by Thomas W. Judson
November 2001
In Japan, as in the United States, calculus is a gateway course that students must pass to study science or engineering. Japanese educators often voice complaints similar to those that we made about students' learning of calculus in the 1970s and 1980s. They believe that many students learn methods and templates for working entranceexamination problems without learning the concepts of calculus. University professors report that the mathematical preparation of students is declining and that even though Japanese middle school students excelled in mathematics in TIMSSR, these same students expressed a strong dislike for the subject.
Japan has a national curriculum that is tightly controlled by the Ministry of Education and Science. In Japan, grades K–12 are divided into elementary school, middle school, and high school; students must pass rigorous entrance examinations to enter good high schools and universities. After entering high school, students choose either a mathematics and science track or a humanities and social science track. Students in the science track take suugaku 3 (calculus) during their last year of high school; most of them take a more rigorous calculus course at the university.
The course curricula for AP Calculus BC and suugaku 3 are very similar. The most noticeable differences are that Japanese students study only geometric series and do not study differential equations. The epsilondelta definition of limit does not appear in either curriculum.
In the spring and summer of 2000, Professor Toshiyuki Nishimori of Hokkaido University and I studied United States and Japanese students' understanding of the concepts of calculus and their ability to solve traditional calculus problems. We selected two aboveaverage high schools for our study, one in Portland, Oregon, and one in Sapporo, Japan. Our investigation involved 18 students in Portland and 26 students in Japan. Of the 16 Portland students who took the BC examination, six students scored a 5. We tested 75 calculus students in Sapporo; however, we concentrated our study on 26 students in the A class. The other two classes, the B and C groups, were composed of students of lower ability. Each student took two written examinations. The two groups of students that we studied were not random samples of high school calculus students from Japan and the United States, but we believe that they are representative of aboveaverage students. We interviewed each student about his or her background, goals, and abilities and carefully discussed the examination problems with them.
Since we did not expect Japanese students to be familiar with calculators, we prohibited their use on the examinations. However, the students in Portland had made significant use of calculators in their course and might have been at a disadvantage if they did not have access to calculators. For that reason, we attempted to choose problems that were calculator independent. However, some problems on the second examinations required a certain amount of algebraic calculation.
We used problems from popular calculusreform textbooks on the first examination. These problems required a sound understanding of calculus but little or no algebraic computation. For example, in one problem from the Harvard Calculus Project, a vase was to be filled with water at a constant rate. We asked students to graph the depth of the water against time and to indicate the points at which concavity changed. We also asked students where the depth grew most quickly and most slowly and to estimate the ratio between the two growth rates at these depths.
We found no significant difference between the two groups on the first examination. The Portland students performed as expected on calculusreformtype problems; however, the Sapporo A students did equally well. Indeed, the Sapporo A group performed better than we had expected. We were somewhat surprised, since the Japanese students had no previous experience with such problems. The performance of Japanese students on the first examination may suggest that bright students can perform well on conceptual problems if they have sufficient training and experience in working such problems as those on the university entrance examinations.
The problems on the second examination were more traditional and required good algebra skills. For example, we told students that the function f(x) = x^{3}+ ax^{2} + bx assumes the local minimum value—(2 )/9 at x = 1/—and asked them to determine a and b. We then asked them to find the local maximum value of f(x) and to compute the volume generated by revolving the region bounded by the xaxis and the curve y = f(x) about the xaxis. The Sapporo A students scored much higher than the Portland students did on the second examination. In fact, the Portland group performed at approximately the same level as the Sapporo C group and significantly below the Sapporo B group. Several Japanese students said in interviews that they found that certain problems on the second examination were routine, yet no American student was able to completely solve these problems. The Portland students had particular difficulty with algebraic expressions that contained radicals. Several students reported that they worked slowly to avoid making mistakes, possibly because they were accustomed to using calculators instead of doing hand computations.
Students from both countries were intelligent and highly motivated, and they excelled in mathematics; however, differences were evident in their performances, especially in algebraic calculation. One of the best Portland students correctly began to solve a problem on the second examination but gave up when he was confronted with algebraic calculations that involved radicals. On his examination paper he wrote, "Need calculator again."
Perhaps the largest difference between the two groups lies in the different high school cultures. Japanese students work hard to prepare for the university entrance examinations and are generally discouraged from holding parttime jobs. In contrast, students in the United States often hold parttime jobs in high school, and many are involved in such extracurricular activities as sports or clubs.
Thomas Judson is a visiting assistant professor at the University of Puget Sound in Tacoma, Washington. He is interested in mathematics education and has spent parts of the last eight summers visiting Japan to learn Japanese and study mathematics education there.

