Applied Mathematics of Teaching

A new aspect of math teacher development of using a mathematical perspective to look at pedagogy.

Recently Dr. Deborah Ball (Univ. of Michigan), came to our campus to give a presentation and some of us were invited to meet with her earlier to discuss the education of prospective math teachers. This post and the next (and maybe others?) are related to ideas that came up in that conversation. I didn’t take detailed notes of who said what, so it is best to assume that anything that is awesome belongs to Dr. Ball and anything less so belongs to me.

First a little background. When there is talk about teacher development it is usually broken down into various aspects depending on perspective, source, etc., and usually some diagram similar to this shows up:

(source http://tpack.org)

The ideas of Content Knowledge (CK) and Pedagogical Knowledge (PK) are fairly easy to understand.  Pedagogical Content Knowledge is a newer idea, and, as the diagram indicates, is a blend of PK and CK.  Usually this blend is in the form of taking ideas from PK (e.g. how students learn and effective classroom practices) and looks at what those say about teaching in the particular content area.  It usually addresses questions like “How should we teach topic X?”, and so is usually taught from the education (PK) perspective.

The interesting part of the conversation with Dr. Ball was her idea of turning PCK around (and it thus might not really be best called PCK), by looking at what mathematical practices and ideas can say about particular PK in the mathematics classroom.  For example, given a specific curriculum, we would evaluate it in terms of whether or not it is correct mathematically. Or we could look at different arguments or ways of solving problems and determine which ones are valid mathematically. The goal of this approach is to encourage the math teachers to keep thinking mathematically in their classrooms and to keep the mathematical level of the pedagogy high. This approach would not be taken in isolation from the other perspectives and one can be mathematically correct and have it end up wrong in the students’ minds. For example, Dr. Ball mentioned a situation where properties of numbers were being defined and discussed and the point that zero is neither positive or negative came up. This is mathematically correct, but this fed into an early misconception that students had that zero is not a number and so they started skipping zero in various number line calculations.  So one would still need an awareness of the tradition PCK.

What I find exciting about this idea is that it provides a natural entry point for mathematicians into discussions of teacher development and a way to bring education materials into a traditional classroom. I can imagine asking students to evaluate arguments and curriculum materials for mathematical correctness and having that contribute to both their understanding of the mathematical topic and their understanding of student learning.