# Perspectives Overview

Interactive Lectures

The purpose of the “Perspectives” series is to provide enrichment and deepen understanding for students who:

• are performing at the top of their calculus classes
• need more material
• desire better preparation for upper class courses or graduate school
• are eager to learn how mathematicians go about proving new theorems

The Lectures assume and require considerable participation from the students. We are not just talking to them from the front of the class, but involving them in the process of developing the material, by asking them questions before presenting each step. Further, many students are proactive in asking questions of their own, so they can fit their own ideas into the material covered.

For this reason, the Perspectives series is not suitable for internet presentation. Live students are needed to make the lectures work. And live students ensure that each lecture is original and spontaneous, because their questions and responses cannot be predicted.

Why “Perspectives?”

The lectures do not consist simply of theorems and proofs. They are called “Perspectives” because that is what they present — a viewpoint within which a large amount of material can be organized. The prepared lectures are interspersed with answers to the students’ questions, as well as additional, unplanned comments by the lecturer in response to the students’ input.

How the Topics are Organized

There are two approaches. The first is to discuss a fundamental idea that applies to many different situations. From the one idea flow numerous theorems, which may seem disparate but in fact are profoundly connected. Often, what seem to be separate theorems turn out to be different manifestations of some underlying principle. Further, this kind of lecture offers a method of proof which will work on many varied problems, so that students begin to develop a “toolkit” of ideas for constructing proofs.

The second approach is through a lecture which concentrates on just one theorem, proving it in several completely different ways. Then we compare and contrast these proofs, considering the strong and weak points of each. These lectures demonstrate the richness of calculus and show that there are many ways to think about a problem. The student will also learn that many times there are easier and harder ways to go about things.

How the lectures were developed

The proofs offered in “Perspectives” were developed by Dr. Mock* specifically for these purposes. The ideas for many of them arose out of classroom situations, where it became clear that fundamental material was not understood. Each lecture has been field tested on real students, and refined to be as simple and clear as possible. The goal is to show why a theorem should be true, not just how to prove it.

Student reactions

A student who attends one “Perspectives” lecture may think simply that this is nice or cute. But after several lectures it becomes apparent that mathematics is not an endless list of topics to be learned, but a set of fundamental ideas from which all the rest flows.

Students who get this far tend to get quite excited, and start bringing in their own topics for discussion and analysis. They start to find approaches of their own to tackling proofs, and develop a craving to understand how things fit together. This is the beginning of becoming a mathematician.

* Almost all of the ideas and approaches are out there in the literature somewhere. Dr. Mock did her own work because the she has a clear perspective of her own and wanted proofs which reflect it; and because the sources are scattered, hard to find, can be difficult to understand, or in some cases contain errors that had to be fixed. Some of Dr. Mock’s proofs were developed without reference materials; others were revised from a variety of books or internet sources. A few proofs, certainly among the most interesting and illuminating ones, were suggested by Dr. Peter Lax.