Teaching Mathematical Thinking
in Highschool

Judy Nesbit

The Montclair Kimbeerley Academy
201 Valley Road
Montclair, NJ 07042
phone: 201-746-9800
email: jnesbit@mka.pvt.k12.nj.us

I teach a course titled Precalculus and Discrete Mathematics. the textbook is part of the University of Chicago School Mathematics Project, published by Scott, Foresman. (Susanna Epp is one of the authors.) Chapter 1 is titled, "Logic". The teachers edition describes the chapter as follows.

One of the themes of this course is mathematical thinking. This chapter deals with one aspect of that thinking, formal logic, and applies that logic to computer logic networks, to the analysis and writing of proofs about integers, and to the logic of everyday thinking.

The content of Chapter 1 is a blend of new and familiar material. The formalization of logic probably will be new for most students. In geometry, students dealt with if-then statements and the contrapositive, inverse and converse of such statements. However, they probably have not dealt with universal or existential statements and the relationship of these statements to conditionals. Nor have they applied these ideas to the examination of computer logic networks. Throughout the chapter, students will need to pay attention to detail; such attention is critical to success in computer science and in calculus.

Both English sentences and mathematical sentences are used to illustrate the logic. The mathematical sentences are taken from geometry and from standard precalculus topics such as properties of exponents, logarithms and trigonometric functions.

The NCTM Standards and the New Jersey Framework emphasize the importance of Discrete Mathematics(including logic) and of applications. Students become proficient at doing proofs such as the following examples:

Show that the integer t is even if t = 6a +8b for some integers a and b.

or

Prove: If m and n are any odd integer, then mn is an odd integer.

They are also quite convinced that these proofs have no applications. It is at this point that I reenact the David Copperfield illusion known as Murder on the Orient Express. Then, my challenge to the students is to figure out how it works. Students, working in groups of 2 or 3, discover that this illusion is based on mathematics. The only illusion is that the participants have free choice about the moves that they make.