Draft
Gathering circles, it is the teaching material which keeps creativity and variety.
Koji Nakagomi
Introduction
My idea was born from a combination of a student's notebook doodles and a tile pattern in a rest room. When I was teaching third year junior high school students in Japan, a student was ignoring my lesson and was instead drawing a picture similar to figure 1 with a compass. It's the kind of situation to which I always pay attention, and even though he wasn't doing well in math class I praised him because the picture was interesting.
I soon forgot about him. However, a few months later while sitting on the toilet at my relatives' house, I looked down and noticed that the tiles were similar to those in figure 2. It was then that I noticed that starting from a hexagonal tile all the tiles would spread outward in a hexagonal pattern. Starting from a triangular tile the tiles would spread outward in a triangular form. A square would follow a square. A pentagon would follow a pentagon, etc.... . More-over they have many laws. It was then that I combined the student's doodles with the tile pattern. I thought that here was an idea for teaching material which would contain a wealth of variety, creativity, and would make the class interesting.
Figure 2
In the Third International Mathematics and Science study(TIMSS), the point of mathematics test of the junior high school 2 graders in Japan is 605 point and that is 3rd place (the international average point is 513). Also, as for 47 % of the students in Japan, they dislike mathematics and as for 53 % of them, they like mathematics (as for the international average, the former is 32 % and the latter is 68 %.). In this way, it is possible to say that even though grades are very good in my country, many students do not like math. One can think of the following reasons for this. First, math classes may not be taught in an interesting way. The problem would then be that students have no interest in the mathematics. Second, students may not have been able to continue getting good grades. They may have falling behind. Thus, insufficient knowledge and a weak grasp of the subject matter would be seen as the reason. Moreover, as a public junior high school math teacher I would say the problem is that students are not good at expressing themselves and their opinions. Mr. Okamoto, author of Starting Classroom Debate, says the following:
It's becoming increasingly important that people have the ability to express themselves in order to communicate effectively so that people can communicate with each other as human beings. Teaching methods within the education system that teach ways in which to express oneself effectively seem to be increasingly sought after.
I believe the above quote not only applies to my students but that it is a normal characteristic of most Japanese students. Many students are passive. In normal lessons, emphasis is placed on the acquisition of the answer method and memorising the knowledge. Students are rarely asked to put their knowledge to use in a creative way.
Study Objective
The first objective is to develop teaching material and methods which, while maintaining students' interest, will encourage students to think creatively to solve problems, will develop mathematical thinking , and will create within the students the ability to look at problems from many different angles.
The second objective is to research ways in which to teach classes that encourage students to express the logic of their thoughts and to discuss them with each other.
The teaching method, which is called the open-ended approach, has been studied by Mr. Shigeru Shimada et al., are quoted as saying:
In the teaching method that we call an "open-ended approach", an "incomplete" problem is presented first. The lesson then proceeds by using many correct answers to the given problem to provide experience in finding something new in the process. This can be done through combining students' own knowledge, skills, or ways of thinking that have previously been learned.