Mathematica-Based Calculus and Cognitive Science

An analysis of many articles, letters from AMS Notices, and other sources clearly indicates a wide-spread belief concerning the need to incorporate technology into mathematics instruction. Cognitive research studies are providing information to support such beliefs.

Consider the question, perhaps iterated to a higher degree than any other question in calculus reform, should students study techniques of integration?

Alan Schoenfeld's studies of calculus students suggest that even when students study techniques of integration, they do not develop good cognitive strategies for problem solving. Instead they are likely to acquire a strategy by default, such as trying the various techniques, substitution, integration by parts, rational fractions, and partial fractions, in the same order the techniques were presented in the course. As Schoenfeld points out, "For obvious reasons, this particular strategy, trying a series of techniques in a particular order, can result in remarkably inefficient problem-solving performance." [22]

Consider how the context of this question changes if the student has a tool like Calculus WIZ available. The convenient size of the Calculus WIZ Solver windows allows a student to keep a set of these tools open while completing a homework assignment. The student could also paste a set of solver buttons into a customized palette.

In this example, the student has selected solvers for four different integration techniques. Instead of a serial search through a sequence of techniques, the student can simultaneously view an array of techniques.

Until recently, most mathematics faculty relied on their intuition and past experience to inform their teaching practice. Now we are seeing the emergence of systematic approaches to research in mathematics learning at the collegiate level. Ed Dubinski has developed a framework for research and curriculum development in mathematics that he calls the Action-Process-Object-Schema (APOS) theoretical perspective [23]. Students encountering a concept for the first time are limited to an action conception of that concept. For example, beginning calculus students may understand differentiation as an action on polynomials, in which rules are applied in sequence. As the student reflects upon a particular action, she begins to view the concept as a process. In the case of differentiation, the student would understand that it is a more general process, not limited to a set of rules applied to individual functions.

With further reflection, a student begins to grasp a process as a cognitive object. Eventually, a student builds a schema that links actions, processes, objects, and other schema into a coherent framework. For a complex subject such as calculus, this is not easily described, and no two schema would be alike. Furthermore, the connections within any student's mind include both conscious and unconscious links. What we should expect is that the student would understand that an important class of functions have associated with them derived functions and that derivatives and integrals have an inverse relationship.

The APOS perspective suggests a teaching cycle, based on Activities, Class discussion, and Exercises [ACE]. Such an approach is radically different from previous ideas of course development based on a sequence of topics, lectures, homework, and tests.

Calculus WIZ is a natural tool for the ACE teaching cycle, given an appropriate course design. One possibility that is increasingly common among college students is to extend "discussion groups" to electronic discussion. Calculus classes are approaching a time when students could not only be sending Calculus WIZ notebooks to each other, but using white board and screen-sharing software to describe problem solutions to each other as they work at remote locations.

The coming years will quite likely see mathematics educators probing further into the domain of cognitive science. The resulting discoveries will reinforce what practice is already showing; for example, the importance of written and visual representations of concepts. Consider this premise by George Lakoff: "natural language semantics requires mental imagery. He continues:

We have been looking at image schemas. There seems to be a fixed body of image schemas that turns up in language after language. We are trying to figure out what they are and what their properties are. I noticed that they have topological properties and that each image schema carries its own logic as a result of its topological properties, so that one can reason in terms of image schemas. That is a kind of spatial reasoning. [24]

Mathematics will also consider the very nature of mathematical thought, and in particular show that "thinking mathematically" is not just high-level computation. According to Roger Penrose:

[Is mathematics] a computational activity par excellence? Indeed, it is not! It is one of my purposes here to emphasize that there is a great deal of what is essential in mathematical thinking that is not of a computational character. Indeed, it turns out that it is possible actually to demonstrate that there is something in our Mathematica understanding, in our insights as to mathematical truth, that eludes any computational description whatever. [25]

Again returning to classroom practice, we see the connection between theory and teaching application. The MAA volume Visualization in Teaching and Learning Mathematics contains a large number of examples, including a realization that visualizing is not always a natural activity for students.

Thinking visually makes higher cognitive demands than thinking algorithmically, and thus it is quite natural for students to gravitate away from visual thinking. [26]

As real-time imaging of the brain continues to improve in resolution, we will encounter studies of students whose brains are being scanned while they study their calculus problems. For steps in that direction, the reader should consult some of the innovative work of French neuroscientist Stanislas Dehaene. [27]