Develop

Mathematical Thinking) and has been one of the subjects of interests, together

with lesson study, in different comparative studies and teacher training

programs for developing countries since 1980’s (Isoda, 2010). The Japanese

Problem Solving Approach has been influenced by U.S. research on problem

solving (Takahashi, 2006), however the U.S. problem solving approach and the

Japanese problem solving approach were found to be different. The former

focused on process of solving a problem and not on developing concepts and skills.

Traditionally, problem solving is used in lessons where the main goal is

developing problem-solving skills and strategies, taught after teaching the

lesson content. It can also be included at the start of the lesson as a

motivation, as recreational problems to make math more fun, as means to

introduce new topic, or as a real-life problem to make the students realize the

value of mathematics (Schoenfeld, 1992). In this case, problem solving is not

the goal itself, but just a means to reach other goals. However, in Japan,

problem solving is not only used for lessons wherein the goal is to develop

problem-solving skills and strategies. It is used throughout the curriculum

mainly to develop mathematical concepts, skills and procedure.

Based on

Stigler and Hielbert’s classroom video study (TIMSS video 1999), this pattern

in Japanese Mathematics teaching was observed: reviewing the previous lesson; presenting

the problem for the day; students working individually or in groups discussing

solution methods; and highlighting and summarizing the major points (Hino,

2007). Similarly, Isoda (Isoda, 2012, Introductory Chapter: Problem Solving

Approach to Develop Mathematical Thinking) described the structure of the

Japanese lesson with the following phases: “posing the problem”, followed by

“planning and predicting the solution”, “executing solution/independent

solving”, “explanation and discussion/validation and comparison, and

“summarization/application and further development”. In another setting, it was

observed that a typical mathematics lesson starts with the students’ rising and

bowing, followed by reviewing previous day’s problems or introducing a

problem-solving topic, understanding the topic, problem solving by students (by

pair/group), comparing and discussing, summing up by the teacher, assigning

exercises (to be done outside class hours), and students rising and bowing at

the end of class (Becker, et. al, 1990). A similar pattern was observed also in

a junior high school class in 2000. The teacher started by posing a problem

which the students first solved individually. The class was then divided into

groups and the students worked on the problem as a group. Then it was followed

by a class discussion and at the end, the teacher compared, synthesized

different solutions presented and summarized the lesson to the whole class.

During the period, the teacher just played the role of a facilitator, giving

guidance and instructions to the students (Kunimune, Sa’dijah, 2000). This is

in contrast to the lesson pattern identified in US, where students work on

problems after the teacher’s demonstration, and in Germany, where students work

after the teacher directs the students to develop procedures for problem

solving (Hino, 2007).

Math

lessons in Japan were observed to follow common theme (Becker, et.al, 1990).

First is having a single objective (It can also be observed in most of the

Japanese math textbooks). In this way, all class activities are focused on the

objective (working on 2-4 carefully selected problems only) and really reaching

the objective of the lesson at the end of the period. It is said that teachers

do not return to the same discussion the next day. The second is having

extensive discussion, with the teacher highlighting the different solutions to

a problem. They also have an intensive curriculum and instruction is intensive.

Lastly, teacher-student interaction does not distinguish gender differences and

there seems to be no differences between boys’ and girl’s attitude toward

learning.

Given the nature of Problem Solving Approach,

the Japanese mathematics lesson has the following major characteristics: (1)

carefully selected word problems and activities, and their cohesiveness, (2)

extensive ‘discussion’ (neriage), and (3) emphasis on blackboard practice

(Bansho). (Becker et. al, 1990; Stevenson & Stigler, 1992; Stigler and

Hiebert, 1999; Stigler, 1987; Stigler et. al. 1999, as mentioned by Takahashi,

2006). It is through the “excellent use of figures (colored chalk and posters)

on the chalkboard, the classroom management of at least 40 students the

purposeful work of students in groups, and the deliberate pausing and

explaining by the teacher” that makes a Japanese math lesson effective (Becker,

Silver, et. al, 1990). Isoda emphasized the difference between tasks and

problems; tasks are given by the teacher while problems come from the students,

what the children would like to do next must also be taken into account.

Through Problem Solving Approach, students will develop and appreciate

mathematical concept, able to generate ideas of mathematics, and develop ways

of thinking through exploring problems posed by them. “As students solve

problems, they can use approach they can think of, draw on any piece of

knowledge they have learned, and justify their ideas in ways that they feel are

convincing. The learning environment of teaching through problem solving

provides a natural setting for students to present various solutions to their

group or class and learn mathematics through social interactions, meaning

negotiation, and reaching shared understanding. Such activities help students

clarify their ideas and acquire different perspectives of the concept or idea

they are learning” (Cai, Lester, 2010)”. Finally, another important element of

the Japanese teaching is the effective use of blackboard containing a summary

of children’s ideas and presentations. In Japanese lesson, comparison