The lesson of "Division containing the remainder "
- Creating the rule of division having the remainder by the pupils themselves-
(3 grade)
(Sorry, this lesson might not be translated into English well.)
1.About this teaching material
The pupils learned division not having the remainder,its meaning and algorithm, in the unit in the beginning of 3 grade. Then they looked for the quotient applying multiplication table a time. But this unit is one that enlarges the learning and thinks out algorithm having the remainder, applying multiplication table a time. It is important that the pupils learn not only the skill of division having the remainder but also think about the relation between the remainder and the divisor to be thought as its principle.
In this lesson, I made the pupils think about ( the remainder )< ( the divisor ) inductively from the results of calculations. Next, I made them talk about the results of calculations of (the remainder)>( the divisor ), as counter examples. I aimed at adding
their learning to create mathematics by themselves, not stopping at being taught something by the teacher and using them. In other words, I wanted the pupils to create
the rule of (the remainder) < ( the divisor ) through discussing about it each other. Therefore it was very important whether I could pull their thinkings out and conduct them well.
2.The record of the lesson
First of all, I presented many multiplication problems to the pupils and indicated them
to solve them (fig.1).
44, 48, 412, these three calculations were easy for the pupils because of ones learned yet. But other calculations were naturally difficult for them. These are indivisible. The answers that the pupils found were the bellow. For examples, 104=
2 remainder 6, 104=2 remainder 3, etc. The answer that I was interested in was 84=2 remainder 0. This answer was one that Takashi(boy) thought out. If we divide any
number by 4, we would get 1 or 2 or 3, as the remainders. But he thought of the remainder 0 when the calculation was divisible. I think it is very big discovey to mathematics learning. Because the remainders have not only 1,2,3 but also "0".
Then Hiromasa(boy) answered as 124=2 remainder 4 . After I asked them whether
anybody wanted to ask why, Jun(boy) raised his hand and said ,"Why does its calculation have the answer, 2 remainer 4. It can be divided by 4." He insisted that the remainder 4 was strange. And Kanatsu(girl) explained that as 104=2 remainder 2, 114=2 remainder 3, thus continued, I guessed he had thought next calculation 124=2 has remainder 4. Her explanation was so logical that other pupils could understand Hiromasa's thinking well. She found The remainders stand as 1,2,3,4-----,orderly.
Nex I asked them what number they would like to change the divisor 4 into. Then they
answered 5 and I presented many problem including 5 as divisor(fig.2).
The pupils answered , 55=1, 65=1...1, 75=1...
2, 95=1...4. (...¨remainder) But 105=2 and 105=1...5. The latter was one
Yoshitugu(boy) answered. Yoshitugu answered 115=1...6 too. Then I asked them whether anyone answered the same and none. About 125, there were several answers. These were 125=2...6, 125=2...2. Yoshitugu answered 125=1...8. But
he is not low ability boy.
Takashi's thinking, the remainder having "0", is very interesting. Adding to it, Kanatsu said, " The largest remainder is 4, this number is one smaller than 5, the divisor." To her thinking, I replied ," Her thiniking is so fantastic, is'nt it? " Then moreover she said, "Comparing 7 and 5, 7 is 2 larger than 5, and we get the remainder 2. And 8 and
5, the remainder 3." To my surprise, Kanatsu saw through the rule of the remainder,
which can be gotten by subtraction between the dividend and the divisor.
And I tried tham to discuss about Yoshitugu's answer, 115=1...6, saying "Let's think about this." Jun said , " 5 and 6 makes 11." Yoshitugu explained, "First 11-5=6, the remainder 6. This is by subtraction not addition." I said, "Compare this answer with
115=2...1 , Is there anyone wants to talk about it ? "(fig.3)
Then Kanatsu murmured,
"The remainder, 6-5=1" And Jun added to it, " 5~2=10, so the answer is 2, but
he gets 1 the remainder 5. It's strange." But both thinkings are right concerned with
ascertaining, multiplying tha divisor by the quotient and the remainder makes the divident.
That is, the former 5~1+6=11, the latter 5~2+1=11. These look both correct. And
I asked them whether both thinkings were right. Then Yoshitugu answered, "I think
both are well." Kanatsu answered, " I think the lower idea is better." To her respon,
I asked again, "Why do you think so? Can you explain the reason? " and " Let's talk
about this around you. " After a while , I asked the pupils which thinking was better.
Then 7 pupils( of 10) answered that 2 remainder 1 was better.
The lesson had began to the last and the most important matter. I told them, "Please
look at the remainders and the divisors. How about both size? " And Yoshitugu answered, " one smaller---than the divisor". Moreover Kanatsu said, " The remainder 6 is more than the divisor 5, so I think the lower thinking (fig.3) is better."
Then I gathered two opinions, saying "In division, we get the remainder less than the divisor. We found gathering Yoshitugu's idea and Kanatsu's idea was better. So the remainder might become less than the divisor."
But Jun asked me," How should we do when the remainder become all means more
than the divisor? " Indeed, we might think out such division.(e.g. 115=1...6 )
I thought that Jun's question was important to us and mathematics learning. Then I
asked tham to name such division. And Yoshitugu answered, "Bigger Remainder- Division( Amadeka-Warizan in Japanese)" It's naming was very interesting and proper
to mathematics learning. Because mathematics, I think , is like a building constructed by somebody freely. "Bigger Remainder-Division" has remained as our fortune now.
3. Conclision
Main purpose of the past teaching was to make the pupupils remember the fact,
the remainder is less than the divisor. However I thought even such rule should
be found out and created by the pupils. And I wanted to construct such learning in
mathematics education.
Of course, you might insist that it is easy to teach it for them. Certainly starting from children's simple ideas like this lesson might be not efficiently because of much
time. But If we hope to foster children real thinking power and expressive power,
such constructive learning would be indispensable for mathematics learning. To foster "Independent Learner", we need starting from children's simple recognitions
in mathematics lesson.
By the way, we will think about the subjects on such lessons.
(1) When we would start from individual subjects in math learnig, it is necessary that the pupils need to obey the truth through interacting their opinions and deepning and refine their own thinking. If only they should insist on their own opinon, it should stop at mere own-rightness and not deepen their learning, moreover be not helpfull for their human-constructure. Fundamentally, one's simple thinking contains any failure and so it is not right that one sticks to it.
(2) Before the pupils are undergone training to obey the truth, they need to foster communication-ability, hearing other person's thinkings and deepening one's thinking with interacting their thinkings each other. In other words, they must not think as absolute thinking . Rather they need to respect it as relative one
and foster ability to compare with other thinkings. Therefore, the teacher must
make the pupils interactive their thinkings and obtain teaching techniques to improve their thinkings.
(3) In math curriculum, We should make a plan how to attach great importance to constructive learning and how to set time to fix fundamental learning. It is the problem of cognition of importance and balance, so we should arrange both learnings appropriately. But I think that we should aim at instructive learning and could add such elements as possible as we could. It would apply to every teaching materials.
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