SOME OBSERVATIONS OF MATHEMATICS TEACHING IN JAPANESE ELEMENTARY AND JUNIOR HIGH SCHOOLS

SOME OBSERVATIONS OF MATHEMATICS TEACHING IN JAPANESE ELEMENTARY ANDJUNIOR HIGH SCHOOLSAuthor(s): Jerry P. Becker, Edward A. Silver, Mary Grace Kantowski, Kenneth J. Travers andJames W. WilsonSource: The Arithmetic Teacher, Vol. 38, No. 2 (OCTOBER 1990), pp. 12-21Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194689 .

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SOME OBSERVATIONS OF MATHEMATICS TEACHING IN

JAPANESE ELEMENTARY AND JUNIOR HIGH SCHOOLS Jerry P. Becker, Edward A. Silver, Mary Grace Kantowski, Kenneth J. Travers,

and James W. Wilson

U.S. -Japan Seminar on Math- ematical Problem Solving was held at the East- West Center

in Honolulu 14-18 July 1986 (Becker and Miwa 1987). Among the seminar's proposals was that cross-cultural research on American and Japanese students' problem-solving behaviors be organized and carried out. The authors were in Japan in the fall of 1988 to meet with their Japanese counter- parts, plan research, and make visits to mathematics classrooms prelimi- nary to conducting a two-year program of research. In addition to planning the research, we were on a fact-finding visit to classrooms to better acquaint ourselves with mathematics teaching and learning in Japan.

Introduction Clearly, education has a high priority in the Japanese culture; both mathematics and the national language are central to this enterprise. The entrance of children into grade 1 of the elementary school (grades 1-6) is a very serious step, which is preceded

by mothers' visits to their children's school to learn of its expectations in terms of both knowledge and behavior. Preparations, such as arranging study space and buying supplies, are

made in the home before the important and ceremonial first day at school, which involves a welcome by school officials, community leaders, and fifth and sixth graders.

Jerry Becker teaches and does research in the curriculum and instruction department at Southern Illinois University at Carbondale, Carbondale, IL 62901-4610. His main interests are students' problem-solving behaviors and international mathematics education. Edward Silver conducts research on mathematics learning and problem solving at the Learning Research and Development Center, university of Pittsburgh, Pittsburgh, PA 15260. He's also a professor of mathematics education and cognitive studies in the School of Education, University of Pittsburgh, where he teaches doctoral courses. Mary Grace Kantowski is a professor of mathematics education at the University of Florida, Gainesville, FL 32611, and serves on the Editorial Board of the Journal for Research in Mathematics Education. Her professional interests include research in problem solving, using the computer to teach mathematics, and training middle and secondary school mathematics teachers. Kenneth Travers is professor of mathematics education at the University of Illinois at Urbana-Champaign, Champaign, IL 61820. His professional activities include cross-national research in curriculum and instruction. He was chair of the International Mathematics Committee for the Second International Mathematics Study. Jim Wilson is professor and head of the department of mathematics education, University of Georgia, Athens, G A 30602. His interests are mathematical problem solving and teacher education. He is a former member of the Board of Directors ofNCTM.

But even before entering grade 1, children are made "ready" for school in the home. The importance of school education is stressed in the home during the preschool years, which helps form children's attitudes and habits of mind. This emphasis

This article is based on work supported by the National Science Foundation under grant no. TEI-87 15950. Any opinions, findings, and con- clusions or recommendations expressed in this article are those of the authors and do not nec- essarily reflect the views of the National Sci- ence Foundation.

И ARITHMETIC TEACHER



carries over into formal schooling and is a foundation on which schools build in fostering a desire to try to learn and an attitude that hard work will lead to progress in learning.

Even the casual observer realizes that all students are regarded as capa- ble of learning mathematics and other subjects. One also sees that proper behavior, being prepared for class, and diligence and seriousness of pur- pose are characteristic of students. The Japanese assume that learning is the product of effort, perseverance, and self-discipline rather than of ability. Consistent with this philosophy, the schools have no ability grouping in elementary and junior high schools and virtually no individualized classroom instruction. Promotion to the next grade is automatic. Students re- quiring additional help receive it at home or their parents arrange for individual instruction in other schools (Stevenson, Lee, and Stigler 1986).

The overriding values in Japanese public elementary and junior high school education are egalitarianism and homogeneity (Kitamura 1986). Closely connected is loyalty to one's group, which, in fact, is characteristic of the larger Japanese society. The objective is to foster group harmony, and relatively little importance is at- tached to individuality. For a detailed description of how this fundamentally important trait of education is elabo- rated in the schooling process, we suggest the excellent book by Duke (1986).

Unlike junior high school teachers, who must prepare students for the high school entrance examinations, elementary school teachers experience no pressure from entrance examinations in Japan. Thus, they have more freedom in organizing class learning while still following the Na- tional Course of Study (NCS) estab- lished by the Ministry of Education, Science, and Culture (Monbusho). Starting in grade 1 , great importance is placed on proper behavior, the routine of class, attitudes toward learning, and respect for teachers and others. Thus, each class begins and ends with students standing to bow to the teacher. Good attitudes and behavior are taught and emphasized while

OCTOBER 1990

^^^^^^^H

teachers follow Monbusho' s ap- proved textbooks, teachers' guides, and school policy. In grade 1 students learn to maintain a notebook for each subject. Teachers are required to teach all the material in the textbook each year. These early years (i.e., grades 1-3) are important in fostering constructive learning habits that are considered fundamental to school, that are reinforced in the home, and that are essential to life as an adult (Duke 1986).

The importance placed on mathematics can be seen from table 1. In elementary school, students learn that mathematics is an important subject and that it is of central importance for entrance to junior and senior high school, as well as to college. Unlike needlessly repetitive American textbooks (Flanders 1987), Japanese textbooks are concise, and, as we have observed, teachers must cover all the

material. Moreover, much content is presented earlier in Japan than in the United States (Fuson, Stigler, and Bartsch 1988). Teachers must introduce and elaborate the concepts, and students must then diligently study and drill themselves on the material (gambare, which means persevere, is a word well understood by Japanese students).

Just as pupils' transition to school at the beginning of grade 1 is accompanied by great ceremony (Duke 1986), so students' entrance to junior high school is viewed as exceedingly important. The students wear uni- forms, and good manners and appear- ance are emphasized, as well as attitudes, behavior, and gambare (U.S. Department of Education 1987). Aca- demics are emphasized even more at this level than in elementary school, and at the end of junior high school (the end of compulsory education),

13



students must take entrance examinations for senior high school. Junior high school students are also required to study English.

Table 2 shows the full curriculum for junior high school. Mathematics and science receive great emphasis, along with Japanese. But the Japanese mathematics curriculum differs from the U.S. curriculum. Topics from elementary school are not repeated. Al- gebra and geometry occupy a promi- nent place in the curriculum. Much of U.S. ninth- and tenth-grade mathematics is learned in the eighth and ninth grades in Japan, with probability and statistics and some solid geometry also included (cf. Hashimoto and Sawada [1979]). The pace at which mathematics is taught to all Japanese students at this level is "roughly equiv- alent to the fast track in a good subur- ban school system in the U.S." (U.S. Department of Education 1987, 35).

A typical school day begins about 8:30 a.m. with a fifteen-minute class meeting. Two classes separated by a ten-minute break follow, and then students are given a twenty- to twenty- five-minute break. Two more classes separated by a ten-minute break are held before lunch and recess, which last until about 1:40 p.m. After recess, students spend about twenty minutes cleaning the rooms and hallways. Two more classes follow, with a ten- minute break between, and the school day ends about 3:50 p.m. after another class meeting. On Saturdays, the three class periods end about noon. (A minimum of thirty weekly hours is required, but many schools exceed it.) Teachers may use some or all of the ten-minute breaks to finish up and pol- ish understanding. Students take this routine in stride and are attentive until the lesson is finished. Mathematics classes are generally scheduled during the first periods of the day.

Observations of Japanese Classrooms We observed fourteen classes in nine elementary and junior high schools. A typical visit began with a short brief- ing by a school official (usually the

14

^^^^^^^^H

principal or head teacher), followed by visits to classrooms, where we observed entire lessons of forty-five to fifty minutes' duration. We were free to move about the room while students worked. Although we video- taped these classes and moved about a great deal, students seemed undis- turbed by our presence. After each class we had an open discussion with the teacher whose class we observed. When we inquired about instances in which the class did not proceed smoothly, the teachers were not de- fensive but frankly commented that they would handle the lesson better next time. They were also open about considering our suggestions. These discussions were usually informal, accompanied by tea and snacks, with

good communication in English or through translation.

We noted various classroom char- acteristics common to most, if not all, the classrooms we visited.

The school and classroom atmosphere Schools are lively and give the ap- pearance of places of learning. Ap- proaching a school, the visitor may see flowers and plants. A statue re- flecting learning and thinking may be placed in the courtyard. Students and adults (including visitors) remove shoes and put on slippers when entering the school building, which is im- maculately clean and orderly. While classes are in session, hallways are devoid of students. Students can be seen erasing chalkboards, cleaning erasers, sweeping floors, washing windows, and taking out garbage at designated times of the day. All schools have a swimming pool and both indoor and outdoor athletic facil- ities. Physical fitness ("a healthy mind in a healthy body") is emphasized throughout schooling.

Classrooms are equipped with a large chalkboard at the front, an over-

ARITHMETIC TEACHER

Good attitude and behavior are taught.



head projector and screen, and a very small desk (podium) for the teacher. Some chalkboards have a curtain that can be drawn to prevent chalk dust from blowing into the air. Erasers are cleaned, usually by students before class begins, by a little vacuum ma- chine at the front. Teachers use colored chalk to advantage. Bulletin boards and poster boards usually display information for students or ex- amples of students' work. Rooms are well lighted by a bank of windows and artificial light. In most instances, students sit in a boy-girl configuration at desks with benches (sometimes in rows of single desks). Forty or more students in a class is typical.

Little chalkboards (approximately 2' x 2' or T x 3'), which can be hung from the top of the big chalkboard, are very effectively used by teachers and students. These small chalkboards frequently have problems or solutions written on them for all to see. Teach- ers also commonly use variously sized sheets of poster board held to the chalkboard by magnetic buttons. Stu- dents write solutions on transparencies that are displayed on the over- head projector (OHP) and accompanied by an explanation by a student or teacher. Some classrooms even have a copier on which transparencies can be made quickly so that teachers can show students' work on the OHP.

Students always show respect for the teacher - they quickly quiet down when the teacher signals attention, and the rising and bowing before and after a class lesson is ubiquitous. Stu- dents are always prepared for class with notebooks and little boxes that contain pencils, a pen, an eraser, and a straightedge. They are well man- nered, sit erect, and display politeness in interaction. We never observed a student being called out of class nor instruction interrupted for administra- tive reasons. Overall, students' diligence in and out of class Sgambare) is combined with high teacher - and par- ent - expectations to produce effective learning.

Organization of instruction

A typical mathematics lesson was organized as follows:

OCTOBER 1990

• Students' rising and bowing • Reviewing previous day's problems

or introducing a problem-solving topic (5 minutes)

• Understanding the topic (5 minutes) • Problem solving by students, work-

ing in pairs or small groups (cooper- ative learning) (20-25 minutes)

• Comparing and discussing (students put proposed solutions on small or large chalkboards) (10 minutes)

• Summing up by teacher (5 minutes) • Assigning exercises (only 2-4 prob-

lems, to be done out of class) • Sounding of soft gong, indicating

end of class. Students rise and bow.

Figure 1 shows an edited copy of the lesson plan furnished by one teacher. We chose this sample plan from cjnong many that reflect the organization of lessons described above. It also represents how class activities move toward a generaliza- tion by generating data and searching for a pattern. What cannot be cap- tured in writing, unfortunately, are the excellent use of figures (colored chalk and posters) on the chalkboard, the classroom management of at least forty students, the purposeful work of students in groups, and the deliberate pausing and explaining by the teacher, which seems to be an important difference between American and Japa- nese teachers (Stigler 1988).

Common themes within lessons

Theme 1: Single objectives. In virtually all classes we saw a single objective in the topic for the day. The in- tention was clearly to focus all class activities on the objective and "knock it off" by the end of the period. The teacher does not return to this work

the next day or year - students are re- sponsible for learning as much as possible during class and finishing up outside of class, if necessary. Students keep notebooks that are examined pe- riodically by the teacher, and in some classes, the teacher handed out a single worksheet that clearly reflected the single objective. Students put their work on the sheet, and it was collected at the end of the period. The teacher assigned only two to four carefully selected problems, which were almost always more complicated than problems assigned in comparable U.S. classrooms. In some classes, the teacher finished the lesson by asking students to state what they had learned, either orally or in writing, and they did so (see fig. 1).

Theme 2: Extensive discussion. While students are working after the problem or topic is introduced, the teacher moves about observing students' work. During this purposeful "scanning" of the class, the teacher may select the work of several students or groups to be presented on the chalkboard or OHP and explained to the class. Later the teacher enters the picture again, usually near the end of the period, and "pulls together" what has been done and explains how it re- lates to the original problem or topic. We regard the discussion and cooper- ative work by students and that of the teacher as extensive and effective in achieving the lesson's objective (cf. Stigler [1988]).

In two classes, the teacher's lesson plan indicated the skill and concept levels of the students. In many classes, we observed the teacher high- lighting the different approaches to the solution of a problem, which, in many cases, were numerous. We also observed students' solutions that we had not thought of, and the teacher's explanation was very thorough.

In two other classes we observed, the lesson was taught by a demonstration teacher, with the classroom teacher and other teachers observing. It is not uncommon during the course of a year for teachers to observe demonstration teachers conducting lessons (see fig. 2 for an example of a demonstration lesson plan). Such

15

Teachers must cover all the

material.



ITQiQbB Fifth-grade mathematics class (edited version of the lesson plan given to us) Topic: Explaining the number of diagonals in a polygon Time: Fifty minutes Introduction: Display a decagon with all diagonals drawn connecting its vertices.

Students' activity Remarks on teaching

Question: How many diagonals are drawn in the decagon?

Understanding 1. Make sure I understand the notion of diagonal. Review the term and concept of diagonal. the problem:

2. Think about the diagonals that can be drawn from Look at simpler cases; use charts on chalk-board one vertex in the following polygons. for each polygon and use worksheet.

Е^1ЩШ РЯУЩШ ^1ЩШ ^К1ЩШ ~1и1ЩШ

a) Draw the diagonals.

b) Record the number of diagonals.

3. Read the table and write down what is discovered Let students know two viewpoints in reading the about the number of diagonals from one vertex. table.

Polygon

Triangle Quadrilateral Pentagon Hexagon Heptagon Number of diagonals 0 I 2 3 4

a) As the number of vertices of a polygon increases by one, the number of diagonals at one vertex increases by one each time.

b) Subtracting three from the number of vertices of a polygon, we get the number of diagonals at one vertex.

Problem solving 4. Draw all diagonals of the five polygons and Let students know that previous table doesn't by students: actually count them. illustrate all diagonals.

a) Draw and count all the diagonals. Treat carefully pupils' incorrect solution, such as

b) Verify whether all the diagonals are drawn. ' "'

c) Explore the way of counting effectively to find the correct number of the diagonals.

Comparing гт

and Po|yg°n discussing: Triangle Quadrilateral Pentagon Hexagon Heptagon

Number of diagonals 0 2 5 9 14

5. Present the pattern based on the table after verifying the number of diagonals. • As the number of angles increases by one, the difference in the number of diagonals increases by

one each time. • The number of diagonals can be found by the formula (the number of diagonals from a vertex) x

(the number of verticesJ/2.

Polygon

Triangle Quadrilateral Pentagon Hexagon Heptagon Number of diagonals

from a vertex 0 12 3 4 Total number of

diagonals 0 2 5 9 14

6. Guess the number of diagonals of the octagon, nonagon, and decagon.

Summing up: 7. Have students write down what they learned through today's lesson. • Making a table is useful in finding a pattern.



demonstration lessons give teachers opportunities to see new content presented; to observe a model of excellent teaching and effective classroom management that keeps students on task (Cummings 1989); and, in gen- eral, to witness a pace of instruction that permits careful examination of students' work, whether correct or incorrect. We viewed no loss of self-es- teem when students made errors that were publicly corrected by other students or the teacher.

Theme 3: Intensive curriculum. Jap- anese classrooms are places for serious learning. The curriculum is extensive, and the content is taught intensively. We call what we observed "lesson intensive." Clearly, an attitude prevailed that the class does today what it intends to do and moves on to new material tomorrow without repetition and that students have the responsibility to learn. The description of classroom organization illustrates how this plan is accom- plished. The teachers pace the instruction in a well-organized and dis- ciplined manner. In fact, the same content is taught at about the same time and pace throughout the country. Although the curriculum makes no systematic provision for individualized instruction, we encountered a few teachers who gave additional problems to students who were learning at a faster pace and others to the slower learners.

We were impressed by the emphasis on the mathematics of the situa- tions discussed in class. We wonder whether this is an important difference between typical American and Japanese classrooms; for example, figure 2 shows a problem situation with many solution paths, followed by an evaluation of them (e.g., discuss each way; which is the best way) and extensions. Such problems help to take students to the heart of the mat- ter and have them doing mathematics.

Theme 4: Gender differences. We observed few or no overt gender differences in teachers' interaction with students. We also saw no discernible differences between boys' and girls' attitudes toward learning. For exam-

OCTOBER1990

pie, when students put their solutions on the chalkboard, those of girls were certainly as good as those of boys. Boy-girl interaction was common when students worked together, and both boys and girls responded with no hesitation when called on. Teachers showed a nice manner of handling wrong answers from students of either sex with patience, no anger, and very good "wait time," by which we mean further time for the student to think out his or her response, correct his or her response, or consider a comment from a fellow student before the teacher called on another student or "took charge."

Open-ended problem solving

During our U.S. -Japan seminar a discussion was held of the use of "open- ended," nonroutine problems in teaching mathematics. Figure 2 shows an example of such a problem. Although we cannot say precisely how widely it has been adopted, an "open-ended approach" to teaching mathematics is currently being used in Japanese elementary and junior high schools. In Japan, this teaching methodology has its origin in attempts in 1971 by Japa- nese mathematics educators to evalu- ate higher-order thinking in mathematics learning (Shimada 1977). In contrast to traditional classroom problems that are formulated to have one and only one correct answer ("closed problems"), the open-ended approach uses "incomplete" problems, which have a multiplicity of correct answers or approaches to solutions. The problems offer experience in learning something new in the process of resolving a problem that is combined with previously learned knowledge, skills, or ways of thinking (Shimada 1977, 1-2).

This teaching methodology focuses on mathematical inquiry. By working individually or in small groups on such problems then discussing the various approaches used by students in finding solutions and various extensions of the problem, students de- velop an understanding of mathematical inquiry and an appreciation for the depth and complexity of the mathematics involved. Students also get an

appreciation that a variety of approaches and strategies can be used in solving a problem, depending on how they think about or view the problem.

We had an opportunity to observe several classes that used an "open- ended approach." Without exception the problems were carefully chosen and lent themselves to different solutions. Students also generated other problems that followed more or less naturally once students got into the open-endedness of the original problem. The problem in figure 2, for example, lends itself to a variety of solutions and can lead to several related problems: looking at the converse problem (i.e., given a number of matches, how many squares can be formed?); developing extensions to a larger number of squares; changing squares to triangles, pentagons, and so on; and generalizations to n matches or squares.

This open-ended teaching methodology is a result of the influence of Japanese researchers (Shimada 1977; Nohda 1983, 1986, 1987; Hashimoto 1987; Sawada 1980) and has a clear relationship to certain high-priority goals in our own current reform movement as reflected in NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). We think that it is important to get a better understanding of the explicit nature and requirements of this instructional method and to consider further how the method can be adapted for teachers and students in the United States. We believe that this approach has great potential for engaging students in mathematical thinking, that students will find it interesting, and that at the same time, it will help them to get a better understanding of mathematical inquiry. Our informal tryouts of open-ended problems with fourth- and ninth-grade teachers and students alike look very encouraging (Becker and Foland 1989). We expect that this approach will continue to be an area of collaboration in research between our two countries . ,

Open-ended problem solving is an aspect of curriculum development and instruction that will be of interest to American teachers and may have great potential for school mathematics

17



НТЕХ^^^З Teacher's plan of problem solving (grade 6) (edited version of the lesson plan given to us)

1. Today's lesson: To solve the problem by finding a rule

i I I i i Problem

How many matches do we need to make 10 squares?

Find the answer in as many ways as possible.

Teacher: How many matches do we need when we have two squares?

Student: Seven matches

Teacher: What about when we have four squares?

Student: Twelve matches

2. Think out a variety of ways of counting using the worksheet.

2 8 7 3 7 3 7 (a) il з1

I I I I (b) I I I I I I (c) I

1

I

1

1

5 7

ó One to one 7x3 + 3x2 = 27 4x10 = 40 (wrong)

ld) |ф|"|©|"|©|

(e) |°|°|°|°|°|

(fl |ТУ|'ЧЧ

Ve-J - IxJ^j^bzl I о I о I о I о I о I T i T и T о Т x'a

rifj 4 x 6 + 3 = 27 5x3 + 6x2 = 27 5x5 + 2 = 27

191 Olli' С) О

7 + 5 x 4 = 27

L(5 - 1)

3. Discuss each way of counting.

a) Which one do you think is the best way? Why?

b) What happens when the number of squares increases?

4. Which is the easiest way when we have 20 squares?

(2 squares with 7 matches) + (5 matches) x (sets of 5 matches) = total number of matches.

+ x (10-1) = 52

7 + 5 x 9 =52 5. Practice: How many matches do we need when we have the following shapes?

a) 8 equilateral triangles

b) 7 squares

c) 15 squares



in the United States. It involves problems that stand in sharp contrast to the "closed-ended" problems characteristic of instructional materials in the United States. The method also stands in significant contrast to the typical U.S. routines of explanation by the teacher, excessive reliance on textbooks (though the Japanese also rely on textbooks), and students' working separately (Silver et al. 1988).

The open-ended approach may also help to disabuse our students of the beliefs that they too commonly hold that mathematics is static and non- functional and that learning it involves mainly memorizing rules (Silver et al. 1988). What's worse is that the NAEP data (Silver et al. 1988) indicate that little has changed since earlier reports (see, e.g., Fey [1981]) and that the data stand in "stark contrast to the vision of mathematics instruction por- trayed in NCTM's Standards" (Silver et al. 1988, 725). The approach has potential for putting students at the heart of the learning process, has them doing mathematics, and may help in reducing the gap between present classroom realities and the vision represented by the Standards.

Juku Outside the regular schools is another very large group of schools called>Á:¿/ ("joo-koo" or "jook") by the Japa- nese. Although we did not visit any juku, any consideration of Japanese mathematics education must take juku into account. The juku are pri- vate, profit-making schools run inde- pendently of the public schools. Still, they play an important role in developing scholarship in students, and they depend in various ways on publicly oriented schools (Sawada and Kobayashi 1986).

Juku furnish opportunities for students to get tutoring, remediation, and enrichment, as well as preparation for entrance examinations. Students attend juku outside regular school hours (i.e., afternoons, evenings, weekends, or vacations). Juku exist throughout the country, with heavier concentra- tion in large urban areas. Insight into

OCTOBER 1990

the relationship between juku and regular schools is displayed by the Japa- nese scholar Kitamura (1986, 161): The dominant values of the Japanese public pri- mary school are egalitarianism and uniformity: pupils are not classified according to their academic ability because all pupils are supposed to keep up with the progress of the class. They are taught by means of a nationally controlled, uni- form curriculum. Despite its principles of egalitarianism and uniformity, however, the school inevitably must produce high achievers and low achievers. The school and its teachers are un- able to counter these disparities because they are bound by the two mandatory principles.

Accordingly, high achievers can attend a juku that offers more advanced classes appropriate for them, whereas students experiencing learning diffi- culties can attend a school offering re- medial opportunities; then, owing to the existence of these juku, the formal school can continue to function according to the two principles of egalitarianism and uniformity (Kitamura 1986, 161). Leestma and his colleagues (U.S. Department of Educa- tion 1987) acknowledge the validity of Kitamura's perspective but also comment that it may be somewhat over- stated.

Juku can be nonacademic as well as academic. The former offer instruction in such areas as music (e.g., vo- cal, instrumental), Soroban (abacus), calligraphy, the arts, and physical education. Elementary school pupils commonly attend these juku. The academic juku generally supply supple- mentary schooling in mathematics, science, English, and the national language. Academic juku are more numerous, and the number of students attending increases by grade level: 40 percent for grade 5, 47 percent for grade 6, 49 percent for grade 7, 55 percent for grade 8, and 67 percent for grade 9 (Sawada and Kobayashi 1986, 2). Seventh-grade students, for example, attend juku 2.1 hours per week

(Sawada and Kobayashi 1986, 2). Sawada and Kobayashi also state that the reasons for the steadily increasing numbers of students in juku from up- per elementary into junior high school are that (1) juku give help to students who fall behind in regular schools; (2) high parental expectations are met by juku; and (3) juku help students prepare for the high school entrance examinations in grade 9. A special type of juku offers preparation for entrance tests to both senior high school and the universities.

The realities of juku and their rela- tive merits and demerits are topics of present research in Japan. The research is attempting to assess how the regular schools and the juku interface. This inquiry is taking place in a con- text in which the whole of Japanese education, as well as the organization of Japanese society, is being reas- sessed (Sawada and Kobayashi 1986).

Technology Japan is a highly developed techno- logical society, so we expected to see hand-held calculators and microcomputers used more commonly than we observed in mathematics classrooms (or than was reported to us in discussions). In fact, we saw no use of calculators in classes. We learned, however, that (1) what goes on in the classroom must closely adhere to Monbusho's NCS, and the NCS does not yet provide for calculator use; (2) attitudes of parents and elementary school teachers do not yet favor their use, although university professors overwhelmingly do (Nagasaki 1987); (3) the use of calculators is not al- lowed on entrance examinations for senior high school; and (4) studies are under way to assess the merits of their use in teaching mathematics (Na- gasaki 1987). Still, with thirty million calculators used domestically and their use expanding, with mathematics educators more widely recognizing their potential for mathematics instruction, and with mathematics educators recommending to Monbusho their expanded use, a trend toward integrating the use of calculators at all school levels predominates (Nagasaki

19

Learning is the product of effort.



1987). Perhaps we will see the situation change when Monbusho's new NCS is implemented, beginning in 1992. The situation is very similar with re-

spect to microcomputers. Although microcomputers are widely used domestically and with great success in experimental schools, a large number of teachers and educators still make no use of them (Nagasaki 1987). Moreover, the present NCS, in place since 1980, makes little reference to computers. However, the situation will likely change, since the newly formed NCS includes operation of computers, flow charts, program- ming, structures of programs, and various computational algorithms for senior high school courses (Sekiguchi 1989). Information remains scarce as to when and how computers will be used more widely at the elementary and junior high school levels.

Nagasaki and Senuma (1987, 15) comment that the use of microcomputers in mathematics teaching is being promoted in response to social de- mands. Although in 1986, 2 percent of elementary schools, 14 percent of junior high schools, and 81 percent of senior high schools were equipped with more than one microcomputer, many problems remain to be solved (e.g., attitudes of teachers, software, and hardware). The use of microcomputers is now being studied in two di- rections: (1) In what areas of the curriculum can microcomputers play a key role (e.g., graphics)? and (2) How can the curriculum be reorganized so that students can conduct mathematical activities using microcomputers? (Nagasaki and Senuma 1987).

In Japanese education, where differing viewpoints are held by teachers, the potential of newly proposed innovations is commonly researched before implementation. If consensus cannot be reached, no changes are made. Data are collected through ex- perimentation, and differing viewpoints are heard and discussed in meetings throughout the country. Once the various views are mediated and if consensus emerges, the Japa- nese move quickly to implement changes. This process is under way now regarding technology. We think

that changes are likely to occur, with more widespread use of technology in the offing, and it will be interesting to follow and study these developments in Japan and their possible implica- tions for teaching mathematics in the United States.

Final Note A visit such as ours requires a great deal of planning and logistics. We wish to acknowledge our Japanese colleagues' excellent preparation and organization for our visit. In particu- lar, we wish to express appreciation to the following people for making our visit both personally comfortable and professionally useful: Tatsuro Miwa (University of Tsukuba), Shigeru Shi- mada (Science University of Tokyo), Yoshishige Sugiyama (Tokyo Gakugei University), Toshio Sawada (National Institute for Educational Research), Nobuhiko Nohda (University of Tsu- kuba), Yoshihiko Hashimoto (Yoko- hama National University), Tadao Ishida (University of Hiroshima), Eizo Nagasaki (National Institute for Educational Research), Hanako Sen- uma (National Institute for Educa- tional Research), Shigeo Yoshikawa (Joetsu University of Education), Ju- nichi Ishida (University of Tsukuba), Toshiakira Fujii (Yamanashi Univer- sity), Toshiko Kaji (Tokyo Gakugei University), Minoru Yoshida (Univer- sity of Tsukuba), Katsuhiko Shimizu (University of Tsukuba), and graduate students of Tsukuba University: Koi- chi Kumagai, Minoru Otani, Yoshi- nori Shimizu, Keiko Ito, Kazuhiko Nunokawa, and Shigeo Tsukahara.

We also wish to thank the National Council of Teachers of Mathematics and the Mathematical Association of America for their donations of pam- phlets, monographs, and books. We also thank Lawrence Erlbaum Asso- ciates for a very generous donation of excellent books dealing with problem solving, thinking skills, and cognitive science.

Finally, we wish to express appreciation to the principals, head teachers, and classroom teachers - too numerous to list - who invited us into their classrooms. They patiently and skillfully answered and asked ques-

tions and treated us in a manner that promoted exchange, communication, and friendship. The authors thank James Flanders for his useful suggestions in finalizing this manuscript.

References

Becker, Jerry, and Neal Foland. Integrating Problem Solving into Mathematics Teaching: A Model and Demonstration Project. Report of an NSF-funded project. Carbondale, 111.: Southern Illinois University at Carbondale, 1989.

Becker, Jerry, and Tatsuro Miwa, eds. Pro- ceedings of the U.S. -Japan Seminar on Mathematical Problem Solving. Carbondale, 111.: Southern Illinois University, 1987.

Cummings, William K. "Some Reflections on National Differences in Mathematics Achievement." Paper presented at the 67th Annual Meeting of the National Council of Teachers of Mathematics, Orlando, Florida, 12-15 April 1989. (Available from the author at Graduate School of Education, Harvard University, Cambridge, MA 02138.)

Duke, Benjamin. The Japanese School- Les- sons for Industrial America. New York: Praeger Publishers, 1986.

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Flanders, James R. "How Much of the Content in Mathematics Textbooks Is New?" Arith- metic Teacher 35 (September 1987): 18-23.

Fuson, Karen C, James W. Stigler, and Karen Bartsch. "Grade Placement of Addition and Subtraction Topics in Japan, Mainland China, the Soviet Union, Taiwan, and the United States." Journal for Research in Mathematics Education 19 (November 1988): 449-56.

Hashimoto, Yoshihiko. "Classroom Practice of Problem Solving in Japanese Schools." In Proceedings of the U.S. -Japan Seminar on Mathematical Problem Solving, edited by Jerry Becker and Tatsuro Miwa, 94-112. Carbondale, 111.: Southern Illinois University at Carbondale, 1987.

Hashimoto, Yoshihito, and Toshio Sawada. Mathematics Program in Japan. Tokyo: Na- tional Institute For Educational Research, 1979. (Note: the program was implemented in 1980.)

Kitamura, Kazuyuki. "The Decline and Re- form of Education in Japan: A Comparative Perspective." In Educational Policies in Cri- sis, edited by William K. Cummings, et al. New York: Praeger Publishers, 1986.

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National Council of Teachers of Mathematics, Commission on Standards for School Mathe- matics. Curriculum and Evaluation Stan- dards for School Mathematics. Reston, Va.: The Council, 1989.

20 ARITHMETIC TEACHER



Nohda, Nobuhiko. Open Mind of the Child in Arithmetic Learning. Tokyo: Kobunshoin, 1986.

. "A Study of the 'Open-Approach' Strategy in School Mathematics Teaching." Ph.D. diss., University of Tsukuba, 1983. Abstracted under the title "The Heart of 'Open- Approach' in Mathematics Teaching" in Proceedings of the 1CMI-JSME Regional Conference on Mathematics Education, 314- 18.

. "Teaching and Evaluating of Problem Solving Using the 'Open-Approach' in Math- ematics Instruction." Tsukuba Journal of Ed- ucational Study in Mathematics 6 (1987): 117-26.

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Sekiguchi, Yasuhiro. Computers in Japanese Mathematics Education, rev. ed. Athens, Ga.: University of Georgia at Athens, De- partment of Mathematics Education, 1989.

Shimada, Shigeru, ed. Open-End Approach in Arithmetic and Mathematics - a New Pro- posal toward Teaching Improvement. Tokyo: Mizuumishobo, 1977.

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L. Kouba, and Jane O. Swafford. "The Fourth NAEP Mathematics Assessment: Performance Trends and Results and Trends for Instructional Indicators. Mathematics Teacher 81 (December 1988):720-27.

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U.S. Department of Education. Japanese Edu- cation Today. A report from the U.S. Study of Education in Japan, prepared by a special task force of the OERI Japan Study Team, directed by Robert Leestma. Washington, D.C.: U.S. Government Printing Office, 1987.

Bibliography Azuma, Hiroshi, Kenji Hakuta, and Harold W.

Stevenson. Child Development and Educa- tion in Japan. New York: W. H. Freeman & Co., Publishers, 1986.

Becker, Jerry P., et al. Mathematics Teaching in Japanese Elementary and Secondary Schools - a Report of the ICTM Japan Math- ematics Delegation. Carbondale, 111.: South- ern Illinois University, 1989. (Available from the author: Curriculum and Instruction, Southern Illinois University, Carbondale, IL 62901-4610; $4.)

Horvath, Patricia J. "A Look at the Second International Mathematics Study Results in

the U.S.A. and Japan." Mathematics Teacher 80 (May 1987): 359-68.

McKnight, Curtis C, F. Joe Crosswhite, John A. Dossey, Edward Kifer, Jane O. Swafford, Kenneth J. Travers, and Thomas J. Cooney. The Underachieving Curriculum: Assessing U.S. School Mathematics from an Interna- tional Perspective. Champaign, 111.: Stipes Publishing Co., 1987.

Stigler, James W., Shin-Ying Lee, G. William Lucker, and Harold W. Stevenson. "Curric- ulum and Achievement in Mathematics: A Study of Elementary School Children in Ja- pan, Taiwan, and the United States." Jour- nal of Educational Psychology 74 (June 1982):315-22. *

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SOME OBSERVATIONS OF MATHEMATICS TEACHING IN JAPANESE ELEMENTARY AND JUNIOR HIGH SCHOOLS