*In "Knowing Mathematics for Teaching," Deborah Loewenberg Ball, Heather C. Hill, and Hyman Bass describe a program of research on the link between student achievement and teachers’ mathematical knowledge for teaching. Across all sides of the debate over how to strengthen mathematics education, there is general agreement that teachers’ knowledge of the mathematical content to be taught is the cornerstone of teaching for proficiency. There is also general agreement that many American teachers, particularly those at the elementary and middle school levels, do not know enough mathematics and do not know it deeply enough to provide effective instruction to students. The mathematical education they received, both as K–12 students and in teacher preparation programs, has not provided them with sufficient opportunities to learn mathematics. But exactly what and how much mathematics they need to know remains a matter of controversy. One of the reasons is that, despite many years of attention to the subject, the research into teachers’ mathematical preparation and knowledge—that is, what teachers need to know and be able to do to raise students’ math achievement—remains surprisingly thin. The following excerpt from *Adding It Up*, the National Research Council’s 2001 report on mathematics education, provides an overview of the state of the research on this subject. (To read the full report, go to www.nap.edu/catalog/9822.html.)*

*–EDITOR*

For the better part of a century, researchers have attempted to find a positive relation between teacher content knowledge and student achievement. For the most part, the results have been disappointing: Most studies have failed to find a strong relationship between the two. Many studies, however, have relied on crude measures of these variables. The measure of teacher knowledge, for example, has often been the number of mathematics courses taken or other easily documented data from college transcripts. Such measures do not provide an accurate index of the specific mathematics that teachers know or of how they hold that knowledge. For example, a study of prospective secondary mathematics teachers at three major institutions showed that, although they had completed the upper-division college mathematics courses required for the mathematics major, they had only a cursory understanding of the concepts underlying elementary mathematics.^{1}

Teachers may have completed their courses successfully without achieving mathematical proficiency. Or they may have learned the mathematics but not know how to use it in their teaching to help students learn. They may have learned mathematics that is not well connected to what they teach or may not know how to connect it. Similarly, many of the measures of student achievement used in research on teacher knowledge have been standardized tests that focus primarily on students’ procedural skills. Some evidence suggests that there is a positive relationship between teachers’ mathematical knowledge and their students’ learning of advanced mathematical concepts.^{2} There seems to be no association, however, between how many advanced mathematics courses a teacher takes and how well that teacher’s students achieve overall in mathematics.^{3} In general, empirical evidence regarding the effects of teachers’ knowledge of mathematics content on student learning is still rather sparse.

In the National Longitudinal Study of Mathematical Abilities (NLSMA), conducted during the 1960s and still today the largest study of its kind, there was essentially no association between students’ achievement and the number of credits a teacher had in mathematics at the level of calculus or beyond.^{4} Commenting on the findings from NLSMA and a number of other studies of teacher knowledge, the director of NLSMA later said,

It is widely believed that the more a teacher knows about his subject matter, the more effective he will be as a teacher. The empirical literature suggests that this belief needs drastic modification and in fact suggests that once a teacher reaches a certain level of understanding of the subject matter, then further understanding contributes nothing to student achievement.^{5}

The notion that there is a threshold of necessary content knowledge for teaching is supported by the findings of another study in 1994 that used data from the Longitudinal Study of American Youth (LSAY).^{6} There was a notable increase in student performance for each additional mathematics course their teachers had taken, yet after the fifth course there was little additional benefit.^{7}

Data from the 1996 NAEP on teachers’ college major rather than the number of courses they had taken provide a contrast to the general trend of this line of research. The NAEP data revealed that eighth-graders taught by teachers who majored in mathematics outperformed those whose teachers majored in education or some other field. Fourth-graders taught by teachers who majored in mathematics education or in education tended to outperform those whose teachers majored in a field other than education.^{8}

Although studies of teachers’ mathematical knowledge have not demonstrated a strong relationship between teachers’ mathematical knowledge and their students’ achievement, teachers’ knowledge is still likely a significant factor in students’ achievement. That crude measures of teacher knowledge, such as the number of mathematics courses taken, do not correlate positively with student performance data supports the need to study more closely the nature of the mathematical knowledge needed to teach and to measure it more sensitively.

The persistent failure of the many efforts to show strong, definitive relations between teachers’ mathematical knowledge and their effectiveness does not imply that mathematical knowledge makes no difference in teaching. The research, however, does suggest that proposals to improve mathematics instruction by simply increasing the number of mathematics courses required of teachers are not likely to be successful. Courses that reflect a serious examination of the nature of the mathematics that teachers use in the practice of teaching do have some promise of improving student performance. Teachers need to know mathematics in ways that enable them to help students learn. The specialized knowledge of mathematics that they need is different from the mathematical content contained in most college mathematics courses, which are principally designed for those whose professional uses of mathematics will be in mathematics, science, and other technical fields.

Why does this difference matter in considering the mathematical education of teachers? First, the topics taught in upper-level mathematics courses are often remote from the core content of the K–12 curriculum. Although the abstract mathematical ideas are connected, of course, basic algebraic concepts or elementary geometry are not what prospective teachers study in a course in advanced calculus or linear algebra. Second, college mathematics courses do not provide students with opportunities to learn either multiple representations of mathematical ideas or the ways in which different representations relate to one another. Advanced courses do not emphasize the conceptual underpinnings of ideas needed by teachers whose uses of mathematics are to help others learn mathematics.^{9} Instead, the study of college mathematics involves the increasing compression of elementary ideas into the more and more powerful and abstract forms needed by those whose professional uses of mathematics will be in scientific domains. Third, advanced mathematical study entails using elementary concepts and procedures without much conscious attention to their meanings or implications, thus reinforcing the making of prior learning routine in the service of more advanced work. While this approach is important for the education of mathematicians and scientists, it is at odds with the kind of mathematical study needed by teachers.

Consider the proficiency teachers need with algorithms. The power of computational algorithms is that they allow learners to calculate without having to think deeply about the steps in the calculation or why the calculations work. That frees up the learners’ thinking so that they can concentrate on the problem they are trying to use the calculation to solve rather than having to worry about the details of the calculation. Over time, people tend to forget the reasons a procedure works or what is entailed in understanding or justifying a particular algorithm. Because the algorithm has become so automatic, it is difficult to step back and consider what is needed to explain it to someone who does not understand. Consequently, appreciating children’s difficulties in learning an algorithm can be very difficult for adults who are fluent with that algorithm.

The necessary compression of ideas in the course of mathematical study also shortchanges teachers’ mathematical needs. Most advanced mathematics classes engage students in taking ideas they have already learned and using them to construct increasingly powerful and abstract concepts and methods. Once theorems have been proved, they can be used to prove other theorems. It is not necessary to go back to foundational concepts to learn more advanced ideas. Teaching, however, entails reversing the direction followed in learning advanced mathematics. In helping students learn, teachers must take abstract ideas and unpack them in ways that make the basic underlying concepts visible.^{10 }

*National Research Council. (2001). *Adding it up: Helping children learn mathematics*. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, D.C.: National Academy Press.*

### Endnotes

1. Ball, 1990, 1991.

2. Mullens, Murnane, and Willet, 1996; see Begle, 1972.

3. Monk, 1994.

4. Begle, 1979.

5. Begle, 1979, p. 51.

6. The Longitudinal Study of American Youth (LSAY) was conducted in the late 1980s and early 1990s with high school sophomores and juniors. Student achievement data were based on items developed for NAEP.

7. Monk, 1994, p. 130.

8. Hawkins, Stancavage, and Dossey, 1998.

9. In fact, it appears that sometimes content knowledge by itself may be detrimental to good teaching. In one study, more knowledgeable teachers sometimes overestimated the accessibility of symbol-based representations and procedures (Nathan and Koedinger, 2000).

10. Ball and Bass, 2000; Ma, 1999.

### References

Ball, D.L. (1990). The mathematical understandings that prospective teachers bring to teacher education. *Elementary School Journal*, 90, 449–466.

Ball, D.L. (1991). Research on teaching mathematics: Making subject matter knowledge part of the equation. In J.Brophy (Ed.), *Advances in research on teaching, Vol. 2: Teachers’ knowledge of subject matter as it relates to their teaching practice* (pp. 1–48). Greenwich, Conn.: JAI Press.

Ball, D.L., and Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J.Boaler (Ed.), *Multiple perspectives on the teaching and learning of mathematics* (pp. 83–104). Westport, Conn.: JAI/Ablex.

Begle, E.G. (1972). *Teacher knowledge and student achievement in algebra (SMSG Reports No. 9)*. Stanford, Calif.: Stanford University, School Mathematics Study Group.

Begle, E.G. (1979). *Critical variables in mathematics education: Findings from a survey of the empirical literature*. Washington, D.C.: Mathematical Association of America and National Council of Teachers of Mathematics.

Hawkins, E.F., Stancavage, F.B., and Dossey, J.A. (1998). *School policies and practices affecting instruction in mathematics (NCES 98-495)*. Washington, D.C.: National Center for Education Statistics. http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=98495.

Ma, L. (1999). *Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States*. Mahwah, N.J.: Erlbaum.

Monk, D.H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. *Economics of Education Review*, 13, 125–145.

Mullens, J.E., Murnane, R.J., and Willett, J.B. (1996). The contribution of training and subject matter knowledge to teaching effectiveness: A multilevel analysis of longitudinal evidence from Belize. *Comparative Education Review*, 40, 139–157.

Nathan, M.J. and Koedinger, K.R. (2000). An investigation of teachers’ beliefs of students’ algebra development. *Cognition and Instruction*, 18, 209–237.

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