In this paper we study quadrilaterals via the study of the diagonals. We think that the study which observes diagonals is important for the study of figures. The unification view will be brought to the curriculum by it. Furthermore, we discuss lessons "which acts to the heart and soul so that soap may act on clothes exactly".
　　A quadrilateral has two diagonals. The character of the quadrilateral is closely reflected in the relation of these two diagonals. It is important to constitute the curriculum from this viewpoint. The fundamental relations about the diagonals of a quadrilateral are the following:
　　(a) The length of two diagonals is equal.
　　(b) Two diagonals cross perpendicularly.
　　(c) Two diagonals bisect others mutually.
Table A summarizes character of diagonals of four typical classes of quadrilaterals.

Table A

　　We think that the viewpoint as which we regard a curriculum and lessons as follows. quadrilateral from its diagonals can be harnessed in the
(1) In elementary school mathematics, it is important to understand Table A by operational activities. Furthermore, it is possible to combine the viewpoint of tiling by a quadrilateral and comparison of area. We give and examine such an example.
(2) In junior high school mathematics, it is important to understand the proof of Table A. Moreover, it is important to utilize it. We give the examples of quadrilaterals which satisfy the properties (a) and (b), but not (c), and then examine several properties of them by using Table A.
(3) In high school mathematics, it is important to regard the diagonals as vectors. Moreover, by using the inner product of vectors and trigonometry, we can study general quadrilaterals. Ptolemy' s theorem is taken up from this viewpoint.

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In Japan, from a viewpoint encouraging the idea of extension, the extension of meaning is emphasized in teaching and learning the multiplication of decimals. In the lesson, specifically, children make sense of multiplication with an idea of proportion mediated by a number line. However, it is pointed out that children cannot cognize the necessity enough nor does the number line function enough. The purpose of this paper, therefore, is to obtain implications for the lesson of the multiplication of decimals through a semiotic analysis of sense-making of a child in an ordinary lesson. The results are followings. (1) It would be effective to consider a method of calculation mediated by the number line for connecting this method to the meaning of expression. (2) It is important to give a continuous image after the proportionality and sets of smaller units in the number line are clearly recognized. Moreover, it is pointed out that the semiotic frame in this paper is able to analyze lessons in more detail from translational and transformational points of view.

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The purpose of this research is to describe various modes of knowledge students construct in mathematics classroom. For this purpose I have focused on the objectification-justification process in this paper, which is the view of mathematics classroom based on the findings from the series of my research, and described various modes of justification and the value and status of propositions students used. Below, I described main findings from qualitative analysis and global consideration: (1) There are two views supporting justifacation by indivisual students. One is the perceptual view, which has the role of visually verifing results from conceptual thinking and making trial and error for grasping clues of problem solving. The other is the conceptual view, which has the role of recalling and using possible theorems or propositions by thinking the meaning of sentences in problem situation. These views are complement each other, because their functions are appeared alternately in problem solving process. (2) Students have the strong belief of the value and status used in argumentation. If they once use very efficient and concise way of thought, they do not develop it but tend to relate different ways of thought to it. (3) Significant or not, all students does not always take part in argumentation. If they could not been convinced of ideas others presented, they often consider those as special ones.

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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 she/he refers to her/his own solving processes or products during or after problem solving. For example, in study (I), we proposed the theoretical framework for analyzing self-referential activity. And, in study (II) and (Ill), we elaborated the variable "OG/NOG" and "M-SE/SE-C" respectively. In studies (VII), we theoretically examined the role of "looking-back" activity in the phase after problem solving, and we identified six roles of "looking-back" activity. In addition, we investigated "looking-back" activity after solving "Telephone-Line Problem" to examine the effectiveness of some kind of treatments to develop solver's solution. The purpose of this article is to investigate the development of solution through some looking-back activities after solving other kind of problem ("Paper-Rock-Scissors Problem"). In order to investigate whether there is any development of solution after a specific looking-back activity, the control group and the three experimental groups were set up. All groups solved two types of paper-rock-scissors problems; the two problems (Problem 1 and 2) had same problem structure, but the second Problem 2 with more broad problem space was more complicated and relatively difficult. And, the subjects in each experimental group had to reply a question between solving Problem 1 and Problem 2. The question statement was intended for implementing a specific looking-back activity with the corresponding function to "checking your own solution" (Check-Solution Treatment), "inquiring into better solution" (Better-Solution Treatment), and "examining generalization of your own solution" (Generalization Treatment) respectively. As a result, we could find the following points. (1) It seemed to be difficult to improve solver's misunderstanding or inappropriate problem representation with/without treatments. (2) "Unsophisticated solution" founded in the control group had a tendency to be fixed even if the problem became more complicated. (3) There was not a significant difference among control group and treatment groups, but each of the treatments seemed to contribute to the development of solution. This tendency suggests that some general and content-free treatments such as ones treated in this article may play an important role in mathematics classroom situation in terms of whole-class problem solving.

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We conducted the design experiment in the teaching unit "plane geometry" which was concieved as the transitional material from geometry in primary mathematics to geometric proof in secondary school, and designed as three subunits of (1) the discovery game of geometric figures, (2) the jintori game, and (3) the discovery game of geometric constructions. This paper reprorts 7th grade students' initial aspects of logical justification of geometric figures in the first subunit "the discovery game of geometric figures". The findings in this paper are the following four points. ・Students first tended to attend to the larger figures visually in the hemp leaf situation of the discovery game of geometric figures. However, when they began to consider the criticism by others and the counter examples in their minds, they could change their object of justification to the smaller figures, with their consciousness of insufficiency of the empirical explanations. They then developed their idea of trying to justify the constituent geometric figures by using the smaller figure (the right triangle). ・Therefore, we may indicate that the idea of placing the constituent figures by the right triangle was produced in the situation of formulation in which the assumption of the others' criticism was added to the situation of action. ・When students tried to justify the constituent figures by one geometric figure (right triangle), they began to consider the order of placing the figures and the justification of the right triangle itself. The justification could be successful when they reconstructed steps of geometric construction as conditions of proof. Furthermore, we suggest that the logical proof was produced as a feedback of the others' criticism and based on the logic of justification of the right triangle. ・We also found students' tendencies of complementing their logical explanation with the empirical means and of approving the empirical explanation in some cases, although they enhanced their abilities of explaining the geometric figures in the logical manner.

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This paper is a part of "Research on Cultural Value in Mathematics Education". The author has aimed to clarify the problem that the present school mathematics education has by catching the mathematics development as the organic whole, clarifying the culture seen there, and applying this aspect to historical development of the education of mathematics of our country. In the process, the following problems have come to the surface. Does the handling of the algebra expression that does the base of high school mathematics really make the best use of cultural value of the algebra expression now? For instance, let's take up the content in the third grade of junior high-school "Square root". When the treated number is limited to the rational number, there is no solution in the second equation x^2 = 2. So, the square root of 2 is thought, it is made to the sign with √<2>, and then it is explained that √<2> is "Number that does the second power and becomes 2". However, the former second equation doesn't request the expression of the solution "√<2>". Doesn't it request the Number that does the second power and becomes 2? Impatience like this making to the sign and formalization makes the essence of the teaching material mistaken. It is necessary to verify the meaning and the significance of the algebra expression that does the base of high school mathematics from the aspect that catches the culture. In this paper, it has approached a part of details how the algebra expression that does the base of high school mathematics was formed, how it developed, and how our country accepted the European calculation when our country took the European mathematics. In the present school mathematics, it is admitted that it are supposed also to speed up making to the sign and formalization that are considered the means too much speeding up the acquisition of "Algebra" and "Analysis". This paper insisted that the following is necessary now. We must catch "Sign algebra" only as no "Means" as "Purpose" again in not losing sight of the essence of the teaching material.

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In this research, mathematical literacy is assumed to have fore components: "the nature of mathematics", "mathematical contents", "mathematical method", and "attitude". The purpose of this paper is to consider "mathematical method" as a component of mathematical literacy. "Mathematical method" has been told in various words and various ways. In this paper, in order to explore the essence of "mathematical method", it is focused the researches that are comparatively at an early stage and have affected the researches of "mathematical method" in Japan. According to the researcher's theoretical background, "mathematical method" is identified by three greatly positions: (1) The way that the main focus is construction of mathematical concepts and developing mathematics by mathematizing from a phenomenon (horizontal mathematization) and mathematizing within mathematics (vertical mathematization). (2) The way that the main focus is applying mathematics to reality by mathematizing from a phenomenon, making a mathematical model, and solving a problem. (3) The way includes the first and the second. The first is the stance of mathematization theory (e.g. Freudenthal, 1968, 1983; Treffers, 1987). The second is the stance of mathematical modelling theory and theory of application of mathematics (e.g. Pollak, 1970, 1997, 2003). And the third is the comprehensive stance which includes the both stance (e.g. Lange, 1987, 1996). Thus, "mathematical method" has two directivities: one go toward the development of mathematics and another go toward the application of mathematics to reality. Mathematical literacy should comprehend both. On that basis, it is necessary to identify "mathematical method" that is required today, and to foster this.

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The purpose of this paper is to address the following question: what kinds of viewpoints should be taken into account in characterizing conceptual change in mathematics education? For attaining this purpose, at first some theoretical problems of conceptual change research in mathematics education are summarized. Secondly "the global perspective" and "the local perspective" are introduced as methodological points of view for characterizing conceptual change, and the needs for these perspectives are discussed. Furthermore from these perspectives we make illustrations in the case of development from discrete to continuous quantity. And in the final place, some issues as didactical implications in developing from discrete to continuous quantity can be shown. On above considerations we have come to the conclusion that "multiplication with decimal numbers" and "irrational numbers (incommensurable magnitudes)" can be identified as problematic situations for conceptual change. As a result of the preliminary analysis, the following issues as didactical implications in developing from discrete to continuous quantity can be shown: ・In the problematic situation "multiplication with decimal numbers", learners need to become aware of inconsistency in terms of existing meaning of multiplication. ・In the problematic situation "multiplication with decimal numbers", the proportional meaning of multiplication that has been implicit status can become more explicit. ・In the problematic situation "irrational numbers (incommensurable magnitudes)", learners need to become aware that the familiar notation (i.e., place value system of decimal notation) cannot represent the quantity (incommensurable magnitude) in question precisely. ・In the problematic situation "irrational numbers (incommensurable magnitudes)", the awareness of incommensurability can lead to advance the degree of rigor on the infinity. As the future tasks, we need to build some theoretical framework/model in order to describe and/or assess learner's conception (status of knowing) about above issues, and to design the mathematics classrooms of "multiplication with decimal numbers" and "irrational numbers (incommensurable magnitudes)".

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Today, we have confronted with the difficulties to utilize the educational evaluation of the target conformity type in schools and the complicated work to develop and revise the criteria and indicators for use in the assessment. In this paper, we discuss the problems of the assessment from the historical point of view. According to this discussion, we try to implement Backward Design for the improvement of the problems of the educational assessment under the present conditions. This approach has three stages: Identify desired results, Determine acceptable evidence, Plan learning experiences and instruction. We show the efficiency of this approach based on Performance Task and Rubric to develop children's insights toward number, quantity and geometric figure in mathematical phenomena through design experiment.

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The purpose of this study is to research the process of mathematics instruction at junior high school through the examination of the effectiveness of materials made to improve mathematical inquiry. The effectiveness was determined by the observation of students in the classroom, their performance in a post-test, and their responses to a consciousness survey. The materials utilized were as follows: "The Method of Multiple Solutions Using Negative Remainders", made for 7^<th> graders to address a particularly difficult subject in the compulsory mathematics curriculum; and "The Study of Polygons", made for 8^<th> graders by reconstructing an existing unit of the compulsory mathematics curriculum. "The Method of Multiple Solutions Using Negative Remainders" was a little too difficult for most of the 7^<th> graders in the study; however, the material seemed to change their unfavorable image of negative numbers and raise interest in them. It also enabled students to enjoy expanding into, and making use of, negative numbers. "The Study of Polygons" deepened student interest in the comprehensive relationships among quadrangles. In particular, the material facilitated student discovery of the properties of parallelograms and the conditions that must be met for quadrilaterals to be parallelograms.

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The aims of this paper are to consider the causes of difficulty in the learning of geometric proofs in junior high school, and to obtain a basis for the improvement of related teaching and learning. Despite much early research into the teaching and learning of geometric proofs, difficulties of them have not been overcome. In this paper, the author considers the causes of such difficulty under the light of didactical contract, a major component of Brousseu's Theory of Didactical Situations. Especially focused on are two areas of requiring improvement to mitigate difficulties in the teaching and learning of proofs: A) Understanding the meaning of proofs B) Doing proofs in didactical situations For improvement of the difficulty of A), the didactical contract of "Generality of proof' is considered, and the proposal is that applied devolution is effective in addressing the difficulty. In respect to improvement of the difficulty of B), the didactical contract of doing proofs in didactical situations is considered, and the proposal is that teaching and learning situations that employ the negotiation of the didactical contract oriented devolution is also effective in its turn.

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The purpose of this study is to examine whether having students pose problems is effective in changing calculation drills, which often tend to become mechanical, into something more creative. Within the scope of differential calculus covered in the course Mathematics III, students individually posed problems for calculation. The teacher classified and arranged these and set them in a format called a small test. The following results can be pointed out. Students obtained a deeper understanding through this approach than through textbook-based instruction. This approach also provided an opportunity for students to try on their own to review what they had studied. This form of posing calculation problems in differential calculus has the advantages of suitability to conducting at a relatively high frequency and of making it possible to address the problems of a greater number of students. Posing problems within the scope of integral calculus, on the other hand, should be done slowly and carefully, at low frequency and in special classes. In calculation drills, in which students are involved deeply for reasons including grading, there were many students who were able to practice with more eagerness under this approach than with conventional calculation drills. In using differential calculus to calculate trigonometric functions, in some cases final expressions may vary due to differences in solution methods. Students were able to broaden their points of view by comparing their own answers to the results of the problem posers and other students and also were able to understand various alternative methods well. For these reasons, it may be said that calculation instruction in which the method of posing problems is introduced increases students' creativity and encourages proactive activity.

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The purpose of this paper is to discuss effective method of mathematical modelling by using computer. We practiced such mathematical activities by university students who were the prospective teachers. In study (I), we reported the practice of mathematical modelling by using computer in which students make mathematical models on Ecology. In this paper, activities for mathematical exploration by using computer are focused. A feature of the method is to provide situations in which students make mathematical models on Epidemics which is different from study (I). And another feature is to give students enough time to get the numerical calculation and make conjectures on some results by using computer. As in the previous study, the practice shows that some students had difficulty in making and solving mathematical models by using computer, but gradually felt interested in considering about some phenomenon. We observed the positive learning activities which might not be observed in the usual classes. It is asserted that the opportunity of mathematical modelling by using computer is very important, in particular for the prospective teachers.

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　　In this research, I have two aims:
　(1) Are there differences for viewpoints of whole by changing the numbers of parts which construct whole?
　(2) When two quantities relate to "double and half" for "part-whole", can pupils deal with relations? In view of two aims, I have carried out the investigation for 5 Lottery Problems. The investigation was carried out in the second term.
　　The subjects are 5th grade pupils (N=63) of two elementary schools in Kobe and Himeji. They aren't taught ratio in this term. They are divided two group [Experimental Group (N=32) and Control Group (N=31)] by difference in the numbers of parts which construct whole. The following ploblems of Experimental Group and Control Group were carried out:
　Question of Experimental Group.
　　There are the A box and the B box with red marbles, blue marbles and white marbles, respectively.
　　The A box has P red marbles, Q blue marbles and R white marbles.
　　The B box has S red marbles, T blue marbles and U white marbles.
　　Which box will have a greater chance of drawing a blue marble?
　　(1) Please mark the one among 3 choices.
　　　A box will have a greater chance. Both boxes will have the same chance. B box will have a greater chance.

[figure]

　　(2) Why?
　Question of Control Group.
　　There are the A box and the B box with red marbles and white marbles, respectively.
　　The A box has X red marbles and Y white marbles.
　　The B box has Z red marbles and W white marbles.
　　Which box will have a greater chance of drawing a white marble?
　　(X=P+R, Y=Q, Z=S+U, W=T)
　As results of analyses, I have clarified the following contents.
　　(1) There are differences for viewpoints of whole by changing the numbers of parts which construct whole.
　　(2) When two quantities relate to "double and half" for "part-whole", pupils can't deal with relations.

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The purpose of this study is to collect the fundamental data on the children's probability concept which provide information for teachers and other developers of curriculum. In this paper, the author adopted a roulette as a continuous event and investigated the Common Cognitive Path (CCP) in the sixth grade children's probability judgments. As a result, the following CCP were found through the analysis of the reaction to the problem which compare likelihood of coming up black on the roulette boards divided into white sectors and black sectors. (1) The ratio of the area of the black sectors to the white sectors is different → The ratio of the area of the black sectors to the white sectors is the same (2) The size of the roulette boards is the same → The size of the roulette boards is different (3) The roulette boards are divided into a black sector and a white sector → The roulette boards are divided into two or more black sectors and white sectors. Then the author proposed the methods and techniques of instruction based on the CCP, and also constructed the rubric to evaluate grounds for children's probability judgments. As a result of instruction based on the CCP, the ratio of children who judge likelihood objectively based on the idea of the ratio of the area of the black sectors to the white sectors increased. This shows the achievement of criteria on the rubric assessment. This study shows the necessity and the possibility of the curriculum development based on the realities of the probability concept of children in the elementary education stage, and it also gives the suggestions for the curriculum development that connects with the curriculum of probability in secondary education in Japan.

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The purpose of this paper is to examine the possibility of the teaching via problem solving in high school mathematics. In this paper, we focused on the learning of the "addition theorems of trigonometric functions" in MAHTEMATICS II, and following was examined. 1) Through quantitative and qualitative investigations, the actual conditions of the activities of the students who worked on the theorems for the first time. 2) Through teaching practice, the effect and validity of the hypothetical ways of support based on the investigations. As a result, it has been understood that the hypothetical ways of support set in this paper worked effectively in the following three points: to understand the problem, to devise the plan for solving in the classroom, and to understand the proof method or meanings of the addition theorems of trigonometric functions. In a word, it is the main result of this paper to have suggested the possibility and the effectiveness of the teaching via problem solving in high school mathematics.

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Once I saw a handout which was used at a secondary school in 1945, where the real shape of the cross section of a cone was asked The handout contains the projection chart of a cone and a plane, which means the space geometry and the descriptive geometry were tightly connected in math classes at that time. From the Meiji era, they were taught separately in mathematics and drawing. Under the 1942/43 syllabi, descriptive geometry was widely adopted in "Daini-rui" (the 2nd part) of mathematics course. Through the handout and real notebooks at that time, I found that the projection skills were actively used in the study of conic sections. Although the projection was left in the mathematics in middle schools for a while in the post war secondary school system, useful contents were very much limited such as real length of a segment. Finally, descriptive geometry was completely deleted from secondary school mathematics when the course of study was revised in 1969. It is important to investigate the relationship between space geometry and descriptive geometry, and use the latter effectively to teach the former.

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This paper mainly focuses on how students in Zambia worked on continuous activities of SLE, which possibly integrates two aspects of calculation practices and some other mathematical abilities such as finding patterns, communicating, guessing, and creating, in the framework of design experiment. In this research, a 23-time short activity in class was continuously introduced in two classes of grade 9 corresponding to grade 3 in junior high school in Japan and three ways of assessment were accordingly put into practice; analysis of pre and post test; formative assessment; and students' actual work through activities. The result of the tests shows that the scores on recognition of patterns and communication increased well although the whole score increased slightly after the activity. The formative assessment indicated that students had difficulty with explaining their mathematical observation in word. The analysis of students' activities shows that a student in the high performance category could improve both calculation skill and his communication skill through continuous activities where he repeatedly goes back to mistakes or inadequate answers in the process of his development, whereas another student in the poor performance category only could manage the calculation of it. The author leads the three characteristics of students' work through SLE; slow improvement divided into several small stages of understanding; return to the previous stage of understanding; and instability of understanding. It is noticed that the language problem in the second language and social problems are also influential to students' understanding of mathematics. Consequently, the author concludes that SLE can be a possible practice in class for students to foster these two mathematical aspects; the basic calculation skill and other mathematical skills; however, the challenge for slow learners on this teaching material relating to language difficulty and social problems is also pointed out.

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