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A history of the calculus reform movement is given, with emphasis on the roles of computers and computer algebra systems (CAS). The implications of CAS for the curriculum are considered. Early advocates of CAS in the calculus classroom stressed the opportunity to focus students' attention on concepts, with the computer taking on the burden of the algebraic manipulations that occupy so much time and attention in a traditional calculus course. Along with this change of emphasis comes the opportunity to alter the traditional sequence of topics in many ways, some of which are discussed here. In addition, CAS can be used to quickly create accurate graphs, allowing a more geometric approach to calculus. Calculus courses using CAS often involve much more student writing than traditional courses. Courses based on CAS generally use a constructivist approach, and frequently incorporate some form of cooperative learning. Evaluating these courses has been difficult because not only the methods, but also the goals, are different from those of a traditional course. Some of the evaluation techniques used by researchers are discussed. Evaluations comparing reformed calculus courses with CAS to traditional calculus courses generally favor the CAS courses, although there is some variation among CAS courses. (***Rewrite the abstract according to APA guidelines.***)
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In the early 1980s, many computer scientists and some mathematicians began to call for replacing calculus with discrete mathematics as the core undergraduate mathematics course (***Get a few references for this.). Although calculus retained its place, the controversy brought attention to the way the standard freshman calculus course had become bloated with topics, obstructing the student's view of the central concepts. This realization spawned the calculus reform movement, which seeks to pare the syllabus down to the essentials and to find creative ways to help students gain a deeper understanding of the ideas of calculus. Some reformers have used computer algebra systems (CAS) in their efforts to achieve these goals.
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Factors Leading to Reform
The early CAS ran on large mainframe computers. The cost of these machines limited access to CAS. In 1979, Soft Warehouse released muMATH, the first computer algebra system for microcomputers. MuMATH, together with the increasing availability of microcomputers, soon made possible the first experiments using CAS in calculus instruction (Heid, 1984; Freese, Lounesto, & Stegenga,1986; Hawker, 1986).
The potential of CAS was just beginning to be felt when the calculus reform movement started in the mid-1980s. Several influential reformers seized on computers and CAS as the vehicles to implement their visions for calculus. This fortuitous timing was not coincidental, for in a sense the computer revolution created the calculus reform movement.
Advances in computer hardware enabled computers to perform an increasing array of tasks, but hardware alone was not enough. Each new advance required new software, thereby increasing the demand for computer scientists. The nature of computers demanded that computer scientists think in discrete, rather than continuous, terms, since computers operate on binary data, not continuous functions. Thus, the growth of computer science created a growing demand for discrete mathematics. By the early 1980's, many computer scientists and some mathematicians were advocating a change from calculus to discrete mathematics as the core undergraduate mathematics course. (**** I need citations here, but don't have them right now.) Even the mathematicians who defended calculus (Lax, 1984; Douglas, 1985a, 1986a) acknowledged the problems in calculus instruction as then practiced.
Market pressures had caused the standard calculus text to expand. Although the inclusion of some arcane topic might sell a text to a professor who specialized in that area, the exclusion of a topic was not likely to sell a text to anyone. One by one, new topics crowded into the standard textbook and gradually gained expanded coverage. Attempts to plow through the resulting "fat" textbook frequently led to breadth at the expense of depth, so that students worked a few each of countless types of problems without understanding the fundamental concepts and interconnections. Against this backdrop, Ronald G. Douglas and Steve Maurer organized a panel discussion, titled "Calculus Instruction, Crucial but Ailing," at the Joint AMS/MAA Anaheim Meeting in January of 1985. Several hundred people attended the discussion.
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The Tulane Conference
Following the panel discussion, Douglas sought funding from the Sloan Foundation for a conference on calculus reform (Douglas, 1985b). The result was the "Conference/Workshop To Develop Curriculum and Teaching Methods for Calculus at the College Level" held at Tulane University in January of 1986.
The Tulane Conference is often credited with being the birthplace of calculus reform. While speakers and writers at the Conference referred to the competition with discrete mathematics, their focus clearly was on improving calculus. They certainly had their work cut out for them: the task of the Conference was nothing less than to chart the course of a complete redesign of the standard university calculus course, both in content and in pedagogy. As the calculus reform movement has grown, the conference report, "Toward a Lean and Lively Calculus" (Douglas, 1986b), has been cited in numerous papers.
The Tulane Conference included workshops on content, methods, and implementation. The Content Workshop created annotated syllabi for a general, two-semester sequence in single-variable calculus. After the conference, some workshop members added two alternative syllabi for the second semester, one of which emphasized CAS. These syllabi were "lean," covering fewer topics than the standard "fat" text, in hopes of deepening student understanding of the core concepts (Tucker, 1986). Members of the Methods Workshop established goals for calculus instruction and recommended techniques for teaching and testing to achieve the goals. They encouraged the use of computers and CAS in calculus instruction (Schoenfeld, 1986). The Implementation Workshop identified activities that would be necessary to create a reformed course and bring it into widespread use (Barrett, 1986).
At the Tulane Conference, Donald Small and John Hosack (1986) presented a paper on using computer algebra systems as tools for calculus reform. They claimed that CAS could be used to: (a) improve conceptual understanding, (b) teach approximation and error bound analysis, (c) improve exercises and test questions, and (d) overcome limitations imposed by poor algebraic skills. In the decade since their paper was presented, teachers and researchers in calculus instruction have given particular attention to their first and last points. Several writers (***reference a bunch of them here, including Paul Zorn, 1986) contend that, by freeing students from the computation and algebraic symbol manipulation that many of them do so unreliably, CAS allows students to explore the big ideas of calculus and to use these ideas to solve more realistic problems.
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Calculus for a New Century
Shortly after the Tulane Conference, the Mathematical Association of America (MAA) appointed a Committee on Calculus Reform to plan the next steps. Conference organizer Douglas chaired the committee. The following year the National Science Foundation proposed to Congress a major curriculum initiative in the reform of calculus. The MAA Committee on Calculus Reform then recommended that the National Academies of Science and Engineering sponsor a national colloquium on calculus reform. This recommendation coincided with the start of a joint project of the Board on Mathematical Sciences and the Mathematical Sciences Education Board. Both are Boards of the National Research Council, which is the principal operating arm of both the National Academy of Science and the National Academy of Engineering. The project, titled Mathematical Sciences in the Year 2000, started with the selection of the Task Force on Calculus, chaired by Douglas. The purpose of the Task Force was to direct the planning of Calculus for a New Century, a colloquium to be held in October of 1987 (Steen, 1987a).
According to an MAA survey (Anderson & Loftsgaarden, 1987), in the academic year 1986-87 only about 3% of US calculus students were required to use a computer for homework assignments. Nonetheless, the impact of computers on the Colloquium was much broader and deeper than that figure would suggest. In addition to the two discussion sessions on CAS (Kenelly & Eslinger, 1988; Zorn &Viktora, 1988), speakers and report writers in other sessions frequently referred to the effect of CAS and related technology on the teaching of calculus, even though computer use was not their main topic. The very existence of CAS, which do reliably what students so often do erratically, affected the entire Colloquium (Steen, 1987b).
Colloquium participants generally agreed on several points: calculus courses should teach students the big ideas of calculus, rather than some arbitrary collection of manipulative skills; the ability to read and write about mathematical ideas is essential; and students need to learn how to use calculus to solve word problems that go beyond the typical "template" problems found in most standard texts. Many writers explicitly connected the increased emphasis on concepts, communication, and applications to the freedom that CAS were expected to provide from tedious manipulations. They gave less attention to the prospect of increased intuitive understanding which the graphical capabilities of computers might help students to develop.
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Priming the Pump
In 1988 the National Science Foundation launched a major initiative to support calculus reform projects. In the first two years, the program made 43 awards, totaling nearly seven million dollars (National Research Council, 1991; Tucker & Leitzel, 1995). Ten of these projects were featured in the 1990 MAA report "Priming the Calculus Pump: Innovations and Resources" (Tucker, 1990), created by the Committee on the Undergraduate Program in Mathematics, Subcommittee on Calculus Reform and the First Two Years.
Of the ten featured projects, computers were essential to seven, and used in an eighth. Another project was based on sophisticated, programmable graphics calculators, which are in some sense small computers. Only one of the ten featured projects did not involve computers or graphics calculators. Another 68 projects were described in abstracts. Nearly all of these projects used computers or graphics calculators, with computers essential to over 75% of the projects.
Five of the ten featured projects were based on CAS. Duke University used MathCAD and Derive (Smith & Moore, 1990); University of Michigan-Dearborn used MicroCalc (Hoft & James, 1990); Purdue University used Maple (Schwingendorf & Dubinsky, 1990); St. Olaf College used SMP (Ostebee & Zorn, 1990); and University of Illinois at Urbana-Champaign used Mathematica (Brown, Porta, & Uhl, 1990b). Although the projects used different software, they had significant features in common. All five project reports mention: (a) the ability to explore more complicated problems because the students no longer have to do all of the manipulations by hand; (b) the use of the graphing capabilities of CAS to give students a geometric view of calculus concepts; (c) students writing about their work; (d) students exploring and creating meaning for themselves; and (e) some form of cooperative learning or group work.
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Resequencing Skills and Concepts
Computer capabilities make possible many sequences of topics that have not previously been used. For example, one computer-based text introduced integration before differentiation, because the computer made Riemann sums so much more convenient and accessible than they were with a pencil-and-paper approach (Stenberg & Walker, 1970). Similarly, CAS provide the opportunity to teach concepts and applications before manipulative skills. Students in the traditional course must differentiate to solve application problems, so they are able to consider most applications only after developing differentiation skills. With CAS to differentiate the functions, students can solve application problems much earlier; some manipulative skills can be taught later, and others not at all. Thus, CAS give authors of courseware new freedom to restructure the course. This raises the question of how best to take advantage of this new flexibility.
Although several early attempts to use CAS in calculus instruction kept the traditional syllabus and simply added computer-based homework and/or demonstrations (Freese, Lounesto, & Stegenga, 1986), it quickly became apparent that there were other options. Some writers insisted that CAS forced a dramatic restructuring of the traditional calculus course, shifting the emphasis from calculation to interpretation (Hodgson, 1987; Brown, Porta, & Uhl, 1990a). While others doubted that restructuring was inevitable, most agreed that CAS did make it possible (Hosack, Lane, & Small, 1985; Zorn, 1986; Boyce, 1987).
M. Kathleen Heid(1984, 1988), at the University of Maryland, was the first to use a computer algebra system to reorganize a calculus course, teaching concepts and applications before manipulative skills. Her 1984 dissertation on this landmark project, which predates the official start of calculus reform, had implications that still have not been fully realized. Heid hypothesized that students could learn the concepts and applications of calculus without first acquiring the traditional skills in computation and manipulation. She believed that students could quickly learn the necessary skills once they had a firm grasp of the concepts.
For the first twelve weeks of her first-semester applied calculus course, Heid used a conceptual approach. Her students answered questions and solved application problems using derivatives, integrals, and graphs produced by muMATH. At the end of this time, Heid's students performed much better than students in the traditional class on measures assessing understanding of the concepts of calculus, such as the meaning of the derivative.
Heid devoted the last three weeks of the semester to developing traditional, manipulative skills, such as use of the chain and product rules and some techniques of integration. Experimental and traditional sections of the same applied calculus course took the same skills-oriented final exam, written by the traditional instructor. There was no significant difference in the students' scores. Heid's students, who had spent most of the term concentrating on the concepts, demonstrated a superior conceptual understanding without sacrificing mastery of the traditional skills.
Ed Dubinsky and Keith Schwingendorf (1991) at Purdue University used an approach similar to Heid's. In the first semester of their experimental course, they spent most of the semester on concepts and applications, devoting only the last two weeks to preparing their students for a departmental final exam on manipulative techniques. Their students averaged 75% on that exam, slightly higher than the department average of 69%. In the second semester, they cut the time devoted to manipulative skills to one week, and obtained results on the final exam that were almost exactly the same as the department average. Like Heid, Dubinsky and Schwingendorf demonstrated that a restructured calculus course using CAS can increase the emphasis on concepts and applications without incurring a loss in manipulative skills.
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A Geometric Approach to Calculus
Since CAS can be used to quickly produce an accurate graph of almost any function, they lend themselves to a graphical or geometric approach to calculus. Charlene Beckmann (1988) evaluated four different first-semester calculus courses at Western Michigan University. She discovered that, while students in courses using a graphical approach and students in the traditional course had similar skill levels, the students using the graphical approaches had greater understanding of calculus concepts and were much more likely to complete the course.
Many courses using CAS have stressed visual and numeric representations in addition to the traditional symbolic representations (Heid, 1984, 1988; Ostebee & Zorn, 1990; Brown, Porta, & Uhl, 1990a, 1990b, 1991a; Graves & Lopez, 1991; Muller, 1991; Small, 1991). Donald Porzio (1994) compared three first-quarter calculus courses at Ohio State University; one was a traditional course, another used graphing calculators, and the third used Calculus&Mathematica. He found that the Calculus&Mathematica students made stronger connections between graphical and symbolic representations than did students in the other two sections. Kyungmee Park (1993) also found that Calculus&Mathematica students showed greater understanding of the relationships between graphical and symbolic representations than did students in the traditional course. Use of multiple representations, particularly when interconnections are formed, has been shown to increase students' understanding (***Get a better reference for this.).
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Writing About Calculus
Many CAS courses are patterned after the laboratory model used in the physical sciences. One feature of this model that has been used in reformed calculus courses is the lab report, in which students write about their work and their conclusions (Smith & Moore, 1990, 1991; Child, 1991; Hoft & James, 1990; Small, 1991). Other reformed calculus courses do not use formal lab reports, but do ask students to explain concepts in their own words (Ostebee & Zorn, 1990; Schwingendorf & Dubinsky, 1990; Brown, Porta, & Uhl, 1991b; Penn & Bailey, 1991). Writing is believed to develop communication skills and deepen understanding of the subject. (***Expand this section. Particularly, cite references about the value of writing across the curriculum.)
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CAS and Constructivism
Because they provide students with the tools to explore mathematical ideas, CAS lend themselves to a constructivist approach. Several developers of calculus courses using CAS have chosen this direction. Some, such as Dubinsky and Schwingendorf (1990, 1991), write explicitly of epistemological constructivism, and refer to Piaget. Others, having studied mathematics rather than educational theory, do not use the terminology of constructivism. Still, the ideas of constructivism are evident in their work. They describe a shift from a teacher-centered classroom, where lecture predominates, to a student-centered laboratory, where students make and test hypotheses and discover mathematical truths for themselves. This constructivist perspective was evident in the report of the Tulane Conference (Davis, 1985), and continued to be prominent in later calculus reform work involving CAS (Smith & Moore, 1990; Child, 1991; Brown, Porta, & Uhl, 1990a, 1990b, 1991a).
Deborah Crocker (1991), studying a calculus course using Mathematica, discovered that students who ranked low or middle in ability at the beginning of the course were more likely to experiment and try varied approaches to problems. In contrast, higher-level students exhibited the most difficulty, often failing to attempt problems or use multiple strategies. This suggests that students who have been successful in situations where passive absorption was the accepted mode of learning are reluctant to try a different approach, while students who have been less successful with previous methods are more willing to try new ones. Since constructivist methods are likely to be new to most students, some attention will have to be given to the problem of how to persuade students who were winning the old game to try playing with the new rules.
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CAS and Cooperative Learning
When CAS were first used in calculus instruction, few students owned computers, so their computer-based homework had to be done on machines provided by their college or university. Usually these machines were located in computer laboratories. Students in the same course, working on the same problems, were now obliged to work in the same room, frequently at the same time. This meant that the change from pencil-and-paper homework to electronic homework was accompanied by a shift from working in isolation to working in community.
Instructors found that conversations between students became a valuable part of the course, whether or not cooperative learning was part of the original course design (Smith & Moore, 1990, 1991; Dubinsky & Schwingendorf, 1991; Beers, 1991; Child, 1991; Small, 1991). Students who worked together in computer labs began to work together on other assignments as well (Hoft, 1991). Describing the calculus course at Rollins College, where Maple is used, Doug Child (1991, p. 94) wrote, "It is important to have an instructor in the lab to facilitate student discussions." Although this is certainly not the purpose of an instructor in a traditional course, it is a key part of some reformed courses.
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The evaluation of reformed calculus courses is challenging. Not only do the methods of reformed calculus differ from those used in standard calculus, the goals have also changed. Standard calculus courses emphasize manipulative techniques, with exams usually calling on students to integrate and differentiate given functions and solve equations. Reformed calculus emphasizes concepts more than manipulations, so the traditional, skills-based exams are not appropriate.
Reformers have had to design new testing strategies to assess students' performance relative to the new goals. This is especially true of the courses based on CAS, since the skills traditionally tested on standard exams may be largely relegated to computers in CAS courses. With two different emphases, and two correspondingly different forms of assessment, the standard and reformed calculus courses are often difficult to compare.
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Evaluating with Skills-Based Tests
Some researchers have used traditional, skills-based exams to compare the performance of students in reformed calculus courses and students in traditional courses. One would expect students in the traditional courses to have an advantage, since the test is designed for their course; the students in the reformed course have devoted most of their time and effort to learning concepts not included on the test. In many cases (Heid, 1984, 1988; Hawker, 1986; Park & Travers, 1996) the two groups have performed about the same on these types of tests, which is a victory of sorts for the reformed courses.
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Designing New Evaluation Instruments and Techniques
Using traditional exams, the best that can be said of reformed calculus is that it is no worse than standard calculus. It may be a great deal better, but this is difficult to measure. New exams are needed, to assess the extent to which students in reformed courses attain the increased conceptual understanding that is the primary goal of calculus reform. Researchers have devised a variety of methods to evaluate whether reformed calculus courses in general, and calculus courses with CAS in particular, have met this goal. These researchers often have been limited by severe constraints placed on use of class time. While there may be more latitude in an experimental course, particularly if the researcher is teaching it, the instructor in a traditional course has a full syllabus to cover and may not be willing to give up class time for someone else's project. (***These last two sentences don't really belong here, but I haven't decided where to put them.)
Heid (1984, 1988), evaluating the muMATH course that she designed and taught, audiotaped nearly all experimental class sessions, observed several meetings of the traditional class, interviewed students from both classes, and made extensive notes on meetings with students outside of class. She also designed some conceptual questions that were used on quizzes and exams given to both classes.
Kyungmee Park (1993, 1996), evaluating Calculus&Mathematica at the University of Illinois, assessed students' conceptual understanding by means of classroom observations, attitude surveys, interviews, pre- and post-tests, and concept maps drawn by the students. She developed an elaborate scoring system for the concept maps. Park was able to give both pre- and post-tests to all students during class, although the lengths of the tests were limited by time constraints. The concept maps were assigned as homework, and completed by about 3/4 of the students.
Jack Bookman and Charles Friedman (1994), evaluating Project CALC at Duke University, used a retention study, a study of students' performance in later calculus-related courses, and an attitude survey and problem solving test given to selected students in both the experimental and control groups during the second semester of the calculus sequence. In addition, matched pairs of students in seven majors were interviewed after completing the sequence.
Porzio (1994, 1995), evaluating Calculus&Mathematica at Ohio State University, used classroom observations, interviews, and written tests to assess students' preferences for numerical, graphical, or analytical representations and their ability to make connections between types of representations when solving problems. For the written tests, he designed two Representations Tests to determine which types of representations the students preferred. A Representations Test on non-calculus concepts was used at the beginning of the course; one on calculus concepts was used at the end.
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The evaluations generally favored courses using CAS over traditional courses. Students taught using CAS were found to complete problems based on manipulative skills at about the same level as other students (Heid, 1984, 1988; Hamm, 1989; Hawker, 1986; Schrock, 1989). On conceptual problems, students from CAS courses performed slightly better than students from traditional courses in some studies (Hawker, 1986) and significantly better in others (Heid, 1984, 1988; Palmiter, 1986; Schrock, 1989). In one study (Bookman & Friedman, 1994), students in a course using MathCAD and Derive were found to be significantly better at problem-solving than students in the traditional course.
In some studies, the students in traditional courses and in courses using CAS were found to have similar attitudes toward mathematics (Hamm, 1989; Hawker, 1986). In others, the students using CAS held more positive attitudes, viewing mathematics as more enjoyable (Judson, 1988) and useful (Bookman & Friedman, 1994), and themselves as more capable (Schrock, 1989; Bookman & Friedman, 1994). One study (Park, 1993) found that students using Calculus&Mathematica developed significantly more positive attitudes toward cooperative learning, while students in the traditional sections showed only slight changes in this area.
Cheryl Hawker (1986) noted that the achievement levels of the two groups in her study were similar, even though the students using muMATH had scored significantly lower than the students in the standard course on an algebra test given at the beginning of the semester. Since poor algebra skills would be expected to lead to lower performance in calculus, the fact that the two groups performed at the same level in the calculus courses could be taken to mean that using muMATH helped the students to compensate for their lack of algebraic skills.
Jeanette Palmiter (1986), studying two courses in integral calculus, found that the students in the course using MACSYMA learned the material much more quickly than the students in the traditional course. She tested the students in the standard course after ten weeks, but she tested the students in the MACSYMA course after only five weeks of study. The students using MACSYMA performed significantly better on conceptual problems, without using a computer. They also performed significantly better on the computational problems, using the computer on that portion of the test. Palmiter noted that, after only five weeks, the students in the MACSYMA course were able to compute integrals that remained far beyond the scope of the techniques used in the traditional course in ten weeks.
Porzio (1994) discovered that students using Mathematica were better able to use different forms of representations (symbolic, numerical, and graphical), particularly where combinations of different representations were involved, and were better able to make connections between representations than either of the other two groups in his study. The other two groups, a traditional calculus class and a class using graphics calculators, were not significantly different from one another in this regard.
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By performing many of the more tedious calculations, CAS can help shift attention from manipulative skills to concepts. As part of this shift in emphasis, the traditional sequence of instruction in concepts and skills can be altered, with beneficial results. The use of CAS in calculus instruction can facilitate a constructivist approach, with students forming, testing, and revising hypotheses, to construct their own understanding of mathematical concepts. Students using CAS can easily generate graphical representations of problems, manipulate algebraic representations, and compute numerical representations. This can help the students to build interconnections between the various forms of representations, thus increasing their understanding. Computer laboratories are natural settings for cooperative learning. Students can benefit from explaining their thought processes to their peers in the laboratory, as well as from more formal explanations written as lab reports.
Although all of this is possible with the use of CAS, the simple introduction of a computer into the classroom does not create instant understanding. In their analysis of Project CALC, Bookman and Friedman (1993) describe the ongoing process of evaluating the project, discovering its flaws, making improvements, and evaluating again. While some results at this point are encouraging, it is clear that the process is not yet complete. As one researcher (Porzio, 1995) wrote, "The tool alone is not enough!" Use of CAS merely creates opportunities. The problem remains for instructors and designers of courseware to realize this potential.
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