Curriculum Changes on Lower Secondary School Mathematics of Japan - Focused on Geometry

Curriculum for lower secondary schools and the change

Outline of curriculum for lower secondary schools

Initiative for developing curriculum

Factors and results of changes of national curriculum

The fourth change occurred in 1977. Newspapers condemned that excessive content was taught and the modern-math was too difficult. After all, almost modern-math content was deleted and time-allocation to mathematics was also decreased in the change. And back to basics was emphasized.
The fifth change occurred in 1989. Personal individualization and computers in education were gradually valued in society. And result of IEA Second International Math Study that many students disliked mathematics was also noticed. Appreciation of mathematics was emphasized, problem situation learning and selective learning were introduced and use of computer was strongly expressed in the change.
In 1995, a committee for changing national curriculum was established in the ministry in 1995. Main issue for the change is 5-day school week system.

Prospective factors for the future changes are as follows:
diminish of number of students, active use of calculators and computers, teachers' understanding of students, general suspicion for schools, and increasing number of students who go away from science and mathematics, etc.
And there seems to be several factors to prevent success of changes as follows: entrance examination, preparatory institutions -juku- and uniformity in social atmosphere, etc.

Recent trends in mathematics curriculum

1. Abstract mathematics oriented
   Mathematics teachers tend to emphasize theory, logic and computation in mathematics. By this, practical aspects of mathematics such as probability, statistics, and use of calculators and computers are not so much valued. However, mathematics in context such as problem situation learning and learning for individual personality are recently emphasized.
2. Emphasis of mathematical way of viewing and thinking
   It is said that thinking mathematically and developing mathematical skills through learning mathematical content is important. However, the meaning of mathematical way of viewing and thinking is interpreted in several ways among university mathematics educators. Among school teachers, there is some confusion about the meaning.
3. Function as an integration concept
   Mathematics content in the present national curriculum is divide into three categories: numbers and algebraic expressions, geometrical figures and quantitative relations. Function is not included in algebra but in quantitative relations, and is regarded as an integration concept.
4. Little differentiation
   One grade is composed of the same age cohort and the same content is nationally taught to the same grade. Though selective learning at 3rd grade is introduced in order to cope with a variety of students, learning that responds to students interests or aptitude is mainly put into teachers hands.
5. Intensive and linear style
   Curriculum is made in intensive and linear style, for example, linear equations with one variable at 1st grade, simultaneous linear equations at 2nd grade and quadratic equations at 3rd grade. And probability is taught only at 3rd garde.

Changes and characteristics of geometry curriculum of lower secondary schools

Changes of geometry curriculum of lower secondary schools

1. The first period: Geometrical figures in our life
   Accepting instructions by the occupation army, Mathematics in our life was emphasized. Deductive geometry was taught only at upper secondary schools. Since this period until now, the Pythagorean theorem has been taught at lower secondary schools because of its regularity and wide applicability.
2. The second period: Euclidean geometry revived
   Euclidean geometry was adopted as content for lower secondary schools. This can be regarded as a revived curriculum of lower secondary school mathematics in selective system before the war. Since this period, theorems on circles have been taught at lower secondary schools because of its beauty and a variety of learning situation that students can explore intuitively and deductively.
3. The third period: Introduction of modern mathematics in addition to Euclidean geometry
   Reflecting world-wide Modern-math movement, ideas of transformation and some topics from elementary topology were added to national curriculum. Euclidean geometry was taught from earlier grade, first grade, but trigonometry has gone to upper secondary school curriculum since then.
4. The fourth period: Withdrawal of modern mathematics and review of Euclidean geometry
   Modern mathematics was obliged to withdraw. There were several reasons for the withdrawal as follows: teachers' complaint about excessive content and students' indigestion; criticisms from mass communication that modern mathematics, particularly concept of set, was not useful; confusion of mathematics teachers who were uneasy about modern mathematics; and effect of Back to Basics, etc.
   At the same time, Euclidean geometry was reviewed. Through the review, induction, analogy, and deduction were firmly placed as the way of mathematical reasoning, and importance of harmonizing intuitive way and deductive way, without emphasizing only deductive way, were expressed.
5. The fifth period: Euclidean geometry unchanged
   Major change on geometry curriculum did not appeared. However, treatment of Euclidean geometry became lighter in some textbooks.

Characteristics of geometry curriculum of lower secondary schools

1. Adoption of Euclidean geometry as an integral part
   Euclidean geometry, which was not the Euclidean geometry but a Euclidean geometry adjusted for lower secondary students, was adopted in national curriculum in 1958 for all students at lower secondary schools. Because school teachers were not satisfied with intuitive treatment such as geometrical figures in our life, through their teaching practice. Since then, Euclidean geometry has been an integral part of geometry curriculum. Today, there are some students who think geometry is fun, and there are another students who think geometry is difficult.
2. Starting by learning intuitively
   Students learn geometry intuitively or empirically from the first grade at elementary schools to the first grade at lower secondary schools. In the process, logical thinking is gradually emphasized.
3. Euclidean geometry with additional axioms
   From the eighth grade, students learn Euclidean geometry deductively. However, statements on the following geometrical properties are not proved deductively but treated as basis for proof like axioms, since these properties are already studied intuitively or empirically in the former grades:
   • properties on parallel straight lines
   • three conditions for congruent triangles (SSS, SAS, ASA)
   • three conditions for similar triangles.

1. Content to be taught
   Then, students start to argue some propositions on triangles, quadrilaterals and circles deductively. The following propositions are taught at the second grade:
   • sum of inner angles of an n-sided polygon is (2n-4) right angles
   • sum of exterior angles of a polygon is four right angles
   • bisector of vertical angle bisects the base at right angle in an isosceles triangle
   • line joining mid-point of two sides of a triangle is parallel to the third side and equal to half of it.

1. Teaching and learning process
   A recommended process in teaching and learning geometry in mathematics classroom is as follows: students start with concrete and empirical activities, discover properties through inductive or analogical thinking and finally prove these properties deductively.
2. Further leaning
   From historical view, value of Euclidean geometry has declined in geometry curriculum of upper secondary schools (Nagasaki, 1992). Instead, analytical geometry, trigonometry, vectors are introduced into upper secondary curriculum. However, Euclidean geometry was introduced as a selective course into upper secondary curriculum in the latest revision. Its intension would not studying product of the Euclidean geometry but enjoying Euclidean geometry as process of inquiry. But number of students who take this course is a few.

Here is a problem on teacher education. Prospective teachers' experiences to attack Euclidean geometry are almost the same as lower secondary students', if they do not take such courses in teacher education. Do they teach Euclidean geometry adequately?

Issues on geometry curriculum

1. Harmony between intuitive way and deductive way
   As mentioned above, students treat the same propositions repeatedly both at elementary schools and at lower secondary schools. Therefore, some students are bewildered. For example, let us consider a proposition, Sum of inner angles of a triangle is two right angles. At the fifth grade of elementary schools, students recognize this fact by empirical way, measuring angles or cutting and gathering angles on paper. At the second grade of lower secondary schools, they learn the same proposition. In the learning, they prove it by deduction based on propositions about parallel straight lines. Students learn not only the proposition itself but also new type of inquiry method. Therefore, learning level is higher than before. But, if students cannot understand the significance of difference of treatments, then they feel such learning meaningless and dull. Some students said to a teacher: I know this. Why do you force to study the same thing once again. I will not study this. (Kunimune,1987) Not only for curriculum development but also for teaching practice in classroom, harmony between treatment on geometry intuitively and deductively is still a big issue.

1. Harmony between students and Euclidean geometry
   In geometry, students are generally required to have higher abilities beyond solving simple routine problems like computations or equations. Students first think inductively or analogically about a property, and guess what statements may be true, and then prove these deductively. This process makes not only pleasure for inquiry in some students but also difficulty for learning in other students. Particularly there are some students who have difficulties in proving mathematical statements deductively. Though there is some problems to teach Euclidean geometry as mentioned above, many mathematics educators and mathematicians in Japan believe that Euclidean geometry should be taught in lower secondary school. The reasons are as follows: first, we have historical experience in 1950s that aim of geometry education was not satisfied by only intuitive treatment for geometry; second, if Euclidean geometry is omitted from geometry curriculum, there is no other field that students explore, prove and explain truth of propositions. Euclidean geometry is one of the most suitable mathematical content which is constructed systematically. How to cope with geometry curriculum without differentiation is a big issue.
2. Harmony between Euclidean geometry and other geometry
   In 1960-70s, transformation and some topics from topology were introduced into geometry curriculum as mentioned above. Further, research on introduction of vectors into lower secondary school mathematics was conducted, too. However, there is no enthusiasm to discuss geometry other than Euclidean geometry now. In other geometry, mathematics educators believe, there was few background that students could understand by deduction, and those geometry could be studied only intuitively. Here we will face a very big issue: what is geometry and what is mathematics for lower secondary school students.

Future of geometry curriculum

Conclusion

Generally speaking, the importance of Euclidean geometry at lower secondary school mathematics is appreciated among almost mathematics educators. However, it seems to us that answers to the following three types of questions will be big issues for changing geometry curriculum: needs of mathematics, needs of society and needs of students. The first, what type of geometry should be treated in curriculum is still an open question. But there is scarcely discussion in our society of mathematics education. The second, geometry is often taught without connection to others, for example, other subjects and our life. Though it may be essential feature of geometry, is there any way to connect? The third, all students must learn proof in Japan. How can we cope with a variety of students abilities? Probably, this may be the biggest factor for changing geometry curriculum in Japan.

References

Howson, G., Keitel, C. & Kilpatric, J. (1981). Curriculum Development in Mathematics. Cambridge. 288 p.

Japan Society of Mathematical Education (JSME) (1995). Mathematics Education of Japan for the Last Fifty Years after the War, What shall we aim at in the future? Journal of JSME, Vol. 77 (6/7). 157 p. (Japanese)

Kunimune, S. 1987. The Study on Development of Understanding about the Significance of Demonstrations in Learning Geometrical Figures. Journal of JSME, Reports of Mathematical Education. Vol. 47/48. pp. 3-23. (Japanese)

Nagasaki, E. 1992. Changes of Teaching Plane Geometry in Secondary School Mathematics of Japan. Tokyo Gakugei Journal of Mathematics Education. No.4, pp.133-141. (Japanese)

The National Council of Teachers of Mathematics(NCTM) (1973). Geometry in the Mathematics Curriculum. NCTM 36th Yearbook. 472 p.

Whitburn, J. (1995). The Teaching of Mathematics in Japan: an English perspective. Oxford Review of Education. Vol. 21, (3), pp. 347-360.

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