At first the author suppose that if a structure-mapping consist between two things, the source system of one side thing act on forming the target system of the another thing. In this paper the author urged as followings (1)〜(6). (1) revalued "counting" based on epistemological viewpoint. (2) suggested that there are degree of difficulty in situations which form the cardinal number concept. (3) poited out that the order number system is analogy to the cardinal number system. (4) showed that there are two different recognition, that is, "these do exist" and "these have equal quantity", in recognizing counting objects. (5) showed that there are two ways as followings for generalizating "counting". (1) expand the domain of count objects. (2) transform "exist" and "equal qantity" in (4) to "equal value". (6) indicated that epistemological obstacle occure on (5)(2).

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This paper is intended as a consideration of metaphorical understanding of fraction in terms of both its form and content. As for the representation of fraction in European style, it is composed of cardinal number and ordinal number like as m-nths. They are extremely different each other in the epistemological genesis. Ordinarity is the index of discrimination. On the other hand, cardinarity is the product of categorizing same. The combination of such an opposite concept reflects the rhetorical intention rather than supports the logical understanding. The fraction in Europe was not originally invented as the object for mathematics but as the means for trade and commerce in the ancient Greece. It, therefore, asked for the metaphorical understanding from the beginning. For instance, "two" means "2/3", and "third" means "1/3" in the line of Iriad K,253 by Homer: "Two parts of the night are past, the third part remains." (Van der Waerden, 1954, p. 50) The former trope by cardinal number could be considered as metaphor because of similarity or analogy relation between two numbers. The latter trope by ordinal number could be regarded as metonymy because the part represents the whole. The point is that the fraction is in itself figurative form, therefore is metaphorical concept. As for the meaning of fraction, there is difference between the literal meaning and the mathematical meaning. It causes metaphorical understanding, which helps to get the sound comprehension of the mathematical concept. This is the conclusion of the didactical analysis based on the Transcendental Recursive Model of Understanding Mathematics by Pirie and Kieren (1994a). They insist that Image Having is characterized by the speech of metaphor and Property Noticing by the speech of simile. Mathematical understanding, therefore, proceeds to simile from metaphor. This seems to be the correct insistence on the development of metaphorical understanding. Their assertion, however, could be criticized by the complementarity in mathematical understanding implied by Dorfler (1984, p. 261) as follows: P1. The transcendent recursive model does not give us any explanation about the system of modification in cognitive paradigm from metaphor to simile. It just corresponds metaphor with image having level and simile with property noticing level. P2. The consideration on understanding mathematics by Pirie and Kieren is inclined to the analysis of relation i.e. die Beziehungsschem and as a result leaves the analysis of activity i.e. die Handlungsschemas in terms of Darner. In the learning process of subjects Don and Sam in their paper, they just focus on the former. The author has introduced the viewpoint of metonymy based on contiguity to answer those problems P1 and P2. This introduction is justified by the above complementarity. A1. The intellectual renovation from image having to property noticing could not be clarified without the consideration on Handlungsschema. The roll of it is analogous to the cognitive roll of metonymy and synecdoche according to Zawatowski (1985, pp.262-263). Activity of children, therefore, should be analysed from their viewpoints. A2. The activity of subjects, Don and Sam, in the paper of Pirie and Kieren could be analysed clearly by metonymy. Their cognitive level should be consequently similical rather than metaphorical although the paper say that it is metaphorical.

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In mathematics teaching we declare to bring up student's logical thinking abilities. But in general, people say that mathematical taled one like mathematicians should not be excellent to perform logical argument in ordinal conversation. Answerng to this criticism against mathematics teaching, I want to point out three factors of logical thinking ability, indeed, logical power, logical knowledge and logical awareness. Logical power is the one which people can perform logical argument in ordinal conversation or mathematical argument at now. Comparing to this, logical awareness is the one which people can aware to logical affairs in our ordinal conversation. For example, in ordinal conversation, I can aware that some sentence is related to the difference between 'conditional sentence' and 'biconditional sentence', but I can not perform completely logical correct argument. And I want to say, mathematics teaching is contributed to bring upchild's logical awareness. And I discussed the problem of creativity of logical thinking. People nowadays respect creativity and criticize logical thinking for its no creativity. Certainly, logical thinking is not creative from a logical point of view. But in practice like human mental activities or natural science studies, logic is contributed to create new knowledge to us. Then we mathematics teachers might be able to declare that logic thinking is sufficiently creative.

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Social Constructivism which Ernest (1991) suggested apart from Radical Constructivism, is an important direction in research of mathematics education. The social context or situation in learning mathematics have been often neglected. For the first time, Gattegno pointed out the significance of learnig situation. But it was almost physical. Sociocultural one was little considered. Social Constructivism suggests the research on sociocultural factors in learning mathematics. And it is another possibility of Social Constructivism that it implys the consideration of learning aims. Therefore the main probolem is what are the basic theories in Social Constructivism. Ernest (1994) proposed two theories, Piagetian thery of mind and Vygotskian theory of mind. But I suggest adding situated learnig theory, especially "Legitimate Peripheral Participation" by J. Laveand E. Wenger. It 'takes a relatively radical position by attempting to avoid any reference to mind in the head (Cobb, 1994, p.18)', which is so different from Piagetian theory and a developed form of Vygotskian theory. Therefore it considers a learner and learners holistically and socially and regards their identity to learn as important. And it could contain the new third direction against indoctrination and constructivism in mathematics education.

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How can we help teachers to communicate their metaknowledge, especially, the nature or character of mathematical knowledge or activity to the students? We regard this as the central research question of our study. The present paper can be understood as an attempt to try to ask the question through consideration of practices in the classroom. To do so, a cooperative teaching experiment was conducted over a four-week period including 4 lessons with an 8th grade class on June 9, 12, 13, and 16 in 1995. The analysis of interaction between the teacher and the students in the lessons revealed that when (1) the teacher helped students to develop their thinking tools which they already have used to approach the task concerned, and further (2) the developing their thinking tools enabled students to be aware or discover what they did not know, the teacher's metaknowledge could be communicated to students.

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It is obviously necessary to educate the talented students. Because all students have equal opportunity of education and a talent is the public property. But Japanease consider that the talented students' education is a discrimination. It is a cause of a delay of the talented students' education. We know that there are the acceleration, the enrichment and the special grouping for the methods of the talented students' education. The way of the general class in Japan is simultaneous learning, so that Japanese have studied the enrichment for their education. I think the studies of the acceleration and the special grouping are necessary, because the acceleration education is being done in the famous private junior high schools. Stanley et al. contend that acceleration may be the only way to provide the best match for the educational needs of extremely and severely gifted children.

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On evaluating structural concep maps which are drawn by students, most teachers must often get into the following difficulties. 1)The teacher is confused by evaluating each map when the structural arrangement of learning elements and the oriented connected line among learning elements in the map are too complex. 2)The teacher spends a lot of time when he evaluates a great many maps and/or the maps consisted of many learning elements. This paper presents a new measure and its scale to evaluate the maps which are drawn by students. On making up the measure, I examined a kind of properties of four measures, namely consistent coefficient, phi coefficient, symmedian rate and new transfer coefficient in order to evaluate the maps. As the result, it became clear that the measure with transfer coefficient was better scale than all the measures with other coefficients. An agreement rate between a value of evaluation on the maps judged by teachers and a value of evaluation on the maps with transfer coefficient was about 86 percent.

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The purpose of this study is to explore experimentally the role of strategic ability which is ability to make use of our mathematical resources in mathematical problem solving. In this study, strategic ability are captured by two dimension, "Ability to Use Problem Solving Strategies" and "Metacognitive Ability". The present paper aims to explore the rate of contribution of the strategic abilities and mathematical resources toward problem solving ability by multiple regression analysis. A total of 300 graders 6 in elementary school were administered the prpblems to evaluate children's problem solving ability, mathematical resources, and ability to use problem solving strategies. And, they were administered "A Questionnaire-Type Instrument for Measuring Metacognitive Ability in Mathematical Problem Solving" (Shimizu, N., 1995). The following were mainly found out through the various multiple regression analysisses. (1) The variables of Mathematical Resoure and Strategic abilities accounted for about 60% of the variance of problem solving ability. (2) The order of the variable which has higher contribution toward problem solving ability were "Mathmatical Resources", "Ability to Use Problem Solving Strategies" and "Metacognitive Ability" in turn. (3) To evaluate children's "Ability to Use Problem Solving Strategies" by all-or-nothing method seems to lead to underestimate the ability. (4) There were a gender difference with respect to the aspect of contribution toward problem solving ability, though there were not a gender difference with respect to the means and variance of the all variables significantly. Although the results of this research may be of limited generality and validity, the following are mainly suggested. (1) We Should not make light of basic mathematical resources. (2) With respect to strategic ability, first of all, we should emphasize teaching of the problem solving strategies, and second, teaching of the metacognition. (3) We should evaluate children's "Ability to Use Problem Solving Strategies" by multiple viewpoints. (4) We should consider the issue of gender difference in problem solving in detail.

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　　The purpose of this study is to investigate the way of developing students' metacognitive skills in mathematical problem solving. As the first step of it, the author established "a framework for investigation" to investigate and analyze the metacognitive skills (Table 1). The poupose of this paper is to grasp with the framework the actual states of metacognitive skills which fourth and sixth graders could use in mathematical problem solving. For that purpose the author investigated and analyzed them.
　　The findings of this investigation were the following:
　　(1) The number of pupil's behaviors with metacognitive skills at the 【Q1】 and at the 【Q2】 were positive correlation.
　　(2) As the pupil becomes older, the number of his/her behaviors with metacognitive skills increased.
　　(3) The number of pupil's behaviors with metacognitive skills and marks at his/her work-sheet were positive correlation.

【Table 1】 The Framework for Investigation
1. the way of investigation
　(1-1) work-sheet
　(1-2) stimulated-recall questionnaire
2. the way of analysis
　(2-1) marking at pipul's work-sheet
　(2-2) identifing pupil's behaviors with metacognitive skills

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The purpose of this study is to make it clear the types of imagistic representation and the roles of Imagistic Representation System in mathematical problem solving process. Imagistic Representation System is a construct in "A Model for Competency in Mathematical and Scientific Problem-solving" which is presented by G. A. Goldin. In this paper, the following four objectives are discussed to fulfill the above purpose; i) to prescribe the types of imagistic representations borrowing Presmeg's types of visual imagery and Dorfler's types of image schemata; ii) based on the prescription at i), to make it clear roles/functions of Imagestic Representation System in problem solving process (translation process); iii) to observe and interpret a problem-solving situation using some types of imagisitic representations as a concrete example of utilization of the prescription i) and ii); iv) to give some feedbacks for the author's serial studies in consideration of above all discussions. In the discussion i), seven types of imagistic representations are prescribed; i.e. (I1)concrete/pictorial imagery, (I2)pattern imagery, (I3)symbolic imagery (and image of formulae), (I4)operational/kinaesthetic imagery, (I5)dynamic/moving imagery, (I6)relational imagery, and (I7)auditory/rhythmic imagery. In the discussion ii), the following three roles/functions are identified; i.e. (1) making sense of problem or grasping problem situation, (2) serving organization of information and analysis of solution, and (3) generalization/abstraction. In the discussion iii) and iv), it is suggested that the abstruction (or syntactic transformation) of imagistic representation and solver's making sense of problem or problem situation should be introduced to a problem-solving model to be intended by author.

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Multiplication and division of decimal fraction is one of the most difficult contents for 5th graders in the primary school. This study focuses on the division of decimal fraction, especially on the generalization from quotitive division to first application of ratio. The purpose of this study is to clarify the understanding process in generalization of quotitive division, especially the factors of difficulties in understanding the division of decimal fraction and the factors for overcoming the diffuculties. The main results are folloing; 1. The generalization of quotitive division is made by extending the idea of 'measurement'. 2. In the generalization, it is effective to use the operative material that can generalize the idea of measurement and that function as the semi-concrete material for number-line expression. 3. The primitive model on quotitive division include the properties of 'quotient is integer' and 'dividend is larger than divisor'. 4. Children feels difficulties in such situations that quotient is decimal fraction and that dividend is smaller than divisor. These situations are related with primitive model. 5. The stages that the idea 'how many times' is abstracted from how many pieces 'are folloing; (a) The idea of 'how many times' is primitively constructed in such a division that dividend is larger than divisor and quotient is decimal fraction. (b) The idea of 'how many times' is realized in such a division that dividend is smaller than divisor. (c) The merit of the idea of 'how many times' is realized when all division of decimal fraction is conceived by the idea.

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The propose of this paper is to consider the reflective thinking to promote constructing the abstract and general knowledge or the problem solving schema to solve the problems. Referring to past research, it follows that the relationship between the construction of problem solving schema and the recognition of similarity of problem structure is inseparable, that is to say, the recognition of structural similarity of problems or structure mapping among the problems is very concerned with the construction of problem solving schema. From Reed, Ross, and Novick's suggestions, I have come to the conclusion that the reflective thinking based on mapping of numerical values among the problems is effective in order to construct the abstract and general problem solving schema consciously. So, I proposed the reflective thinking based on mapping, since I consider that students could construct the abstract and general problem solving schema to solve the problems through the mapping of numerical values between two isomorphic problems after solving them.

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The purpose of this paper is to examine the relationship between proving ability and creative ability according to measuring creative ability in geometric proof problems posed by students. Examination indicated that students who could solve a given proof problem advantageously gained high fluency scores and flexibility scores in comparision with students could not solve the proof problem. But, Examination indicated that the originality scores differs little among three groups, too.

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The purpose of this paper is to prepare good examples for promoting spatial cognition in the geometric education. Two concrete teaching materials for geometry are offered and analyzed from both points of view, mathematics education and mathematics itself. One material treats a problem which asks the number of domains given by subdividing the Euclidean space by hyperplanes. The solution includes various aspects involving topology and combinatorics, and the answer enlightens us on a beautiful combination of geometry and arithmetic. Another material is concerned with transformations in the space, and it is illustrated that the affine method works well for understanding of compositions of isometric transformations. The problems given in the contents are related with thought on the group theory, and also promote spatial cognition on movement of objects.

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