LESSON FOR THE FUTURE

1.Opening
They say it is important that we must see scholastic ability with new point of view. But, though it passed for long time after they said so first, we can't see such practices that concrete it now. In other words, we can't see pupils' figure learning lively and teachers' figure instructing as to concrete it. Where does its reason exist in ? How must we reform mathematics education ?
I present my reform plan through my lesson and its analysis.
2. Lesson of understanding concept of square(1).
(1) About [Triangle and square]
In this teaching materials, we teach pupils understanding concepts of triangle and square and then understanding concepts of rectangle and square. The author of text book devises to learn concept of square in the lesson, contrasting concept of rectangle. But it can't attract them consciousness of right angle in 4 angles and equality in 4 sides, though it can aim at difference in 4 sides by constracting rectangle. In the other words, it is not sufficient for learning concept of square.

(2) Lining up square, rectangle, rhombus, learning to search a shape not in a group
To conbine with understanding concept of square, I had pupils search which, square and rectangle and rhombus, is not in a group. Rhombus is not known by pupils as a name and is only one of squares.
In the begining of the lesson, to ascertain pupils what is searching "not in the group", I presented pupils five papers which is drawn three kinds of objects and they searched "not in the group" among them. This is warming up to learn and pupils answered vigorously. Among them, it was a group that contains "a ruler, an alarm clock ,a tape measure". Most pupils answered " an alarm clock", but Yuta(boy) only answered "a ruler". I asked him why he answered so, he told me that only ruler is straight figure. He may attention to figures.
Next, I said to pupils, " let's think about not in the group among the following figures" and boarded the below figures.

And I instructed pupils as the follows, " which is not in the group among them? Write that I think ○ is not and why." Seeing pupils' describing walking among the desks, I found 「ウ」(the third) are almost. Then I quivered pupils' thinking, saying "I am rather dull because all answers are same." Pupils looked like having thought again for the teacher's saying. I reminded them a lesson that all are not correct though all their answers are same. I have taught them that all the member agreement is very dangerous.
And I asked them. The result are the bellow.

               Maya(girl): "ア" is a square and "ウ" is a shape that we see a square 

                                from the above of a square.

               Hiroki(boy): "イ" is only one rectangle.

               Yuto(boy):  "ウ" is a shape as "ア"is crushed.  "イ" is a shape as 
                            double "ア"s.

               Yuta(boy):  Though "ア" and "ウ" are the same long in sides, "イ"is

                             not.

Then Yuto(not Yuta) came to the blackboard and explained that both "イ"and "ウ" are made of "ア". In a word, he says that "イ"and "ウ"are both based on "ア".
Moreover, Yuta came to the blackboard once again, explaining that "イ"and "ウ" can be transformed to "ア" but not "ア" any more. He explained with use of a word "transformation" that "イ"and "ウ"can be transformed to "ア".
And just then, Yuta murmured, "Perhaps ア may be not in a group・・・". And Yui said the same things, too- "Yes, ア may be not in a group." Other pupils linked to such idea as Yuta and Yui.
So, I commented that ア might be not in a group. And moreover I told them " We learned that we can think about one thing in various ways, not showing the correct answer."
And I asked them what name we can call this shape "ア", one of them answered as a shape not to be able to tranformed. Then I told them that the shape can be named so and explained that the shape is named "square". Moreover, I wrote on the blackboard ,"Have 4 right angles, 4 same length wides."
The lesson ended after pupils made a square from a rectangle.

(2)About this lesson
Pupils learned "Not in a group" in order to make groundwork of understanding concept of square. Through this learning, I think they could understand concept of square rather naturally. While they are looking for "not in a group", thier thinking are deepening and thier discussion are gotten active , with clearifying concept of square. Then we named the shape and one more time we disposed of difinition of square.

3. About Learning Task Theory
You may feel weird to hear learning task theory after I wrote about a lesson of concept of square. After al you may hear first of "learning task theory". I think that it is important to mathematic learning that we clearify learning task and pupils have clear task-consciousness. But how important is learning task for mathematic learning? I have doubted about this matter since before. Though I can't explain of this question, even this lesson was not clearified learning task or learning goal. If it has learning task, it's task would be "which is not in a group?" And it would not connect with difinition of square. With importance of learning task, I think we must acknowledge lessons not having clear learning task on new mathematic lesson in the future.