Treating Children as Mathematicians
The concern that the United States is falling behind in mathematics compared to other countries such as Japan led the directors of TIMSS to look at aspects of mathematics teaching and learning outside of achievement scores. They wanted to know what other factors were behind this seeming disparity between U.S. students and those of other countries. Toward this end, the study included an intensive videotape survey of 231 eighth-grade mathematics lessons in the United States, Japan, and Germany. TIMSS was the first attempt to collect a nationally representative sample of videotaped observations of American classroom instruction. According to TIMSS, “The purpose of gathering this information was to understand better the process of classroom instruction in different cultures to improve student learning in our schools” (Third International Mathematics and Science Study, 2004a, 2004b; Greene, Herman, Haury, & ERIC Clearinghouse for Science, 2000).
By using the videotape study and other TIMSS data, we can begin to examine and compare instructional methods in the United States with those of Japan. One conclusion drawn from the videotape study is that the Japanese do a much better job of treating their students as mathematicians. In U.S. classrooms, however, the tradition is to teach mathematics through memorization and practice (Elkind & Piaget, 1979; Kamii, 1984, 1990; Wenglinsky, 2004; Wood, Nelson, & Warfield, 2001). Ultimately there is little difference in the teaching philosophies that inform how primary, middle grade, and high-school students are taught in the United States (Greene et al., 2000). The study found that the main goal in the U.S. classroom was “teaching children how to solve a problem and obtain a correct answer.”
Mr. Gerhig said to his class, “I am going to show you how to figure out the number of degrees in any figure. First you take the number of sides, then you subtract two and multiply by 180. Juan, how many degrees would a square have?” Juan answers, “360?” as more of a question that an answer. “Right,” says Mr. Gerhig. “All you have to do is remember this formula and you can compute the answer.”
You may ask, “Is it not the same in Japan?” In Japan, students are given the tools to solve problems; the emphasis is on concepts rather than answers. The study found that in Japan the goal is to support conceptual understanding—in other words, less memorizing of formulas and more thinking about concepts.
Mr. Okawa draws a polygon on the board and says, “Using what we know about the area of a triangle, can you change this 4-sided figure into a 3-sided figure without changing its area?” Students then work in groups and present their solutions to the class. Mr. Okawa asks one student, “Can you tell me how you know the area is the same?” The student replies, “If the height and the base are the same it must be the same area.”
When we begin to think of children as competent mathematicians who, while working on age-appropriate problems, are using the same thought processes as advanced mathematicians, it changes the way we think about curriculum development. To do this, we must know what mathematicians do when they are presented with a problem. We can then apply these principles to design mathematics curricula for young children. Throughout this discussion, we will step inside the Japanese and American classrooms of the TIMSS study to examine how these instructional methods can be applied.
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