1. Introduction

Katsuro Tejima worked at the Elementary School attached to University of Tsukuba in Tokyo for 20 years and after that, he has been a professor of the university until now. Tejima has influenced elementary mathematics education in Japan so much. His books that were published one after another, his first book was 'Lesson of problem solving in elementary mathematics', have had impact upon Japanese elementary mathematics education as much as its base would be overthrown. A few years ago, I happened to talk with Tejima and then he talked that it was at the risk of his life for him to write 'Lesson of problem solving in elementary mathematics' and publish it. I asked him to whom it was at the risk of his life and he replied that it was against elementary mathematics and mathematics education world in Japan then. It was true, I thought. If one would face something and reform it earnestly, one's work could not help being at the risk of one's life. As teacher or practical researcher, I felt deep impression and sympathy. Then, I will introduce his work, but I have not ability enough to introduce his whole picture. So, I first will introduce his typical lesson and its record, and in addition, I will state some comment for the lesson and his work.

2. Lesson Record -How many children are there lining?- (the 2nd Grade)

This lesson delt with 'orderly number' and Tejima presented his research theme in the lesson plan, which represents mathematics appreciation on the axis of common teaching material as possible.

As soon as the lesson started, he made students sing a song, with such words-Teacher is a friend of mine----. And then, he began to write the problem on the blackboard. Students also began to write on their notebooks as the same. The problem statement is as follows.

Children are lining.  Masako is the 8th from the top, and the 7th from  the back. 

He made students read the problem statement all together. And then, he said that this problem was continuing and let's make a story. T: Are there students who can make a following story? C: (18 children raising their hands) T: Now, stand up students raising your hands. I would like you to present your story but sit down students who would think that your storys were the same as your classmates'. O.K?

 C1: Children are lining. Masako is at the 8th from the top ,and the 7th from the back.
     How many children are there all ?
                                       ~~~~
 C2,C3,C4:  (almost the same story as the story of CI)

Approving of each expression, he adopted the response of C2 as common task in the lesson, writing on the blackboard as follows.

  Children are lining.  Masako is the 8th from the top, and 7th from the back. 
       How many children are there lining all?

And then, he asked students whether they understood the problem. For his inquiry, students cried, "I understand!" or "I can solve it!". And he further asked them the bellow. T: Write your own answer on the memo pad and next write about the reason why you thought such an answer. Then, you may write it as figure or as expression or story. Students begn to write their thinkings on the memo pads. For a while, they turned in their papers to him. And he talked as follows, T: I guess, you think that your answers are the same. But, the answers are different. I will write your answers on the blackboard and so you should write it on your notebooks.
And then, he wrote the answers presented from students and the number of each response. The number is the bellow.

       Students' answers
                       A   16 children............ 3 students
                       B    15 children...........26 students
                       C   14 children.............5 students
                       D   don't understand....4 students

When he wrote the answer of D, "don't understand", naturally the voice of "It is just an answer that I do not understand." raised up from students. Tejima replied for the voice, " Yes, the fact that we do not understand is fine answer". Next, he inquired them as follows, T: There are students who answer as 16children, and there are students who answer as 15 children, and moreover, also 14 children and "don't konow". Are there any answers you wish to inquiry among these answers, thinking that it is strange. For this inquiry, the students who responsed as 15 children first raised their hands and soon Satoru(temporary name-the bellow same) asked. Satoru: I would like to ask mates who think as 16 children. Why are there 16 children? And then, Tejima asked Satoru, T: Satoru, which is your answer? Satoru: 15 children of B. T: Then, why are there 15 children? Let's hear Satoru's thinking. And, Satoru went to the blackboard and made a expression, explaning the reason.

 Satoru's writing      
                                8+7=15
                                    15 children
                                    ~~~~~~~~~~~~~~~

Next, inquired students who responsed as 16 children presented. 2 students explained their reasons. One of them, Yoshio presented as follows,

Yoshio's presentation

There are 8 children in front of Masako and 7 children in the back of her, and Masako is the center of them. So, 7 add 8 equals 15, then, adding Masako equals 16 children.

He insisted that Masako is between 8 children and 7 children. Hearing his remarks, Tejima talked to students, T: Yoshio has an awful thinking.......but you can not understand what he says, can't you? For such provocative inquiry, other students reponsed intensely. First, Nozomu explained, Masako is at the 8th from the top, so she is th 8th from the top when we count them. Then there are 6 children in the back of her, therefore, 8+6=14 children.

Next, Hideharu who had the same thinking of 14 children as anaswer raised up his hand , and presented the following, writing 14 circles on the blackboard.

Hedeharu: As Masako is the 8th, there are 7 children in front of her, and there             are 6 children in the back of her.  7 add 6 equals 13, and in addition to
           Masako, all are 14.

And next, Tutomu presented as follows.

Tutomu:  If we add  Masako into the front number, there become 8 children.
         If we add  Masako into the back number, there became 7 children, so,
         8 add 7, further subtract 1 equals 14 children. 

Seeing Tutomu's and Hideharu's presentations, students who answered as 15 children cried, "Ah! it's so!" or "Understood!""I got it!". Ichiro and takeshi answered the same one, either. Ichiro says, "If there were 14 children....", pasting 14 red magnets on the blackboard. He chanted,"1,2,3,4, 5,6,7,8" ordrely, and changed the 8th magnet into other colored magnet. Seeing his doing, Tejima said, Tejima: Oh, he did a wonderful thing! You looked at it? Ichiro arranged 14 red magnets and changed the 8th magnet into a white one. And he explained that the white magnet is the 8th fron the top and the 7th from the back. Next, Osamu presented as follows,

Osamu: If there were 15 children(saying so, pasting 15 square magnets on the            blackboard), Masako is the 8 th from the top, so she is here.  And, she is         the 7th from the back, so she is here. Then, Masako becomes be 2       Masakos!  So, we sould subtract 1, the answer is 14. 

In that time, alomost students leaned toward the thinking of 14 children. And, Tejima asked students, Tejima: You had better change your thinkings comparing the former ones. What do you think about it,now? The result is the bellow.

                             A   16 children...........0
                             B    15 children..........3
                             C   14 children.........35
                             D    don't know.........0

This was so different result much as the former. However, all sudents did not answer as 14 children. 3 of them still insist on 15 children. Among 3 students , there was Eiji.
Eiji raised his hand and asked,
Eiji: Masako is between "8th" and "7th", isn't she? But, classmates who answer as 14 children insist that 8 add 7, and subtract 1, doing so is strange. But, Tejima did pursue his idea more, on the contrary, he handed small 20 building blocks(enveloped) out to each student. Building blocks included 19 yellow ones and 1 red one. became to arrange blocks. 1 red block represents Masako. Arranging blocks, students cried here and there, " Look here, 14!". These 3 students who insisted on 15 children also seemed to be convinced that the answer was 14 children, through operating activity with building blocks.
And lastly, Tejima had a girl read her composition.
Tejima: Kazuko, how many legs do you have?
Kazuko: 2.
Tejima: Well, Makoto, how many legs do you have?
Makoto: 2
Tejima: Good, if we add Kazuko's legs to Makoto's legs, how many legs become all?
Students: 4
Tejima: You may think so. But, in Ikumi's composition, she write that if we add 2 men's legs, we will get 3 legs.
Students: Eh!!
Tejima: How do you think about it?
Students: I understood!!
Students: It is three-legged race!!
Thus, Ikumi read her composition and the lesson was over.

3. Various opinions about this lesson

I can not put down to the small details of lesson, so the above is outline.
About this lesson, many researchers and teachers expressed their opinions. A Korean researcher wondered why Tejima did not explain about orderly number at the end of lesson. It seemed strange that Tejima did not state any explanation about the conclusion. Against this question, he answered that math lesson needs not always a conclusion and after the lesson we might have any problem. Problem solving should be successive in math lesson, he insists. That is, students hold further problems after lesson.
And Abe(1989) analyzed this lesson by minute in order to search Tejima's teaching technique, for example, time of his writing, waiting, etc. Certainly, teachers need teaching technique to teach students math. And Teaching technique may be based on the lesson. But I am not so interested in it that I will deal with thinkings toward math lesson.
Next, Yamamoto(1989) studied Tejima's lesson plan carefully to improve math lesson plan. In his study, he stated that he found the conditions that math lesson plan should include. But, about what we should think about math lesson plan, there are various views in Japan. Therefore, Yamamoto's study and thinking is just one of them and it is possible that we might think about it differently.
Lastly, I will introduce one more opinion, that is, Shozi's(1990)theory. Shozi analyzes Tejima's lesson in his writing "Mechanizm of Lesson" in detail. Shozi, who is a scholar of pedagogy, reconsidered Tejima's lesson cognitively or philosophically. He aimed at "spreading related worlds" and persuaded that it is important with spreading image and interpretation in understanding. The bellow is the original by Shozi.

Before we study Tejima's lesson concretely, I make reference to his basic thinking with related to stream of math lesson. In a word, his thinking consists of how teacher should shake students' intellectual conflicts as most important task. To do so, he set up his concrete research theme how teacher should organize math lesson at the axis of students' questions. ... And, the fact that should be noticed first in the process of lesson stream, is how to present a problem in the stage of lesson introduction. About this point, Tejima himself explained in "How to see lesson video" and "Relation with lesson research theme". Main teaching task in the lesson introduction stage, is whether students can grasp the problem presented by teacher as learning task which they should wrestle with eagerly. It is said that lesson comes into existence only when contents that teacher wishes to teach students change into contents that students wish to learn. In the view of point of lesson that thinks much of spreading related worlds, the first important condition is to clarify students' task conciousness on the base of spreading them, and it connects with frutefull learning activities and its stream. ...... Understanding or catching is feeling or becoming aware of. By doing so, interpretations very come into existence. Understanding is also interpreting. ........They say that in the science lesson or math lesson explanation is main role and there is no space to join interpretaion. But according to recent interpretative study, it is not always true. seeing tejima's lesson, it proved well that interpretation is not used only in literary art.........Students' subtle different deversity and explanations that we could find in this lesson, bring about all from students' interpretations........ This is real picture that students's mathematical view or thinking improve (growing,deepening, spreading). Indeed, "reconstruction of experience" points to it. Here exists in educational significance that teacher makes students explain their own thinkings one after another, although their answers are all the same, 14 children. Therefore, teacher(Tejima), even if students' answers were wrong, makes much importance of students' reasons or interpretations that they have had themselves. So, he assessed students who answered as 16 children, saying "He has a fine reason."or "She explained it beautiful." This is surely to make use of students' failures...........Originally, understanding is to spread related worlds. ...... if so, students' interests and expectations might increase by thinking that it would relate to other matters not noticed now. Then, students' interest and concern would increase more and more.....

4. Concluding Remarks

I introduced Tejima's lesson and various opinions about his lesson. There are many math lessons by Tejima but I am sorry for not being able to introduce them.
Tejima is very creative and radical, therefore,he has much influenced upon Japanese mathematics education. His view of teaching materials, techniques, are revolutionary to us. if you would like to study this lesson more, you could get the video tape from "Tosyo-Bunkasya.co.."(Japan). If you would watch the video tape, you could learn more details or find other things. And you might understand Japanese elementary school and education.
Certainly, his teaching techniques is excellent, however, the matter that I would like to know more is how he brought up students. His students, only 2nd graders, could interpreted with the problem variously, diversely. We might bring up 2 to 3 students who can explain in such words, but it must be impossible that we would bring up 10 excellent students. In the above lesson, 10 students did explained their interpretations about their answer,14 children. And diverse thinkings were announced there. His practices and insistence are a challenge for us. But if we would continue to pursue his practice, we might catch up with him someday. And we should create new math lessons more than Tejima's, I think.

5. Tejima's recent article