Calculus&Mathematica: 
Instructors' Perspectives on Continuing Controversies

Concentrating on the computer-based course, Calculus&Mathematica, the author considers the merits and addresses the concerns raised by critics of technology-based instruction.

by Judy Holdener

In the mid-eighties the National Science Foundation launched a major initiative to improve the quality of calculus instruction at the national level. This initiative resulted in the development and implementation of numerous innovative calculus projects. Many of these projects incorporate technology, and now, over 10 years later, math departments world-wide are using graphing calculators and/or computers in their calculus courses. According to a survey by the MAA [Tucker and Leitzel 1995, pp.29-30], 80% of responding institutions make substantial use of calculators, and over half use computers in some way. Essentially all reform efforts appear to use technology. 
Amidst the increasing use and enthusiasm for computers in the classroom, however, not all mathematicians proclaim the joys of technology in education. As illustrated in a "mini-forum" of four articles in the American Mathematical Monthly discussing calculus reform [Askey 1997; Kaput 1997; Knisely 1997; Ostebee and Zorn 1997], mathematicians do not agree on the merits of technology in undergraduate math education. One article discusses the "modern perspective of calculus" stating:

"Technology should be a tool we welcome with ex- citement—no more rigging problems so the algebra comes out right… . Our debate should be about how we should use technology, not if we should use technology." [Knisley 1997, p. 726]

Another author expresses his concern that technology will do more harm than good:

"Technology has a place in mathematics instruction, but it needs to be used carefully. Until much more is learned about the drawbacks, it should not be pushed too much." [Askey 1997, p. 742]

In this article we address some of the concerns regarding technology-based education. We concentrate on the Mathematica-based course, Calculus&Mathematica, developed by Uhl, Porta, and Davis [Davis, Porta and Uhl 1994]. We will focus primarily on learning issues. Our discussion is based on the compilation of opinions and findings of many experienced (past, as well as current) instructors of Calculus&Mathematica. More specifically, instructors were asked, in light of their experience with Calculus&Mathematica, to address issues such as proof-abuse and students' lack of hand skills. For results of comparative studies done on Calculus&Mathematica, the reader is referred to [Alarcon and Stoudt 1997], [Crocker 1991], [Holdener 1997], [Lefton and Steinbart 1995], and [Porzio 1995].

CALCULUS&MATHEMATICA: BACKGROUND INFORMATION

Calculus&Mathematica is a computer-based course which uses Mathematica's symbolic, graphic, numeric, and word-processing capabilities to illustrate the ideas of calculus. The "textbook" consists of a series of interactive Mathematica notebooks, and students learn calculus by working through tutorial problems, experimenting with "built-in" Mathematica routines, and explaining graphical and computational output. Each lesson follows the same format with four components:

◼ The Basics section introduces the fundamental ideas in the lesson by presenting problems along with their electronic solutions. Presentations are often visual in nature, and students are encouraged to experiment with the problems by changing functions and numbers and rerunning the Mathematica code.

◼ The Tutorial section, like the Basics section, presents problems and solutions in parallel. Tutorial problems concentrate on techniques and applications related to the basic ideas.

◼ The Give it a Try section is the heart of the course providing a list of problems to be solved by the student. Like the Tutorial section, problems are often application-based, incorporating models from economics, physics, chemistry and biology. Information needed to solve the problems are found in the Basics and Tutorial sections.

◼ The Literacy Sheets present problems to be worked out with paper and pencil. This section is typically completed after students have seen the concepts with the computer.

Although format may vary with the instructor, most Calculus&Mathematica classes are conducted in a computer lab. Typically, every third or fourth session meets in a classroom for a "traditional" chalkboard discussion.

Common Concerns

Opponents of technology-based instruction cite a wide range of concerns, but they are generally worried about lowering standards. In the context of Calculus&Mathe- matica, lowered standards are often attributed to one or more of the following three categories: gadgetry over intellect, "proof abuse", and the students' lack of necessary hand skills. We will address these concerns based on the reponses of the Calculus&Mathematica instructors.

Gadgetry over Intellect

Instructors were asked to respond to the following characterization of this concern:

"Computers reinforce the fascination with gadgetry, as opposed to intellect… . Most software is based on immediate gratification and does not encourage sustained mental effort. While physically playing an active role, the pupil is intellectually passive, and has little opportunity to be creative; that is, the pupil is programmed to follow a path already laid out in detail by others." [Koblitz 1996, p. 13]

Response

A majority agreed that "most software" had the downfalls Koblitz states but took exception in the case of Calculus&Mathematica.

"I think I agree with the letter of the statement, but probably not the spirit. The item I want to dissect is most software. Most software is junk. Yawn. But Koblitz does provide nice criteria for good software: it requires sustained intellectual activity and creativity. I am not familiar with all the available software, but Calculua&Mathematica and Project CALC make excellent use of technology. I haven't seen a supplement to a traditional text that makes decent use of technology. In fact, let's use these criteria to evaluate the traditional calculus curriculum taught using a traditional lecture. I don't think it fares particularly well. The students aren't even physically active." 

(David Olson, Michigan Tech.)

Several instructors emphasized the teacher's role in the successful implementation of technology.

"I think this depends on the way that a particular teacher chooses to use the technology. If you make an assignment which requires little more than clicking, cutting, pasting, and printing, I agree with Koblitz. But I don't blame the computer; I blame the teacher. The computer should be used as a tool to help better understand mathematics (and perhaps better understand the computer). It is not a panacea. As always, the amount of student intellectual activity is determined by the teacher's expectations. If you ask the student to do an intellectually demanding project or assignment and then provide guidance with technological as well as mathematical obstacles, then I disagree with Koblitz. Such assignments are hard to craft and can take several iterations. This is the challenge of good teaching, and I think the Calculus&Mathematica materials have a lot of good starting points for precisely these kind of assignments. With proper guidance, I don't believe that using the computer will stifle student creativity. In fact, its novelty can provide a jolt of energy which could even stimulate creativity if properly harnessed. But, of course, it depends on the teacher." 

(Lew Lefton, University of New Orleans)

"Proof-Abuse"

Proof-abuse occurs when heuristic arguments replace precise proofs [Wu 1997]. In his paper titled "The Mathematics Education Reform: What is it and why should you care?", Wu writes:

"The cavalier manner in which the reform texts treat logical argument is nothing short of breath-taking. Heuristic arguments are randomly offered or withheld and, in case of the former, whether these are correct proofs or not is never made clear. Rather than being the underpinning of mathematics, logical deduction is now regarded as at best irrelevant." [Wu 1997, pp.3-4]

Wu continues the debate in the light of technology. In reference to Calculus&Mathematica he writes,

"Problems of a different kind arise when heuristic arguments based on appeals to technology are made with increasing frequency: the computer or calculator begins to assume the role of arbiter of pure reason." [Wu 1996, p. 1533]

Wu then cites several examples to illustrate his point. One of his examples comes from Calculus&Mathematica's coverage of approximations. Wu writes:

"On p. 31 of Approximations of Calculus&Mathematica, the discussion of finding the power series expansion of 1/ (1 - x) goes like this: One uses the computer to print out the first 50 terms, and of course the print-out reads:(∑_(n = 0)^50) (x^n). The comment that follows is (p.32): Ain't no doubt about it. The expansion of f  [x] = 1/ (1 - x) in powers of x is \(TraditionalForm
      \`1 + x + x\^2 + x\^3 + ⋯ + x\^k + 
        ⋯\). By using computer print-outs to replace the elementary derivation of the geometric series, Calculus&Mathematica and other like-minded texts send out the unmistakable message to K-12 that learning about the geometric series is no longer something of consequence." [Wu 1997, pp. 3-4]

Response

Calculus&Mathematica veterans responded similarly to this criticism. Agreeing that Calculus&Mathematica didn't always provide complete proofs to mathematical statements used in the course, they indicated that the omission of complete proofs is sometimes pedagogically sound. Although a few instructors expressed concern regarding the preparation of math majors, most shared the belief that students at the introductory calculus level often gain more from heuristic arguments than from formal proofs.

"For everything there is a season, and a time to every purpose under heaven: a time for proofs, a time for heuristics, a time for noticing patterns, a time for finding exceptions, a time to test hypotheses on the computer, and a time to doubt the output. 

First semester calculus students come to us with little or no understanding of proof. Many do not even know that a few examples do not constitute a proof, let alone understand more subtle errors. We can teach them what it means to prove things, but we have to acknowledge that this will be a difficult and time-consuming task, which will reduce our opportunity to teach other things. We cannot teach all of mathematics in one course or sequence; we have to decide what is worth our time, and let the rest go. Depending on the goals of the course, it may be more important to give students an intuitive understanding of the derivative and the integral of "nice" functions than to develop precise proofs of general cases or to examine pathological counter-examples. Detailed proofs may appeal to mathematicians, but if the student is not functioning at the necessary level, time spent on proof is wasted. If it is truly important that the students understand a particular proof or group of proofs, the instruct or will have to start at the level where the students are, and take them to the level where they can grasp the proof. This may mean spending weeks on basic ideas of proof. 

I once taught a one-week unit on these ideas, emphasizing, among other things, that examples cannot make a proof unless all possible cases are considered. We considered mathematical examples along with practical examples like: "It rained last Monday. It rained last Friday. Does this prove that it rained every day last week? If it did rain every day last week, how could you prove that? How about last year? Could you prove that it always rains on Monday? If so, how? If not, why not?" At the end of the unit I gave a quiz which included several "proofs," all but one of which were invalid. I asked the students to identify the one valid proof, and find the errors in the invalid ones. A substantial portion of the class was not able to find the error in the "proof" which consisted of a short list of examples. If these are the students to whom Wu wants me to present many detailed proofs, then I think he is deluding himself about the value of this experience for the students." 

(Lisa Murphy, University of Illinois)

Regarding Wu's criticism of Calculus&Mathemat- ica's coverage of approximation, Sidney Graham writes,

"The objection to this is logically correct but pedagogically unsound. The only decent learning tool that most students bring to the study of infinite series is a rudimentary ability to recognize patterns. Wu wants to tell students that their intuition about patterns is wrong and thereby further undermine what little confidence they have learning this concept. This is unconscionable, unless he is also prepared to teach the students what they need (e.g. mathematical induction) to make their pattern guesses rigorous. Personally, I think at this level, it's better to help the students improve their pattern recognition, and save the rigor for some other time." 

(Sidney Graham, Michigan Tech.)

Lack of Necessary Hand Skills

Perhaps the greatest area of concern with increasing use of technology is that students will leave calculus courses lacking the hand skills necessary for success in later coursework. Andrews explores the "concerns about the tendency toward inbalance in the current reforms of mathematics instruction" [Andrews 1997]. He poses five questions to the mathematics community, including: "Are standards being maintained?" and "Has the use of calculators impaired basic skills?"

Response

Student attainment of necessary hand skills was also the most cited concern among Calculus&Mathematica instructors. Most respondents indicated that they were careful to ensure that their students learned the basics. 
Some used gateway exams, others collected written assignments regularly, and virtually everyone tested basic skills using written examinations. Here are some responses:

"It is up to me to craft the class so that this (lack of necessary hand skills) does not happen… .I assign hand work and correct it, work on it in lecture, and put students up at the board, etc." 

(Elizabeth Covington, Casper College)

"I felt at the time that the students' hand skills suffered a little; that is why I believe that we need to take the approach that computation must be used wisely. Students can appreciate the idea that some things are not worth turning on the machine for, like solving simple equations, taking non-chain-rule intensive derivatives and integrals, etc. BUT, having taught calculus both ways, I also know that just because students can do chain rule problems quickly and correctly doesn't mean they can "abstract" the chain rule for arbitrary functions, f(g(x)), as needed to understand and do proofs in differential equations. In fact, the students having intuitive understanding of the ideas but slower computational skills handle the abstraction better!" 

(Holly Hirst, Appalacian State University)

"This is a question of some concern. We do not, in fact, know what the role of skill at hand computation is in the natural history of the mature mathematical mind. We do ensure that our students can find derivatives by hand by use of "gateway tests." Otherwise, we will have to wait and see if the Calculus&Mathematica approach leaves things out of balance. If so, then readjustments will be necessary. We do not currently believe that our students leave our (Calculus&Mathema tica) courses at any significant disadvantage. 

(Lou Talman, Metro State College, Denver)

Many instructors felt that the gain in conceptual understanding and the exposure to interesting applications offered a preferable trade-off to the concerns regarding hand skills.

"I believe students are better prepared with Calculus&Mathematica. Their perceived disadvantage pales compared with the fact that they understand what is going on. They will not remember calculus as one more course needed for graduation, but as an interesting scientific experience. The necessary hand skills are practiced and tested in the gateway exams. 

(Juan Manfredi, University of Pittsburgh)

On the other hand, at least one instructor felt that software packages such as Mathematica, can improve students' skills in symbolic manipulation. For example, David Olson noted that students often misinterpret function notation in compositions such as c  o  s (l  n (x)), and apply the product rule instead of the chain rule when differentiating.

"Perhaps students can learn to understand function notation by working with a computer program. For example, Mathematica uses "[ ]" to enclose the argument of a function and "( )" for algebraic grouping. In order to use the program, students must learn to make the distinction between function notation and algebraic grouping.

He goes on to say

"Calculus&Mathematica might do more for a student's symbol skills than one would conclude based on superficial considerations. If students learn some key conceptual ideas, they might develop better symbol skills than the students who spent their entire course working on symbol skills." 

(David Olson, Michigan Tech.)

Finally, many people indicated in their responses that more thought should be given to the issue of symbolic manipulation. For example, how much symbolic manipulation is necessary at the calculus level? How much is necessary to ensure preparation for future coursework?… and future employment? Which hand skills should be included in a calculus course? What should be left out?

"Before one can address the issue of necessary hand skills, it is necessary to define what they are; they will most definitely differ from person to person. C&M (Calculus&Mathematica) students may be at a disadvantage in later courses if the emphasis of these courses is proficiency at hand skills, although I believe they will do fine on everything but the more complex symbolic manipulations that do require lots of practice in order to know the right "trick" to use. I have no problem with my C&M students being at a disadvantage of this type because I do not believe this to be what is important in calculus." 

(Donald Porzio, Northern Illinois University)

"Is an accountant who can't add two columns of numbers at a time, as my father could, considered incompetent these days? Or is it more important that he know how to organize a company's records, identify the most significant factors in its operations and develop accounting PRACTICES for tracking them? Does the student who can do ten pages of pointless "math" (i.e., manipulation of symbols) by "hand" have a greater potential than the student who knows which symbolic calculation is worth doing, and then uses Mathematica to do it? Are "hand" skills more important than "brain" skills?" 

(William MacDonald, University of Maryland)

Conclusion

The concerns and responses presented here are indicative of the opposing viewpoints expressed within the math community regarding the use of technology in undergraduate education. Nevertheless, both "sides" of the dialogue are thinking about the same issues. For example, most agree more time and effort should be spent on determining which hand skills are necessary. As with any ongoing innovative process, further discussion and debate will prove valuable.

Acknowledgements

Many thanks to the following people who provided valuable input: Peter Cazazza, Elizabeth Covington, Sidney Graham, Holly Hirst, Jeff Holt, Lew Lefton, William MacDonald, Juan Manfredi, Tom Morley, Lisa Murphy, David Olson, Don Porzio, Allan Struthers, Lou Talman, Lee Wayand.

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About the Author

Judy Holdener
Department of Mathematics
Kenyon College, Gambier, OH 43022
holdenerj@kenyon.edu