# The curious--and crucial--case of mathematical knowledge for teaching.

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 "rule" for subtracting negative numbers--"subtracting a negative is the same as adding a positive." 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 "matching" 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 "taken 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. As in many of her other lessons, Ms. Gonzalez encourages students to construct solutions and explain their answers. Many of her students' answers reveal that they are confusing addition and subtraction of negative numbers. Recognizing this as a common struggle can help Ms. Gonzalez plan for and even prevent this confusion.

We are not criticizing Ms. Gonzalez. In fact, once she saw the difficulty, she focused clearly on the procedure. She also actively involved students in the content of the lesson and trying to make sense of the material. She tried creating a more realistic example with money and debt to help her students understand, but the story she told did not match the problem and was confusing. How do you represent 1 - (-3) using money and debt? It can be done, but it requires some care and involves an understanding of "net worth." What are other situations that correspond to the subtraction of negative numbers? Examples like this show the mathematical demands of making mathematics comprehensible to students, and make clear that the mathematical knowledge involved is more than being able to solve the problems oneself. The simple instinct to "make connections to students' lives" turns out to be more complicated mathematically than it seems.

Teachers' Mathematical Knowledge

What must teachers know and be able to do? Despite years of research and a wide variety of methods for measuring teacher knowledge, the answer to this question has been surprisingly elusive. Some have used teacher certification as a simple proxy measure for teacher knowledge and quality (Ball and Hill 2008). Only a handful of these studies show that high school teachers certified in mathematics produce somewhat higher student gains than those certified in other subjects. Many studies, some at the elementary level and some at other levels, show no effects of teacher certification on student outcomes (National Mathematics Advisory Panel 2008).

What about courses taken or degrees attained as a measure of teacher knowledge in mathematics? This indicator is closer to what teachers may actually know. Not surprisingly, this indicator is somewhat more consistent in showing effects on students' achievement, but only in some studies. Such effects show up only at the secondary school level; these same effects are not present in studies of elementary teachers (National Mathematics Advisory Panel 2008).

These studies suggest that, though it would be foolish to say that mathematical knowledge is not important to teaching mathematics, conventional content knowledge seems to be insufficient for skillfully handling the mathematical tasks of teaching.

Because the evidence from this body of research puzzled us, we began to study teaching practice. We sought to identify common tasks of teaching that require mathematical skill and insight. We observed many classrooms. As we made progress in identifying and describing these teaching tasks, we began to appreciate the mathematical demands of ordinary teaching. We saw the mathematical understanding involved in posing questions, interpreting students' answers, providing explanations, and using representations. We saw it in teachers' talk and in the language they taught students to use. We realized that the capacity to see the content from another's perspective and to understand what another person is doing entails mathematical reasoning and skill that are not needed for research mathematics or for bench science.

As we investigated Mathematical Knowledge for Teaching (MKT), we also began to notice different domains (Ball, Thames, and Phelps 2008).

It was clear that some of the mathematical resources that teaching requires are similar to the mathematical knowledge that other professionals use. We labeled this common content knowledge; it informs such teaching tasks as knowing whether a student's answer is correct, the definition of a concept or object, and how to carry out a procedure. But we also saw that teachers required some specialized mathematical knowledge--for example, being able to model integer arithmetic using different representations. We also noted that some MKT was more of a blend of mathematics with other kinds of knowledge, such as knowledge of students or knowledge of teaching or curriculum. These blended forms of content knowledge--knowledge of content and students or knowledge of content and teaching and knowledge of content and curriculum--appeared as finer-grained categories of what Shulman and his colleagues termed "pedagogical content knowledge" (Shulman 1986; Shulman 1987; Wilson, Shulman, and Richert 1987). Recently, we've begun to see signs of another sort of MKT that we are calling "horizon knowledge" to describe a kind of mathematical "peripheral vision" needed in teaching, that is, a view of the larger mathematical landscape that teaching requires (Ball and Bass 2009).

We tested our emerging theories by investigating whether these ways of knowing and using mathematics matter. We focused on several key principles that we hoped would set our questions apart from conventional multiple-choice assessments. First, we wrote items to represent the specialized knowledge that our studies had led us to hypothesize were crucial to teaching, such as being able to:

* Interpret and analyze student work;

* Provide a mathematical explanation that's intelligible to young learners; and

* Forge links between mathematical symbols and pictorial representations.

Second, we wrote items to represent the mathematical tasks of teaching that recur across different curriculum materials or approaches to instruction. These included such tasks as:

* Analyzing student errors;

* Encountering unconventional solutions;

* Choosing examples; or

* Assessing the mathematical integrity of a representation in a textbook.

We refer to our items as the Mathematical Knowledge for Teaching (MKT) measures, and they have been used by dozens of researchers and professional development projects.

Administering these questions to large groups of teachers has helped us and other researchers learn more about what kinds of knowledge are related to student outcomes. For instance, a group of economists recently administered a survey to 418 beginning teachers. This survey included our measures, measures of general cognitive ability, and measures of several personality traits, including conscientiousness. Of all these variables, only MKT was a significant predictor of student outcomes, with an effect size almost double that of the general cognitive ability (Rockoff et al. 2008). We have found similar results in our own work involving over 300 teachers (Hill, Rowan, and Ball 2005). We also have found that teachers' MKT is strongly related to the mathematical quality of their instruction, including their use of mathematical explanation and representations, responsiveness to students' mathematical ideas, and ability to avoid mathematical imprecision and error (Hill et al. 2008).

Developing Mathematical Knowledge for Teaching

How do teachers develop and use MKT? Strong MKT seems to correlate with certain habits of mind, such as careful attention to mathematical detail and well-explicated reasoning, as well as agility with a variety of mathematical productions from textbooks and students. In other cases, teachers report developing their own knowledge through extensive mathematics-focused professional development. In one of our own studies, we found that summer professional development sites that focused teachers' work on mathematical representation, explanation, and communication outperformed similar sites with less focus on those topics (Hill and Ball 2004). Much work remains to be done in this arena.

Given this progress in understanding the nature of the mathematical knowledge needed for teaching, several key questions and problems lie ahead. For example, can MKT be better built into useful instructional guidance? Could Ms. Gonzalez' teacher's guide have supported her understanding of the key mathematical ideas involved in subtraction of integers? Could it have offered her other representations--the number line, for example--and not only showed exactly how to use them but also compared their merits to the chip model? More generally, can materials be designed that better support teachers' work? Tools and resources typically support professionals' work in other fields, yet in teaching we have left most of the reasoning to the individual teacher, based on the view that teaching is a creative act that depends on context. Given the intricacy of the work as well as the the size of the teaching profession, this has been an inefficient and ineffective way to support high-quality instruction (Ball and Forzani in press). Ms. Gonzalez does not need to invent how to represent integer arithmetic; this can be more closely supported, leaving her the discretion to make localized judgments about a host of other important teaching issues.

Another important question is to identify those aspects of MKT that show the greatest potential for improving learning. Is detailed knowledge of place value of particular utility? Are some representations--the number line, for example--more vital for teachers to have command of than others? Our studies suggest that knowledge of mathematical explanation and representation may be especially important.

Also, in order for teachers to have opportunities to learn MKT, those who prepare teachers and provide professional development will themselves need to have adequate support. Better materials, more specific guidance focused on the teaching of MKT, and better design of opportunities to learn from practice are essential.

Teaching--helping others learn to know and do--requires specialized ways of knowing the domain. As we begin to appreciate the special kind of content knowledge that it takes, along with other kinds of knowledge, skill, and commitments, we will be better able to support teachers to do this important work. And in the end, if skillful teaching is better and more systematically supported, the beneficiaries will be young people, who will get the instruction they deserve.

REFERENCES

Ball, Deborah L., and Hyman Bass. "With an Eye on the Mathematical Horizon: Knowing Mathematics for Teaching to Learners' Mathematical Futures." Paper prepared based on keynote address at the 43rd Jahrestagung fur Didaktik der Mathematik in Oldenburg, Germany, March 1-4, 2009.

Ball, Deborah L., and Francesca M. Forzani. "The Work of Teaching and the Challenge for Teacher Education." Journal of Teacher Education (in press).

Ball, Deborah L., and Heather C. Hill. "Measuring Teacher Quality in Practice." In Measurement Issues and Assessment for Teaching Quality, ed. Drew H. Gitomer, pp. 80-98. Thousand Oaks, Calif.: Sage, 2008.

Ball, Deborah L., Mark H. Thames, and Geoffrey C. Phelps. Content Knowledge for Teaching: What Makes It Special?" Journal of Teacher Education 59, no. 5 (2008): 389-407.

Hill, Heather C., and Deborah L. Ball. "Learning Mathematics for Teaching: Results from California's Mathematics Professional Development Institutes." Journal of Research in Mathematics Education 35, no. 5 (November 2004): 330-351.

Hill, Heather C., Brian Rowan, and Deborah L. Ball. "Effects of Teachers' Mathematical Knowledge for Teaching on Student Achievement." American Educational Research Journal 42, no. 2 (Summer 2005): 371-406.

Hill, Heather, Merrie Blunk, Charalambos Y. Charalambous, Jennifer M. Lewis, Geoffrey C. Phelps, Laurie Sleep, and Deborah L. Ball. "Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction: An Exploratory Study." Cognition and Instruction 26, no. 4 (2008): 430-511.

National Mathematics Advisory Panel. Foundations for Success. Washington, D.C.: U.S. Department of Education, 2008.

Rockoff, Jonah E., Brian A. Jacob, Thomas J. Kane, and Douglas O. Staiger. "Can You Recognize an Effective Teacher When You Recruit One?" NBER Working Paper 14485. Cambridge, Mass.: National Bureau of Economic Research, 2008.

Shulman, Lee. "Those Who Understand: Knowledge Growth in Teaching." Educational Researcher 15, no. 2 (February 1986): 4-14.

Shulman, Lee. "Knowledge and Teaching: Foundations of the New Reform." Harvard Educational Review 57, no. 1 (Spring 1987): 1-22.

Wilson, Suzanne, Lee S. Shulman, and Anna Richert. "150 Different Ways of Knowing: Representations of Knowledge in Teaching." In Exploring Teachers' Thinking, ed. James Calderhead, pp. 104-124. London: Cassell, 1987.

R&D appears in each issue of Kappan with the assistance of the Deans' Alliance, which is composed of the deans of the education schools/colleges at the following universities:

Harvard University, Michigan State University, Northwestern University, Stanford University, Teachers College Columbia University, University of California Berkeley, University of California Los Angeles, University of Michigan, University of Pennsylvania, and University of Wisconsin.

Deborah Ball and Algebra Project creator Bob Moses talk to Kappan editor Joan Richardson about equity and math education.

See Pages 54-59 in this issue of Kappan.

The mathematics class for rising 5th graders is collectively planned and studied by a diverse group of professionals, including teachers, researchers, teacher educators, student teachers, and mathematicians. The group meets each day before class, observes the lesson, and then debriefs together.

http://sitemaker.umich.edu/eml2009/home

HEATHER HILL is an associate professor at the Harvard Graduate School of Education, Cambridge, Massachusetts. DEBORAH LOEWENBERG BALL is dean of the University of Michigan School of Education, Ann Arbor, Michigan, and also the William H. Payne Collegiate Professor of Education there.
COPYRIGHT 2009 Phi Delta Kappa, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.