In the 50s of the Showa Era (1975-1985), on the basis of a scientific view of teaching and classes, learning objectives were classified and lesson planning was discussed in terms of its validity, reliability and objectivity. However, this development, contrary to its original purpose, led to "standardized classes" and a "choking atmosphere in class". According to the author, it is suspected that this result was caused by the fact that the scientific nature of classroom teaching was supplanted by one based on natural science. In fact, it is commonly accepted that according to a system-based approach, such a complex phenomenon as teaching and activity in a classroom should be formed through a "systemic-progressive" approach, not an "mechanical-technical" approach. In particular, as far as mathematics education is concerned, there are problems with teaching and classes that are based upon behaviorist theories.
(1) Behaviorism-oriented classes originally derived from a "factory model" common in industry and its purpose is to develop the labor force.
(2) Behaviorism-oriented classes are based on atomism, which results in the lack of the 'ability to apply', learning continuity and a strong dislike of mathematics.
(3) Behaviorism-oriented classes do not fit the nature of mathematical activities (i.e., "continuous development of patterns from patterns").
(4) Behaviorism-oriented classes neglect the nature of human existence (i.e., they focus on "we", not on "I").
(5) The standards for classroom evaluation (i.e., generalizability, reliability, replicability, and so on) have been changing.
　　In order to solve these problems, the present paper argues for the development of mathematical learning on the basis of teaching principles. This suggestion presupposes that mathematics education is a type of design science.
　　Teaching principles, which are developed from various teaching-learning theories, are testified in real live classrooms. Thus, these teaching principles can be regarded as a set of hypotheses and also the sum of what has been learned by researchers, such as examples of whom are Piaget, Bruner, Dienes, Freudenthal, Dewey, and Vygotsky.
　　Here are some examples of the teaching principles: genetic principle; operational principle; principle of activity; principle of dynamic interaction; principle of social learning; wholistic principle; principle of the spiral learning; principle of interaction of representation; principle of focus on central ideas; principle of progressive mathematizing.
　　 The structure of these principles is summarized by Wittmann (1998) as follows;

Figure: Principles of learning and teaching

　　In this paper, the author suggests that in accordance with these principles, decisions on how the Course of Study and virtual classes should be developed and evaluated must be left to those who are expert in mathematics education (i.e., scholars and teachers). This is because each expert has a different view of "mathematics", "mathematics education", and "children as learners". The fundamental nature of mathematical learning is freedom (openness).

View full abstract

　　There are two types of conceptual metaphor, that is, grounding metaphors and linking metaphors. Grounding metaphors allow you to project from everyday experiences (like putting things into piles) onto abstract concepts (like addition). Linking metaphors link two different branches of mathematical concepts, for instance linking geometry to arithmetic, as when you conceive of numbers as points on a line.
　　The purpose of this paper is to analyze mathematics learning from the view of conceptual metaphor. Two types of conceptual metaphor yield two types of mathematics learning, that is, mathematics learning by grounding metaphors and mathematics learning by linking metaphors.
　　The following figure and table show some differences between mathematics learning by grounding metaphors and mathematics learning by linking metaphors.

[figure]

[table]

　　Through several specific topics of mathematic learning, the author shows some features of the two types of mathematics learning. The results are as given below.
　- In the mathematics learning by grounding metaphors
　　[G-1] Concrete experiences give a meaning to mathematical concepts.
　　[G-2] The proficiency of mathematical concepts yields the reification of symbolic representations of mathematical concepts.
　- In the mathematics learning by linking metaphors
　　[L-1] Meaning of a mathematical concept will come from the other mathematical concept by linking metaphor.
　　[L-2] Mathematical problems can be solved by linking metaphors in which original problems are metaphorically replaced by other problems.
　　[L-3] The linking metaphor may encourage the emergence of new mathematical ideas.

View full abstract

The purpose of this paper is to suggest theoretical basis towards a model of conceptual change in terms of the phases of conceptual change. For attaining this purpose, this paper consists of two main parts: in the former part some earlier theoretical framework of conceptual change are critically reviewed in terms of phases of conceptual change and an alternative characterization are shown; in the latter part the alternative framework are examined by means of preliminary analysis of one conceptual change situation, that is "multiplication with decimal numbers". Fundamentally speaking, the notion of "conceptual change" can be characterized as two different phases based on the frameworks of theory change in philosophy of science: T. Kuhn's "paradigm theory" and/or I. Lakatos's "scientific research programme". From the viewpoints of two phases, some earlier researches (e.g., Posner et al, 1982; Vosniadou & Verschaffel, 2004; Merenluoto & Lehitinen, 2004; Morimoto et al., 2006) can be summarized. On the other hand, such earlier researches on conceptual change tend to overlook the crucial differences between different claims about theory change in philosophy of science. This study tries to identify the differences and to show alternative views on the phases of conceptual change, particularly in the case of mathematics. That is to say, the phases of conceptual change in mathematics learning can be characterized as three different phases: "extensional enlargement", "internsional reduction" and "conceptual reconstruction". The three phases of conceptual change showed in this paper are explained and examined by means of preliminary analysis of one conceptual change situation: multiplication with decimal number. The teaching and learning of this content can be a problematic situation in the development form discrete to continuous quantity.

View full abstract

The purpose of this paper is to capture how children do cognitive activity in process of polishing up, and to see about possibility of buildup of metacognitive knowledge. In this paper, the research on teaching of metacognition was considered, and I checked to show that involvement of others (teacher and children) is desirable to do both during and after individual solving. Then, process of polishing up provided two categories: Stage of process of polishing up, individual activity in stage of process of polishing up. I considered cognitive activity in process of polishing up from aspects such as sender of information and receiver of information, and I tried to capture cognitive activities. In the result, I captured some cognitive activities of sender of information and receiver of information. It is cleared that children works metacognition for cognitive activity then buildup of metacognitive knowledge.

View full abstract

Globalization is now in progress based on information technology in our society. Education is a social endeavor from the preceding generation to the following generation, and this social change demands inevitably for change in education. Mathematics is embedded in the basis of information technology, but the more it relies on information technology the more impossible it becomes to grasp the whole image. Human beings increase dependency on information society, and on the other hand, it becomes more impossible to understand its basis. Keitel (1997) calls this imbalance "mathematized society and de-mathematized people", and attracts attention to the fragile relation between human beings and society. This research analyzes formation of mathematical literacy, which is required in such a society, from open-end approach and values, and clarifies its possibilities and issues.

View full abstract

The purposes of this research are to develop mathematics lesson which makes a point of connections with society for the high school student, and to foster competence to deal with mathematically the phenomena and the problems in the society. As one of the work to achieve the purposes of this research, this paper makes an analysis of "the degree of fictional nature" that is observed in mathematical word problems that were made by students. In this paper, the author described what is "the degree of fictional nature" with relation to mathematical modeling process, and how it is related to the purposes of this research. The investigation was executed with a concrete teaching material, and mathematical word problems that students made were analyzed in terms of the degree of fictional nature.

View full abstract

We conducted a design experiment for clarifying the transition process from geometry in primary mathematics to geometric proof in secondary school for the seventh grade's teaching unit "plane geometry" which was designed as three subunits of (1) the discovery game of geometric figures, (2) the jintori game, and (3) the discovery game of geometric constructions and the geometric proof. In this paper we examined the students' thought processes when they could develop the geometric constructions through the discovery game and explore the geometric proof on the superposition of two triangles which are placed at the arbitrary positions, and aimed at abstracting the factors of students' development towards geometric proofs. The factors of the transition towards logical proofs that we abstracted from our analysis of the design experiment are the following points. 1. Cognitive aspects are ・to find the geometric constructions by using and combining the figures and their properties, ・to confirm and limit the extent where the law works using inductive reasoning and empirical explanation, ・to make the proposition so that inductive reasoning can be changed to deductive reasoning, and ・to reinterpret the procedures of geometric constructions as the conditions for justification. 2. Aspects of the recognition of figure and shape are ・to find out figures and use them for geometric constructions and inferences, ・to conceive shapes included in the figure in terms of various relations and correspondences, ・to see figures as the variables which can be transformed dynamically, and ・to see figures as the representations of relations through showing reasonings diagrammatically. 3. Social aspects are ・to learn based on conjectures, refutations, and consensus among the participants, ・to make conjectures while assuming the criticism by the other students, ・to develop conjectures by examining the criticism and the counter-example by the other students, and ・to reinterpret and express the others' explanations more explicitly. 4. Aspects of the teaching unit structure are ・to envision the fundamental situation and the unit structure as a series of the situation for action, the situation for formulation and the situation for validation in the theory of didactical situation, ・to pose a proof problem so that the learned procedures of geometric constructions can be reconstructed as the conditions for justification, and ・to make geometric transformations and geometric constructions interact with each other at the final stage of the unit.

View full abstract

The purpose of this research is to develop instruments to measure pupils' potential ability for learning mathematics and to study theoretically and empirically the ways to utilize the instruments for improving mathematics education at elementary and secondary school levels. In the series of our study (Nakahara, 2002; Nakahara et al., 2008), we has elaborated the concept of "potential ability" and we constructed the theoretical framework for potential ability which is constituted of two aspects, "Mathematical Thinking" and "Mathematical Content". "Mathematical Thinking" is categorized into four components: logical reasoning, pattern recognition, manipulation, and flexible thinking and "Mathematical Content" is categorized into three components: number and quantity, shape and space, and function and relation. According to the theoretical framework, we developed the instruments to measure pupils' potential ability for learning mathematics (Nakahara et al., 2008). These instruments for 4th graders and for 8th graders have 20 items each, and the reliability of the instruments were validated. The purpose of this article is to investigate and consider the relationships between potential ability and mathematical attainment of 4th grade pupils. 4th graders (N=363) from three public elementary schools were administered the instrument of potential ability and mathematical attainment test. As a result, the following findings were mainly found out. (1) There were relatively strong correlation between potential ability and mathematical attainment. From the viewpoint of the "Mathematical Thinking" aspect, pattern recognition, manipulation, and flexible thinking correlate relatively strongly with potential ability, but logical reasoning correlate relatively weakly with potential ability. (2) By introducing "the subtracted attainment (attainment-potential ability)" and "the divided attainment (attainment÷potential ability)", we could characterize relatively the difference of the effect of teaching mathematics conducted in each school. (3) It was found that manipulation and logical reasoning components have relatively strong effect toward mathematical attainment and have a certain effect toward pattern recognition component.

View full abstract

There is the case that the argumet does not function well in talks activity truly even if their communication is active. I think that the appearance of the argument might be different depending on the development of the learner. The purpose of this study is that I identify a characteristic of the talks activity to come out of the development of the learner. Therefore, I worked with the following observation investigation. I divided it into groupuscule every university student, junior high student, primary schoolchild and showed some mathematical problems to include the same problem in each group and directed it to solve problem cooperetively. All groups are only children, and there is not the teacher. When they reached solution to understand together, I finished their talk activity of the group. After the problem solving, I let a self-commentary investigation while watching the thing that a video recorded the state of the solution of the problem in all groups. I analyzed those activity and considered it. Through those example observation and interpretatin, I was able to derive various approach of the agreement to reach from the following order characteristic. (1) Communicative comfort by having been the same idea and perfomance (2) Communicative adjustment by having been conscious of a difference (3) Uptake (awareness) of the difference and the unconsciousness of the difference (4) Consciousness of the difference (5) Acceptance of the intersection of ideas and non-acceptance of the difference (6) Acceptance of the difference However, these were derived by the observation of a certain group and may be peculiar to the group. There is generally the aspect that is hard to say that I expressed the general characteristic of the agreement process. I increase examples more and must push forward the inspection in future.

View full abstract

In this paper, we develop an assessment sheet for improving the class practice power of mathematics teachers and students who study mathematics. Furthermore, we propose the method of raising a student's class practice power by making them understand knowledge of results of a trial lesson. We evaluate and analysis class practice power of undergraduate students who study mathematics by using the assessment sheet in a trial lesson. The results are as follows. - For the students who had little class practice power, they could not discriminate about whether the teaching method of the mathematics class is good or bad. - Although the students found by themselves in little power about plan and knowledge of teaching method, but they did not find in little knowledge about special contents and understanding of the situation of learners. - By method of factor analysis, we found four factors "consideration to the student", "understanding about teaching resource", "grasp of a student's understanding" and "selection of teaching contents and teaching materials" in the student's trial lesson. - The students could not understand the difference clearly between special knowledge who should be learned before their class practice and instruction technique which should be required of class practice. - The students could consider the remedy of a trial lesson concretely by acquiring the knowledge of results which were found out from the assessment sheet.

View full abstract

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 mathematics education 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? In the preceding paper, it 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 this paper, the focus was applied to the receipt of the European mathematics in the age when the mathematics education of Japan did shape. Especially, the focus was applied to the receipt of the elementary algebra that was the content that which was related to the algebraic representation previously described and related to the secondary education. And it searched for the modality of mathematics and the mathematics education at that time. As a result, the following has been understood. The elementary algebra was digested to Japan through the movement of the mathematics technical term unification, the enhancement of the translation book, and the maintenance of the textbook, etc. in about the middle of the Meiji era. It arrived at "Receipt" of the elementary algebra by completing Japanese original "Textbook" in around 1897. And, it is deeply taken part by there was a tradition of Japanese mathematics "Wasan" in Japan in the process of the "Receipt". In addition, when the process from relations with Japanese mathematics "Wasan" to "Receipt" of arithmetic and the elementary algebra is caught, the process is roughly divided into the next three stages. The first stage: Stage of "Conversion" from expression of Japanese mathematics "Wasan" to expression of West mathematics. The second stage: Stage of "Translation" from original of West mathematics to Japanese. The third stage: Stage of Japanese "Textbook" compilation of original Japanese it. It is future tasks to consider details afterwards of received arithmetic and elementary algebra, and to consider the receipt of other fields of European mathematics to which light has not been applied yet.

View full abstract

The purpose of this article is to devise teaching materials using the lattice points in the space as a part of enrichment of the mathematical activity. The authors are considering that lessons using the lattice points have possibility to bear a cycle of the mathematics activity, which is typical to the plane lattice using the geoboards. In the case of the space lattice, there may be some obstacle to make and handle some good models for the space lattice, but the material is considered to be appropriate to promote the space recognition of students with enjoying the discrete thinking. In this article, we first show some characterization of the teaching materials relating to the lattice points. In particular, the materials using geoboards are discussed, and after Pick's theorem for the plane lattice figures is introduced, the impossibility of the direct generalization of the formula to the lattice polyhedra in the space is revealed. Although there are some formulas created by Reeve or Ehrhart, which can be regarded as generalizations of Pick's theorem, these formulas use some fractional lattices. We design several practical teaching materials relating to Pick's formulas using only lattice points for the lattice prisms or trees of lattice parallelepipeds. Through such teaching materials, we show some rich contents of the space lattice to enhance the space recognition of students.

View full abstract

The purpose of a series of our studies is to discuss more effective method of mathematical problem posing by using computer. We have reported in the previous studies such mathematical activities by university students who were the prospective teachers. In study (I), we practiced the problem posing by using computer after solving original problem, and through study (II) to study (IV) we examined various ways for the effective problem posing. The features of our method is to give students enough time to create problems, and 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 the same problem as study (I). In addition to the above features, we employ the method to devide the problem posing into two stages. That is, we provided situations in which students made problems after solving original problem without using computers at the first stage and made a mathematical inquiry about their problems by using computer at the second stage. From the activities, we examined how they made problems from original problem and how they made problems with mathematical inquiry by using computer, through the analysis of problems posed by the students. Results of analysis indicated that such activities enabled students to extend mathematical inquiry step by step. As in the previous studies, the practice shows some tendency that students who tackled making problem by using computer get some deep understanding for the mathematical properties related to the problems. Students solved problems to each other, and the solver and the student who posed a problem commented on problems each other. As a result, there was communication between students. We observed more positive learning attitude of students than the usual classes. It is asserted that the opportunity in a class to discuss problems posed by using computer is very important in such activity, in particular for the prospective teachers.

View full abstract

The purpose of this study is to propose a way for changing the current state of teaching and learning mathematics from memorizing knowledge and skills to cultivating students' mind, for example their interest or attitude toward mathematics in high school. In this paper, I suggest the so-called "open-activity" and develop cultivating concrete instances of it in high school mathematics. "Open-activity" is to work on an open problem that Pehkonen defines. Strictly speaking, it is a set of students' divergent and convergent acts. Students chose some documents, pose conditions for their starting situation and goal situation of an open problem, compose and solve their questions. I set three frames of "open-activity" based on this conception and developed three concrete instances; "filling circles problem", "the cross section problem of the tinplate" and "triangular five centers problem".

View full abstract