Keynote Part I

Students do not only produce errors. They also devise and experiment with other procedures and representations. So, next, I'm going to show you a mathematical procedure with which you may not be familiar and the questions will be: What is this method? and: Would it work in general for all whole numbers? If it's not a valid method, what seems to be lacking? And, finally: If you think it is indeed a valid method, what are the mathematical advantages or pitfalls of it? This task, analyzing mathematical procedures, arises commonly in teaching.

Here's the same problem we just looked at, and a different method for its solution:

You might notice in this case that the answer is correct. Your task right now is to figure out the method. What method is producing this answer? (This solution has actually come up in second and third grade classrooms.) After you have figured out the method, then try to work on these two questions at the bottom. They are similar to questions that teachers regularly confront. Again, this is all in service of illustrating a kind of mathematical thinking that's central to teaching.

The series of questions posed above unfolded in three stages. It is important to highlight these three stages for you as we think about teaching as mathematical work. The first question is asking, "What is going on here? Is there a method and, if so, what is it?"

This the same task we've seen a couple times already: You're presented something with which you may not be familiar, and are asked to reason about what is going on mathematically. That is step one. Step two is something we also looked at briefly once: When presented with something you haven't seen before that seems to involve a method of some sort, it's useful to try to figure out how general the method is - would it work in all cases? And again, this is something one would want to reason about prior to making any pedagogical decisions about what to do next. The third step involves evaluating the method. A teacher may see the method used, decide it works for any two whole numbers, but she might want to think about the advantages and pitfalls of the method. For example, in discussing this method with other people, some of us feel it's less error prone than the traditional place value algorithm because it doesn't require a lot of "crossing out." If you were pretty skilled with the subtraction, it's somehow more straightforward. You're less likely to make all those cross-out errors that are quite common among students. Imagine trying this method with a problem that involves subtraction across a number with many "zeros" - a type of problem that typically gives students a great deal of trouble:

With a problem such as this one, this different method might be less error-prone. On the other hand, using this method requires the person who gets an intermediate result (2-6-5-4) to be pretty facile at reconciling this intermediate notation for the numerical result to a standard representation of the answer (2000 - 600 - 50 - 4 = 1346).

Mathematically, it is important to think through the potential advantages and pitfalls of the method. The considerations reflect aesthetic as well as practical mathematical issues. Did you try it with other examples, or with larger numbers? Testing a method like this involves mathematical judgment about what numbers to try. As the numbers get larger, we might find this method to be equally error-prone or complicated. Does it seem more or less elegant than some other algorithm, perhaps than the U.S. regrouping (or "borrowing") algorithm? Does it involve more or fewer steps, more or less mathematical knowledge? None of these questions is, as yet, pedagogical. Rather, these are mathematical issues, arising in the course of a pedagogical situation.

Again, this example illustrates kind of mathematical work that arises in teaching. This problem is something not typically faced by others who use mathematics professionally. Physics, engineering, accounting, nursing -- each of these professions involve mathematical reasoning and applications. But the problems we see here are central to teaching. The sort of mathematical work that teaching requires is not trivial.

Let's turn to one more example before we move to consider what this would mean for professional development.