"Das Zahlenbuch (A Book of Number)" is a textbook of mathematics in primary school (1-4 grade), written by E. Ch. Wittmann and the others. Konstruction priciples of this book are follows. (1) Focus learning contents on basic ideas. (2) Stressing on active-discovery learning and social learning (3) Saving a means of learning and expressions (4) A productive exercise (5) A natural individualising The feathers of a learnig content are "basic of a thousand", "power of five", "pattern of number", "lightning calculation" and "exploration of number realm". The feather of a teaching method is a holistic approach. This is characterized by "vague whole → analysis → sophisticated whole", "connecting a learning content" and "a progressive schematising (mathematising)".

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　　The popular teaching pattern of multiplication of division following textbooks in Japan is making an expression at first and solving its answer. For example in division by decimal a problem are shown as follows:
"A ribbon is 72 yen for 2.4 m. How much is it per 1 m?"
The teaching stages are based on principle of the Permanence of Equivalent Forms and like these:
(1) making the expression 72 2.4 by "the word expression"
(2) seeking the solution of 72÷2.4 with thinking the price of 24 m ribbon using a line segment figure.
(3) calculating 72÷2.4; 72÷2.4=(72×10)÷(2.4×10)=720÷24=30
　But this way isn't necessarily natural and understandable to students, as they mistake like this: 72÷2.4=72÷24÷10=0.3
　　I propose a hypothetical learning trajectory for division by decimal based on the chain of signification as follows:
(1) understanding the problem using the picture of a ribbon or a ribbon itself.
(2) estimating the price, if it is 72 yen for 2 m or for 3m, using a belt figure.
(3) trying how much for □m, based on 72 yen for 2.4 m, using a line segment figure
(4) understanding 30 yen per m, on 720 yen for 24 m.
(5) expressing the formula of division by decimal based on a expression to figure out the price per m at 2m, 3m.
　　The chain of signification and hypothetical learning trajectory suggest us a mathematics teaching based on constructivism. I would like to investigate a teaching experiment under the learning trajectory.

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The purpose of this paper is a study of the construction of lessons of mathematics in which the aspect is applied to mathematical activities. Many children do mathematical activities in the lessons of arithmetic and mathematics. I apply the aspect to these mathematical activities and have carried out a study on lessons of mathematics. Then, it is thought that there are the following four factors in the children's mathematical activities; a) knowledge that has already been studied, b) mathematical thinking, c) the base of creativity, d) applications of mathematics. Moreover, it is thought that the class can be composed of the following three levels by applying the aspect to this mathematical activity. The first level is a stage in which one thinks about the mathematical problems from the presented problems. The second level is a stage where these problems are solved by mathematical reasoning. The third level is a stage where the solved problems are adapted for introduced problems or are developed into new connected problems. I will advance the study on this class composition method further through the practice of lessons and the study of lessons in the future.

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The purpose of this paper is to clarify the framework of teacher's listening. There are three forms of listening of teacher; the evaluative, interpretive and transformative listening of teachers, which I adapt for analysing the listening of teachers. (1) Evaluative Listening: If a teacher is listening in an evaluative manner they will characteristically have an evaluative stance. For a teacher, students' contributions are judged as either right or wrong. This listening is primarily of the responsibility of the learner. (2) Interpretive Listening: Interpretive listening is characterised by an awareness of the fallibility of the sense being made. If a teacher is listening in an interpretive manner they will characteristically have an active interpretive stance. (3) Transformative Listening: There is an attempt to interpret and make sense of what the speaker says, but always from the point of view of the listener. If a teacher is listening in a transformative manner they will characteristically have an open stance to the interrogation of assumptions they are making. (a)To entertain other points veiw (b)To connect with the present to the past, the future The transformation of experience: the connection with the present to past The seeing of a new world: the connection with the present to the future There are three forms of mislistening of teacher; fallible, Unsuccessful and selective listening of teachers, which I adapt for analysing the mislistening of teachers.

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It is pointed out that solving a word problem is difficult for the elementary school children. In order to overcome this difficulty, it is considered that improving their problem solving ability is needed. Then the auther consider that incorporating estimation activity into problem solving process is effective for it and choosing operation. Estimation activity is to anticipate and calculate roughly an approximate answer before calculating the exact answer of problem. It has been pointed out that estimation activity in the arithmetic eduation is very important. Rvewing the previous studies about estimation activity, there are many studies about computational estimation on one hand. On the other hand there are few studies that relate estimation activity with problem solving and choosing operation. The purpose of this paper is to review the previous studies on estimation of the answer of a word problem, to consider about the signification of estimation activity, and to clarify the effect of it through the investigation with the 6th grade children. Consequently, the following results are found out: (1) Estimation activity promotes an analysis of problem and provides an appropriate orientation basis. (2) It seems that estimation activity can improve the problem solving ability of elementary school children. (3) Estimation is effective on the simple word problem involving decimal number and fraction. (The simple word problem mesns a problem that can be solved by one operation.) (4) If we incorporate estimation activity into daily arithmetic lesson, it will help the problem solving ability grow up.

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The main purpose of this article is to verify the teaching process with "interpretation discussion" that author proposed in order to promote the constructive condition of mathematical problem solving strategies. And so, experimental lessons were conducted in 8th grade. There are three issues of the investigation of the teaching process in 8th grade: 1) Can interpretation discussion be introduced in the lessons, and function as an instructional method to promote the constructive condition of mathematical problem solving strategies? 2) Can students construct mathematical problem solving strategies in everyday lessons by using "interpretation discussion"? 3) Can the constructive condition of students on mathematical problem solving strategies improve by the teaching process with "interpretation discussion"? These issues were verified by comparing experimental group that practiced the teaching process through 7 lessons with control group that was not instructed strategies at all. Pre-test and Post-test problems are not similar with problems which used in experimental lessons. Through the lessons and these analyses, first and second issues are verified positively. But authenticity of third issue could not be verified because of some factors. And this article showed some suggestions for the effective introduction of mathematical problem solving strategies instruction.

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This research is related to the group discussion method to promote the formation of the mathematical concept. Especially, this research applies the focus to the proposal and the evaluation of the model of a mathematical concept as a basic research. The main contents of this research are as follows. (1) It is mentioned that there is a tendency that the teaching of elementary school mathematics leans toward instrumentalism as a result of the preceding research. (2) The characteristics of the mathematical concept are discussed from the aspect of "instrument", "representation", "definition" and "relationship". Moreover we classified "relationship" into "functional relationship", "conceptual relationship" and "social relationship". (3) The characteristics of the formation of the mathematical concept are discussed from the point of view of "development" and "hierarchy". We described the lesson as the transformation process of private mathematical concepts into public mathematical concepts. In this discussion "relationship" played an important role. (4) We proposed the model of the mechanism of the formation of the mathematical concepts and referred "teacher", "learning materials" and "children" as factors to promote the formation of the mathematical concepts. (5) Finally we tried to evaluate the effectiveness of the model proposed in this paper by appling it to a lesson.

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The continuous research on mathematical attainment is a part of the International Project on Mathematical Attainment (IPMA) in which such countries as Brazil, Czech Republic, England, Hungary, The Netherlands, Ireland, Japan, Poland, Russia, Singapore and USA are participated. The aim of this project is to monitor the mathematical progress of children from the first year of compulsory schooling throughout primary school and to study the various factors which affect that progress, with the ultimate aim of making recommendations at an international level for good practice in the teaching and learning of mathematics. In Japan, the total of eight different public primary schools have agreed to participate in the project. We asked all two-cohort children and their classroom teachers from these schools to be involved and to take mathematical attainment tests for six years. At the present we have carried out five tests, i.e. Test 1, Test 2, Test 3, Test 4 and Test 5 to about 300 children of first cohort for four years. The purpose of this paper is to analyze the data of three tests, i.e. Test 3, Test 4 and Test 5 to investigate children's progress of mathematical attainment in order to find out some suggestions for improving the teaching and learning of mathematics at these primary schools. In our previous paper (Koyama et.al., 2002), according to the percentage of correct answer to each test item, we made such categories as high [H], medium [M] and low [L] attained items. We defined the fixity of mathematical attainment such that for three tests if a child's changing pattern of correct (1) or incorrect (0) on an item is [1→1→1] or [0→1→1] then the child's mathematical attainment on the item is fixed. As a result of analysis in terms of these categories and the fixity of mathematical attainment, we found out the following: ・There were five different types of [H→H], [M→H], [L→H], [L→M] and [L→L] from Test 3 to Test 4 based on the progress of each some test item in Test 3 which is learned at the third grade. ・There were two different types of [L→M] and [L→L] from Test 4 to Test 5 based on the progress of each some test item in Test 3 which is learned at the fourth grade. ・We found that the only basic item in Test 3 which is learned at third grade was insufficiently fixed among children. These results suggest that the teaching and learning of mathematics at the third grade was more effective than that one at the fourth grade in these schools and that more efforts should be made in the teaching and learning of contents such as decimal, fraction and division with remainder.

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In this paper, I describe the degree of creativity development of students by using "the quadrilateral" that is a basic figure as the teaching materials in arithmetic education. The experiment involved 742 students from elementary school grade 3, 4, 5, 6, junior high school grade 2, senior high school grade 1 and second and third year university students. I set four evaluation viewpoints to measure creativity ability test. These are divergency, fluency, flexibility and originality. I examined the degrees of creativity ability development, the correlation coefficient and the causality relation among the scores of each evaluation viewpoint by using the creativity ability test. The results are as follows: - The difference of the creativity ability development is seen between elementary school students grade 4 and grade 5, and senior high school students grade 1 and second and third year university students. - As for scores of divergency, fluency, flexibility and originality of the creativity ability test, the score of elementary school students grade 5 and 6 is the higher compared with the score of junior high school students grade 2, senior high school students grade 1 and second and third year university students. - The development about flexibility and originality is hardly seen even if the grade rises. - The students who get higher flexibility score achieve higher fluency score and originality score. - The students who get higher originality score achieve higher fluency score and flexibility score.

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In this paper, we describe the relationship among the creativity power, the creativity attitude and the scholastic achievement in mathematics learning. The experiment involved 70 participants from junior high school students grade 2. For the creativity power, we made a creativity test of "the linear function" and used the test score. For the scholastic achievement, we made the test of "the linear function" which has three categories. Three categories of the scholastic achievement test are composed by the problem about the understanding of a basic computation skill, the understanding of the procedure when the students solve problems and the understanding of the structural relation of the learning contents. We call scholastic achievement test "the three dimension test". For the creativity attitude measurement, we used a Creativity Attitude Scale (CAS) that was developed by Saito, N. We examined the correlation coefficient and the causal relationship among the creativity test score, the scholastic achievement test score and the creativity attitude scale score. The results are as follows: - The students who get higher the creativity attitude scale score get higher the creativity test score. - The students who get higher the creativity test score get higher in the three dimension test. - The students who get higher score in the three dimension test get higher the creativity attitude scale score. Especially, the understanding of a basic computation skill has a very big influence on the creativity attitude. - The understanding of the structural relation of the learning contents is a basis of the understanding of a basic computation skill and the understanding of the procedure when the students solve problems. - The understanding of the structural relation of the learning contents in three categories has the biggest influence to the creativity.

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　　In this paper, I have described findings in investigations after lessons on velocity, which stand on the ground of a graphical schema (Dorfler, W., 1991, p.74) of processes of generalization.
　　For connecting an item with another item on this model, I have made use of 6 stages on problem solving of Toda, K. (1954) and processes of the thinking experiment of Mach, E. (1971).

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　　By the way, I have set up the following 3 hypotheses and examined to verify them. The participants in examinations include 170 pupils of 3 elementary schools' grade 6 in Kobe. "
If I carry out lessons on velocity as planned which raise task-analyzation power on velocity and are based on the general model proposed by Dorfler, W.
Hypothesis (1)
　　For task-analyzation power, there maybe exist meaningful differences between pupils whom I taught and ones whom another teachers taught.
Hypothesis (2)
　　For emotional aspects, there maybe exist meaningful differences between pupils whom I taught and ones whom another teachers taught.
Hypothesis (3)
　　For pupils whom I taught, there maybe exist correlation between task-analyzation power and emotional aspects, but there unlikely do for pupils whom another teachers taught.
　　As the result of the examination on task-analyzation power, F-test and t-test verified "Hypothesis (1)" (i.e. F (34, 134)≒44.423 > F (30, 120), t≒6.33^* (^*p≦.05)).
　　As the result of the examination on emotional aspects, F-test and t-test verified "Hypothesis (2)" (i.e. F (34, 134)≒1.175 < F (30, 120), t≒3.48^* (^*p≦.05)).
　　Also Pearson's product-moment correlation coefficient was 0.746 in pupils whom I taught, and was 0.051 in pupils whom another teachers taught, so that this verified "Hypothesis (3)".

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In this paper, the possible treatment for the students who have highly competence for mathematics is examined. In order that, the cases from the United States, where they have different sense of equality or equity from Japan, are analyzed for some implications to Japanese mathematics education. First of all, the case of Advanced Placement Program is discussed, as the case of mathematics education, where they teach college level mathematics in high schools. This is a nation wide project which is conducted by The College Board. As the second, another case of University of Minnesota Talented Youth Math Program is discussed. This is the program where the students come to the university to take high-level mathematics courses, and it provides a challenging mathematical education for highly motivated, talented pre-college mathematics students (grades 5-12) in Minnesota. From the analysis of these two cases, the following themes are pointed out. (1) How is the teacher training for such programs supposed to be in Japan? (2) How can we cope with the continuity in high school and college mathematics? (3) How is the approval of college credits for high school students applied? (4) The possibility of having classes for high school students in universities. (5) The relation to other school subjects.

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In this paper, we discussed the formation process and the characteristics of the arithmetic workbook edited by Zingo Shimizu in 1933 and 1934. He was a leading arithmetic teacher of Sakumon-centered Arithmetic Education which had been put into practice under the leadership of the director Takezi Kinoshita at the elementary school attached to Nara Women's University since the middle of the Taisho period. The arithmetic workbook was made to answer questions and requirements form elementary school teachers who worried about how to implement Sakumon-centered Arithmetic Education into their classes. The major problems were as follows: (1) Children had a tendency to make the same arithmetic problem; (2) The problems posed by children were on the whole easy in the higher classes of elementary school; (3) It was very difficult to treat the national textbook with reference to the problems posed by children; and (4) Teachers were worried that they failed to make children obtain satisfactory results on the performance of computation. In response to these requests, Shimizu developed a new curriculum which was intended to coordinate children's mathematical activities in daily life and the contents of the national textbook. But, this curriculum required a great deal of teacher's efforts in implementing into classes. Then, he set himself to make the arithmetic workbook which was in conformity to ideas of Sakumon-centered Arithmetic Education and could be used together with the national textbook in class by teachers. Through discussing the characteristics of the arithmetic workbook, we pointed out that firstly it was the utilizable learning resources for teachers to encourage children's qualitative thinking, secondly it covered the demand of the national textbook, thirdly it was affected by the spirit of that time, totalitarianism. The Shimizu's attempt should be evaluated on the point that he developed it as a mediator between the two national textbooks, Black Cover and Green Cover.

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The purpose of this paper is to study the transfiguration of the education of function in text books compiled by the Ministry of Education, founding on higher elementary schools from 1904 to 1935. We can classify the teaching content into four periods, as follows. In the first period, from 1904 to 1909, textbooks did not teach functions. In the second period, from 1910 to 1918, there are only three questions about functions but there is a conscious awareness of the existence of functions. In the third period, from 1919 to 1925, compared to the progress in the education of function from the first to the second period, rapid progress is made. The teaching contents in this period are substantially similar to the fourth period teaching contents. The 1919 revision was aware of the improvement movement of mathematics education in secondary school and anticipated the changes needed for the 1926 revision. In the fourth period, from 1926 to 1935, functions are formally dealt with. The teaching contents from the third period are reconstructed and quantity of functions taught is increased. Characteristically, falling movement is dealt with using geometry. The transfiguration in this period reflects the mathematics education improvement movement which had been rapidly improving. The transfiguration in this period reflects indirectly the demands of those who were concerned about education. The demands tried to approximate the higher elementary school to the preceding term of the secondary education. The 1926 revision also included the innovation of the subject/teacher system and the introduction of compulsory industrial subjects. We can find the following as a whole. The graph is a useful way to represent the relationship and variety of the quantity. The transfiguration of education in the higher elementary and secondary schools occurred at about the same time frame, but characteristically, the change in the higher elementary schools went a step ahead of the change in the secondary school.

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The objective of this paper is to clarify the role of Japanese mathematics books translated into Chinese in Modern China. To attain this, I overlook and examine the situation of mathematics education in China before the publication of Japanese mathematics textbooks translated into Chinese, the necessary and sufficient condition for the introduction of Japanese textbooks, and the real state of the textbooks published in China. The results are as follows. (i) The promotion of the translation of Japanese mathematics textbooks was underpinned the appeal by pioneers, the cooperation by Japanese educators, and the domestic circumstances such as the urgent necessary of the compilation of mathematics textbooks and the favorable reception Japanese mathematics textbooks. (ii) Chinese students in Japan and Japanese were concerned in the translation of Japanese Mathematics books. (iii) The translation of Japanese Mathematics books were of good repute in quality as well as numerous. As it became widespread, signs of Chinese mathematics was replaced with those of western mathematics.

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The purpose of this paper is to consider the influence of medium of instruction on both cognitive and affective sides in learning mathematics. The index for cognitive side in this paper is "pupil's everyday concepts of fraction" and "the influence of everyday concepts on the formation of concepts of fraction in the school". Here, everyday concepts are defined by Vygotsky's theory. And the index for affective side is "pupil's preference for mathematics" and "How pupils look English and Filipino as a medium of instruction". The survey was conducted for 157 fifth graders in two schools. In one school, medium of instruction for mathematics is English and in the other it is Filipino. Cognitive side is examined in terms of test items and affective side is in terms of questionnaires. As a result, the following 5 points were found out. (i) The pupils of both schools regard "kalahati", the everyday concept of half in Filipino, as the word to represent the half of continuum. (ii) The concept of "kalahati" have an effect on the formation of concepts of "1/2". (iii) Pupils tend to switch English to Filipino when they are talking a conceptual matter. (iv) Most pupils in both schools like mathematics. Even after medium of instruction has been changed to English, this tendency doesn't change. Pupils accepted this change positively. (v) Most pupils in both schools prefer English as a medium of instruction in learning mathematics. The affective side was underpinned by not only the educational reason such that "it is easier to learn numbers in English", but also the social reason such that "English is an international language". Here, we can see the gap between the fact of code-switching in cognitive side and the preference in affective side.

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　本研究は,バングラデッシュの初等数学教育のための現職教員研修の効果を評価することを目的とする。この目的を達成するために,本研究は,その可否および質的改善に必要な諸条件の分析に統計的な手法と質的な手法を用いている。質問紙,授業の観察とその評価シート,フォーカス・グループ・ディスカッションを通して,データが収集され,分析された。その結果,教員研修後,教師の教科内容に関する知識と教授技術は主として理論面において改善されていることが明らかとなった。そして,研修後の教師の授業において若干の研修内容の応用が観察されており,生徒の成績には肯定的に影響していた。

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In this paper, we discuss an improvement of teaching the vector in the high school mathematics. First, we compare the current objectives of studying the vector with those of 1960's when the vector was introduced in the high school as a teaching material for the first time. Then, we point out that current students do not have so much experience to see the efficient use of vectors other than mathematics and that the role of the vector to cooperate with the space analytic geometry is currently insufficient. However, our assertion is that an improvement of teaching the vector along with a mathematical activity is important, and that students should learn to expand the mathematical methods through the utilities of the vector, the utilities to represent movements and sites of figures and to apply the vector operation in various ways. We show three teaching materials of the vector to illustrate the above points. The first material describes the efficient use of vectors in explaining the function of a contraction tool, the second suggests the valid use of the orthogonal projection when we introduce the inner product of vectors, and the third is a vector expression of the inner center of a triangle, which shows a typical use of the vector operation in analyzing figure properties.

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The purpose of this paper is to discuss effective method of mathematical problem posing by using computer. We practiced such mathematical activities by university students who were the prospective teachers. In study (I), we reported the practice of problem posing by using computer after solving original problem. In study (II), we reported the practice of problem posing by using computer which assign students planning problems freely from the first. A feature of these methods are to give students enough time to create problems. And another feature is to provide situations in which students make conjectures on results and get the numerical calculation by using computer. In this paper, we report the practice of problem posing by using computer after solving original problem which is different from study (I), and examine the effective use of problems posed by students. As in the previous study, the practice shows the students who tackled making problem by using computer learn many mathematical contents related to the problems. Each student solved two problems posed by another students and commented on problems each other, which established an effective use of problem posing and communications between students. We observed more positive learning activities than the usual classes.

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