The purpose of this study is to clarify the structure of the teaching unit of addition and subtraction with positive and negative numbers from the holistic perspective both theoretically and practically. We think that the difficulties on teaching and learning algebra result mostly from some fundamental factors such as the gap between elementary mathematics and secondary mathematics, the epistemological character of mathematical expression, and the paradigm behind the teaching of algebra. In particular, it seems to us that the mechanistic and atomistic teaching more or less deprives students of their significance to do mathematics. Therefore, we need to discuss the teaching unit in terms of the alternative educational paradigm and the didactical theory that realizes it as well as students' cognitive development on mathematical expression. In the first half of this paper we overlooked the holistic perspective of education, discussed the theories of the didactical situations and the algebraic cycle of thinking, and set up the provisional framework for designing and analyzing the teaching unit. And in the second half we analyzed the teaching and learning activities of the unit "addition and subtraction with positive and negative numbers" and discussed the structure and characteristics of the teaching unit based on the framework. The results are as follows. 1. The teaching unit of addition and subtraction with positive and negative numbers can be designed as the three stages which consist of the situation for action, the situation for formulation or communication, and the situation for validation. These can be also explained as the process of optimizing equilibration in terms of students' knowing on the one hand, and conceived as the phenomena in which the methods for thinking in the previous stage are successively transformed into the objects for thinking in the next stage on the other hand. 2. If we expect that students realize the significance of algebra during their continuous activities, the classroom lessons should be designed as they can constitute the tight links between their ideas and the milieu in the situation for action, produce the addition and subtraction simultaneously in the situation for formulation, and realize that transformation of the expression makes the link clear in the situation for validation. 3. We can convey the following educational ideas in the teaching unit; The algebraic nature as mathematical language, the humanistic view of mathematical learning, and the constructivist attitude of learning in the classroom lessons.

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The objective of this research is to consider the relationship between mathematics education and society, because mathematics is highly intertwined with principle of modern society and also because mathematics education is believed to cultivate ability which promote the present status of modern society. This principle was deliberated in the light of concept of development. The word 'development' etymologically had a meaning of opening or unfolding what is wrapped and it has no positive or negative sense in it, but with influence of theory of evolution, it gained a positive momentum of opening up any possibility of society and human being. After reviewing some past researches, the relationship between mathematics education and society was summarized into three components such as universality of mathematics, promoting function of mathematics education toward economic development and towards development of science and technology. Combination of these components gives strength to the status of mathematics education within the framework of development. Thus, the relationship between mathematics education and society should be appreciated but at the same time should be carefully examined. Especially for mathematics education in the developing countries, these components were analyzed critically from the perspectives of ethnomathematics and critical mathematics education. Finally, analysis of these proposes to seek mathematics education, which can give an opportunity for children to consider alternatives and to form an opinion.

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We believe that the more adults become engaged in their children's education, the greater the chances that children will succeed. School has to find a way for parent and child to spend time together doing math that's fun, challenging, and it is also important to build a bridge between Parents and Teachers Association, community and children for math mainly through mathematical life-long learning of the child as well as their parents' participation under the activities of Education Ministry. We hope school to start planning assignments that can be done by students and their family members together on their holidays, or by deciding the specific time. We made sure that School math teachers can manage to help parents and children, regardless of their prior math experience, and all children should be expected to study and to be successful in mathematics under "Mathematics for All". My research is concerned on the role of school math teachers and the status of mathematics education within families in Japan. We surveyed how family concerned with their child's math study, including helping homework, knowing school mathematics curriculum. The measurement is to show whether it is possible or effective that school math teacher can cooperate with family and if so, how they could be encouraged. Secondary sources are taken from books mainly FAMILY MATH concerned, public documents and private documents. Primary sources are planed by giving questionnaire to 80 parents and 120 children, using records of attainments and comments on assignments and model practical teaching at elementary and lower secondary university-attached schools and one prefecture lower secondary school in Japan. Results indicate that 80% (64 people) of parents have experienced in teaching math to their children and 70% of them (45 people) used school math texts while teaching. On the other hand, 50% (40 people) of parents were frustrated not knowing enough their children's math program or not understanding math their children were studying. Although 72% of parents believe that they can help children if they're introduced with school math curriculum, but 75% of children don't have good impression with their parents. However, after finishing home assignment for family and child, both of them made positive comments that they noticed how important studying math in a family for upgrading student's problem solving skills and attitude but also for solving family's difficulty not knowing appropriate methods. They asked to school more references, effective and fun method for doing math. This research has also proved that school can work with family members who intend to know the process of learning math and school math extension program, by getting aid and cooperation with school authority and community, and using hands-on materials. According to data, we can see even though their education levels are different; families can work cooperatively to analyze and solve problems with children supported by school math teachers. This is of great benefit not only for busy parents and those who live in rural area have to be upgraded the ways of thinking with respect to their recent world children, but also for reliability and possibility of school teachers.

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Through the work in mathematics and science education, it has been observed that students react in a similar way to a wide variety of conceptually nonrelated problems which share some external, common features. Tirosh, D., Stavy, R. who have studied students' comprehension of specific notions have also found that often their responses are in line with intuitive rule. The intuitive rule SameA-SameB relates to comparison tasks. Typically, the students are asked to relate to two systems or two entities which are equal with respect to one quality or quantity (A1=A2), but may differ with regard to another quality or quantity, either B1=B2 or B1=⃥B2. The students are asked to compare the two systems or entities with respect to quality or quantity B. It was found that students often claim that B1=B2 because A1=A2. In the present article, I describe the model of intuitive rule. I consider how intuitive rule has something to commit an error and what meaning intuitive rule has in mathematics teaching. The result of this research are as follows: (1) The missconception is sometimes caused by intuitive rule. (2) Intuitive rule revitalizes in various domain. (3) Intuitive rule is relevant to constitute conservation and proportion. (4) Intuitive rule restricts and facilitates children's recognition. (5) The presentation of the opposite opinion has an effect to restrain from revitalization of intuitive rule. (6) It is necessary to explore the way of coexistence together intuitive rule. (7) Intuitive rule has a strong predictive power about children's responses to the task.

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The purpose of this paper is to practically examine the effectiveness of the method of teaching and learning of mathematics with inductive and analogical reasoning through the lessons dealing with the division of decimal fractions. This method is constructed of the teaching and learning process and its means in each phase. This process is as follow: Phase 1. Cognition/Understanding of new mathematical knowledge utilizing inductive or analogical reasoning; Phase 2. Solving the problems similar to problems used in the Phase 1 utilizing analogical reasoning; Phase 3. Constructing a schema through the problems in the Phase 1 and 2 are made an object and schema induction is utilized; Phase 4. If analogical reasoning was utilized in the Phase 1, comparing the bases in the Phase 1 with the schema in the Phase 3. As a result, the following points are showed: (1) It is possible to enhance children's understanding of contents with the method; (2) It is possible to enhance children's ability of reasoning, especially of analogical reasoning with the method; (3) In both inductive and analogical reasoning, it is a important role to recognize not only the similarities but also the differences, especially recognize the relations of alignable differences.

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The purposes of this study are to explore the actual condition of the child about an understanding of the order of calculation based on investigation and to consider the influence on a literal expression. Moreover, it is also the purposes to investigate the actual condition of the child about the cognitive obstacle over an numerical expression and to consider the state of instruction of an numerical expression on the base of these. The main findings of this investigation are the followings: (1) It turns out that many children do not understand about the order of calculation and it is caused by the cognitive obstacle. (2) Overgeneralization can be considered to the cause of a cognitive obstacle. By encouragement of "an elegant procedure", a child generalizes over the associative law realized only in addition and multiplication with you may somewhere calculate. (3) The child who made the calculation mistake by the numerical expression makes the same mistake also in a literal expression. Consequently, Following three are mentioned as the state of future instruction. In an elementary school 1. You should guide the agreement of the order of calculation finely. Especially, you should brace an understanding of the meaning of 'usually' of "usually calculating sequentially from the left". 2. It is making a child understand the law of associative law correctly. 3. By expansive study, it is taking in mostly about the order of calculation and more than three digits calculation in expansive study. In a junior high school you should teach algebra by bearing in mind these results of an investigation. Such a thing is considered to lead to cooperation of a junior high school and an elementary school.

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The purpose of this paper is to show the previous research results by the author, and to construct the principles for mathematics class to promote students' number sense. In this research, I was able to bring the activities together which became the references of number sense by using the research results in this area, but was not able to answer what the number sense was. Then, I clarified the inner nature of number sense by using phenomenology which searched for the nature of thing. Being based on these suggestions, the principles for mathematics class to promote students' number sense were constructed. By using these principles, I analyzed an actual arithmetic class. As a result, the following points became clear. First of all, in the class where the number sense is promoted, the teacher always does pay attention to make students' consideration turned to the problem solving process and their outlooks on value. And, the teacher makes the discussion among students' active by using students' surprise opinions. Simultaneously, it was also suggested that others and the class environment were important to promote number sense.

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There are many occasions that geometrical sense plays essential roles in the process of recognizing geometrical concept. For example in the process of recognizing the figural concepts, we identify the shape of the objects and see the figure as geometrical figure by use of geometrical sense. In the process of geometrical problem solving and in the reasoning process, we use geometrical sense freely to deal with geometrical figure. But we can't clarify the pedagogical meaning of geometrical sense and as a result we don't have any programs to promote geometrical sense in the teaching of geometry yet. The main purpose of this research is to solve the pedagogical problems about geometrical sense in the teaching of geometry: what is the meaning of recognition of geometrical sense in the teaching of geometry, how is geometrical sense promoted in the teaching of geometry, and so forth. To satisfy the requirements I firstly clarified the characteristics on the recognition of geometrical sense. The role of imagery on the geometrical figure must be respected to make geometrical sense work actively. Secondly I pointed out the importance of the intentional experiences on geometrical sense. We must direct our intention toward objects, geometrical figure, and the geometrical concept to recognize geometrical sense. Thirdly I indicated the possibility of teaching geometrical sense. Finally I possessed some examples of the teaching materials to promote the recognition of geometrical sense. For example intentional activity of tessellation is useful for recognizing internal sense of geometrical sense, and tangram is effective for promoting perceptual functions of geometrical sense, and so forth.

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The purpose of a series of my study is to show the concept of Symbol Sense and propose the methed of instruction to nurture Symbol Sense. In this paper, I investigate both interesting behaviours and the achievement situation of 9th grade students when working on algebraic investigation problems. These investigation problems are based on the framework by which we can grasp Symbol Sense (Hayashi, 2002a). I was able to observe many interesting behaviors as manifestation of Symbol Sense, and check the validity of this framework from this investigation. As a result of this investigation it was found that many students didn't fully nurture Symbol Sense, conversely were dependent on an algorithm, a formal procedure and algebraic processing.

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The main aim of this paper is to analyze the process of problem-solving by changing an equation with letters from point of Analogy, Mathematical Generalization and Abstraction Theory and to clarify how the process contains difficulties in understanding. In order to attain this aim, at first I divide the process into 4 parts as follows. the process(1); constructing the equation. the process(2); changing the equation. the process(3); reading the equation. the process(4); verifying that the values don't change or some values add. By the analysis of this process, I could obtain the following results. In the process(1) that the students construct the equation from the word-problem given, the their mind functions toward so called Primitive Abstraction and Removal Generalization, while the values remain to be invariable or some values add. In the process(2) of changing the equation, the values of the equation don't decrease so long as we follow the valid deductive rules. In the process(3), the values are specialized or materialized, and they are checked whether the values are fit for word-problem or not. In process(4), it is very difficult that teacher make students to understand logically the values remain to be invariable or some values add, then I propose that there can be the teaching method using the Valance-Model based on Analogical Understanding as a kind of understanding modes.

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In this paper, a case of "Reflexive Writing" is analyzed from the viewpoint of "Semiotic Chaining", which is a concept from Presmeg (2001). First of all, the concept of "Semiotic Chaining" is examined through Peirce's Semiotics and Saussure's Semiologie. Then, the idea of Semiotic Chaining is applied to analyze Reflexive Writing Activity model, and two types of Semiotic Chaining are identified. In the case analysis on the view from Semiotic Chaining, the effective learning activity is identified as the one concluded by Reflexive Writing. Since the connection is regarded as the essential part of mathematics learning, it is concluded that Semiotic Chaining is a remarkable phenomena for the ideal learning activity, as well as the effective tool to analyze students' learning.

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The purpose of a series of our studies is to investigate the role of "self-referential-activity" in mathematical problem solving. The term "self-referential-activity" means solver's activities that he/she refers to his/her own solving processes or products during or after problem solving. In study (I), we proposed the theoretical framework for analyzing self-referential activity. And, in study (II) and (III), we elaborated the variable "OG/NOG" and "M-SE/SE-C" respectively. In these series of our studies, we have investigated self-referential-activity focusing on the problem-solving phases until the end of problem solving. In this article, we theoretically examined the role of "self-referential-activity", especially "looking-back" activity in the phase after problem solving. And we conducted an experiment to investigate "looking-back" activity after problem solving. At first, we identified six roles of "looking-back" activity. These are as follows: Checking answer or solution, exploring different or more sophisticated solutions, understanding the problem deeply, expanding or elaborating the solution and applying the solution to other problems, thinking critically or creatively, and considering validity or generality of the solution. Secondly, we conducted an experiment to investigate "looking-back" activity after problem solving. Junior high school students (2nd grade, N=117) were divided into two groups. One group was given some questions to encourage solvers to look back their solution of a problem and think other approaches, and another group was not given such the type of question. Subsequently, all subjects were asked to solve more difficult problem. As a result, we found that various "looking-back" activities occurred by simple instructions and these "looking-back" activities were effective to improve solvers' solution. These finding supports our theoretical assumptions and the value and possibility of studying the kind of "self-referential-activities".

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　　When people think and deal mathematically with various real phenomena to solve the real problem, 'hypothesizing' is indispensable for the problem solving. However, the activity that requires simplification of the activity of real situations and also selection of essential variables in the situations is so hard that mathematics teachers have avoided dealing with it in mathematics class. Therefore, in this paper, we tried to construct mathematics class focused on 'hypothesizing' and to investigate the effectiveness of the class by practicing it.
　　The following is the mathematics class focused on 'hypothesizing' that was suggested in this paper.

1. Show the real problem → 2. Solve the problem mathematically ((1)select some essential variables, (2)collect information from the distributed references → (3)hypothesize and solve the problem mathematically → (4)present the ideas and solutions each other → (5)correct each of the ideas if it is mistake) → 3. Investigate each of the solutions and write a report on the problem solving

　　As a result of this investigation, most of the students could hypothesize and solve the real problem mathematically. However, in '(4)present the ideas and solutions each other', the students didn't discuss each of the ideas actively. In addition they couldn't investigate each of the solutions, because they had no time to do it.

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　　Recently, learning with the portfolio assessment is paid attention to in Japan, because it focuses on the processes of children's learning. The portfolio assessment is one of the way of assessing children's activities and learning processes with their portfolios.
　　The purpose of this study is to discuss the portfolio assessment on mathematical learning to develop the metacognitive ability. For the purpose of this study, this article proposed the framework of a rubric on mathematical learning with metacognitive perspectives. The rubric of portfolios refers to the scale of assessment of portfolios. Using this framework, you will be able to identify children's metacognitive activities, and use the portfolio assessment to develop the metacognitive ability. The characteristics of this framework are the followings;
(1) it is two dimensions which are criteria and standards.
(2) its criteria include four aspects.
(3) its standards include the metacognitive aspects, and 5 levels. :
　　Metacognition requires targets because of its definition (Flavell, 1976). So metacognitive aspects refer to its standers on this framework.
　　After this article, I will use this framework to the portfolio assessment on mathematical learning, and analyze the characteristics of their metacognition. It will be related to development of children's metacognitive activities.

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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, Holland, 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 that could affect the 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 three tests, i.e. Test 1, Test 2 and Test 3 to both the about 500 children of first cohort and the about 440 children of second cohort for two years. The purpose of this paper is to analyze the data of these tests, investigate children's progress of mathematical attainment and compare two-cohort children's progress 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 followings. First, there were five different types of [H→H], [M→H], [L→M] and [L→L] based on the progress of each some test item from test 1 to test 2 or from test 2 to test 3. For example, the type of [H→H] means that for those test items in this type children had done well at the first test and did so at the second test a year later. The type of [L→H] means that for those test items in this type children had not done well at the first test and became to be well at the second test a year later. It reflects a positive effectiveness of the teaching and learning of mathematics for one year. As a result of comparative analysis by using these types, we found it common to two-cohort children that the teaching and learning of mathematics at the first grade was more effective than that one at the second grade in these schools. Second, as a result of comparative analysis in terms of the fixity of children's mathematical attainment, we found it common to both cohorts children that four items in test 1 were insufficiently fixed among children and suggested that more efforts should be made in the teaching and learning of mathematics related these items. As a final result, we can identify there is a very similar tendency in the progress of two-cohort children's mathematical attainment for two years. It could be interpreted as a reflection of the similarity in the practices of teaching and learning of mathematics at eight primary schools for two years. We could say that such similarity would be a characteristic of the teaching and learning of mathematics in Japan.

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In this paper, we describe the degree of creativity development of students by using "the sum of the interior angle of the quadrilateral" that is a basic figure as the teaching materials in mathematics education in junior high school grade 2. The experiment was participated by 316 students from junior high school grade 2, senior high school grade 1, second year university students and first year master course students. We set four (4) evaluation viewpoints: divergency, fluency, flexibility and originality to measure a creativity ability test. By using the creativity ability test, we examined the degrees of creativity ability development, the correlation coefficient and the causality relation among the scores of each evaluation viewpoints. The results are as follows: - The higher grade students achieved lower divergency score in the creativity ability test as compared with lower grade students. For instance university students (higher grade students) were compared to senior high school students (lower grade students). On the other hand, senior high school students (higher grade students) were compared to junior high school students (lower grade students). - There were causality relation among the fluency score, flexibility score and originality score in the creativity ability test to all students. For example, the students that got higher originality score achieve higher fluency score and flexibility score. - The higher grade students achieved higher fluency score, flexibility score and originality score in the creativity ability test as compared with lower grade students. However, all students got low originality score in creativity ability test.

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In this paper, I describe the degree of creativity development of students by using "the linear function", which is the teaching materials in mathematics education in junior high school grade 2. The experiment was participants by 170 students from junior high school grade 2, senior high school grade 1 and second year university students. I set five evaluation viewpoint to measure a creativity ability test. These are divergency, logicalness, 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: - As for total score of the creativity ability test, the score of high school students is lower compared with the score of junior high school students and university students. - As for divergency score in the creativity ability test, the score of high school students is lower compared with the score of junior high school students and university students. - The difference of the development about logicalness and fluency is seen between students of high school and university. - The development about flexibility and originality is hardly seen even if the grade rises. - In all grades, the structure of the correlation coefficient and the causality relation among the scores of five evaluation viewpoints was similar. - The students who get higher originality score achieve higher logicalness score, higher fluency score and flexibility score. The students who get higher flexibility score achieve higher divergency score, higher logicalness score and higher fluency score.

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In this paper we describe relation among the creativity power and the creativity attitude in mathematics learning. In experiment involved 107 participants from junior high school students grade 3. For the creativity power, we made a new creativity test of "The figure and Congruence" and used the test score. For creativity attitude measurement, we used a Creativity Attitude Scale (CAS) that was developed by Saito N. We examined the correlation coefficient and the causal relation among the creativity test score and the creativity attitude scale score. The results are as follows: - The score of the flexibility and the originality of factors that composed the creativity is much lower than the diffusion and the fluency of ones. - The correlation coefficient among the flexibility and the originality of factors that composed the creativity is middle. - The correlation coefficient among the diffusion and the fluency of factors that composed the creativity is high. - As for students that get higher creativity attitude scale score, the scores of the diffusion and the fluency get higher. - As for students that get higher score of the diffusion, the score of the fluency get higher. It occurs conversely. - As for students that get higher score of the originality, the creativity attitude scale score get higher.

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The purpose of this paper is to study how Motoji Kunieda (1873-1954) understood the concept of function and how he thought about the education of function. He considered that the goal of mathematical education was to cultivate mathematical common sense, which gives the person knowledge and the power of judgment in mathematics and arithmetic, and the concept of function was one aspect of mathematical common sense. He thought that the basic functional relations could be learned at elementary school, but that the real leaning of function should be cultivated in junior high school. This is because the comprehending of function was depended on the understanding of variables, and variables are represented by algebraic symbols. We can classify the teaching contents in his textbooks into four periods, as the followings. In the first period, from 1912 to 1921, the education of function began to be taught to students. But, some of the contents were omissible in teaching. In the second period, from 1922 to February 1926, the teaching contents have been developed. Especially in algebra, the teaching contents which had been omissible in the first period were formally treated. In the third period, from March 1926 to 1932, the teaching contents were enriched, quantitatively and qualitatively. In the forth period, from 1933 to 1937, the teaching contents have been edited a little and ordered.

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In the paper "A Study of Number System in School Mathematics (III)", I defineed the framework of systematic understanding of numbers. The purpose of this research is to reconsider this framework, and to investigate systematic understanding of numbers under the framework. I considered following framework of the systematic understanding of numbers. 1. Understanding the structure of a set of numbers ・Understanding a calculation rule ・Understanding a definition of numbers, and obtaining a image of numbers ・Understanding the operation in a set of numbers ・Understanding the linear order relation in a set of numbers, and understanding the density of a set of numbers 2. Understanding the density of a set of numbers 1 and 2 are complementary rather than independent. Under this framework, I investigated the degree of achievement of systematic understanding of numbers. And I analyzed how the understanding of the structure of set of numbers influenced the Understanding of a relation between sets of numbers by discriminant analysis. As a result, the following was clarified. Understanding the closure properties of set if numbers and the density of a set of numbers are effective to the understanding of the density of a set of numbers. In response to this result, I introduced Concrete teching methods that focus on the closure properties and the density of a set of numbers, and considered these teaching methods. A future subject is to consider systematic instruction of the number in school mathematics on the basis of these results of an investigation.

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The purpose of this paper is to discuss more 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 by using computer. We gave students enough time to create problems and assign students planning problems freely from the first. A feature of the method in this practice is to provide situations in which students make conjectures on results and get the numerical calculation by using computer. Students who tackled making problem by using computer were seemed to be interested in considering around various problems and get some deep understanding for the mathematical properties related to the problems. Some students showed much originality in their problems, which might not be made without using computer. We observed more positive learning activities than the usual classes. It is asserted that the making opportunity of problem posing by using computer planned freely from the first by students is very important, in particular for the prospective teachers.

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The purpose of this paper is to discuss more effective method of mathematical problem posing. We practiced such mathematical activities by students in high school (Grade 3). A feature of the method is to proivde situations in which students plan problems freely from the first. And another feature is to give students enough time to create problems. Actually, we report two practices of problem posing. In the first practice, some students had difficulty in making problem from the first, but many students were interested in considering around various problems and showed much originality in their problems. We observed the positive learning activities in problem posing, which might not be observed in the usual classes. In the second practice, many students were more interested in considering around various problems. We observed that many students got some deep understanding for the mathematical theories related to their problems and found that two practices of problem posing were more effective than one practice of that. Two practices show that opportunity in a class to discuss problems planned by students is very important in such activity.

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