1.Beginning
Generally speaking, it is said that to recognize spatial figures is difficult,especially for younger children. So I tried to search in the math lessons whether it is true ot not. Teaching material, the title is "Let's make a dice." In this lesson, I had the students draw some plans(figures) to make a dice. And I had them discuss about the plans whether the plans was proper to make a dice. Lastly we thought of the rule that is common with plans which are able to make a dice.

2. Lesson record and some plan
First I passed a plotting paper to each student and asked them to draw a plan to make a dice. They began to draw it, seeing the picture of dice put on the text book. Of course they drew it using a ruler(almost 30cm-ruler). For a while, they drew 4 kinds of plans and showed them as the below.

Speaking exactly, we had one more plan(fig.x). We first discussed about this plan whether it is proper. It has really 6 surfaces that need to make a dice. But a student said, " It has not another bottom. It can be a dice." So the plan was denied by them. To make a dice, the plan needs 2 bottoms. Then, I asked them, "Could you make a dice when you would draw a plan?" That is, I asked them the condition to make a dice. And Hidechika answered, " They all have 4 surfaces horizontally, and 2 surfaces longitudinally." Certainly they have 6 surfaces and they can be made a dice. Horizontal surfaces are лллл, and longitudinal surfaces л л. Longitudinal surfaces connect above horizontal ones and below by each one. But Yuichi's plan was different from others and has 3 horizontal surfaces, 3 longitudinal ones. We were at a loss. How should we overcome the obstacle? We became to notice that the problem was not the number about the surfaces. Then we payed much attention to these figures and thought well if there were any rules among them. After short minutes, I thought that there might be an important rules in the sides. Therefore, I watched closely these figures, especially their sides(boundary lines). And I did find that there were common horizontal sides and longitudinal sides among them. That is, the plans that can make a dice involves 2 horizontal sides and 3 longitudinal ones!! This is a condition of plan to make a dice. So I told the students to watch the plans once more again.

As long as the condition is filled in the plan, they can draw any figures. We could think many plans besides these plans. Then we changed our attention to other problem. A dice has 6 surfaces and 12 sides , 8 vertexes. I tried to discuss about the numbers, 6, 12, 8. Because I thought that there might lie any rules among 3 numbers. I asked the students if they could find any rules among them. They gazed at them, and a boy answered, "12 is doubled 6." Surely that was right but we could not understand what he meant. What does the fact that the number of sides is doubled the number of surfaces show to us? We thought of it more and more. A surface has 4 sides, so 6 surfaces have 24 sides-----12 is a half of 24------. At the first time, I found that when we make a dice, the number of sides reduces into the half of all sides. That is, it shows mathematical fact as the below.

3. Conclusion
When we teachers discover teaching materials to cultivate children's mathematical thinking, we need some ideas for them to grasp themselves. We had not to impose on the students such discoveries. Impressions by discovery causes from sufficient activities and after that reconsidering them. We had not to be impatient and more teach it to them. To create mathematics, it takes long time and it is important for us to wait well. Even from natural materials we can discover any mathematics rules or meanings. It is real mathematics learning, I think. By piling up such learnings, students will become to be interested in mathematics and increase their will for math learning. We ourselves have to find out interest of mathematics and appriciate it with students. Certainly students are looking forward to such math lessons.

Reference ; A lesson of geo-board (private file)

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