Excellent Japanese Teachers and their Works(2)
Kozo Tsubota and his Work
He is now a teacher of the Elementary School attached to Tsukuba University in Tokyo.
In a word, I think, he is just an elementary school teacher who is good at teaching students and has excellent teaching theory. I have ever seen his mathematics lessons some times and always felt his teaching techiniques and skills so excellent. He can deal with students very well in order to deepen their discussion in math lessons. In his book, he says, "In the lessons of mathematics, we teahcers should set the situations that students would think "Why?" or "Oh dear!"and consider about why the matters become so each other. And we should be a consultant or a emcee in the math lesson and praise students'thinkings. Furthermore, it is the most important role of teacher to set the situation that students would think "I got it!!". " He has endeavoured the development of elementary mathematics education in Japan for about 30 years and got much results and written many articles and books. He now influences us so much as a active teacher and a researcher .
His career of research on teaching is the bellow.
1) Students' making problems
2) Open-ended approach,and so on.
He won the 32th Yomiuri Education Award.
And, he wrote many books about elementary mathematics education.
I will present a few of his practices here and state my opinion.
2. From the math lessons for the 2nd graders(Be freer, mathematics lesson)
First he showed students a large sheet of paper written a figure. (fig.1)
Students guessed the number of circle,"About 20" or "About 30", looking at the paper. Then Tsubota asked them why they thought thus in order to grasp
their grounds. Next he handed them one by one the paper written a figure. And soon, students counted the number of circle in the figure and shouted,"25!". The answer became clear already, so it is not the purpose that students will search the answer-the number of circle.
The problem after that is that how students would count the number of circle easily. Each student began to think hard. After a while, he made students present their thinkings, telling them to express their own thinkings as a formula. Then students began to present their thinkings. But Tsubota did not make them explain about their thinkings themselves, on the contrary, made the thinkings judge by other students. He thought
that by doing so students would be able to interpretate it variously and it is the meaning
for students to learn mathematics in the classroom.
Presentation1 A(boy): 5*2=10
3*2=6
7+10+6+2=25
We can understand the meaning of the formula soon. But he made other students think about it.
7+(5*2)+(3*2)+(1*2)
Other student explained, "I think, he saw the figure having 2 sets of 5 and 2 sets of 3 thus."(fig.2)
And another students explained, "7 is the center, and 2 is the both side."
And he made students name this method, as "Standing method in line".
Next presentation2. B(girl): 3*3+4*4=25
Many students shouted, "Ah, it is the same as mine!".
Then, Tsubota said, "I will ask other people explaine this formula."
And a student explained, "Its mehtod shows that she counted the number of colored circles and white circles looking diagonally."
(fig.3)
But, some students had other views from the same formula. A student presented as the
bellow.(fig.4) 
presenation3: I thought thus, though the formula is the same.
(3*3+4*4=25)
This was a wonderful thinking. Other students all were very frightened. Certainly, we can find that there are 3*3 in the center area and 4 times 4 in the four corners.
Students named these method soon,the former as "Changing View Method", and the latter as "4 Dividing Method".
By the way, the formula thought by next student brought about much discussion.
presentation4: 5*5=25
* Can you guess how the student thought about it?
Other students wondered the reason why. Because the presentation was expressed
in the simplest formula, too simple! All the students could not understand why it was expressed as 5*5. We must see that there are 5 times of gathering of 5 in it. But we can not see it like such formula, even if viewing from various points. For the while, we thought about presentation4. Then, a student said, " As he knew the answer was 25, he made the formula, 5*5, didn't he?"
But, another student came to the blackboard and said," I wonder, perhaps he moved
this O(circle)?"(fig.5) 
Other students marvelled at the explanation, "Ah! that's true! Great!" For the method, a student named as "Revolving Method". The student who presented the formula(5*5)
showed a pleasant expression.
The left figure shows the formula 5*5 surely!
And yet, there was another point of view for the marvellous formula. The teacher(Tsubota) also was very surprised. A girl came to the blackboard and said,"
I could find 5 5-spots of dice." Saying so, she drew the figure as the right one.
I think this is wonderful viewof point. All the students admire the thinking, naming as "
Dice-Spots Method".
Then, fun to think hard on counting the circles spread to the students, and they said,"
We would like to present our thinkings more and more!",furthermore, the lesson became more lively. Some students became to present "My thinking is oo-Method.".
That is to say, it shows that the matters which could not be thought out by oneself have appeared in the lesson as various kinds of interesting thinkings through reading the formula. Saying in the other words, even the matter that one student can find only few
idea in it, through discussing by many students, they could find some ideas which have not be appeared. This is the very lesson to appreciate pleasant to learn together. This lesson must be so interesting to students and it comes from Tsubota's ideas. These ideas are not superficial, on the contrary, ideas on the basis of the eyes on students and teaching materials.
3. From the Lesson of "Division of Fraction; Measurement Division with Remainder(6th grade)
Tsubota tried the following lesson in order to search for what understanding students would hold on division of fraction. In general, it is said that the process to instruct division of fraction is understanding the meaning of calculation process to multiplying A
by the reciprocal of B. Certainly, it is important to understand the meaning. However, the calculation procedure is not given students but should be constructed through producing the meaning. And a few lessons after introducing lessons, students would become to be drill-learning that division of fraction might be "Multiplied A by the reciprocal of B". Students would master its procedure and solve the calculation at once.
And the teacher would be pleased to see them. To tell the truth, there is just a pitfall. Tsubo showed the fact cearly by the following problem of measurement division.
In the above problem, he selected the problem number as not 3/4m but 2/3m. This showed that students had to think about the meaning of division again by operating measurement division of fraction, not decimal fraction.
At the first Tsubota made students take notes on the problem and read the poblem statement. And students began to solve the problem.
Several students solved it soon, and Tsubota named four of them, having them write down their answers on the blackboard.
Students were used to applying the procedure of multiplying by the reciprocal to dividing by the fraction, because it was the content of former lessons. Therefore, they
judged that they could use the same procedure in this problem and calculated it soon.
Furthermore, they interpreted the remainder of the answer as the asked part of not integer. But, it was not examined whether the remainder was true answer. They were full
of confidence of their answers.
Then, Tsubota made them explain their answers. (omit this part)
Seeing their presentations, a boy said, "My answer is, 7 ropes and remains 1/3m...".
Tsubota said to him, "Well, you have somthing to tell them, and your answer is different from them, isn't it?" And for the four students, other students argued against them one by one as bellow.
C5; The answer, 7 ropes and remains 1/2m, did not fit with the proof.(7*2/3+1/2)
C6; 1m, the answer of C4, as it is 2 of 15 / 2, shows 1/2.
C7; My answer is also, 7 ropes and remains 1/3m. However, 7and 1/2 is the same. 7 and 1/2 shows, I think, that there is 7 and 1/2 times of 2/3, so
7 times of 2/3 and remains 1/2 of 2/3. Can't you understand this? Please
understand me!
C8; The fact that there is 1/2 of 2/3, 2/3*1/2, I think.
C9; C4 explained the answer remains 1m, but if remains 1m, we could get one more 2/3m. So, it is strange.
The discussion changed to the stream how they should solve it correctly. And Tsubota
said to the students, "I wonder that there is 7 and 1/2, .....look, people gape in surprise.." This remark shows that the former explanations from students are not sufficient, because they are all about the formal procedure.
However,another student proposed to explain it using the diagram. (fig.6)
Seeing C4's presentation, Tsubota said, "This explanation was easy to undestand...diagram is great,too. Here remains 1/3 m. But, it can not be gotten from the calculation, 1/3m. Your methods of calculation are wrong? We can look at 7, but
where 1/2 has gone? Many people wrote as the remainder of 1/2m. I wonder what 1/2 was? Explain about it, somebody...". And C10 and C11 answered,
C10; This 1/2 shows, not the unit of meter, 1/2 as the unit of 2/3, so
2/3*1/2.....
C11; Become 1/2 within 2/3, this 1/2 is 1/2 within 2/3, so......
T; Telling only with the words, we can not understand you.....
And, Tsubota asked students, "Is there anybody who wish to speak more? We will hear
form C11 who has insisted on 1/3 at the first." And C11 wrote his thought on the blackboard.(fig.7)
Furthermore, C7 explained as follows.(fig.8)
By thinking his explanation comparing the former diagram, students became to understand the meaning of the formula. And, Tsubota confirmed with students that the unit of 7 and 1/2 is 7 and 1/2 ropes, and the fact that there is 1/2 of 2/3m, is 1/3m. Furthermore, he pointed out that dividing by fraction is not solved by merely multiplying by the reciprocal, following the problems, we should examine the formula and the answer
again.
I introduced Tsubota's two lessons, but could not translate the latter one into English very well. Sorry, please read between the lines.
4. Conclusion(Editor's Comment)
I have been taught from his lessons so much. I might not understand his intention of them, therefore, there might be diverse interpretation about his lessons. How did you think about them? It is not difficult for us to understand the former multiplication lesson, on the contrary, the latter measurement division lesson is difficult. The factor of difficulty lies under teaching materials and the method of teaching. We cannot probe this teaching method and thinkings well. Tsubota, he is now practicing new lessons passionately everyday.
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