Chapter 1: Statement of the Problem

I first encountered calculus in a summer course at a community college in 1978. I remember being totally baffled by a graph of the function y = e x, with various lines superimposed on it and some accompanying text about how this function was its own derivative. I read it several times, but couldn’t make heads or tails of it. I didn’t know what a derivative was, what was special about this function, or why the textbook was treating it like a big deal. Finally, I went on and tried to work some problems, thinking that this might shed some light on the whole matter. In an unexpected way, it did. By working the problems, I learned that I didn’t really need to understand the graph or any of the accompanying text. I simply needed to be able to follow a few rules: the derivative of a xn is na xn-1, the derivative of sin x is cos x, the derivative of cos x is -sin x, and, of course, the derivative of e x is e x. Then there was a chain rule and a product rule, and a whole bunch of word problems that all looked exactly like examples in the book. The mysterious graph didn’t seem to relate at all.

I did prefer to understand math, when that was feasible. I had understood the permutation and combination problems about the different colors of marbles in the bag and the girl with 5 blouses and 4 skirts, even though no one I knew wore skirts and it seemed odd that the girl wouldn’t care what combinations looked good together. I had understood the process of finding solutions to systems of equations, and how those equations and their solutions related to semi-practical matters such as how long is would take two men to paint a fence. I had even understood geometry, where we constructed painstaking proofs of things that seemed perfectly obvious. This derivative thing made no sense at all, so that was a bit frustrating, but at least it wasn’t too hard. The rules were simple enough, and I could follow them quickly, accurately, and reliably, so I got an A in the course. I was “successful” in calculus. It was years later before I developed any concept of the derivative as a rate of change or as a slope.

My experience in calculus reflects a common problem in mathematics education. Students fail to grasp fundamental concepts, and so are left trying to reproduce the forms of mathematics without having any idea of the substance behind them. Not everyone reproduces the forms as well as I did, and many of these students “fail” their calculus course. It might be more accurate to say that the calculus course fails them. When even the “better” students can get through the course “successfully” without grasping the central concept, it is clear that something is seriously wrong.

The growing awareness in the mathematical and mathematics education communities that calculus courses were failing in this respect led to the calculus reform movement1, which began in 1986, about eight years after I first took calculus. The major goal of this movement was to help a larger number of students (ideally, all students who choose to take calculus) to understand the concepts of calculus in such a way that they could use them effectively to address real problems. This is in agreement with the more recent policy statements of several influential organizations, including Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, 1989), Crossroads in Mathematics: Standards of Introductory College Mathematics Before Calculus (American Mathematical Association of Two-Year Colleges, 1995), and Everybody Counts: A Report to the Nation on the Future of Mathematics Education (National Research Council, 1989), all of which emphasize the need to make mathematical power available to all students. This power comes not only through manipulative skills and “successful” completion of courses, but, more importantly, through understanding mathematical concepts and being able to apply them to real problems. The present study participates in the reform movement by attempting to identify an effective way to help all students understand the concept of derivative.

Many calculus courses and textbooks, including the one I used as a student in 1978 and several texts from which I taught calculus in the late 1980s and early 1990s, introduce the derivative by giving graphs of curves and talking about their slopes, or, more precisely, the slopes of lines tangent to the curves at various points. The curves often represent physical motion, such as a car driving along a highway or a bicyclist riding along a path. This seems like a natural approach, one that should get the concept across, and yet many students are left perplexed, wondering what it all means. When the unfamiliar idea of derivative is introduced by means of the supposedly familiar graphs of motion, there are two immediate sources of confusion: 1) the students don’t have a solid understanding of how various features of the graphs represent various aspects of the motion; and 2) the students’ conceptions of the relationships among distance, velocity, and acceleration are incomplete and often faulty. Attempts to build the concept of derivative on this shaky foundation lead to high failure rates and a general belief that calculus is too difficult for most people to learn.

To avoid the first source of confusion, one might introduce derivatives in some other way, reducing or eliminating the reliance on graphs. Although I don’t know of anyone who has taken this to an extreme, I do know one professor, prominent in the calculus reform movement, who refuses to use tangent lines to introduce the derivative because they are so often misinterpreted. However, this approach seems unsatisfactory for two reasons: 1) graphs, slopes, and tangent lines, when properly understood, can be immensely effective in helping the student to grasp the concept of rate of change; and 2) even if we were to eliminate them entirely from calculus instruction, graphs would remain vital to many other areas of mathematics and science, and students would still need to be able to interpret them.

This first point, that graphs can be used effectively to help students understand the concepts of calculus, has been a recurring theme in the calculus reform movement. One of the major themes of reform courses is the use of multiple representations, including symbolic, numeric, and graphical. In comparison to the more traditional courses, which use symbolic representations almost to the exclusion of all others, reform courses rely much more heavily on graphical representations of functions and problem situations. The increased attention to graphs has been coupled with the use of sophisticated technology, particularly computer algebra systems (CAS). 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. Many reform calculus courses using CAS have stressed graphical 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).

Several comparative studies indicate that a graphical approach is effective. 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. 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, a graph-intensive calculus course based on the computer algebra system Mathematica. Porzio 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.

The second point, that graph interpretation is vital to many other areas of mathematics, science, and general numeracy, has been emphasized in many recent mathematics reform documents. The National Council of Teachers of Mathematics (1989) suggests that “the connections among a problem situation, its model as a function in symbolic form, and the graph of that function” (page 126) should receive increased attention in the curriculum of grades 9 through 12. The Standards also state that the curriculum in in grades 9 through 12 should develop in all students, even those who do not plan to take calculus, the ability to do the following: “use tables and graphs as tools to interpret expressions, equations, and inequalities” (page 150); “represent and analyze relationships using tables, verbal rules, equations, and graphs; translate among tabular, symbolic, and graphical representations of functions; analyze the effects of parameter changes on the graphs of functions” (page 154); “apply general graphing techniques to trigonometric functions” (page 163); “construct and draw inferences from charts, tables, and graphs that summarize data from real-world situations” (page 187); and “determine maximum and minimum points of a graph and interpret the results in problem situations” (page 180). Similarly, the American Mathematical Association of Two-Year Colleges (1995) recommends that college mathematics before calculus should develop in all students the ability to “demonstrate understanding of the concept of function by several means (verbally, numerically, graphically, and symbolically) and incorporate it as a central theme into their use of mathematics” (page 13). In short, graph interpretation skills play an important part in the development of mathematical power, as advocated by both NCTM (page 5) and AMATYC (page 12).

The same arguments apply to the second source of confusion, the use of examples relating to motion with students who lack clear concepts of the relationships among distance, velocity, and acceleration. When motion concepts are understood, motion examples can be very effective in explaining the derivative, and motion concepts are necessary in enough ways outside of calculus that it is useful to build an understanding of motion even for students who never take calculus.

For these reasons, I have chosen to continue using graphs of motion events, and even to increase their use in introducing the derivative, but to do this in a way that addresses the need for greater understanding of motion and of the way graphs are used to represent phenomena. The most effective method of teaching graph interpretation that I was able to find in my study of the literature involves microcomputer-based labs using motion sensors. The literature also suggests that work with motion sensors may help students learn motion concepts.

Motion sensor instruction is only one example of new methods of teaching and learning that have become available with the advent of sophisticated computer technology in the classroom. Several organizations influential in mathematics education reform have advocated the use of technology in the classroom. The first of five standards for pedagogy recommended by the American Association of Two-Year Colleges is “Teaching with Technology: Mathematics faculty will model the use of appropriate technology in the teaching of mathematics so that students can benefit from the opportunities it presents as a medium of instruction” (page 15). Similarly, the National Council of Teachers of Mathematics (1989) recommends that “students should learn to use the computer as a tool for processing information and performing calculations to investigate and solve problems” (page 8). The present study implements these recommendations by providing information on the effects of different ways of using computers in teaching the derivative, so that teachers and others can select methods appropriate to their needs and circumstances.

Motion sensor instruction, although effective in teaching graphing and motion concepts, has a few disadvantages. It requires about \$400 worth of special-purpose hardware for each working group, a site license for the motion sensor software, and a large empty space for each group. The space requirements are particularly inconvenient, because all furniture has to be cleared away. The sensor measures the distance to the nearest object, which is supposed to be the moving student. If a chair or desk is between the student and the sensor, the sensor will detect the furniture instead of the student. I have found that a space at least four feet wide and about ten feet long works well. In large classes, trying to clear this much space for each working group may be inconvenient or impossible. Also, the equipment requirements usually mean that motion sensor work must be done in the classroom or lab. In many school situations, it is not feasible to allow students to take equipment home for homework.

If the success of motion sensor instruction could be duplicated in a more convenient form, without the requirement of special-purpose hardware, without the need for large empty spaces in the classroom or lab, and in a form that could be used by students at home as well as at school, this would provide classroom teachers with an effective tool for teaching important mathematical concepts with minimal expense and inconvenience. However, motion sensor instruction has been shown to be much more effective in teaching graphing concepts than many other forms of instruction. Any potential substitute would have to be examined very carefully to ensure that it is equally effective.

In this study, I examine the research on motion sensor instruction, develop a Java applet to simulate the motion sensor, design an instructional unit to use the sensor or applet to teach students the concept of derivative, use this unit to teach the derivative to students using motion sensors and to other students using the applet, and compare the results of the two forms of instruction. I hope to show that the applet is approximately as effective as the motion sensor, thus providing teachers with a less expensive, more convenient, effective alternative for teaching graphing concepts. It may happen that the applet is not as effective as the motion sensor. In this case, I will attempt to explain why this is so, so that a more successful effort can be made in the future. Either way, this study should reveal something about how students learn graph interpretation concepts.

Research Questions

1. Does the choice of motion sensor vs. Java applet affect what students learn about graph interpretation or motion? If so, what are the differences?
2. Does the choice of motion sensor vs. Java applet affect the attitudes that students develop toward their own abilities, mathematics, calculus, graphs, or use of computers in mathematics? If so, what are the differences?
3. How does the performance of students receiving this instruction compare to that of other students in the same course? Does the instruction affect their progress through the course in any way that might be revealed by the normal exams, quizzes, and other work of the course?
4. Do any of the above effects differ according to the sex of the student?

1 For more on the calculus reform movement, see Murphy (1997)