Keynote Part I

I begin with a basic premise: that teaching is a practice, not merely a domain of knowledge. Because teaching is a practice, the problem of teachers' content knowledge is really a problem of teachers' knowledge and use of content. It is not merely a problem of what teachers know or do not know.

Shifting the Focus - From Knowing to Knowing and Using

Criticisms abound about teachers' lack of content knowledge. The evidence of this lack ranges from low scores that teachers receive on different kinds of exams to their poor performance in content area courses. Studies of teachers' content knowledge in science and mathematics also show repeatedly that U.S. teachers lack fundamental understanding in these fields. Added to these sources of data are the ubiquitous anecdotes - for instance, someone announces, "Can you believe what my child's teacher told the class today?" or "Look at this paper -- look at the question and how it was marked." We have all heard such evidence. We have heard of teachers not knowing mathematics or science, and we may have had firsthand experience with this phenomenon, whether in our own work, through our children, or in the communities where we live. However, I argue that the problem is framed in a misleading and incomplete way, and that this mis-framing leads to problematic approaches to its solution.

What is often thought to be the solution? Many tend to think that, given the evidence, the remedy is to either increase the subject matter requirements for teachers or to add a lot of mathematics or science content to professional development. Some think it is a matter of recruiting people to teaching who have stronger backgrounds in mathematics and science.

This reasoning is not without merit; however, by itself without further thought about what it would mean to do these things well, these approaches are unlikely to make much difference. These approaches are unlikely to make much difference because they are based on faulty assumptions about the nature and role of content knowledge in teaching. Indeed, there are examples around the country of people who have thoughtfully tried to develop much more significant opportunities for pre-service teachers and practicing teachers to learn science and mathematics content, and while the results of such programs are often quite exciting in some ways, they have also been questionable in other ways. For instance, teachers who have experienced such learning opportunities in mathematics will frequently report they feel, for the first time, that mathematics is something they enjoy; they will comment that they didn't realize the subject could be so interesting. However, when one follows them into the classroom, the results are less encouraging. That is, while teachers themselves feel more confident, and like the subject more, they may not able to use that knowledge in the work they do. When students devise non-standard methods, make puzzling errors, or ask core questions, the teachers are unable to make sense of the issues skillfully. "Better recruits" - people who enjoy and have been more successful with the subject - face this, too. The problem is not merely that teachers need to be stronger in mathematics. The problem is that they will need to use mathematics in special and non-obvious ways.

Knowing Mathematics Versus Knowing and Using Mathematics for Teaching

As one brief experience of this idea, consider this example of multiplication:

Do this multiplication yourself. Think of this an example of knowing multiplication. I presume all of you can calculate this product but go ahead and work it out because I'd like you to actually have in front of you the method you used to do this.

There are likely a fair variety of methods that people use. It would be interesting to see one another's methods. (You might try this with one or two other people in your setting.)

I presume everyone got the same answer: 875. Hence, we can say that everyone among us knows how to multiply two two-digit numbers. However, this knowledge alone is insufficient for knowing mathematics for teaching.

Now I'll give an example of how a teacher might have to use knowledge of multiplication. Here are three other ways that students might write out the multiplication of that same problem ³:

The question is this: Which of these students is using a method that could be used to multiply any two whole numbers? Please note that I'm not asking anything like, "What would you do next?" "Would you allow this?" "How would you respond to the students?" I'm just asking a mathematical question - one that is similar to questions that teachers face regularly as they teach mathematics: Does this alternative method work for all numbers? Take a moment to think about this question. Look at each solution and try to figure out what's going on and decide whether or not the solution involves a method that would work no matter what whole numbers are used.

When you did the problem originally yourself, did you do something most like method A, B, or C? Or did you do something else entirely? You might pose the original multiplication to a couple of other people, and see what they do.

What about the question of whether or not each of these would work for any two whole numbers?

Consider the difference between simply doing the multiplication yourself and being in the position to answer this question, to stare at something you haven't done before and say, "Wait, what's going on here?" and, "Would this actually work?" This is a kind of mathematical reasoning in which teachers must engage, and is, in some sense, a kind of reasoning engaged in prior to making a pedagogical decision. I am not talking here about whether to encourage students to use alternative algorithms, or what to do if a student presents one. I am not raising pedagogical questions. What I am talking about here are quintessential mathematical questions. They involve mathematical analysis, the use of basic properties of operations (commutativity and distributivity), and understanding the meaning of place value notation.

In other words, a good decision about what one might want to do next as the teacher in such a situation may depend, in part, on being able to do this sort of mathematical problem solving with facility. Teachers need to be able to size up unfamiliar mathematical ideas and procedures with relative comfort, speed, and agility.

I offer this example to illustrate what it might mean to know mathematics for teaching and be able to use it, as opposed to simply what it might mean to know mathematics for oneself. Someone who is not a teacher can be perfectly well off simply by being able to calculate 25 x 35 correctly. Someone who teaches must know more than that. For now, that is my main point. Let's go further.