Mathematics is the study of patterns and structures. Where natural sciences must eventually refer to phenomena which are observable and measurable, mathematics is concerned with structures which are merely possible. Scientists sometimes find in mathematics a useful language for describing and analyzing observed patterns; indeed, it is increasingly difficult to make progress in science without using mathematical analysis at some stage. Modeling a physical situation mathematically sometimes makes it possible to predict complex behavior, and these predictions make it possible to test the strength of the underlying theory. As a prominent mathematician of the twentieth century said (somewhat immodestly) “Mathematics describes the laws that God Himself is obliged to follow.”
The language needed to describe such things can be technically challenging at times, and the teacher of mathematics must work to maintain a balance between technical facility and clarity. On the one hand, one can do little mathematically if one can not correctly manipulate mathematical expressions while, on the other hand, the formal manipulation of expressions can be a rather dry and hollow exercise. Pedagogically, some students appear to take a great deal of comfort from the precise and uncompromising rigor of mathematical algorithms while others rebel at what can at times appear to be an arbitrary game based on arbitrary rules. But one should not have to choose. Technical rigor and understanding should strengthen and reinforce each other. In our classes we will strive to teach students how to use technical mastery to gain understanding and how to use intuitive understanding to sharpen technique.
Students are exposed to problem solving from the earliest ages. It is at the level of concrete problems that the dual aspects of mathematical reasoning confront each other. Asking questions about patterns arising in simple sequences of numbers or in basic shapes quickly leads to questions and conjectures that require new techniques. Learning to phrase these questions in a reasonable way is an essential part of mathematics. To be technically useful, a mathematical statement about a problem must use familiar mathematical objects in a familiar way. To be conceptually useful, a mathematical statement about a problem must suggest ways to analyze the situation further.
At all levels, the goal of our classes is fluency with the basic techniques of mathematics and an understanding of mathematical reasoning sufficient to approach problem solving in a variety of contexts. The laws of basic arithmetic will be presented along with problems of counting and sequence while more subtle laws of algebra flow naturally from elementary problems of probability and simple games. The axioms Euclidean Geometry can be seen in simple drawings and Geometric reasoning can be learned from constructions with straightedge and compass.
In later years, students are asked to confront problems drawn from a much wider set of topics. The introduction of perspective into Renaissance painting brought with it a host of new questions in geometry and algebra. These questions in turn gave rise to a variety of new techniques. Trigonometry didn’t truly flourish until it was needed by the architects, navigators, and geographers of the 16th century and Calculus had to be invented in order to describe Newton’s theory of Motion. In each of the more advanced courses, mathematical techniques are taught alongside the questions, which gave rise to them.