Good teachers know both content and how to get it across to their students. But specifying this knowledge has proven surprisingly difficult. A common approach is to require teachers to major in the fields they will teach and then add knowledge of how children learn and classroom experience. But some argue that the content knowledge that teachers need is different from that needed by mathematicians or physicists. Take the case of something as apparently simple as what knowledge is involved in teaching operations with integers. Most adults remember a for subtracting negative numbers--subtracting a negative is the same as adding a Is knowing this rule enough to teach this material? Note that this isn't the same as asking what students need to learn. Rather, we ask about the mathematical understanding needed to teach this topic. To focus the question, we drop in on Ms. Gonzalez, a 7th-grade mathematics teacher. She begins her lesson by using black chips to represent positive numbers and red chips for negative numbers. Adding one black and one red chip results in zero. Her students have been solving such problems as +4 + (-8) = x by as many black and red chips as possible, then counting the chips left over (in this case, four reds). The model seems to help her students solve addition problems. But the next problem in the text is different: Find the missing part for this chip problem. What would be a number sentence for this problem? Ms. Gonzalez begins by modeling -1 - (-3) on the overhead projector by combining two red chips and one black, or -2 + (+1), which is -1, and then subtracting three reds: The students struggle with this representation. A student ventures that the answer is -1; another proposes that the answer is 5; and a third argues for an answer of -2. Many more note that matching a black with red leaves four reds, or a result of -4. Ms. Gonzalez checks the answer in the teacher's edition; the answer it gives is 2. Unclear how to use the chips to show this, she abandons the model and demonstrates how to solve -1 - (-3) = 2 by showing that the minus sign in front of the 3 and the subtraction sign combine to make addition of a positive. What is the mathematical knowledge needed to teach this material and to interpret and use the text? Knowing the conventional procedure is clearly useful, and Ms. Gonzalez did know it. She is able to easily use it to solve problems involving subtraction of integers. But our analysis of the mathematical demands of teaching this lesson shows that more is involved. Modeling Mathematics in Teaching One of the most easily observable teaching tasks is constructing representations that are both mathematically accurate and helpful to learners. In this case, one of these representations involved using chips to solve subtraction problems. As the teacher and student confusion shows, this task is far from straightforward. The representation Ms. Gonzalez created--while mathematically correct--cannot be easily manipulated to arrive at the solution. A more promising way is to interpret subtraction as taking away (-3) from the initial quantity: However, only two red chips (-2) are present. How can (-3) be away? The solution, as briefly described by the textbook, would be to add another pair of black and red chips: The extra pair of black and red is equal to 0, so the total is still -1, but with this representation, three negative units can be taken away, showing the answer as +2. This representation is similar to what we do in multidigit subtraction when we rename the number (conventionally called regrouping or borrowing) to be able to subtract. Had Ms. Gonzalez seen this connection, she might have been better able to support students' use of the chip model. The lesson also requires facility with understanding and handling the mathematics that students say and do. …