[American Mathematical Monthly 99 (1992), 313-317]
Stimulating problems are at the heart of many great advances in mathematics. In fact, whole subjects owe their existence to a single problem which resisted solution. Nevertheless, we tend to present only polished theories, devoid of both the motivating problems and the long road to their solution. As a consequence, we deprive our students of both an example of the process by which mathematics is created and of the central problems which fueled its development.
A more motivating approach could, for example, begin a discussion of infinite sets with Galileo's observation that there are as many integers as there are perfect squares. This observation seems as paradoxical to today's students as it did to Galileo. Its ingeniously simple resolution (through a better definition of ``size'') is a tremendous educational experience, an example of the kind of education which the German logician Heinrich Scholz characterized as ``that which remains after we have forgotten everything we learned''.
We have designed a lower division honors course aimed at giving students the ``big picture''. In the course we examine the evolution of selected great problems from five mathematical subjects. Crucial to achieving this goal is the use of original sources to demonstrate the fundamental ideas developed for solving these problems. Studying original sources allows students to truly appreciate the progress achieved through time in the clarity and sophistication of concepts and techniques, and also reveals how progress is repeatedly stifled by certain ways of thinking until some quantum leap ushers in a new era. In addition to allowing a firsthand look at the mathematical mindscape of the time, no other method would show so clearly the evolution of mathematical rigor and the conception of what constitutes an acceptable proof. Thus most homework assignments focus on gaps and difficult points in the original texts.
Since mathematics is not created in a social vacuum, we supplement the mathematical content with cultural, biographical, and mathematical history, as well as a variety of prose readings, ranging from Plato's dialogue Socrates and the Slave Boy to modern writings such as an excerpt on ``Mathematics and the End of the World'' from . They form the basis of regular class discussions. Two good sources for such readings are [11,18]. To encourage student involvement, the discussions are led by one or two students, and everybody is expected to contribute. As the finale, each student gives a short presentation of a research paper written on a topic of his or her choice.
Our course serves as an ``Introduction to Mathematics'' drawing good students to the subject. It attracts students from remarkably diverse disciplines, serving as a general education course for some while acting as a springboard to further mathematics for others.
Here are our mathematical themes and original sources.