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.
Mathematicians often work for a time on a single problem.
Mathematicians may spend months and years thinking about and working on a proof to one problem. To enhance their problem-solving abilities students need to be offered fewer problems and more time to complete them. In reality, children are often given many problems and a short time limit in which to complete them. They should be allowed ample time to work on a few meaningful problems, or even just one, rather than a worksheet of 20, 50, or even 100 problems (Kamii et al., 2000).
In the video of the U.S. eighth-grade geometry class, over 100 geometry questions were asked and answered in rapid-fire succession. When students were asked to do problems alone in class, they were given over 40 problems to complete in just 20 minutes (Kamii & Institute for Development of Educational, 1996; Kamii et al., 2003). In contrast, the Japanese class worked on only three problems during the entire 50-minute class period. The students were given ample time to think and experiment with many different methods of achieving a solution. They were also given only one problem for homework, which was derived from their discussion during class time.
Mathematicians collaborate with their colleagues and study the work of others.
Social interaction is one of the most important aspects of being a mathematician. A mathematics classroom should include many opportunities for social interaction (Kamii & National Association for the Education of Young Children, 1982). Piaget felt that the simple act of one child explaining his problem-solving method to another caused the child to understand his own thought process better. Have you ever had the experience of explaing something to someone and realizing you had made an error halfway through the explanation? Vygotsky, on the other hand, felt that the social interaction needed to be based on a peer-tutoring model—a more experienced peer can help a less experienced peer work to solve a problem more efficiently. Both of these views can be developed into actual classroom practice by having children interact with each other, argue and defend their answers, and tutor their peers when they need help. If children are going to be viewed as young mathematicians, they must be allowed to collaborate, argue, consult, defend, ask, explain, and pose questions to and with other students using mathematical ideas (Kamii, Lewis, & Jones, 1991, 1993).
Traditional U.S. mathematics lessons and homework are designed to be solitary acts. Children are not encouraged to defend solutions or collaborate on solving problems. Instead, they are given individual practice worksheets and asked to complete them quietly. Without interaction, children simply memorize how to get the correct solution, without developing a greater understanding. While memorization often helps short-term test scores, it harms long-term conceptual understanding. This deficiency is evident in the TIMSS scores (Kamii, 1990; Kamii & Institute for Development of Educational, 1996).
Mathematicians must prove for themselves that their solution is correct and so should children.
Mathematicians must question assumptions and understand the mathematics behind their answers. They must prove to themselves and others that their solutions are correct, and why. If students are taught merely to memorize answers and constantly rely on their teachers to tell them whether or not they are correct then they are denied this important process of proving a solution (Weinstein, 2002).
In the TIMSS videotape study, the U.S. classroom primarily used a call- and- response model. The teacher would call out the problem and a student would respond with an answer. If his response was incorrect, the teacher would ask rhetorically, “Are you sure about your answer?” He would then call on another student to give the correct answer and reinforce that answer by repeating it and saying, “Right” or “Correct.” By contrast, in the Japanese classroom, after the students had been given time to think and discuss their answers, they were asked to present and defend their solutions to the class. The teacher did not tell them whether they were correct but asked the class if they understood how a student got his answer. They were then required to reason and prove their answers.
The problems that mathematicians work on are complex.
Complex problems promote problem-solving abilities. Children, like mathematicians, should be immersed in complex problems that require mathematical problem solving and complex numerical thinking. Good problems ask students to find innovative solutions without a time limit being set on their thinking process (Kamii et al., 2000; Kamii et al., 2003; Kamii, Knight, & Teachers, 1990). Problems can and should spark discussion and even disagreement among students.
This does not mean that problems have to be overly complicated or contrived, but they should be designed to spark children’s thinking processes, rather than simply promoting rote memorization and repetition. Marilyn Burns (1992) outlined the four criteria to designing a good math problem:
- It presents a perplexing situation that the student can understand.
- The student is interested in finding a solution.
- The student is unable to proceed directly toward a solution.
- The solution requires application of mathematical ideas.
With these simple criteria, teachers can develop mathematics problems that are engaging and promote the mathematical thinking process. The “horse” problem you were asked to present earlier is a good problem from this perspective. While it may seem since there are numerous ways to solve it, that this problem is too advanced for first-graders, they have a great time trying to figure it out.
Mathematicians get satisfaction from the problem-solving process and take pride in their solutions.
Children will understand mathematical concepts and procedures more thoroughly if they are allowed to use their own thinking processes to explore mathematics (Kamii, Lewis, & Jones, 1993). It allows them to make connections to prior knowledge and real life experiences. In the process of discussing and comparing the different methods they use to reach solutions, children strengthen their understanding of both concepts and procedures.
At the end of the lesson in the Japanese classroom the students discussed the days’ problem with the teacher. They asked if the procedures they learned could be applied to other similar situations. The teacher replied by asking them to think about it and try some alternatives at home that evening, and they would discuss them in class the next day. In essence, the students developed their own homework and were excited by both the process and the content of the days’ work. They demonstrated a sense of great accomplishment.
Children can get very excited about a mathematics problem and take pleasure in the process of problem solving. If children are allowed to think for themselves and discuss and defend their ideas, mathematics becomes just as fun as trying to solve a video game or diligently working to put a puzzle together.
Mathematicians use unsuccessful attempts as stepping-stones to solutions.
If we really want children to be treated as mathematicians, we need to encourage them to realize that they may have to try many different approaches, some unsuccessfully, before they reach a solution. We need to place an emphasis on the valuable mathematical thinking going on in the child’s mind rather than on the production of a correct answer. It should be emphasized and modeled to children that unsuccessful attempts and errors can be stepping-stones to solutions.
In the U.S. classroom, students did not have time to go through this process. They were given one chance to get the solution right and the thinking behind incorrect solutions was never discussed or examined. Incorrect attempts at providing an answer would often silence a student’s participation in the lesson. The Japanese lesson, by contrast, was designed in such a way that students could experiment with different solutions and then discuss them with other students or with their teacher. They were given time to use different methods and prove their answers to themselves and others.
Negative responses to answers can have disastrous consequences for the way children interact with mathematics. Math anxiety can develop if failure leads to shame or if students perceive too high a risk factor (Burns, 1998; Kitchens, 1995; Stuart, 2000). Math should be presented as a process where wrong answers are a natural and recurring element, not an end point. Instead of getting a problem “wrong,” children should understand progress is being made towards a correct solution, and they should be encouraged to continue working (Verzosa, 2001; Third International Mathematics and Science Study, 1997).
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