow real-time student-directed data acquisition, display, and / or analysis. in ways that "give students unprecedented power to explore, measure, and learn from the material environment" (Tinker & Thornton, 1992, p. 158). Science educators see MBLs as having the potential for "substantially reorganized and revitalized science curricula in which much more science was covered much earlier in ways that increased both content learning and an appreciation of the process of science" (Tinker & Thornton, 1992, p. 158). The reason for the promise of the MBLs is twofold: they relieve students of the tedious tasks of repetitive operations and they allow them access to a larger set of real-time data. Based on experimental classroom trials of the MBL with elementary and junior high school students, these science educators say: "Qualitative observations confirm that by reducing the time and effort required by repetitive operations and calculations, MBL helps students understand the relation between phenomena and their representations" (Tinker & Thornton, 1992, p. 160).
Students encounter new problems and new emphases in problem solving in a technologically enhanced mathematics environments. Blum and Niss (1991) contend that with the use of computers "More complex applied problems with more realistic data become accessible to mathematics instruction at earlier stages and more easily than before" and that "Relief from tedious routine makes it possible to concentrate better on the applicational and problem solving processes ..." (p. 58). Fey points out that computers allow access to more interesting data and that instruction naturally shifts from execution of routine algorithms to planning problem solving approaches and interpreting results.
As mathematical modeling becomes increasingly important in students mathematical worlds, mathematics educators have examined ways in which different technological environments and tools affect students mathematical modeling. The medium in which mathematical modeling occurs makes a difference in the ways students approach modeling and in their success in various parts of the modeling process, and these modeling environments affect students differentially. Adner (1990) found that students in a computer-programming environment were much more precise in the naming and use of variables than students in a paper-and-pencil environment. In contrast to students in the paper-and-pencil environment, the computer programming students did not spontaneously write algebraic models, equations, to solve other problems. Moore (1993) investigated the effects of three different kinds of physical representations on the ways in which students developed their understanding of variable and function. The researcher found that students who used a physical device or a simulation of that device (as opposed to a computer-generated numerical representation of the device) were better able to use the representation to reason about functions and variables. Zbiek (1992, 1998) found that students developed a range of strategies for using computer-based curve fitters to generate mathematical models.
3. Generating and refining a mathematical model
Recent research has targeted characterizing the ways in which students generate and refine mathematical models when technology is available. Zbiek (1992), for example, analyzed the ways in which students with considerable college mathematics backgrounds used computing tools to develop and validate functions as mathematical models. Her exploratory study generated grounded hypotheses about the nature of understanding of prospective secondary mathematics teachers about function, proof, and mathematical modeling. She observed that her subjects had a view of function heavily influenced by the concept of rate of change, that they tended to look for prototypical functions within function families, that they recognized intricate relationships among variables in the real world but oversimplified these relationships when they tried to mathematize the situation. She observed that "The models they do construct are usually developed using either simple mathematical operations and personal experience or numerically correct but situationally irrelevant computer-generated fitted functions" (p. iii). Nemirovsky (1994) described the case of Laura, who made sense of the notion of negative velocity only after experimentation with the MBL. Through her use of the motion detector, Laura refined her concept of velocity by studying the effects of a dynamic physical situation on its symbolic behavior. In further studies of how students refine narratives that describe graphical representation of motion, Nemirovsky (1996) characterized his students processes of generating and refining a mathematical model as having three parts: repairing, playing out, and idealization. Doerr corroborated this notion of the development of a model as cyclical and successive refinement. Doerrs students used various tools, the physical situation, simulation software, and analytic software, to help their conjecturing and validation of the models they were developing. This cycle of refinement for mathematical models was found in the research literature on the learning of geometry as well. In a study of middle school students using a transformation geometry Logo-like computer environment, Edwards (1991) concluded that "in general, the students used the microworld in a process of conceptual debugging in which their initial partially-correct models of the transformations were reconciled by comparing them to the mathematically-correct models instantiated in the microworld" (p. 13). Not all uses of technology, however, are reported to reflect this successive refinement cycle. Hodgson and Harpster (1997) noted that, in the absence of a mathematical understanding of regression, students anthropomorphized technology, shifting to it the responsibility for reflection and discrimination.
Dynamic geometry tools are increasingly present in secondary mathematics classrooms, affording geometry students increased opportunities to create and manipulate representations of geometric figures - representations which retain the characteristics which defined their construction. In addition to the research on learning in the context of the Geometric Supposer, empirical research on the use of dynamic geometry tools like Cabri and The Geometer's Sketchpad is beginning to appear.
Tools like Cabri, the Sketchpad, and the Geometric Supposer suggest a learning environment in which students generate constructions about which they make conjectures. Because of the ease of generating successive cases, these conjectures may lead only to the gathering of additional empirical data and the further testing of specific cases rather than to mathematical justification. The ability to look at a large number of cases quickly is reminiscent of this capability that graphing tools provide in the context of algebra. Yerushalmy (1990) observed: "Using the Supposer, it is immediately possible for the user to explore the validity of discovered relationships in a large number of equivalent cases. It also allows one to riffle through a large number of equivalent exemplars in the hope of inferring some regularity that belongs to the class of images" (p. 26). Using computer tools in geometry settings allows students to generate more information; a consequence is the increased importance of students learning to make effective use of the information they generate. Yerushalmy (1990) has analyzed the role of information in Supposer-enhanced geometry and has pointed out three areas of potential difficulty or challenge for teachers and students: accuracy, large amounts of data, and representativeness of the data. Students need to decide when a piece of data is close enough to provide an instance and when it is sufficiently different to constitute a counterexample, they need to determine how much data to collect, and they need to develop strategies for increasing the representativeness of their data.
Several questions arise in the context of this ability to generate a large number of cases: What is the nature and quality of students conjecturing? To what extent are students satisfied with conjectures which have been supported by specific cases? Does conjecturing lead to deductive proof?
1. Nature and quality of student conjecturing
The nature of students conjecturing was different in the Geometric Supposer classrooms studied by several researchers (Yerushalmy & Chazan, 1990; Yerushalmy et al., 1986). When classes which used the Geometric Supposer were compared with non-Supposer classes on a paper and pencil test to evaluate students' conjecturing ability, Supposer students were more likely to write conjectures based on a change in their view of the diagram. They were able to focus on different parts of the diagram, and they used auxiliary lines to help them in their proofs and to create new conjectures.
Lesters study (1996) of female secondary geometry students focused on the "cognitive effect of The Geometers Sketchpads capability of dynamically manipulating, transforming, recording and upgrading data on the quality of conjectures written after completing investigation of sketches" (p. 32). Experimental group students were superior in their application of conjectures to problem solutions. Frerking (1994) studied high school geometry classes, one experimental class using the Geometric Supposer 14 times throughout the 24 week study, a second experimental class used The Geometers Sketchpad 22 times throughout the duration of the study, and a control class used a traditional approach. Both experimental classes used a conjecturing approach. The researcher concluded that conjecturing in a geometry class "does not improve van Hiele levels, entering geometry knowledge, proof-writing skills, or achievement of geometry objectives" (p. 97). The researchers suggested that students engaged in certain levels of geometry activities "need to be given activities with questions which escort them into writing appropriate conjectures" (p. 98).
In describing a two-year trial of using the Geometric Supposer in high school geometry classes, Yerushalmy and Houde (1986) described their teaching as resembling that of a science class, "where the primary focus is on the scientific process of collecting data, conjecturing, and finding counterexamples or generalizations" (pp. 421-422). The developers of the Supposers and of the other dynamic geometry tools, like the developers of other microworlds, see their software as providing an environment in which it becomes natural for students to be doing mathematics (Laborde & Laborde, 1995; Schwartz, 1987).
2. Drawing versus constructing what do students learn on their own?
Reports of the use of dynamic geometry tools with students include analyses of how dynamic geometry environments foster new status for geometrical objects and new approaches to known problems (Holzl, 1996). Colette Laborde (1993) discusses specific examples of how a tool like Cabri can influence the mathematics encountered and learned. Balacheff and Kaput (1996) have pointed out that dynamic geometry environments can make the distinction between drawings and figures "a visible part of the geometric activity of the learner" (p. 476). Because of the innate mathematical structure of the program, it is enticing to assume that students will notice and use the inherent mathematics. One study (Foletta, 1994), however, found that students using the Geometer's Sketchpad produced pictures instead of the constructions the teacher had intended them to produce since they saw their tasks as producing a printout instead of engaging in the intended mathematics. Pratt and Ainley (1997) reported similar results from their exploratory study of 2 classrooms of younger children (8-9 year olds and 11-12 year olds). They found that children who explored Cabri on their own gained some understanding of the software but the meanings were connected with drawing instead of constructing. It was encouraging to note that the older students understanding of Cabri was more mathematical and focused on constructions.
3. Conjecturing and proof in a dynamic geometry environment
In light of the ease of generation of examples in a dynamic geometry environment, the question of the role of justification and proof in that environment is a natural one. Would students see a need for proof, and if so, what would be the nature of the proof they would seek? Galindo and his colleagues followed ten high school honors geometry students as they used The Geometers Sketchpad in their geometry instruction. The researchers applied Harel and Sowders broad classification of proofs into authoritarian, empirical, and analytic. Reporting on an in-depth analysis of a series of interviews with two of these students, the researchers noted student progress from a preference for a paper-and-pencil authoritarian proof scheme toward a preference for a computer-based empirical inductive proof scheme. Although the students understood analytic proofs and recognized the inadequacy of empirical proofs, they did not always choose to produce an analytic proof. This tendency of students not to seek analytic proof when empirical evidence was readily available but to appreciate the importance of the analytic proof was also observed by Chazan (1993) in his study of 17 high school students using the Geometric Supposer.
Geometry construction tools offer considerable promise for turning mathematics classrooms into the investigatory laboratories envisioned as part of the mathematics reform movement. Initial studies of their use by students support this promise, with students engaging in what appears to be authentic mathematical activity. Future research is needed to address the nature and role of verification and proof in technological environments as well as the transparency of dynamic geometry tools.
A. Problem Solving/Posing
Perhaps it is because of the ways in which technology can promote problem solving that it has taken such a center stage role in thinking about mathematics education reform. Nonetheless, there are fears about the effects of technology that are grounded in experience. Fielker (1987) feared that the calculator would "simply enable pupils to pursue superficial strategies more efficiently ." Roth recognized that technology could detract attention from problem solving when he commented on the use of the Interactive Physics by some high school physics students: "...students initially spent a lot of time related to the computer and software package. Thus, although for competent users (such as the teacher) Interactive Physics was a transparent tool for testing ideas about motion, for the students in this study it often had the aspects of a broken tool (or a tool unready-to-hand) that draws all of the problem-solvers attention onto itself." Schofield (1995) noted a problematic aspect of some intelligent tutors: "[T]he tutor often took over before they [the students] had a chance to fully explore the routes they wanted to follow. Often they were forced to leave the path on which they were working and move to a correct path without being convinced that they had been irretrievably wrong, even when this was the case" (pp. 55-56).
Research studies have concluded, based on their experiments with students and learning, that the use of technology in the teaching of mathematics has the potential for enhancing problem solving in the following ways:
computer programming on problem solving
Liao and Bright's (1989) meta-analysis of studies that assessed the relationship between computer programming and cognitive skills related to problem-solving abilities suggested that computer programming had a slight positive effect on student problem-solving performance. Based on his review, Blume (1984) observed that it was the nature of the activities with programming and not the programming itself that made the difference. McCoy's (1990) review examined the effects of computer programming on different phases of problem solving: general strategy, planning, logical thinking, variables, and debugging. Although results were mixed in the areas of logical thinking and variables, McCoys concluded that the general strategies of mathematical problem solving and computer programming were related, that planning skills could be improved if specifically taught within a programming context, and that experience in programming seemed to have a positive effect on debugging skills.
Not all research on the effects of computer programming on problem solving is as promising. Adner (1990), for example, found that students did not always transfer what they has learned through computer programming to their work with algebra. Blume and Schoen (1988) found that students with one semester of programming instruction used systematic trial more frequently and checked for and corrected more errors but that programmers did not differ from non-programmers in this study in use of planning processes, frequency and effectiveness of use of variables and equations, and number of correct answers.
B. Reasoning and Proof
Many have argued that technology can change the way students think about mathematics - that it can facilitate deeper thinking in students (Balacheff & Kaput, 1996) and that it can develop an "investigative stance to mathematical enquiry" (Hoyles, 1993, p. 3). At times technological approaches enable students to raise their levels of thinking in the absence of technology. We have already discussed at length the effects of a dynamic geometry environment on students proof-constructions. Johnson-Gentile and her colleagues (1994) instructed fifth and sixth grade students on geometric motions using a Logo motions curriculum, and, in so doing, raised the students geometric reasoning to van Hiele level 2. Battista and Borrow (1997) worked with fifth grade students using the Shape Maker environment of the Sketchpad along with reflection on their processes. They noted that students moved from thinking holistically about a figure to thinking about interrelationships between a shapes part, indicative of Van Hieles level 2.
If we are to help students to learn to think more deeply about mathematics, we can capitalize on technology to provide avenues to that deeper thinking. One strategy for thinking about mathematical things is to experiment with them by conducting successive trials, each informed by the last. Technology can be "patient," allowing students to try continually until arriving at a correct answer. Technology can have a good memory for the steps in a routine, making repeated experimentation easier (Eidson & Simmons, 1998). Technology can insist that a student work within established rules. Observing students working on problems designed to engage them in thinking about asymptotes, Yerushalmy (1997) discovered that the definitions that students developed for asymptotes were evolved "from the use of technology by allowing the definition to be displayed as is; then refined." This allowed the teacher to have the students define the concept at hand then work on a problem that helped them to refine the definition. Finally, technology can promote reflection. Noss (1988) suggests that the computer can facilitate reflection: "... the computer does contain the potential for focusing the learner's attention on selected ideas and concepts by providing feedback in an interactive way which is not available with other technology" (p. 254).
1. How technology affects reasoning
Many have argued that the use of technology can empower mathematics students, and a range of recent research has attempted to characterize how that happens. Nemirovsky and Noble (1997) see anthropomorphizing as one of the ways that in which objects and symbols become internalized. This process is one in which the subject refers to qualities of the computer in terms of human senses. "The computer sees ..." or "the computer feels ..." are examples of this anthropomorphizing process. In one instance, Karen (the subject) had to position herself so that she saw the movement of the computer grapher the way the computer "saw" it. She began with this to reason about the slope of objects by coming to terms with the concept of perspective.
A second way in which use of technology affects mathematical reasoning is that it can promote reflection. Noss (1988) explains how interactive feedback, a forte of technology, can foster reflection: "... the computer does contain the potential for focusing the learner's attention on selected ideas and concepts by providing feedback in an interactive way which is not available with other technology" (p. 254). The capability of particular technologies to foster the externalization of representations serves to enhance the role of those technologies in promoting reflection. Several studies that implemented software designed to encourage students to evaluate and reflect on their reasoning show significant performance improvements (Nathan, 1994; Wertheimer, 1990).
A third avenue for mathematical reasoning embedded in technology is that technology can provide an alternative to rote learning and automatic memorization by supporting a guided inquiry learning environment that allows the construction of definitions and algorithms. Technology like that used by Yerushalmy can make tangible "mental" entities, such as equations or expressions, by treating them as visual objects. Yerushalmy (1997) discovered that the definitions that students developed for asymptotes were evolved "from the use of technology by allowing the definition to be displayed as is; then refined."
Finally, computing technologies can afford students the opportunity to reason about and manipulate mathematical ideas and relationships prior to fully comprehending them. Their engagement in this activity helps promote further mathematical learning (Hoyles & Noss, 1992).
Communication has become an inevitable focus of mathematics education reform, and one of current interest to mathematics education research.
Central to discussions of the role of communication in learning is the pedagogy of small group work. Even in early computer-based mathematics learning studies, the importance of group work was noted. Sheets and Heid (1990), for example, reported on the group structure that naturally evolved in technological environments in calculus classes even when the instructor did not formally plan group work. This collaboration developed as a result of the public character of the computer screen and of the need for a joint decision when students share technology. One student explained:
The computer started the working relationship. It's a way to get to know someone-a way of getting annoyed at someone also! The computer is a sort of ... focus-it gets people working together-that's [also] true for other people I've seen (p. 290).
For students in this study, the computer room became a place to meet and talk about mathematics; one student described the "computer room atmosphere" to the instructor:
The computer room atmosphere-where you or some math student is available-or just being able to work with other students-means a lot more ideas get thrown around, and you're exposed to a lot more ideas. ... It's a community. You get this group of people working on a common problem. And you have to share often because there aren't enough computers, which means it's not just the computers that get shared. It often creates discussions over the keyboard. You can get a good working relationship and a sense of shared responsibility (p. 291).
In ninth grade CIA classes, as students worked in pairs they expanded their views of sources of authority to include fellow students as legitimate resources for learning (Sheets & Heid, 1990). The shift in teacher and student roles as a result of collaborative work with the technology was noted in geometry classes using the intelligent tutor, GPTutor (Schofield, 1995), in several studies of introductory algebra CIA classrooms using the computer as a tool for the exploration of real world situations from a functions perspective (Heid et al., 1988, Heid et al., 1990), in a study of classroom teachers who used the computer software for supplementary instruction (Fraser et al., 1988), in a study of teachers using graphing calculators in a calculator-intensive precalculus course (Farrell, 1989), and in a study of general mathematics students in a computer-intensive problem-solving curriculum (Fiber, 1987).
Technology can also be a catalyst for student-teacher and student-student interaction. Roth (1995) described this phenomenon in his report of an action research study which engaged eleventh grade students enrolled in a qualitative physics class which was using Interactive Physics , a computer-based Newtonian microworld. In the course of the semester, although students began by using everyday language and terminology, their conversations (among themselves, with the teacher, and with the computer) eventually tended toward canonical form. The researcher pointed out some of the affordances of microworlds like these in describing the opportunities as: "Settings for teachers to evaluate students understandings about relevant phenomena by engaging them in conversations over and about the computer displays." and "Means to mediate the conversations between student and teachers so that both parties can make sense of the others understanding" (Roth, 1995, pp. 344-345). Keller and Russell noted improvement in their subjects language as they used CAS calculators to explore calculus: "... the TI-92 students language was increasingly mature and fluent as they were asked to communicate mathematics to a greater extent than is usually done in traditional sections." Other researchers noted the decline of students dependency on teachers as their interactions with teachers rose to a more collegiate level (VanStone 1994, Dugdale, 1986; Dugdale, 1987; Hoyles, 1991).
Researchers have suggested other roles for the computer in classroom communication about mathematics. Tall (1990) corroborated Roths conclusions and suggested that technology could play a role in allowing the externalization of internal representations when he observed that his approach to teaching mathematics provided "an open forum in which the conflict can be brought to the fore and discussed dispassionately through shared phenomena on the computer screen, instead of focusing on the personal hidden recesses of the pupil's mind" (p. 60). Technology is not, however, as flexible as the human mind in its ability to manufacture and operate on representations. Hoyles reminded us of this when she observed: "Sometimes software just does not let you model the situation as you see it even if your model is quite legitimate from the point of view of mathematics!" (p. 209).
The essence of school mathematics as envisioned by mathematics teachers and researchers today is a connected curriculum - one in which ideas from one academic year or one class session have an intimate relationship to ideas from the next year or the next class session. There is, however, another type of connection that is of concern to mathematics teachers. That type of connection is one meant to characterize students thinking. To what extent does or can technology help students make connections between a real world phenomenon and its mathematical representations, between a students everyday world and his or her mathematical world, between a students mathematical world and his or her computer world, between what a student sees and what he or she does, and between a students internal and external worlds (assuming these to be meaningful terms).
An interesting venue for looking at connections subjects make between real world phenomena with which they work and their mathematical representations is the world of banking. By teaching their students to construct and use Logo functions, Noss and Hoyles (1996) studied technology-based ways in which they thought they could help banking employees make connections between their everyday work and the mathematical representations which described that work. They explained the nature of the problem:
There is an interesting contradiction here. In order to make predictive calculations, the bank employs theoretical models based on mathematical relationships - typically functions from Rn R. Yet their models were almost entirely hidden from view. Understanding and reshaping them was the preserve of the rocket scientists; the separation between use and understanding was absolute and the models structures were obscured by the data-driven view encouraged by the computer screens. (p. 17)
They found this way of helping students make connections between mathematical and everyday mathematics to be particularly productive.
A technological world take on a meaning of its own, and students do not always see the connection between the mathematics they are doing on the computer and the mathematical ideas their computer work is intended to represent. Sigal (Hativa, 1988) viewed the arithmetic she did in class as substantively different from the arithmetic she did outside of class on the computer. Folettas (1994) geometry students viewed their Sketchpad drawings as pictures instead of as representations of mathematical entities. Boyd and Rubin pointed out the dilemmas their students faced as they worked both in their everyday world and their mathematical world. When representations are working as intended, a representation and its image should work in concert to inform each other. When a students representation and its preimage match in inappropriate ways, the student will carry the inappropriate image until a conflict arises. The student may still make a connection between the representation and the represented but it may be one that carries some inconsistent information (Noss et al., 1997).
Representations have played a central role in the reform of school and early college mathematics. There is little doubt in the teaching community about the power of different representations, about the ways in which visual or graphical representations enhance understanding, or about the correlation between deep conceptual understanding and the ability to use one representation to inform another one. Mathematics educators sing the praises of multiply-linked representations as ways to deepen students understandings. Even if there is little doubt (and maybe because there is little doubt), however, it is necessary to examine carefully the evidence we have about how students learn from representations. Researchers who have reviewed learning in technological environments have provided evidence from their own studies or from the studies of others that technology can facilitate students development of connections among the graphical, numeric, and symbolic representations.
Because the literature on learning from multiple representations is potentially so broad, venturing into psychology and instructional systems as well as mathematics and science education, I will not attempt to survey the topic. Instead I will relate illustrative findings from the mathematics education literature. The representations to which mathematics educators most often refer are the visual or graphical, the symbolic, the numeric, and the contextual. Much of the research on the contextual has been discussed earlier in the section on mathematical modeling. I will discuss a few results from the literature on technology and visual representations.
There is something enticing about visual feedback. In discussing their research on cognitive style, Nasser and Aabou-Zour (1997) posit that "One of the most powerful ways that schooling and culture can influence mathematical thinking is by providing tools for solving problems through a transmission of a repertoire of conventionalised concrete and visual representations" (p. 369). Yerushalmy attributes her subjects success in thinking about asymptotes to the fact that their prior use of technology had presented a "large repertoire of visual examples ... (emphasizing) the mental operations of dilation as a graphical operation that preserves important properties and the tendency to think about families of graphs ..." (p. 46). The visual seems to be one of the most salient features of learning to a number of mathematics education researchers whose work is based in technology. In his study of 30 ninth grade Algebra students working in a Logo environment, Olive (1991) concluded that the microworld "appears to reinforce visual level thinking by its ability to provide immediate visual feedback to the user. The user can become a successful problem solver in this microworld at the visual level." (p. 109).
Visual feedback is a theme that permeates the literature. Edwards (1992) reported that students interviewed after playing the game Green Globs used the visual feedback to improve their "shots." Noss and his colleagues (1997) refer to the interplay between an action and its visual outcome as supporting the development of new meaning for the student in their study. Yerushalmy posits that technology makes tangible "mental" entities, such as equations or expressions, by treating them as visual objects. Noss and his colleagues (1997) observed that for the subjects in their study: "Computer screen representations of visual objects and even visualised relationships can be acted upon directly, and we can observe the ensuing changes in the existing relationships" (p. 297). They warn, however, not to be too complacent about what we know about learning from diagrams: "when diagrams are made a requirement in a mathematical investigation, they simply take on a ritual character, becoming more appendages to problem solution rather than a part of its process" (p. 205).
Notes1 I would like to thank the CAS-IM graduate students at Penn State and at the University of Iowa for their work in locating these sources. My special thanks go to my CAS-IM co-PI, Rose Mary Zbiek, and to graduate research assistants Linda Iseri, Cynthia Piez, and Walter Deckert for their support in this process.
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