A Closer Look

The "technology" issues are usually teaching and learning issues

The above dialogue is similar to many discussions occurring in math departments since the late 1980's and early '90's. At first glance, these discussions appear to be a debate about "things that must be learned" (techniques of integration) versus "things that must be used" (Mathematica). However there are deeper underlying themes. A characteristic of the debate is that the "technology" issues are quite often teaching and learning issues. Student errors, for example the error in computing an average by entering a + b/2 into a calculator, are cited as a by-product of the use of technology. But such errors certainly predate the use of calculators and computers in mathematics education--there was never a Golden Age of Accuracy in which calculus students did not make errors.

Rather than complain about the student errors, the proper action for teachers is to study and understand the errors students make. Such a study has been successfully done at the elementary school level by Kurt VanLehn, who argues that as children acquire arithmetic skills they often develop cognitive "bugs"--small, local misconceptions that cause systematic errors. VanLehn has described a taxonomy of these bugs in student reasoning. By focusing on these procedural misconceptions, he studies how mathematics students acquire procedural skills in instructional settings and what these errors reveal about the learning process [16].

At the level of college calculus, a less formal but nevetheless interesting and useful collection of possible student errors has been assembled by Barry Cipra [17]. Cipra begins a section on integrals with this advice to students:

If you remember nothing else from calculus, remember this: A definite integral measures the area beneath a curve. In particular, a positive function must have a positive integral. Thus

is clearly wrong because is clearly positive. [page 1].

Cipra actually introduces two errors here: a bookkeeping error, in which the upper limit of integration is wrongly written as -1 instead of 1, and an algebraic procedural error, in which the expression becomes . What is the remedy for the student who makes such an error? To some, the answer is "more drill on definite integration." But the conceptual error here is the failure to note that the definite integral of the positive function cannot have a negative value. Even if drill would improve the student's accuracy with respect to data entry and algebraic manipulation, drill does not get at the conceptual error.

A first example with Calculus WIZ

Here is an example of how a problem might be designed to take advantage of the Calculus WIZ software, with Mathematica running as the underlying support system. Suppose that the previous problem is posed in the following way:

Exercise X.X:
In the context of a technical article, the value of is given as -6.

1. Explain why this must be an incorrect value.
   
2. What sort of human (or machine) error might have resulted in this incorrect value?

The student must deal first with the significant issue--that the function with the given limits of integration cannot have a negative integral. Then the student can bring up the definite integral template and experiment with possibilities to explain the second value. For example, the student could use the appropriate solver to generate formatted answers to the homework.

Answer to Exercise X.X

1. The value = -6 is obviously wrong, because this is an area integral. (See Calculus WIZ Section 4).
   
2. The incorrect value would result if someone entered the limits of integration in the wrong order.

   

Imaginative strategies that students might use for exploring mathematics with Calculus WIZ and Mathematica.

With a few suggestions, students will try things out with Mathematica that they would not attempt, or be able to attempt with traditional approaches. Although the real power of a system like Calculus WIZ comes into play when students begin to study significant problems, the lower-level mechanics of calculus can also be made more meaningful.

In this regard, The Mathematica Book contains a significant message. [p 815]

Particularly if you introduce new mathematical functions of your own, you may want to teach Mathematica new kinds of integrals. You can do this by making appropriate definitons for Integrate....You can set up your own rule to define the integral of a function such as Sin[Sin[x]]  to be, say, a "Jones" function, giving you results such as IntegrateSin[Sin[3x]]ddx =

This changes the rules of the game of learning calculus. The student is handed an unexpected option: Define your own integral and explore the consequences of your definition!

Students can also be given "fuzzy" problems to explore. For example, in the email exchange above, the skeptic states his belief that students should understand the tricks used to find a closed-form for . The answer-- "What  makes more interesting than ?" --suggests the following non-traditional question.

Calculus students soon find out that expressions may "look similar" in symbolic form, but some admit simple closed-form integrals...and some don't. For example, consider dx] and ] dx]. One is easy and the other hard. However, the graphs, say over to , certainly show a difference. The hard integeral has a "more complicated" graph.

In general, does looking at the graphs of a function and its integral give any indication as to how hard it might be to find a closed-form expression for its integral?

The student can begin by verifying the relative complexity of the graphs. Since the syntax of Calculus WIZ is essentially the same as that of Mathematica, the student may choose to work directly in the Mathematica program:

Verifying the statement is just the beginning. The student must now select functions that admit closed-form solutions and those that do not. This is an empirical approach that is not naturally encouraged in traditional mathematics instruction. From these experiments, a student should come to the conclusion that simple graphs do not necessarily imply closed-form solutions.

The role of the instructor is to help the student understand the significance of a "mathematical" answer to the question. Here is one example:

The answer is "No!" Multiply a complicated function by its derivative and integration is easy. The graph will usually be more complicated. Add lots of fairly simple functions and you can get something arbitrarily complicated. But if the terms can be integrated in closed form, so can the sum. For example, construct a polynomial with 100 terms. [18]