Chicago Shimpo
Japanese Mathematics Education Explained
By Professor Akihiko Takahashi

• The way mathematics is taught in Japan holds the key to the high-level performance of Japanese students, a DePaul University professor explained during a seminar in Chicago on August 29.

• The seminar, “Japanese Mathematics Education, How Japanese Mathematics Instruction Creates High-Performing Students” was hosted by the Japan America Society of Chicago. Akihiko Takahashi, Ph.D. and an Associate Professor at DePaul University in Chicago, was the featured speaker.

• According to international studies, math scores of Japanese elementary and middle school students are consistently high. A recent OECD study shows that not only Japanese students score high in math but gaps among them are much smaller than in the U.S. Why is that?

• Takahashi, who teaches mathematics and mathematics education, shared that there exists a substantial difference between the U.S. and Japan concerning the idea about teaching math. The Japanese approach, he argued, fosters problem-solving and reasoning capabilities, “better exemplifying” the current U.S. reform ideas for math education.

• Between the late 1990s and the early 2000s, the U.S. Department of Education conducted a video survey, the Third International Mathematics and Science Study (“TIMSS”), on 8th graders in the U.S., Germany and Japan to see how mathematics was being taught in each country.
• The survey studied 81 classrooms in the U.S., 100 in Germany and 50 in Japan.

Japanese Approach

• The Department found that the teaching approach in the U.S., while it was identical across the nation, was quite different from the other two countries (The Teaching Gap, by J. Stigler and J. Hiebert, 1999).
• According to Stigler and Hiebert, during the survey in Japan, the researchers noticed that the teacher presented a problem to the students “without first demonstrating how to solve the problem,” which the counterparts in the U.S. “almost never” did. In contrast, a U.S. teacher usually demonstrated “a procedure for solving a problem before assigning it to the students.”
• Researchers then suggested that it would be important to let the students work by themselves, but many U.S. teachers “hesitated” to do so.
• During the 1980s, Takahashi said, many U.S. studies encouraged Japanese educators to focus on problem solving as the “main instructional strategy,” and they seriously read the studies and implemented the suggestion. By the early 1990s, they had completed the shift and elementary schools began using textbooks that were designed with a problem-solving approach.

Disasters of “Well-Taught” Math

• Alan H. Schoenfeld of University of California-Berkeley argued in his 1988 book, Learning to Think Mathematically, that “well-taught” students often end up in “disasters,” regarding development of students’ ability on discovery or invention.
• Taught formal math, students often fail to use information from it when they are required to solve a problem. When they can’t solve it in five minutes, they would stop working on it and move on.
• Takahashi said that some teachers have intentionally taught to skip it and go on to the next.

• Schoenfeld also argued that students, when failing to solve a problem, may resort to thinking that “only geniuses are capable of discovering, creating or really understanding math; I don’t understand math, but it’s OK.”
• When students accept what’s passed down “from above” like this, it fosters a passive attitude, without expectations that they can make sense of it for themselves.
• Takahashi said, “That’s why we should give students an opportunity to think more about math.”

Japan’s Shift to Problem Solving

• By the early ‘90s, Japan’s math education “completely shifted” to teaching the problem solving approach.
• According to the 2003 TIMSS survey, the international average of correct answers on math topics taught in school (4th grade), either prior to or during the year of the assessment, was 53% (an average of 73% were taught or learned the topics).
• Among the surveyed, 82% of Singaporean students were already taught the topics, with the correct answer of 74%.
• In the U.S., the correct answer was 58%, while 82% were already taught the topics.
• In Japan, the correct answer was 69%, while 54% were already taught the topics.
• This result indicates, Takahashi noted, that 4th graders in Japan are now used to problem solving – when they were given a problem they hadn’t been taught in class, they tried to solve it for themselves nonetheless. That’s one of the powers of problem solving, he said.

• According to the OECD’s 2015 Education Policy Outlook, Japanese students were among the top performers in reading, math and science. They weren’t the highest in the most recent study of March 2019, but were still high enough.
• The impact of socio-economic and cultural status on student performance in Japan was 9.8%, lower than the OECD average of 14.8%. Such status variance does not seem to have much impact on student performance in Japan.

• In the U.S., the survey showed that the proportion of children from disadvantaged backgrounds achieving at least PISA (the Program for International Student Assessment) level 2 in mathematics was 40% lower than that of the most advantaged peers.
• The gap was also visible in terms of location: the proportion of children from rural areas achieving at least PISA level 2 in mathematics was 16% higher than that of children from urban areas.

“Thinking Mathematically”

• According to the 1982 book Thinking Mathematically by J. Mason, L. Burton and K. Stacey, everybody can think mathematically, and mathematical thinking can be improved through practice with reflection. Furthermore, mathematical thinking is supported by an “atmosphere of questioning, challenging and reflecting.” It can also be provoked by contradiction, tension and surprise.
• Takahashi said that many teachers want to avoid such things in their classroom. But studies show that those elements are important for students to develop mathematical thinking.

• For example, consider a problem like this (see Figure 1).
• You are supposed to come up with a way to cut the shown figure into two equal-sized parts with a straight line. Can you do it?
• You can, as Takahashi suggested, look at the figure as a combination of two squares, one small and one large. A square can be divided into two equal parts with any straight line that goes through the center point (think about folding origami using a square origami paper). Once you picture this irregular shape as a combination of two squares, you may be led to possible solutions, with any straight lines that go through both of the center points of the two squares (see Figure 2).
• It’s not difficult when you are told the solution, but it’s not so easy to come up with it on your own. And just telling the solution doesn’t help students.
• “The purpose of teaching math is to provoke students to develop this type of thinking,” Takahashi explained. “It’s not about memorizing correct answers; it’s about helping students find various different ways to solve problems, and process is more important in this approach than getting answers.”

• Takahashi is critical about U.S. math textbooks that emphasize answers too much, and how to get the correct answer. He wants to spend more time on how you “find and appreciate a variety of solutions.” Once you find them, you can apply them to a new situation.
• In Japan, educators say that when one student has found an answer, real mathematics begins, according to Takahashi.
• That’s when an open discussion starts about how you came to understand the equation. And an equation is a way to express your thinking, a communication tool.
• Takahashi noted that they use a blackboard in Japan for this type of discussion, where all different ideas are written and compared. The writing on the board becomes a visual tool for students to come up with ideas for multiple solutions. In doing so, the students are encouraged to take notes.
• And it is the teacher’s important role to visualize the ideas put forth by the students.

Three Levels of Math Teaching

• Lastly, Takahashi talked about the three levels in math teaching expertise, quoting from Lesson Study Alliance.

• Level 1 is where the teacher can convey to the students important basic ideas of mathematics such as facts, concepts, procedures and practices.
• In this level, anyone can be a teacher if they know math enough – even a high school student can be a teacher of their younger siblings. But you can’t call such a person a professional.

• Level 2 is where the teacher can explain the meaning of and logic behind important basic concepts and practices of mathematics so that a student can understand them.
• In this level, the teacher must be able to explain, for example, why you begin the division of 432 by 3 with the left-most digit of 432 and move on to the right.

• It involves calculation with paper and pencil. Why do teachers spend so much time to teach calculations with paper and pencil, when we can get the answer easily by using a calculator?
• One critical point of using paper and pencil is to develop a “number sense,” Takahashi said.
• Also important is the fact that using paper and pencil helps a student seize the relationship between numbers in addition, subtraction, multiplication and division.
• Level 2 also requires an understanding of the mechanism behind any number manipulation, such as the multiplication table, rather than resorting to simple memorization.

• Finally, Level 3 is the professional level. This is where the teacher can “provide students with opportunities to understand basic mathematical contents and develop mathematical practices, and support student learning so that they become independent learners.”
• A Level 3 teacher does not necessarily teach math, but designs a lesson that will help students think mathematically and collaborate what they have learned.
• This level of teaching is clearly apart from the other two levels, and is difficult to achieve. Takahashi said that the U.S. classrooms are “almost always” in Level 1 – or Level 2, “if you get lucky.”
• In Japan, a shift occurred from Level 2 to 3 during the ‘80s and ‘90s, he added.



kihiko Takahashi is an Associate Professor at DePaul University where he teaches mathematics and mathematics education. Before coming to the U.S., he was a teacher in Japan and nationally active in mathematics education reform. He received his Ph.D. from the University of Illinois at Urbana-Champaign and has published over 80 journal articles and book chapters in English and Japanese. Professor Takahashi has been serving as an Honorary Reader of University College London and Specially Appointed Professor of Tokyo Gakugei University in Tokyo.

Dr. Akihiko Takahashi, Associate Professor at DePaul University, explains Japanese Mathematics Education.

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