Presenting A Strategy for Indefinite Integration.

Introduction and theoretical overview. Students in a first-year calculus class should, with a reasonable amount of practice, be able to evaluate indefinite integrals with much the same facility as their instructor does. At least, practice is apparently the telling advantage teachers have over their students. Unlike subjects such as "related rates," integration is sufficiently straightforward that the instructor's "cleverness," "insight," or "mathematical ability" should play a minor role except in the most complex problems. Equally important, integration is a technical skill in which one is little helped by theoretical knowledge. In many areas-the mean value theorem and its implications, for example-the instructor's background provides a perspective completely inaccessible to the student, so one could not expect the student to deal with the subject matter the way the teacher does. In integration, however, we would expect an instructor's perspective to be of no more help in evaluating an integral than would a knowledge of abstract group theory be of assistance in determining the product of two three-digit numbers. Thus we could reasonably expect to set very high performance standards for our students. Yet they consistently have more difficulty evaluating integrals than they "should," in spite of many hours of working on practice problems. In this article, we argue (1) that these difficulties occur largely because students lack a coherent and efficient means of approaching problems in integration, and not because they have difficulty learning the important techniques; and (2) that we can enhance students' abilities to evaluate integrals, with no extra study time, by providing them with a coherent approach. We discuss our methodology briefly, in the hope that our readers will consider applying it in other areas. It has been used successfully, in areas ranging from artificial intelligence simulations (for example, Slagle's SAINT program for indefinite integrals) to instruction in the art of juggling. In broad outline, the major phases of experimentation are as follows. 1. Determine, as accurately as possible, what your experimental audience can and cannot do. (The next section of this paper assesses student facility with indefinite integrals, both before and after the "standard" presentation of the material.) 2. Examine, in minute detail, the action of "experts" as they perform the desired task. The purposes of this examination are to: (a) discover regularities, strategies, or implicit rules the "experts" may be using-often without being conscious of them-as they solve problems; and (b) decompose the problem-solving process into smaller, more manageable "chunks" which can be taught individually and then carefully combined. (The third section describes the process of isolating an effective strategy for indefinite integration.) 3. Create a model of the problem-solving process based on the "chunks" discovered in (b), making explicit use of the strategies discovered in (a). Design instructional materials which make both accessible to the students. (The integration materials are described in the fourth section.) 4. Test and revise both the model and the instructional materials. Preliminary test results for the integration strategy are described in the fifth section.