1.Introduction
Learning of "Length" is connected with understanding of length to measurement with a scale. They learn indirect comparison with a voluntary unit and then a universal unit because of inconvenience, after that measure a actual object. But if we teach the pupils measurement as such, there will be some pupils that can not measure objects with a scale easily. They can not understand the relation of "cm" between "mm"and numeral continuity on a scale. Unexpectedly it is difficult that they learn length. So we teachers have to introduce much activities, concrete operation activities, for them and make them understand little by little that they can measure a object by knowing how many voluntary units contain. When they have such foundation on understanding, they may understand that measurement with a universal unit is just extension of measurement with a voluntary unit. So I tried a lesson that made much of introduction into learning of length.
First, I made two pupils(Hiroshi and Chinatsu),who sat at the front in the classroom, take out their pencils and put them up as figure1. And I asked them which pencil is longer than the other without moving both pencils. Then one of them expected that Hiroshi's is longer and the pther did Tinatsu's
is longer. I asked them how they compared two pencils. And Shota(boy) said, "I compared two pencils, getting near with eyes." He explained so, performing as fig2.
However he murmured," It does not look well----" and complained of difficulty to get
two pencils near for himself, who sat at the back in the classroom.
And other two pupils, who sat at the front, also complained of its method and told that they could not look at two pencils with the method. Shota's method is interesting but not correct.
Then I told the pupils, " What method can you compare two pencils exactly by? " And
Hidechika(boy) answered," We set two pencils on the desks." As I din not understand
what he said, I made him explain about it at the blackboard. Hidechika stood two pencils
on the desks and look into the pencils from the top of Hiroshi's. (fig.3)
To know clearly what he did, I asked him what method he used to compare two pencils. And instead of him Keima(boy) explained looking into two pencils actually, "If we look into two pencils from Tinatsu's, Hiroshi's will hide behind Tinatsu's. If we look into them from Hiroshi's, Tinatsu's will apear over Hiroshi's."
Then I asked whether they could think other methods out and Keima also answered
that he stood two pencils and drew a line, connecting with a straight long stick at the line.(fig.4)
So I asked him, "After we connect
with a stick, how we compare the lengths of both? " Then Hidechika instead of Keima told that a shorter side would bend. He insist, I guess, that the side of a shorter pencil
will go down.
Next Eri(girl) said, " I will compare them with hands." I asked her how she would compare them and she sandwiched a pencil between her hands, after that keeping the
space she moved from Hiroshi's to Tinatsu's. (fig.5)
But this method contains some troubles. So I asked the pupils whether Eri's method
contained any trouble. Ryou answered, "Her hands looks moving------". In the other words, when she measure the length of one pencil by sandwiching with her hands, while she move to another pencils keeping the space, the space changes merely. Therefore, the method is not accurate.
After that, Keima said once again, "I examine it with Hand-ruler." I asked him how he thought about it and he answered that we first set the pencil to the top of the middle finger, marking the sign how inside the pencil went in the palm. (fig.6)
He told that he judges which longer the pencils are by the length of signs. It is a method that is measured by indirect-comparison with hand. These are natural but just childish ideas.
I asked them again whether there is any method and so Natsumi(girl) said, "I use a pencil-cap." And then I prompted her to compare the pencils by using a cap and she
began to examine Hiroshi's and Tinatsu's caps, about how many caps they were. (fig.7)
Examinning them, we found, Hiroshi's was three caps and Tinatsu's was four caps. Then
I asked Natsumi which pencil was longer and she answered that it was Tinatsu's. The fact, that the lengths of the objects can be compared with a voluntary unit, could be found by the pupils themselves.
But this discovery do not connect with introducing a universal unit. And then I made the pupils think deeply, asking them as the follow. "Though in these methods we compared two pencils walking around two pupils(Hiroshi and Tinatsu), if we were not able
to walk, how ways you could think about comparing them? "
The voice of "Eh---!" rose up from them, for a while, a pupil proposed the idea that lents Hiroshi and Tinatsu caps. Then I asked them, "I see, you lend them caps. Humm, but when we compare the pencils with caps, what trouble will occure? " Naoko(
girl) said, " We can not compare the pencils when the sizes of these caps are different
each other. When Hiroshi's caps are small and Tinatsu's caps are large, the longer one
becomes less(shorter)." Naoko explained that we could not compare them not if the lengths of caps were the same. We made sure that we must measure with the samething (the same voluntary units), when we compare not only the rooms as the classroom but also far more detached objects.
As we began to notice the neccesity of a universal unit, I made them try to answer
the length-problem put on their textbooks. It is one that they have to compare the three lines(red, blue, yellow) without using with a ruler. They began to compare them
measuring each one another, referring these discussions. The methods, which were thought out by them, contained one with an erase, a cap, a pencil, a textbook(use a edge), and so on. Then the lesson was over.