1.Introduction

Mathematics, I think, is a learning that expresses the rules and laws in the nature with a language of number. Therefore, mathematics learning have continuosly new discovery and often even invention. In this lesson, we think how to cope with children's thinking and analyze how children's thinking develope as children find out some rules of multiplication table(3's row).Then, I will soppose that we should remove our frames of thinking in orderto create new mathematics lessons.

2.Instruction of multiplication table

In Japan, we teach our pupils maltiplication table in the 2nd grade, and its order is 5's row, 2's row, 3's row, 4's row---,I think this order is better. So, They have studied 5's row and 2's row since then and studying maltiplication table is going to be usual one for them. But every row has rules peculiar to and pupils await to discover it. Multiplication table has rather many rules as treasurys are hidden in the earth and I promoted a lesson talking to them that let's search hidden treasurys with pupils.

(2)The record of the lesson

The lesson began writing 3's row of multiplication table on the blackboard and reciting it again. Next I directed the pupils to find out the rules of 3's row that was written on their notebooks. After the pupils tought looking at 3's row and write down for a long time, they wrote some rules found out by them on their notebooks. I made them express their discovery in 6 to 7 minutes. First Kayo(girl) told that adding as that (fig1.)makes 30.But 3×5=15 only does not make 30 and same phenomenon ocurred in cases of other rows. The pupils tried to solve it from many points of views. In adding to it, Hiroki(boy) told that adding numbers of first position makes 10. Telling other words, 3 and 7 ( in 27) makes 10. Though his thinking is similar with Kayo's idea , I respected his discovery. Next Tetsuya(boy) told that these answers increases by 3. This fact is proper for 3's row, but a rule might be paraphrased and it was not proper for the pupils so I took it up importantly. Next Ami(girl) told that there are all -1,2,3,4,....8,9 in the first position.(fig2.)Aftter her announcement, Hiroki(boy) told that adding two answers makes 9. Asking him what he said, he answered that he thought as fig2. presents. When we look at numbers without 9 in 1 to 9, we can find out Hiroki's discovery. However I cannot explain what meaning it has. Then Yui(girl) came to the blackboard, telling that subtracting multiplying number from answer(solition)number makes 2's row in the multiplication table. (fig3.) This presentation unrespected to us. It was a idea not seen in the rules of 5's and 2's rows. And we can apply, adding answers to multiplying number makes 4's row, but its idea was not presented then. Then Ikumi(girl) told that adding 2's row to 3's row makes 5's row. Certainly 3+2=5 6+4=10.....so on, 5's row is made. The pupils are becoming to think flourishingly. Next Tomomi(girl) told that numbers in the tenth position consist of 1,1,1,2,2,2.(fig4.) Then I asked her how about 3,6,9 and she answered that numbers of the tenth position are '0'. In other words, though tenth position is empty, she thought, '0' exists there. And moreover Tomomi presented another rule. She told that 1and 2 makes 3, 1and 5 makes 6, 1and 8 makes 9, 2 and 1 makes 3, 2 and 4 makes 6, 2 and 7 makes 9, thus 3,6,9 are repeated. Adding number of first position to number of second position shows some rules. Asking her how about 3×1=3, 3×2=6, 3×3=9, she answered that they become 0+3=3, 0+6=6, 0+9=9. Surely we can understand they repeat 3,6,9. However I asked her more deep question whehter they really continue 3,6,9. She noded at hearing so. Then I made nexts to 3×9=27 with pupils(fig5). They had known how to make 3's row, that is, adding second number to the former answer(3's row). I wrote on the blackboard , 3×10=30, 3×11=33,　3×12=36. 　 And then, they looked showing 3,6,9 continued. But moreover when I wrote 3×13=39 , 3+9 makes 12, and I told pupils it doesn't make 3. Nevertheless Satiko(girl) said, "Teacher! These 1 and 2 make 3!" 　　

                    3×10=30　→　3+0=3
  　　　　　　　　　　3×11=33　→　3+3=6　　　　
 　　　　　　　　　　 3×12=36　→　3+6=9　　
                                  fig5.

　　　　　　　　 3×13=39　→　3+9=12　→　1+2=3

3. Necessity of removing our frames in thinking
I insist on defeating formalizm and uniformalizm on mathematics education. Moreover it consists of authoritarianism. In the other words, it could be called a fixed idea that we have worn without awareness for a long time. They shows that we cannot think and create freely by stiffing our thinking captureed in the frames. Mathematics, I think, is a learning of having freedom and creativity. And to study freely and creatively is to live really for us. How do you think about it?