14AMERICAN EDUCATOR FALL 2005 W ith the release of every new international mathe- matics assessment, concern about U.S. students’ mathematics achievement has grown. Each mediocre showing by American students makes it plain that the teaching and learning of mathematics needs improve- ment. Thus, the country, once more, has begun to turn its worried attention to mathematics education. Unfortunately, past reform movements have consisted more of effort than effect. We are not likely to succeed this time, either, with- out accounting for the disappointing outcomes of past ef- forts and examining the factors that contribute to success in other countries. Consider what research and experience consistently reveal: Although the typical methods of im- proving U.S. instructional quality have been to develop cur- riculum, and—especially in the last decade—to articulate standards for what students should learn, little improve- ment is possible without direct attention to the practice of teaching. Strong standards and quality curriculum are im- portant. But no curriculum teaches itself, and standards do not operate independently of professionals’ use of them. To implement standards and curriculum effectively, school sys- tems depend upon the work of skilled teachers who under- stand the subject matter. How well teachers know mathe- matics is central to their capacity to use instructional mate- rials wisely, to assess students’ progress, and to make sound judgments about presentation, emphasis, and sequencing. That the quality of mathematics teaching depends on teachers’ knowledge of the content should not be a surprise. Equally unsurprising is that many U.S. teachers lack sound mathematical understanding and skill. This is to be ex- pected because most teachers—like most other adults in this country—are graduates of the very system that we seek to improve. Their own opportunities to learn mathematics have been uneven, and often inadequate, just like those of their non-teaching peers. Studies over the past 15 years consistently reveal that the mathematical knowledge of many teachers is dismayingly thin. 1 Invisible in this re- search, however, is the fact that the mathematical knowl- edge of most adult Americans is as weak, and often weaker. We are simply failing to reach reasonable standards of mathematical proficiency with most of our students, and those students become the next generation of adults, some of them teachers. This is a big problem, and a challenge to our desire to improve. Knowing Mathematics for Teaching Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide? By Deborah Loewenberg Ball, Heather C. Hill, and Hyman Bass Deborah Loewenberg Ball is interim dean of the School of Edu- cation and the William H. Payne Collegiate Professor in Educa- tion at the University of Michigan. Her areas of specialization in- clude the study of efforts to improve teaching through policy, re- form initiatives, teacher education, and mathematical knowledge for teaching. Heather C. Hill is associate research scientist at the School of Education, University of Michigan. Her areas of spe- cialization include educational policy, instruction, and teachers’ content knowledge. Hyman Bass is the Roger Lyndon Collegiate Professor of Mathematics and of Mathematics Education in the Department of Mathematics and the School of Education, Uni- versity of Michigan. His areas of specialization include algebra (geometric methods in group theory), teacher education, and mathematical knowledge for teaching. 1 For example, Liping Ma’s 1999 book, Knowing and Teaching Elementary Mathematics, broadened interest in the question of how teachers need to know mathematics to teach (Ma, 1999). In her study, Ma compared Chinese and U.S. elementary teachers’ mathematical knowledge. Producing a portrait of dramatic differences between the two groups, Ma used her data to develop a notion of “profound understanding of fundamental mathematics,” an argument for a kind of connected, curricularly-structured, and longitudinally coherent knowledge of core mathematical ideas. (For a review of this book, see the Fall 1999 issue of American Educator, www.aft.org/pubs- reports/american_educator/fall99/amed1.pdf.) Reprinted with permission from the Fall 2005 issue of American Educator, the quarterly journal of the American Federation of Teachers, AFL-CIO.