# Teachers’ Mathematics

I recently came across an interesting and humorous lesson, intended to generate discussion and reflection about the properties of our number system. It is based on the idea of a fictional cashier who knows absolutely nothing about numbers. The worksheets (Tarzan 1 | Tarzan 2) were created by mathematician Robin Pemantle for use in a course for prospective teachers. [Find more material from his preservice courses on his home page.]

Those worksheets are an example of how coming at familiar concepts in unexpected ways can help deepen one’s understanding, or at least trigger a need to put that understanding into words. Here are two more ways to get at the properties of the number system:

– In Rachel Chou’s guest post on Teaching the Distributive Property, she proposes looking at cases where the property does not apply.

– In my Introduction to Abstract Algebra, I share an entertaining set of lessons about mathematical structures that share some important features with our number system.

Such sideways approaches can be used with students, of course, but they are also important for us as teachers. If we only see lists of properties of the number system, year after year at the beginning of a textbook, we lose sight of meaning, and feel no need to think about the topic. Varied explorations of familiar concepts should be part of teacher education, both preservice and in-service. They are a part of teachers’ mathematics, which in my view consists of three components:

• Concept Exploration: Look at familiar curriculum topics from unfamiliar angles, as in the above examples.
• Problem Analysis: Start with a problem that can be posed to students, and end with an analysis at a deeper level, generally by seeking generalization and/or proof. For example, the McNuggets problem.
• Formal Development: Prove familiar results that are usually accepted without proof. For example: the graph of y = mx + b is a line.

I share activities in all three categories on my website. You may decide to dedicate a department meeting to work on one of those topics! (And if you do, let me know how it goes!)

— Henri