CHARACTERISTICS OF JAPANESE MATHEMATICS LESSONS
Akihiko Takahashi
DePaul University
Japanese mathematics lessons, especially for elementary grades, include a significant
amount of problem solving. This instructional approach, called structured problem
solving, is designed to create interest in mathematics and stimulate creative
mathematical activity in the classroom through students’ collaborative work. The
lesson usually starts with students working individually to solve a problem using theirown mathematical knowledge. After working with problems, students bring various
approaches and solutions to classroom discussion. The teacher then leads students in awhole-class discussion in order to compare individual approaches and solutions. This
whole-class activity provides students with opportunities to develop theirmathematical abilities including conceptual and procedural understanding.
THE REFORM MOVEMENT IN TEACHING AND LEARNING
MATHEMATICS
Teaching mathematics through lectures may be an easy instructional method forteachers. When students are passively listening to teachers, however, theiropportunities to understand mathematical concepts and procedures are not maximized.
Rather than just listening to teachers talk, students need to be actively involved inmathematics and to do mathematical activities (Brown, 1994).
In Japan, the major reform movement in teaching and learning mathematics occurred
during the 1970s and 1980s (Takahashi, 2000). One of the major aspects of this reformmovement was to shift from a traditional classroom that focuses on teachers’
instruction, to a student-centered classroom that focuses on students’ engagement inmathematical activities. During this reform movement, Japanese mathematics
educators and teachers worked collaboratively to find ways to implement the ideas ofreform mathematics teaching and learning by referring to various documents publishedin the U.S. Those references included the National Research Council’s (1989)“Everybody Count: A Report to the Nation on the Future of Mathematics Education,”and the National Council of Teachers of Mathematics’ “Curriculum and Evaluation
Standards for School Mathematics” (National Council of Teachers of Mathematics,
1989). A basic assumption of the reforms is that students can learn by constructing
their own conceptions of mathematics (National Research Council, 1989). In otherwords, students are viewed as active constructors of knowledge, rather than passive
recipients of it (Brown, 1994). Based on the TIMSS videotape classroom study (1997),
Stigler and Hiebert argue that Japanese mathematics lessons better exemplify current
U.S. reform ideas than do typical U.S. mathematics lessons (1999).

CHARACTERISTICS OF JAPANESE MATHEMATICS LESSONS
Student-centered instruction using problem solving as a foundation
Although images of good practice might not be exactly the same among all Japanese
educators, most of them would agree that Japanese school mathematics has beenstrongly influenced by the emphasis of problem solving as a good practical applicationof reform mathematics.
Stigler and Hiebert described Japanese mathematics lessons as “structured problem
solving” by summarizing several characteristics of Japanese mathematics lessons.
In Japan, teachers appear to take a less active role, allowing their students to invent theirown procedures for solving problems. And those problems are quite demanding, bothprocedurally and conceptually. Teacher, however, carefully design and orchestrate lessons
so that students are likely to use procedures that have been developed recently in class. An
appropriate motto for Japanese teaching would be “structured problem solving” (J. Stigler& Hielbert, 1999, p. 27)

Similar characteristics of Japanese mathematics lessons were also reported in theproceedings of the U.S.-Japan Seminar of Mathematical Problem Solving (Jerry P.
Becker & Miwa, 1987; Jerry P Becker, Silver, Kantowski, Travers, & Wilson, 1990).
Japanese structured problem solving was built on the firm foundation of emphasizing
story problems in mathematics teaching and learning. Historically, Japanesemathematics teaching and learning has been focused on developing mathematicalthinking skills by using a variety of story problems. In fact, teaching patterns similar tothe ones that Becker reported (Jerry P Becker, Silver, Kantowski, Travers, & Wilson,
1990) were found not only widely throughout Japan but also in earlier documents.
Those earlier documents include publications as early as 1937 (Minoru Yoshida, 1992).
Based on the existing resources of story problems and of lesson plans focusing on
promoting mathematical thinking, Japanese teachers, researchers, and administrators
worked collaboratively through Lesson Study, a professional development approachthat is popular in Japan, to develop mathematics instruction by referring to Polya’s(1945) four phases of problem solving work (Takahashi, 2000). Studies of U.S.
documents of mathematical problem solving that focused on teaching mathematicalthinking skills (Lester & Garofalo, 1982; Schoenfeld, 1985) also influenced theJapanese mathematical reform movement.
Although Japanese structured problem solving has been influenced by U.S. research on
problem solving, it is not the same as the problem solving approach used in the U.S. In
the U.S., problem solving is often viewed as an approach to develop problem-solving
skills and strategies. As a result, U.S. mathematics lessons employing the problem
solving approach are usually focused on the process of solving a problem and notnecessary focused on developing mathematical concepts and skills. These
problem-solving lessons often end when each student comes up with a solution to the
problem.

In Japan, on the other hand, problem solving is often viewed as a powerful approachfor developing mathematical concepts and skills. Thus, Japanese teachers use problem
solving not only for lessons that focus on developing problem-solving skills andstrategies but also throughout the curriculum in order to develop mathematicalconcepts, skills, and procedures.
Structured problem solving
Structured problem solving is a major instructional approach in Japanese mathematics
education. Structured problem solving is designed to achieve two extremely importantgoals that are essential to the Japanese mathematics reform curriculum: to create
interest in mathematics, and to stimulate creative mathematical activity in theclassroom during the collaborative work of students.
This instructional approach emphasizes the process of problem-solving activities andprovides students with opportunities to re-invent mathematical ideas and concepts bythemselves. This is why the lesson usually starts with students working individually tosolve a problem using their own mathematical knowledge. After working withproblems, students bring to classroom discussion several different approaches andsolutions. The teacher then leads students in a whole-class discussion in order tocompare individual approaches and solutions. This whole-class activity provides
students with opportunities to develop their mathematical abilities including
conceptual and procedural understanding (Figure 1).

Figure 1: Structured Problem-Solving
Using structured problem solving as a major instructional approach, Japanesemathematics lessons bear notable characteristics (Jerry P Becker, Silver, Kantowski,
Travers, & Wilson, 1990; Stevenson & Stigler, 1992; J. Stigler & Hielbert, 1999; J. W.
Stigler, 1987; J. W. Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1999). Amongthese characteristics, the following three major characteristics will be discussed in thispaper.
•••

Carefully selected word problems and activities, and their cohesiveness
Extensive discussion (Neriage)
Emphasis on blackboard practice (Bansho)

Carefully selected word problems and activities, and their cohesivenessTypically, each Japanese mathematics lesson is designed around solving a single
problem to achieve a single objective in a topic. In order for students to achieve theobjective, the teacher carefully selects a problem an activity for the day. It is rarely
seen that a lesson includes two or more problem-solving activities, although teachers
might give students a couple of extended problem or exercises after a major problem
solving. Thus, the selection of a problem for the problem-solving activity in each classis extremely critical for teachers when they plan a lesson.
When we look closely at Japanese mathematics textbooks, the use of carefully selected
problems and activities and their cohesiveness emerge. The following examples arefrom one of the elementary grade textbook series1 in Japan.

Figure 2: Japanese fourth grade mathematics textbook (4B) pp.22-23
To develop the concepts and skills for finding the area of basic figures, the Japanese
textbook includes an introductory activity in the first grade. The page is designed to
introduce the concept of direct comparison, indirect comparison, and comparison using

1

The textbook series that is referred in this paper is the English translation of the textbook series for 1st grade to 6th grade
published by the Tokyo Shoseki based on the 1989 National Course of Study translated by the Global Education Resources,
NJ (www.globaledresources.com).

an arbitrary unit. The page is also designed to provide mathematical situations fromstudents’ daily life in order for them to become familiar with comparing area.
Building upon the activity in the first grade, the fourth grade textbook includes a unit
designed for students to develop the concepts of finding the area of rectangles andsquares by using multiplication formulas. This unit begins with a problem fromstudents’ familiar context. The rectangle and the square are given to students tocompare which is bigger (Figure 2). The sizes of these two geometric figures arecarefully chosen. For example, the perimeters of these two figures are the same but
their areas are different by the area of a square with 1 cm sides. This is used because
students tend to be confused by a typical misunderstanding that the areas of shapes canbe determined by their perimeters. In order for students to overcome thismisunderstanding, the textbook intentionally chose two shapes with the same
perimeters but different area.
Through this problem-solving activity, students are expected to use their priorknowledge to compare the area of the rectangle and the square by using the area ofsquare with 1 cm sides as an arbitrary unit. Then, measuring the areas of the rectangle
and the square by using 1 cm2 is introduced as measurement by a universal unit.
This unit also includes problem-solving activities to extend their capacity to use theseformulas on irregular shapes (Figure 3). This problem-solving activity is designed toprovide students with an opportunity to apply their previous learning, which is theformula for finding the area of a rectangle and a square, to a new situation. This
experience will be the foundation for developing the formulas for finding the areas ofother basic shapes in fifth grade.

Figure 3: Japanese fourth grade mathematics textbook (4B) p.28

Building upon what they learned in the fourth grade, students in the fifth grade are
given opportunities to develop the formulas for finding the area of a parallelogram,triangle, trapezoid, rhombus, regular polygon, and circle. Throughout the fifth gradestudents are expected to develop formulas for finding these basic figures throughhands-on activities such as cutting and rearranging shapes on grid papers.
The sequential order of the problems and activities in each unit is carefully designed inorder for students to develop their concepts and skills. A unit in Japanese mathematics
textbooks is considered as a series of problems and activities rather than a set ofproblems and activities.
Extensive discussion (Neriage)
In the structured problem-solving approach, Japanese teachers emphasize that one of
the most important roles of the teacher during a lesson is to facilitate mathematical
discussion after each student comes up with a solution. When the teacher presents a
problem to students without giving a procedure, it is natural that several different
approaches to the solution will come from the students. Thus, the textbooks include
examples of students’ typical approaches and ideas. Because the goal of the structured
problem-solving approach is to develop students’ understanding of mathematicalconcepts and skills, a teacher is expected to facilitate mathematical discussion forstudents to achieve this goal. This discussion is often called Neriage in Japanese,
which implies polishing ideas. In order to do this, teachers need a clear plan for the
discussion as a part of their lesson plans, which will anticipate the variety of solution
methods that their students might bring to the discussion. These anticipated solution
methods will include not only the most efficient methods but also ones caused by
students’ misunderstandings. Thus, anticipating students’ solution methods is a majorpart of lesson planning for Japanese teachers.
Towards the end of a lesson, a teacher often lead the lesson to pull all the different
approaches and ideas together to see the connection. Then, he or she summarizes thelesson to help students achieve the objective of the lesson. The teacher often asks
students to reflect on what they have learned during the lesson.
Emphasis on blackboard practice (Bansho)
Another notable characteristic of Japanese mathematics lessons is the use ofmathematical expressions, figures, and diagrams on a large-size blackboard (Jerry P
Becker, Silver, Kantowski, Travers, & Wilson, 1990; J. Stigler & Hielbert, 1999).
Japanese classrooms are equipped with a large blackboard at the front. Yoshida (2005)
summarizes how Japanese teachers use the blackboard during mathematics lessons asfollows:•••

To keep a record of the lesson
To help students remember what they need to do and to think about
To help students see the connection between different parts of the lesson andthe progression of the lesson•
To compare, contrast, and discuss ideas that students present•
To help to organize student thinking and discovery of new ideas•
To foster organized student note-taking skills by modelling good organization
Discussion that compares and synthesizes several different solution methods demands

Figure 4: Use of blackboard (Bansho)that students not only clarify the idea behind each method and justify the adequatenessof each one. It is also important to examine the limitations of each method. To facilitatesuch discussion, Japanese teachers use the blackboard as a visual aid for students toparticipate in discussion throughout all the grades while considering student level ofunderstanding of mathematics as well as their communication skills (Figure 4). Since
this blackboard practice is an important skill for teachers to develop, Japanese teachers
use a special term, Bansho, to discuss issues regarding the board writing skill. Thus,
developing a plan for using the blackboard is another major component of lesson
planning. In fact, some schools choose developing teachers’ skills for using theblackboard effectively as a school goal when conducting school-based professional
development.
CONCLUSION
Japanese mathematics teaching, especially for elementary grades, includes asignificant amount of problem solving in order to provide students the environment toconstruct their understanding of concepts and procedures in mathematics. Although
Japanese textbooks exemplify the characteristics of mathematics lessons, thosecharacteristics do not solely came from the textbook authors’ effort. It is important tonote that the characteristics of Japanese mathematics lessons are the result of
collaboration among Japanese educators, researchers, and policymakers, including
efforts from the lesson study practitioners. Moreover, the effort to improve
mathematics teaching and learning is still ongoing.

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