# Keynote Part I

II. Seeing Teaching as Involving Substantial Disciplinary Reasoning and Problem-Solving: The Case of Mathematics Teaching

Starting from an examination of teaching and working toward a better understanding of the mathematical knowledge and skills required of teachers, my colleagues and I have developed a list of several common kinds of mathematical activities in which teachers frequently engage. The following list of examples is far from complete. Still, it provides a glimpse of some common forms of mathematical problems in teachers' work:

 What Mathematical Problems Arise in Teaching? Examples Analyzing errors Giving and evaluating explanations Appraising unexpected claims, solutions, and methods Choosing and using representations Examining correspondences among representations and solutions Choosing and using definitions Interpreting and responding to students' ideas

When one scrutinizes the list more closely, I think it's easy to see that doing these well requires a great deal of mathematical knowledge and skill. For example, to analyze student error - which we'll do in a moment - requires you to think mathematically as you look at what the student did.

We'll be looking more closely at these items: first in this section of my talk and then later we're going to look at them in relation to designing opportunities for teachers to learn mathematics. In this section of the talk, where we'll explore what it means to see teaching as mathematical work, I'm going to concentrate on three examples of the above problems, namely:

• Analyzing errors
• Appraising unexpected claims, solutions, and methods (we've already had one example of this)
• Choosing and using definitions

I'm going to try to engage you in a bit of work to help you get inside of what I mean when I say that teaching is mathematical work. What I'd like you to pay attention to as you work on the examples is not just the problems themselves, and your solutions, but also the mathematical reasoning in which you're engaged. Try to see if you can become clearer about the knowledge you're drawing upon. Pay particular attention to the sort of mathematical reasoning you are doing.

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