Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. Liping Ma. 166 pp. Lawrence Erlbaum Associates, 1999. $19.95 (paper).
Would kindergarten through 8th-grade students learn more if their teachers knew more? If knowing more means that teachers have a master's degree or have passed courses in advanced mathematics, the answer may well be no. But if it means thoroughly understanding the material one teaches, the answer is very likely yes.
Ma compared the responses of 23 American elementary school teachers and 72 Chinese teachers to the same four questions. Although the American sample was small and nonrandom, Ma presents convincing evidence that the American teachers were better than average. (For example, half were attending a summer institute for math teachers; the others were about to receive master's degrees.) The Chinese sample consisted of the math teachers at five broadly representative Chinese elementary schools.
One of Ma's four questions was to divide 1 3/4 by 1/2 and to make up a story for this problem. Only 9 of the 23 American teachers gave the correct answer in an appropriate form (3 3/4 or 7/2 or 3.5). Several gave incomplete answers like 14/4 or 28/8, whereas six gave no answer. All 72 Chinese teachers gave the correct answer in an appropriate form. Many of the American teachers who answered correctly were merely recalling a procedure (invert the denominator and multiply) that they could not explain. The Chinese teachers not only knew how to divide fractions; most of them could also suggest alternate methods that would have enabled them to connect division of fractions to a student's prior knowledge, or to alert a good student to approaches that might be simpler under some circumstances.
Only one of the American teachers supplied an even technically correct word problem whose solution required dividing 1 3/4 by 1/2. But this teacher's problem was pedagogically dubious since the units were people and the answer thus involved a fractional person. Most of the Americans' problems showed confusion over fundamentals, such as the difference between dividing by 3/4 and dividing by 2 ("dividing into half") or between dividing by 1/2 and multiplying by 1/2.
Sixty-five of the 72 Chinese teachers made up conceptually correct stories. Several made up more than one appropriate story in order to illustrate different applications of division. Ma distinguishes three models for division: the measurement model (how many 1/2 foot lengths are there in 1 3/4 feet?), the partitive model (if half a length is 1 3/4 feet, how long is the whole?) and products and factors (if the area of a rectangle is 1 3/4 square feet and one side is 1/2 foot, how long is other side?). Chinese teachers not only compute accurately; they can present a richer store of applications. They have "pedagogical content knowledge: ways of representing and formulating a subject that make it comprehensible to others."
Judged by credentials the American teachers should have been superior. More than half of the U.S. sample had or were about to complete a master's degree. All certified U.S. teachers have at least a bachelors degree. But Chinese teachers typically complete 9th grade, then attend a normal school for two or three years—which in terms of years of schooling might be equated to a U.S. high school diploma.
Is there a connection between the superior understanding of Chinese teachers and the superior performance of Chinese students (who, for example, scored well above U.S. students in TIMSS, the Third International Mathematics and Science Study)? Ma cites a number of studies (for example, the work of Deborah Ball, who was Ma's mentor at one stage of her career) that demonstrate that students learn more when their teachers understand more.
Ma's work has been well received on both sides of the so-called math wars—the sometimes vitriolic debate on the proper direction of math reform. Supporters of the National Council of Teachers of Mathematics (NCTM) reform agenda are pleased by her stress on real understanding as opposed to mere computational competence. But most NCTM reformers believe that understanding comes from "constructivist," or discovery, learning. They give low priority to computational skills (unnecessary, they say, if students have calculators). Their goal is not to teach traditional mathematical subjects as logically coherent systems but to develop "higher-order" thinking skills that are supposed to be transferable to all subjects.
Ma, in contrast, stresses the importance of mastering facts and procedures. She does not advocate child-centered classrooms ("teach the child, not the subject") in which the curriculum is secondary to fostering juvenile self-esteem. Unlike NCTM reformers, Ma does not see computational competence and real understanding as antithetical ("drill and kill") but as mutually reinforcing. Chinese teachers have both.
What practical steps can be taken to improve U.S. math instruction? Perhaps most important, elementary school mathematics should be taught by teachers who teach only mathematics (as is the case in China). The college curriculum for math teachers should be revamped with a stress not on general teaching skills but on imparting to teachers what Ma calls a profound understanding of elementary mathematics.
Schools of education typically place more stress on pedagogy (which they teach) than on subject matter (which is mostly taught elsewhere). Under Ma's proposal, future math teachers would not be taking the same college math courses that math majors take but special courses focusing on the math they will teach. Such courses could appropriately be taught in education schools. Since Ma's reforms would not undermine the special role of education schools, they might well cooperate. One hopes that Ma's work will help shape a consensus on the direction of math reform.—Malcolm J. Sherman, Mathematics, The University at Albany, State University of New York