Instruction connecting calculation procedure with the principle of calculation

-2nd grade : From the instruction of carrying down in the subtraction "-"(2 place number)-

since 28.Oct. 1997

1. Bridging is an obstacle to the pupils in the lower graders.

Mathematics in the lower grades contains mainly basic addition and subtraction, which are proper materials to the teachers, the adults. Therefore, we apt to think that the pupils could understand and calculate them if we would taught them about calculation procedures once. The typical materials that we think so are addition and subtraction having bridging. When we teach the pupils them, we find that so many pupils don't understand and calculate them. Though the teachers, who have taught the pupils in the lower grades for long time, have a knowledge of difficulty with it, we have to search about how to improve the instruction in order to build up pupils' understandings and calculation ability.

2. Before repeating drills

Many teachers think that the pupils have to solve many calculating problems as possible as they can. It is, what is called, practice by drills. Of course it is an important study, but before it we have to make the pupils understand about why they can calculate with such procedure and the principle of calculation. In other words we have to make the pupils consent to it. It is after the pupils understand the principle of calculation and the mechanism, connecting to calculation procedure, that drill study becomes effective.

3. From the lesson record of "Subtraction of 2 place number"(2nd grade) (1)How do the pupils deal with unlearning calculation when they meet it?
The unit, "The subtraction of 2 place number", has two small units, the first of which is the subtraction not to bridg. And the second unit(3 hours) is the subtraction to bridg. Let's think about the lesson of introduction in the second unit together. The introduction problem is the following.

 "There are 37 children playing.  The number of the girls are 18.
                       How many boys are there?  "

The pupils could understand what they should seek for. And moreover they could understand its calculation, the subtraction, formulating "37-18". And they arranged

this formula into   37     .   I asked them what was different from the calculation that
                   -18

they had learned since yesterday. Then Taiga(boy) answered that the calculations learned since yesterday was one which they could subtract 1 place number,but today's calculation was one which they could not subtract 1 place number soon. And I told them ," Let's study the subtraction that we can't subtract 1 place number soon". Next I asked them challengingly whether they could compute this calculation. For my asking, they responsed vigorously," I can, I can -----". Though it was a calculation unlearned, I made them compute it by themselves. As a result the following answers appeared. 
                     A    21------------24 pupils 
                      B    29------------  1 pupil          
                      C    19------------12 pupils
                      D    20------------  1 pupil
                      E     Don't know------   1 pupil
For the result, I asked them,"Are there some ones that want to ask whether it becomes so?" Then Yui(girl), who's thinking is 19, asked other pupils," Why does it become 21?" For her question, Emmi(girl) answered, "I think we had better turn over 7 and 8. Because we can't subtract 8 from 7, subtracting 7 from 8 is 1 and subtracting 1 from 3 is 2." Other pupils who answered "21" told as same as Emmi's thinking. Perhaps the pupils ,who study mathematics only in the school, would give an answer using the strategy of making do with it. They do use the rule of subtracting small number from large number. It is just a solution to use knowledge learned yet. Testuya(boy) who answered as "21" has a bud of carrying down, but can't treat 2 place number. To my interest, it was Yuta(boy) who answered as "20". I guessed his thinking,

The pupils who answered as "I don't know", I think, are just honest, so their thinkings are fine ones. Anyway pupils' thinkings are as they like, with different from the adults. I would like to recognize their appreciation of thinkings. I did not have the pupils talk about these thinkings more, but to raise their will to research we may have them talk together moreover. Founding thus talk, next concrete operations will be effective.

(2)Carrying farward the lesson on a obstacle to concrete operations

We had the matter what a right answer was, because of differing the answer of "37-18". Then I asked the pupils," With what you can solve it?" The pupils answered, "With blocks" I guessed that they might thought to it soon as they had used sometimes them since 1st grade. However it was not easy for them to operate blocks. Though I had them put "37", 2 place number into "3", 1 place number "7", the next operation was more difficult. In a word we can't get 8 from 1 place number "7".

After I asked them,"Are there anyone introuble?" walking around the pupils, Hironori(boy)told that he could not subtract 8 from7 and also Yuta murmured that he could not compute this calculation. There was a important point. In this lesson I asked them," How sould you do?" and I had the pupils, understood yet because of preparation of lesson, present the idea that borrow from 2 place number. But it might be better that the pupils would solve the difficulty by themselves. In a word this idea is one that they learned in 1st grade, one to borrow 2 place number, in the unit "2 place number - 1 place number", so the pupils introuble might present the idea by think more deeply. But I found that "2 place number - 1 place number" and "2 place number - 2 place number" were quite different as like different kind of calculations for the pupils. Therefore it is very different from "17-8"(which they learned in the 1st grade)and "37-18"(taking up now).

(3)To map block operation into calculation procedure

After the pupils understood that they could soon subtract from 1 place number and presented the idea to borrow 1(that is 10)from 2 place number, next we need to map block operation into calculation procedure. It is important that the pupils operate blocks and create calculation procedure seeing the meaning. In the lesson, as a pupil came to the blackboard operating blocks, I presnted the pupils the calculation procedure mapping the operation.

This learning ,mapping block operation into calculation procedure, has a function that connects calculation procedure with the principle of calculation in pupils' thinkings. Stopping block operations one by one and making calculation procedure, the learning, which have the pupils make sure of it, may be rather sure in understanding calculation procedure and acquiring it.

4. Conclusion

Even if how effective instruction method it would be told, it is not all-round and can't have all the pupils understand it soon. This method told "Mapping instruction" have much possibility but remains several problems in practise. Some pupils may think about it with difficulty. As a suggestive method, I will practise trying it and develop one that will be useful for the pupils. (This article was put on the "The elementary Mathematics" Oct. 1996.)

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