H. Louise Amick, Washington College
I've always suggested to students that they read ahead in the text to have some idea of what is to be discussed in class, but that suggestion hasn't always been successful. Some students have ignored my advice knowing they will survive by coming to class, while others have struggled to comply, but found that the text really wasn't readable. My wish for previews that students could really read and work through came true for one class when we revised our Precalculus course.
When our department decided to require graphing calculators for Precalculus, we were concerned about the financial burden that this (together with the purchase of a text) would place on our students. We decided that we could generate our own handouts for the course so that we wouldn't need a text. A colleague and I agreed to collaborate on this project. As we outlined and discussed the format of our handouts, our summer project expanded into a summer spent writing a text.
Each chapter of the text is composed of three parts: a preview, a lesson, and a problem set. The preview discussion and problems are to be done by study groups prior to the presentation of the chapter in class. These previews act as levelerseveryone comes to class with some knowledge to contribute to the development of the chapter. The lesson section of the chapter is very concise. Examples are included only when essential, avoiding templates from which students can model their solutions to problems. The problem sets contain a variety of problems, including applications whenever possible.
We are not suggesting that it is necessary to write one's own text to initiate class discussion and collaboration. While we are pleased with the entire text and the changes it has produced in our classes, we believe the previews are the key to our success. Consequently, we are advocating that carefully developed previews can be used in any math class to foster collaborative learning.
The nature of the previews we've developed for Precalculus varies over a wide spectrum from reviews of prerequisite material to guided development of formulas and identities. As we began to write each preview, we first asked ourselves what prerequisite knowledge we would assume for the chapter and how we could guide students to recall and review it. Then we focused on how much of the chapter's content could be discovered by students through experimentation with the graphing calculator, by applying geometry and algebra, or through guided stepbystep examples that could be generalized.
One of the most successful previews has been the one shown below, which introduced the chapter on composite and inverse functions. In developing this lesson in class, I only had to provide the definition of a composite function and the notation for an inverse function. Everything else sprang from the students' discussion of their work on the preview.

Given the following pairs of points:
and
and
and

Plot manually the three pairs of points and the line y=x on the same coordinate system.

Describe as explicitly as possible the relationship of the paired points to the line y=x.

If the coordinate system is folded using y=x as a fold line, what relationship can be observed between the paired points?

Show that the line segments and are perpendicular to the line y=x.

Show that y=x is the perpendicular bisector of the line segments , and .


Show that , and in the above problem satisfy the equation .

Show that , , and in the above problem satisfy the equation .

Show that is equivalent to .

Use your graphing calculator to sketch the graphs of , , and y=x on the same viewing window without erasing.
These previews have allowed us to change our classes from lecture classes into classes focusing on problemsolving and discussion. They have allowed us to ``flesh out'' the lessons in class and develop our own examples. Having done this, both the development of the lesson and the solving of the problems in the problem sets become collaborative ventures. In addition to the change in the atmosphere of our classes, we have seen measurable success in another sense. Prior to the use of this approach, each year we had approximately 25% of our Precalculus students either withdraw from the course at midterm or fail at the end. Now only 14% do so.
H. Louise Amick
Washington College
Department of Mathematics
300 Washington Avenue
Chestertown MD 216201197
[email protected]
The Innovative Teaching Exchange is edited by Bonnie Gold.