Teachers' Mathematics

Math Lessons for Math Teachers
by Henri Picciotto

Participants in the debates on math education argue about many things, but they seem to agree on at least one idea: math teachers should know math, the more the better. Unfortunately, many math teachers in the United States have not studied math much beyond the classes they teach. Others have, but found that college-level mathematics courses are not all that related to the material they teach.

One solution lies in developing preservice and inservice education that helps teachers develop more depth of understanding of pre-college math. Zalman Usiskin considers this field a part of applied math, and calls it "Teachers' Mathematics". Ideally, each math teacher and each math department would be engaged in continuous studying of teachers' mathematics, in a way that is intimately connected with classroom teaching. Unfortunately, given how little preparation time teachers have, much of this has to happen at conferences and during summer institutes.

On this page, I link to lessons in teachers' mathematics which I have used with middle school and high school teachers. Most of the lessons are based on ideas that are accessible to students, though not at the same depth.

The lessons fall more or less in three categories:

Concept Analysis

Look at familiar material from unfamiliar angles, in order to increase depth of understanding.

Function diagrams

Parabolas and Quadratics

Iterating linear functions

Abstract Algebra

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.

The "Mc Nuggets Problem" (PDF)

Staircases (PDF)

Pattern Block Trains (PDF)

Formal Development

Put pre-college mathematics in a more formal framework.

Geometry of Linear Graphs (PDF)

Geometry of the Parabola: 2D | 3D

A Different Approach to the Quadratic Formula (PDF)

(background: Constant Sums, Constant Products)