## 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.