International Panel on Policy and Practice
in Mathematics Education: 2001 Report

Issue 4: Tradition and Reform in Mathematics Education

How does your country handle the balance between tradition and reform in mathematics education? What do tradition and reform mean within your mathematics education system?

Japan: Yoshihiko Hashimoto and Miho Ueno

Opening Statements

Yoshihiko Hashimoto
Yokohama National University

Changes in mathematics education are currently taking place in Japan. The school week is changing from six days to five days per week to allow children to develop "competency for positive living." Changes in Japan seem to occur approximately every ten years, although the mathematics content over the last 30 years has been relatively stable. Reform is based on tradition and varies according to the times. Since 1994, Japan has had a "core and optional modules" model for upper secondary school mathematics. An overall objective of the entire curriculum is to foster students' abilities to think mathematically. "Open" methods of teaching - open process, using different ways to solve a problem; open-ended where problems have multiple correct answers; and open problem formulation where students pose new mathematical problems can help meet this objective. Traditionally, one important feature of learning mathematics was to develop the ability to calculate rapidly.

Miho Ueno
Tokyo Gakugei University Senior High School

The goal of mathematics education might be seen as learning the basic idea of calculus by the time students graduate from high school. Beginning calculus is taken by 82 percent of high school students in a traditional mathematics program. It is often difficult to implement a new course of study due to factors such as a lack of teachers in a small school system, which limits the courses offered to those needed for college entrance, or to a lack of technology. Thus, the intended curriculum may not get implemented.

The results of the Third International Mathematics and Science Study (TIMSS) indicated that Japanese students disliked mathematics. The new course of study addresses this by stressing mathematical activities aimed at helping students appreciate the importance of mathematical approaches and ways of thinking. What matters is finding principles in given phenomena in the world and finding materials that will allow students to use mathematics spontaneously. Reform should be realized through teachers' attempts to enrich the contents of the prescribed curriculum.

Participants identified the similarities and differences represented in Table 5.

Table 5 Japan: Similarities and Differences Among the Participating Countries with Respect to Tradition and Reform in Mathematics Education

Country Similarities to Japan Differences from Japan
Brazil   Open-ended problems are not explored much in the curriculum.

Teaching approach not influenced by policies to reform schools. Teaching is teaching, and direct method is used.

Tradition relates to how schools approach failing students. Before if a student failed one subject they repeated everything. Now, things are changing. Teachers and parents are unhappy about this
Egypt Curriculum recommends open-ended problems  
France Reform tends to occur at 10 year intervals

Reform encourages the use of more concrete activities and more student participation

Time in school is also being reduced (from five days to four days per week in France)
Traditional math is very formal and abstract. Does not emphasize computation
India Traditional math focuses on quick computation without technology. Arriving at solutions, not interpretations.

Despite reform efforts, traditional approach still common
Open-ended problems are not explored in the curriculum. One specific method allowed on exam items, but since 1986 there has been more of an effort to promote training in logic, mathematical thinking, and mathematical reasoning
Kenya Emphasis on teaching high level mathematics Do not have core and optional program
Sweden Reform tends to occur at 10 year intervals (since the 1960s)

Changes to classroom practice proceed slowly
 
United States Traditional math focuses on closed problems, and quick computation without technology

Despite reform efforts, traditional approach still common

Some curricula emphasize open-ended problems and flexible thinking

Reform tends to occur in cycles

Changes to classroom practice proceed slowly
Students say that they find math enjoyable (TIMSS)

Observer Commentary

Hiroshi Fujita
The Research Institute of Educational Development, Tokai University

Traditional mathematics, particularly ancient mathematics should be appreciated as a component of the culture of the country, and including these topics in the curriculum is a good motivation for learning. When considering traditional mathematics for inclusion in the curriculum, the value of the traditional mathematics must be examined from the perspective of the purpose of the curriculum. Origami has never been explicit in the mathematics curriculum in Japan; it is regarded as a child's skill, for fun. When the Japanese government needed to establish the education system in Majii Restoration, they chose to go with western mathematics. The Japanese Wasan lost its connection with science and mathematics and was not clearly part of the curriculum any more.

It is not always clear what tradition means within a culture. Tradition and its relation to the existing curriculum can be examined from the perspective of purpose, process, and strategy for reform. One philosophy in Japanese education was to cultivate mathematical intelligence in students with foci on mathematical literacy and mathematical thinking, in a proper balance. The core and option structure of the mathematics curriculum was part of a strategy intended to make depth and breadth compatible.

Mathematics education can be compared to medical science. Mathematicians do mathematics for its own sake. The study of mathematics education is for the practice of teaching. Teachers correspond to practical physicians. Research on mathematics education is similar to basic study of practical education. Researchers must be trusted by practitioners. The role of mathematicians is to help teachers and not to criticize.

Hyman Bass
University of Michigan

Tradition can be interpreted two ways: in a cultural sense and in a sense of habit. We heard interesting cases from Japan and India of ancient mathematical traditions. Some of this is echoed in work on ethno mathematics, where people in various cultures attempt to identify attention to mathematical ideas deeply embedded in culture, sometimes explicitly, sometimes through art and other cultural artifacts. It is important to honor those traditions.

In some countries, opposition between tradition and reform is concerned with the ways we have been teaching mathematics over the recent past. At times, either the teaching or learning is found to fall short of expectations, or the needs of society cause new demands. This is when change is attempted. A typical reaction seems to be to find fault with the old system, discard everything that was done in the past, and replace it with something new. This is at the root of debates about basic skills. Frequently what seems to be wrong about teaching basic skills is not so much with subject matter but with the method of instruction. Typically in the U.S., to learn computation children would be given formal rules and pages of exercises. The conclusion was drawn that teaching these mathematical ideas implied a kind of teaching and pedagogy that led to an oppressive experience. The result was to remove emphasis on these parts of mathematics from the curriculum. Opposition between tradition and reform is somewhat artificial. Change should be made carefully, and serious thought given before abandoning topics.

Themes That Emerged From the Discussion
Theme 1: Patterns of reform

Policies mandating or supporting change can help or hinder actual classroom implementation of the ideas underlying the changes. Changes in curriculum seem to occur in a cyclic fashion, in some instances approximately every ten years. In some cases, such as Japan, this is part of a designed reflection on the status of mathematics education; in other countries these changes occur for reasons ranging from a change in society's structure to advances in technology that influence mathematics education. The group concluded that societal demands drive curricular change and determine the nature of the change, including changes in the way mathematics is taught as well as changes in content. For example, some countries structure their mathematics curriculum using a spiral approach, with a topic revisited each year over several years to develop a deeper understanding of the important ideas.

The role of professional education organizations varies considerably. Some countries, such as the United States, have strong mathematics education associations, and members of these groups are involved in the reform process from the beginning. In other countries, such organizations have yet to be formed or are just beginning to take active roles in thinking about changes in mathematics education.

The group agreed that to make change successful all members of society had to be part of the process. Some reform initiatives involve changing the time allocated for school, which affects the time allocated for mathematics. Japan and France are both shortening their school week; some secondary schools in the United States are redesigning the internal structure for the time allotted to specific content areas. Brazil, to accommodate students who work, has a system of three days of school and three days of work.

Illustrative quotes

"We try something for a year or two, and if it doesn't work we throw it out and start over. For example, we had modern math - no good, threw it out; then back to basics - no good, threw it out." (Burrill)

New curricula are also introduced about every decade in Sweden - from 60s and after, foci are very much the same as in some other countries. It is interesting that this seems to be so worldwide." (Brandell)

Theme 2: Appreciation of mathematics

Policies and programs are needed to link content and practice in classrooms in ways that build an appreciation for the use and power of mathematics. The attitude of students towards mathematics was a common concern. In most of the countries, society at large feels that mathematics is important and supports the role of mathematics as an integral part of the curriculum. It is acceptable, however, to be mathematically illiterate or to proclaim that you are uncomfortable with mathematics. Such an attitude influences students in classrooms and interacts with what teachers do as they teach. This negativism may be due to the topics that are taught or to the way the mathematics is taught. Educators in all countries are searching for approaches that move from punishing students for not succeeding to finding ways to engage and support them as they learn.

One strategy for making mathematics more attractive to students is to emphasize changes in the practice of learning mathematics rather than changing the content (Sweden). Another approach is to link content and activities to make the topic interesting for students (Japan). Another is to link some of the historical elements of a society to the mathematics curriculum. For example, the tradition of Vedic mathematics, rooted in the Indian culture, can help children do calculations quickly. In the United States, some teachers and curricula advocate working from data as a way to motivate students.

Illustrative quotes

"You just don't meet math outside, so either you like it in school or not. History, etc., you meet outside. Maybe pupils want to play mathematics and not do the drill. It's the same problem as faced by those who train soccer players-how to get them in shape without the drill." (Lins)

Theme 3: The role of tradition in reform

Tradition can be used as a platform for improving mathematics education or it can be a hurdle. Each country has a history of mathematics education that has transmuted in various ways into current policies and practices. "Traditional" mathematics is relative and means different things to different countries. In some countries such as France and the United States, tradition means relatively formal and abstract content and teaching. To Japan and India, tradition refers to early mathematical techniques and processes such as Wasan or Vedic mathematics. Tradition to some can also mean strong expectations for certain ways of teaching and for learning certain content based on what was done when the current adults were in school.

Part of the tradition of mathematics education is who has responsibility for student successes and failures. In countries such as Kenya, Brazil, and India, students are responsible for their own success or failure. In the United States, France, and Sweden, failure is not seen as the failure of an individual child but rather as a failure of the educational process.

Illustrative quotes

"We don't have a strong tradition about certain ways of doing things. We have strong expectations among parents, students, and school administrators about what math should be and what students and teachers should be doing. We are stressing open problems, but the word open can bring lots of confusion. We do not necessarily mean problems that do not have solutions. We mean that there are lots of ways to think about the problem and move towards the answer." (Burrill)

"Vedic mathematics is a collection of elegant computational algorithms, but it can be inflated beyond reasonable dimensions as a means to solve all problems in mathematics, like Fermat's theorem." (Shirali)

Key Questions

Key questions about tradition and reform related to:

Motivation and nature of reform

  • Who is in charge of reform? What mechanisms are used to ensure that the principles of reform are clearly understood and implemented by the entire system?
  • Does the system have the capacity to implement reform?
  • What steps are taken to ensure that examinations, including university entrance examinations, are aligned with the directions of reform?
  • What is the role of teachers in bringing about reform?
  • How can some of the rich traditional mathematics be incorporated into the curriculum?
  • What is meant by tradition? Traditional content or traditional teaching? What is meant by curriculum and syllabus? Reforms for whom? Teachers, students, ministry of education?

Implementation

  • What strategies can be implemented at the policy level to support a changing attitude towards mathematics?
  • How do teachers come to know and understand the changes suggested as a country works to improve mathematics education?
  • What process is used to ensure that those at the university and those who teach teachers understand the new directions and emphases?
  • How can teachers access methods and materials to make mathematics more interesting?
  • Will changes in teaching practice enable students to be more comfortable learning mathematics?
  • Is tradition a hurdle to modernization?

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