Using Content Knowledge in Teaching: What Do Teachers Have to Do, and Therefore Have to Learn?^1-2

Deborah Loewenberg Ball
University of Michigan

Abstract

I. Clarifying the Issue of Teachers' Content Knowledge

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 problem³:

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.

In these conversations of teacher knowledge and in our efforts to remediate the problems we see, it's useful to remind ourselves of the issue we're really trying to address. Sometimes we become distracted about what we're trying to accomplish. The problem we're trying to address is not, in the end, teachers' understanding of content. The problem we are trying to address is the quality of teaching and learning. We are trying to improve what students actually have opportunities to learn in school -- in science, in reading and language arts, in social studies and in mathematics.

What is the actual issue that we are trying to address? The quality of teaching and learning Teachers' knowledge of content and their ability to use it in their teaching

In the work that my colleagues and I are doing, we hypothesize, that teachers' knowledge of content, and their ability to use such knowledge in their work - on their feet in their classrooms, with children - is a key factor in the quality of science and mathematics instruction. We suspect teachers' knowledge of content is a key place to intervene if we hope to improve students' learning of content. Typically, however, the remedies tried have not produced the sorts of changes in practice that we'd like to see. Therefore, I'd like to suggest a somewhat different approach to how we might think about the content knowledge - and opportunities to learn it - that teachers need.

Rather than the common approach of starting with an examination of the school curriculum or the discipline to determine the important skills and concepts teachers should know, our research is founded on the premise that we would do better if we started by examining the work of teaching itself. Such an examination could help identify more clearly what teachers have to do when they're teaching mathematics or science, and the ways in which disciplinary reasoning and practice figure in that work. In our work on mathematics teaching we have come to ask: What are the mathematical problems that teachers recurrently face and have to solve? What are the mathematical tasks teachers have to do in the course of their work? Based on answers to such questions, we can begin to analyze the mathematical knowledge and skills, the topics and areas of study, that hold some promise of giving teachers leverage in their work. In science, we might ask, analogously: What are the scientific questions and issues that teachers confront as they teach science? In what ways does the teaching of science require teachers to contend with scientific puzzles, logic, and evidence?

By starting with the work of teaching (instead of the curriculum or the discipline) and trying analyze what is being called upon, we arrive at somewhat different answers about what teachers might need to learn or the knowledge they need to be able to use in their work. It is important to realize that it is not just a list of topics, skills, or orientations to mathematics in which we are interested. Equally important is the need to understand how teachers deploy this knowledge in their work. What are the uses to which they put such knowledge?

Here is how we pursue this:

What is Necessary Mathematics Knowledge for Teaching? Examine teaching, and identify the mathematical work that teaching entails Analyze what mathematical knowledge -- topics and skills -- is needed to do that work Analyze also the qualities of that knowledge -- how it must be understood and known to be serviceable for the work

Consider what we have already seen briefly: One type of mathematical problem solving teachers might have to do recurrently is to confront methods or solutions different from the ones they know, and try to size them up. This is what we did with the three examples of multiplication solutions. When we look closely at what is involved in appraising these three methods, we see something about the quality- the characteristics - of this knowledge. Analyses of these and other typical problems that arise in teaching can enable us to have better answers to the question, "What do teachers need to know about the subjects they teach, and how do they have to use that knowledge?"

This leads to the second part of this keynote in which we are going to spend some time investigating the argument that teaching a subject is embedded with specific disciplinary reasoning and practice. What I will do next is to take you through a way of thinking that will begin to develop this notion a bit. Keep in mind that although I am using examples from mathematics, our discussions over the next couple of weeks will take up science teaching as well.

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.

Problem #1: Analyzing Errors

Let me give the first example of analyzing errors. For many of you, this will be relatively second nature and therefore will not constitute a difficult problem for you but if you've not taught at the grade levels where this tends to come up this may be a more difficult mathematical problem for you. Here's the problem -

Analyzing Errors 1 3 2 - 5 7 1 2 5 What method is producing this answer?

Teachers see a wide variety of answers to problems they assign. And so, they are often involved in analyzing errors. A student produces this answer to the problem, 132 - 57. What method is producing this answer? Take a moment to think about this. See if you can describe to someone else what method produced that answer. Is there more than one possible method?

If you've taught second or third grade you've probably seen this a million times. High school teachers and mathematicians in the audience may not have seen what was going on quite so quickly. It's not a trivial problem, even though for some of you this is obvious. If it was obvious, it is because you've actually learned a kind of mathematical reasoning from your practice that makes this question easy to answer. But I guarantee that not every one finds this easy. That's something to notice about the sort of mathematical reasoning involved in analyzing errors. One of things you were doing was you looked at a result and you tried to think about what operations could produce that result and what, therefore, might have been the algorithm for it. That is a significant and productive form of mathematical reasoning. And that's what you were drawing on.

Let's try one that is, perhaps, a little more challenging.

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:

Analyzing Mathematical Procedures 1 3 2 - 5 7 1-2-5 7 5 What is the method that produces it? Will this method work to subtract any two whole numbers? What are the mathematical advantages and pitfalls of this method?

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.

What is the method being used here? Will it work to subtract any two whole numbers? What are some of its mathematical advantages or pitfalls?

(Note: To engage in a discussion on these questions, click on the link above.)

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:

  3004
- 1658

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.

If you have ever taught any elementary grade, you've probably heard this question. Children often wonder, and ask, "Is zero even or odd?"

Defining Concepts and Terms

So here we will examine a third kind of work that teachers do, one that I'll call defining concepts and terms.

I'm going to take us through two stages of this. One thing one might hope is that teachers, when confronted with questions like this, would appreciate that mathematics is a domain where definitions play a central role. Addressing this question about the parity of 0 requires being able to identify a definition that can help students to resolve the question.

The first thing I'm going to show you is a set of textbook definitions that teachers would likely encounter:

Examining Textbook Definitions How well does any of these help to answer the question? An even number is a number of the form 2k, where k is an integer. An even number is a natural number that is divisible by 2. An even number is any multiple of 2. An even number is a number that has 0, 2, 4, 6, or 8 in the ones place.

The first is an example of a definition a teacher is likely to have learned in a math course for teachers. The second, third, and fourth are examples drawn directly from U.S. textbooks. What I'd like you to think about is the adequacy of each definition for helping the teacher answer the student or to help the students answer their own question. Spend a moment looking through these definitions asking yourself, "How well does any of these definitions help to answer the question about whether 0 is even or odd?" Keep in mind that the context here is, let's say, third grade. In thinking about how well the definition helps to answer the question, you're thinking about both its mathematical features and its usefulness as a definition.

What do you think about these definitions and their usefulness in resolving the question? Do you find any of them particularly helpful? Are any quite useless? What are some of their differences?

One interesting observation is that the definitions differ with respect to which numbers are included in the domain to be considered as even or odd. The second definition does not include integers and the first, obviously, explicitly does. The other two definitions leave it tacit: We don't know exactly what is being intended there.

Let's stretch just a little now. Again, in this stretching, I'm still not moving to teaching but I am moving in the direction of knowledge use. In mathematics, a definition needs to function in a way that's usable by a particular user, in a particular context, in order to help make certain kinds of distinctions. Therefore, if you have a definition that is mathematically correct but incomprehensible to the person who wants to use it, it's not a very good definition.

In many ways, the first definition is fine. It's a perfectly conventional definition and it would be useful to many of us in the audience to discern whether zero is even or odd. It's probably not a useful definition for a third grader, however. It's not useful because it contains terms and notation that wouldn't be usable yet. The kind of thinking a teacher must do here is quite complex. It includes reasoning through the question, "How does one make up a mathematically viable definition that's accurate - correct - and yet usable by students?"

It's a very subtle form of mathematical understanding to realize what mathematical definition is and needs to do. From the perspective of a mathematical user who is only eight years old, each of these definitions has various features: The first is relatively incomprehensible; the second includes the term "natural number" which may or may not be comprehensible; the third relies on knowing what "multiple" means. The fourth is a useful criterion for evenness, but it is not so natural as a definition since it is not conceptually intrinsic, relying as it does on special feature of place value notation. For example, does the number zero have the zero in the ones place? It is a description that focuses on how a number is written using place value notation instead of a concept of "evenness" and "oddness." It doesn't permit you to determine whether numbers are even or odd if they are written using Roman numerals, or in base five.

Let's move to a classroom context where students can - and are encouraged to - offer definitions. For example, the teacher may ask, "So what is a definition of an even number?" I'd like you to look at several student definitions and think about the work of the teacher in sizing up each of these student responses:

Considering Students' Definitions of Even Number How well does any of these help to answer the question? 1. An even number is a number you can split in two equal parts without having to break anything in half.          2. An even number is a number that when you group it in twos, there are none left.   3. The even numbers are every other number on the number line, like 2, 4, 6, and so on.

These are three definitions we often hear students using. Again, the question is still, "Is zero even or odd?" Try to think about how well any of these helps to answer the question.

How well does each of these work as a mathematical definition? How well does it help to answer the question of whether 0 is to be considered even or odd?

(Note: To engage in a discussion on these questions, click on the link above.)

Being able to think through this and how these are similar to or different from one another is important mathematical work. How do these student definitions map back to the textbook definitions? How do they map to each other? There's a lot we could do together about the mathematical adequacy and the usability of these definitions. This kind of detailed thinking about what makes a definition good enough for use is a kind of mathematical judgment that we've not often thought about as central to elementary or middle school teaching but in fact comes up frequently. Rephrased, then, the mathematical problem for a teacher is:

What is a mathematically appropriate and usable definition of even number for third graders?

This question requires subtle use of and knowledge about definitions. Additionally, it is not the kind of knowledge of definitions that would necessarily grow from having a mathematics course where you learned the first textbook definition presented above. Such knowledge alone wouldn't be sufficient to help a teacher work through this question.

So, what would you consider to be a mathematically appropriate and useable definition of even numbers for third graders? If you can make one that you think satisfies the two requirements -- that it be mathematically correct, and also usable by eight-year-olds -- share it with others. As we examine proposed definitions, we can talk about what makes a definition usable -- a mathematical question, to be sure. In so doing, we will see some of the mathematical reasoning and knowledge required for the task.

What is your idea of a mathematically appropriate and usable definition of even and odd numbers for third graders? What is your justification that it satisfies both of these considerations?

(Note: To engage in a discussion on these questions, click on the link above.)

We will move next into the third section of this keynote in which we will talk about the implications of thinking of content knowledge for teaching as comprising the use of ideas and reasoning needed to solve problems endemic to practice.

What have you noticed about the mathematical ideas, habits of mind, and practices that you employed as you worked on these problems that arise in teaching mathematics?

(Note: To engage in a discussion on these questions, click on the link above.)