In "What I learned in Elementary School," Ron Aharoni describes his discovery of the importance of "layering" in elementary mathematics, with each layer of knowledge built upon the previous one. It is an insight that is borne out by research on the educational practices of the world's highest performing nations. Below, William Schmidt describes how TIMSS data demonstrate the importance of a coherent mathematics curriculum, in which the topics are chosen carefully. For a more comprehensive treatment of this subject, including a stunning visual comparison of the topics covered in U.S. and international mathematics curricula, see "A Coherent Curriculum: The Case of Mathematics," Dr. Schmidt's Summer 2002 article for American Educator.
By William H. Schmidt
The data are clear. Recent results from the Third International Mathematics and Science Study (TIMSS) show that U.S. eighth- and 12th-graders do not do well by international standards—ranking below average in both grades and, in fact, near the bottom of the international rankings on a mathematics literacy test at the end of high school. Even our best students, taking an advanced mathematics test, do not fare well against their counterparts in other countries. Those results were obtained in 1995, but retesting in 1999 and 2003 found few gains. Put simply, there is no evidence to suggest that we as a nation are doing better, at least relative to other countries.
People from other nations often ask me why U.S. student achievement does not improve, especially given that we are constantly reforming mathematics education in the U.S. The short answer is that we often engage in reform that is not based on scientific evidence but rather on opinion and someone's ideology. TIMSS offers us a good opportunity to use scientifically collected data on some 50 countries to find a more promising answer to the question of what we can do to improve the mathematics education of all children.
TIMSS results suggest that the top achieving countries have coherent, focused, and demanding mathematics curricula. What would a coherent curriculum look like? A coherent curriculum leads students through a sequence of topics and performances over the grades, reflecting the logical and sequential nature of knowledge in mathematics. Such a curriculum helps students move from particular knowledge and skills toward an understanding of deeper structures, more complex ideas and mathematical reasoning including problem solving. For example, students should be expected to master the basic concept of number and basic computational skills in the early grades before they tackle more difficult mathematics.
What does the U.S. curriculum look like? The U.S. curriculum, as reflected in many of the states' standards and in our nation's textbooks, tends to reflect an arbitrariness, with topics appearing somewhat haphazardly throughout the grades. For example, teachers are expected to introduce relatively advanced mathematics in the earliest grades, before students have had an opportunity to master basic concepts and computational skills. In addition, the curriculum continues to focus on basic computational skills through grade eight and perhaps beyond. Jumping back and forth between basic and advanced topics obscures the logic of mathematics. I would argue that if the logic of mathematics is not transparent to students, then it becomes difficult for them to develop a deep understanding that could lead to higher achievement.
What do curricula look like in the top achieving countries? They are focused and rigorous. The number of topics that children are expected to learn at a given grade level is relatively small, permitting thorough and deep coverage of each topic. For example, on average, nine topics are intended in the second grade. The U.S., by contrast, expects second-grade teachers to cover twice as many mathematics topics. As a result, the U.S. curriculum is accurately characterized as "a mile wide and an inch deep."
Coherent standards move from the simple to the complex. By the middle grades, the top achieving countries do not intend that children should continue to study basic computation skills. Rather, they begin the transition to the study of algebra, including linear equations and functions, geometry and, in some cases, basic trigonometry. By the end of eighth grade, children in these countries have mostly completed mathematics equivalent to U.S. high school courses in algebra I and geometry. By contrast, most U.S. students are destined for the most part to continue the study of arithmetic. In fact, we estimate that, at the end of eighth grade, U.S. students are some two or more years behind their counterparts around the world.
All of this is related to what students learn. If we are serious about providing all students with a challenging mathematics curriculum, it must be coherent and demanding—not by our own sense of what this might mean, but by international standards. It must be focused. It must require our middle schools to expect more of our students. It must be for all children. And it must be taught by teachers who are well-prepared in mathematics and in instructional approaches that are steeped in mathematics, as well as cognitive theories about how children learn.
William H. Schmidt is a University Distinguished Professor at Michigan State University and the director of the U.S. national Research Center for the Third International Mathematics and Science Study. This article was adapted with permission from a presentation to the U.S. Department of Education's 2003 Secretary's Summit on Mathematics.
Helping Children Learn Mathematics
What I Learned in Elementary School
By Ron Aharoni
The Role of Curriculum
By William H. Schmidt
Knowing Mathematics for Teaching (PDF)
Who Knows Mathematics Well Enough to Teach Third Grade, and How Can We Decide?
By Deborah Loewenberg Ball, Heather C. Hill, and Hyman Bass
Mathematical Knowledge for Teaching: A Research Review
Mathematics for Teaching: Then and Now (PDF)
Mathematics Professor Richard Askey Provides Answers to "Mathematics for Teaching: Then and Now" (web extra, PDF)