Goals of Mathematics Instruction: Seven Thoughts and Seven Illustrations of Means

Abstract

The goal of this chapter is to present what the author sees as the state-of-the-art approach to mathematics instruction, and the state-of-the-art use of mathematical Olympiads in bringing instruction closer to ‘real’ mathematics and identifying young talents. One of the principle goals of mathematics instruction ought to be showing in a classroom what mathematics is and what mathematicians do. This cannot be achieved by teaching but rather by creating an environment in which students learn mathematics by doing it. As in ‘real’ mathematics, this ought to be done by solving problems that require not just plugging numbers into memorized formulas and one-step deductive reasoning, but also by experimenting, constructing examples, and utilizing synthesis in a single problem of ideas from various branches of mathematics, built on high moral foundations. The author’s eight recent Springer books present fragments of ‘live’ mathematics, and illustrations of these ideas. The chapter also describes the role of mathematical olympiads in instruction and includes some problems used at the Colorado Mathematical Olympiad over the past 34 years.