LESSON4

The lesson of БuDivision of

and

Бv
Б|Enlarging children's world related to their simple questionsБ|

since 4.Oct. 1997

1.Introduction
We have to overcome two tasks to realize the lesson which has pupils appreciate some pleasure of problem solving, asking the questions for themselves. One is to make chil dren ask their questions, another is to investigate the questions and to give lessons which can make children feel pleasant to solve the problems. The former deals with the instruction of how to ask and the latter is a big matter related to all lesson research. First I will present the lesson record and research the method to solve the tasks.

2. From the lesson record "Division of 0 and 1"
The 3rd graders study the concept of division and the algorithm of it, which contains the small unit-"Division of 0 and 1". If I explain fully, it is "dividing 0 by some number" or "dividing some number by 1". According to the textbook, we have to teach them two contents in one lesson. But I will present you the summary that taught them for two lessons.

Yoshitugu(boy) murmured, "What is '4БА0'?", when he understood '0БА4=0' by dividing cookies in the empty box. It doesn't mean "dividing 0 by some number", but "dividing some number by 0". But it is not the elmetary mathematics and the question that we need not to take up. However I thought it was very interesting. To take up this question ,I thought, would connect with deepening children's comprehension of division and I decided to investigate the question.
I asked them how they thought about it, and first Naoki(boy) answered that 4БА0=0 because 0Б~4=0. But Kanatsu(girl) soon told to him, "I don't know why it became so." Then Naoki came to the blackboard, explaining that the formula of division can be written 0 )4 , so multiply 0 by 4 - 0Б~4.Б@Though anyone did not understand his explanation, I told him ,"You did think so!" Next Jun(boy) answered that 4БА0=0.Б@However to this answer also other pupils asked why it did become so and he said only "I don't know". I guess that he thought so intuitively.

Next Hiromasa(boy)said,"To find out the answer of 4БА0, we have to use 0's row . But muitiplying 0 by any number is 0. Б@And there remains "4". I don't know how I treat this remainder 4 ." It is an idea thought well. After this, Yoshitugu told that he could not understand it though he thought enough. Its speech is frank. Then Kanatsu rised her hand and came to the blackboard. She said, "There are two methods of division(partitive division and measurement division), so the answer is 0 because 4 things are distributedБ@into 0 person.

Б@Б@

She had tried to solve it through operating the figure , thinking that the formula of "4БА0" is one of word problems. Kanatsu told abot not only the thinking of dividing into 0 person but also the thinking of dividing by 0, but she could not explain well. Then Miho(girl) said , " The answer is 0, but I can't explain of it." After I asked all of them about their thinkings(my class has only ten pupils), Yoshitugu spoke about Naoki and Kanatsu's thinkings. "If 4БА0=0, as division is the reverse of multiplication, 0Б~0=0. So it does not become 0Б~0=4, strange." This is Hiromasa's speaking too. If 4БА0ББ0 , it becomes 0Б~0=4, it is irrational. Because multiplying 0 by any number is 0.Б@ The pupils's thinkings were gradually converged on one, Hiromasa told that it could not be settle. It means that its calculation are solve by them. And the same time Yoshitugu said," it does not contain the answer." It looks probably 4БА0 has not the answer. Then I told them that this problem is one not to have a answer or not to be solved. They cried "Oh!", but I explained that it is a problem that they think wether the problem has a settlement.

Then its matter was over, but after 8БА1=8, Yoshitugu moreover questioned, "Well, how about 1БА8? " He might have an interesting about the reverse of 8БА1. However the calculation of 1БА8 is one that can be solved by them after they will learn decimal fraction in 5 grade. It is not one to be thought by the pupils who just finished learning division. But I thought I would take up this question. As the pupils presented their qustion specially, I thought that I tried to make them research it.

Then as I gave them some time, I instructed them to murmur about their thinking. And Hiromasa said," Something remains!" He felt some remainder, seeing 1БА8 .Б@I thought his speech was good. Next Yoshitugu murmured," Nothing to become 1 by 8'row." I saw the pupils think hard. Then Kanatsu raised a hand and came to the blackboard. She said," We take one piece when one block are divided by 8."(fig2.) In other words it is one piece by divided into 8 parts and just an answer of 1БА8. Yoshitugu, looking at the figure, said "Though this is a paraphrase, it becomes 1/8." But he has not yet studied fraction of 1/8. Certainly He must has recognized by Kanatsu's figure.

Then Kanatsu came to the blackboard again, and explained of it redrawing the figure. "I really wanted to divide by 10. But there remains two pieces if we deliver one piece to 8 persons.(fig.3)And she said, "A piece ,one of the figure,is called '1bu'(bu--Chinese character)." I asked her what "1 bu"meant and she answered that it is a character, a chinese one "bu". In other words Kanatsu knew that old Japanese called the name of the first decimal fraction "bu". Then I wrote on the blackboard, "1БА8=1bu remainder 2bu". Just then Kanatsu said , adding to it, "I have just remembered. 1bu is 0.1" Surprisingly she might have learned already decimal fraction(5grade).

As Kanatsu presented her idea that there were two remainders when dividing 1 by 10 and it by 8 persons, I thought that I would like to have the pupils think about two rmainders. I thought the pupils had beter reserch it with the method of using the figure as Kanatsu. In fact the method of her is rational. We decided to divide the remainder two parts into 8 persons. Then enlarging these parts, we divided by 10 too. And dividing it into 8 persons, there remains two pieces in the sme way. I asked them about the remainders, "Well, if we divide two remaindres by 8 persons?" And the pupils answered , "We get only a little." But even if we cintinue to divide it by 10 similary, there will remain the remain for ever and not over.(fig.4) Next this time we divided it by 8. There does not remain by dividing by 8. We can likely solve it. I told the pupils , "It can be likely divided." and Kanatsu replied,"Yes, it can be solve the last." (fig.5)In the word this method represents that the first operation is 0.1, the second one is 0.02, the third one is 0.005. In the lesson I did not show 1БА8=0.125 clearly, however, the pupils might understand quantitive size of the answer. And to my astonishing, Nana(girl)who did never speak in the lesson, came to me, murmuring near my ear" Teacher, it becomes 0.125." It is good that some pupils can present their ideas, but the matter how to take the thinkings of the pupils who do not speak in the lesson, was remained.

3. Conclusion

The questions that was presented by the pupils in the lesson were points of view or point of view changing the condition. It is the time when they think prosperously after learning something that they have their own questions. I expressed it before, it generates when we think about the relation between one knowledge and another knowledge. Of course the phase of the questions generated might be different within grades and progressive stages. And as the method to utilize it in the lesson process, I took the mehod to murmur their fruits and flashes this lesson. I wrote them on the blackboard and had them think deeply, attempting to structurize it. We appreciated the pleasure to discuss in the solving process than the pleasure to solve it out in the lesson. Under the simple questions the pupils' mental world was spread by deepening ideas in the class. I think that the experiences, that create mathematics by bump their own ideas, can change their own questions.(This article will be put on the magazine "The elementary mathematics education"published 13.Oct.)

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