Learning activities begin from asking a question. By occurring the chain of the questions one after another in child's arithmetic learning, investigate activities such as consciousness, clarification, comprehension, internal assimilation, introspection, are done continuously. The purpose of this paper is to verify two points through turning around long-term arithmetic class practices from reflective viewpoint. First point is to clear the characteristics of arithmetic class practices which made child's question axios. Second point is to clear the subjects which it must get over. Arithmetic classes which made child's question axios is composed of the repetition of four learning activities, "(1) occurrence of child's question, (2) sharing as comon question, (3) search and apprication for the question, (4) occurrence of new question". Through analyzing Murakushi elementary school's long-term practice in hamamatsu, five characteristics become clear about arithmetic class which made child's question axios. a. Setup of the situation to raise child's questions by seeing through the whole of the unit. b. Teacher's attitude to clarify child's unsophisticated questions through adapting stage of development such as lower grades, middle grade and the upper grades. c. Indicating arrival target clearly of the lower grades, the middle grade and every upper grades about the standard of learning plan and self-evaluation in making child's question axios. d. Evocation of learning will by accepting the value of child's question into the unit. e. Learning activities which suggest introspection by comparing and contrasting questions at stage of introduction and end in unit. On the other hand, the subject which it must get over is as follows. ・Investigation about the learning curriculum for making use of child's arithmetic question effectively. ・Concrete investigation to identify arithmetical essence through the unit. ・Teacher's technical ability formation in teaching and learning which made child's question axios. ・Investigation how to cope with the child who is interested mainly in the way of making a question. We would like to disscuss and think about what kind of power do we raise, what do you aim at, and put value on what, in arithmetic learning in relation to these subjects.

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　数学の架空性は,数学が発展する上で,重要な役割を担っている。そのため,子どもが数学を学習するときに,理解できなくなる難所になることが多い。数学学習の過程において,どこでどのように架空性が生まれるのかを考察した。中学校1年生が「一次方程式」を学習する中で,「架空性」の問題が生じた事例を取り上げた。「創発モデル」の理論における「意味の連鎖」を用いて,それを分析した。その結果,「記母」が「記子」に結びつく過程,つまり記号化の過程の中で,架空性が生まれることがわかった。つまり,それぞれの記号化の中では,何らかの架空性が生じているのである。したがって,学習内容が現実的か架空的かは,本質的に個々人が,そのことを認識するかどうかに依存している。つまり,相対的なものである。また,架空性そのものが,必ずしも理解困難とは限らない。記号化が理解の助けとなることと同様に,数学的理解の助けになることもある。さらに,数学の学習において必ず現実的なモデルから始める必要はない。創発モデルは,個々の単元の中だけで,構想されるべきではない。数学の長期にわたる学習を全体論的に想定すべきである。

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The purpose of this paper is to construct 'a principle for the design of mathematical learning environment' based on the theory of situated learning and to design the environment based on this principle. First, the theory of situated learning is examined. Secondly, mathematical learning is captured based on the theory; and 'the principle for the design of mathematical learning environment' is constructed. The principle is as follows: DMLE 0. - Decide mathematical-collaborative activities as an aim and design the mathematical learning environment that includes activities, artifacts, and others (DMLE 1/2/3). DMLE 4. - Provide the students with an opportunity to actualize the mathematical-collaborative activities. Finally, a mathematical learning environment' - 'let' s order pizza' - that was designed based on the principle is presented.

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This article considers about the important examples of the cognitive load-obstacle arising in a process of mathematical generalization. In Section 2, I explain the summary of basic theory of mathematical generalization while giving four basic styles of mathematical generalization. In Section 3, at first, I list five basic cognitive load-obstacle arising in a process of generalization. Next, I give six important examples of the cognitive load-obstacle arising frequently in the situations of arithmetic/mathematics teaching-learning. Those examples are as follows. Example 1 is an example of the cognitive load-obstacle arising when substitute a number for a letter of a literal expression among many cognitive load-obstacle arising in a process of expansive generalization. Example 2 is an example of the cognitive load-obstacle arising when we force children to think of the meaning of an equivalent system needed when generalize number concept. Example 3 is an example of the cognitive load-obstacle arising in the process that generalize a=a to b=a or F≡F to G≡F. Example 4 is an example of the cognitive load-obstacle arising from cause that the mental image must tranfigure by generalization of figure concept, but it is not able to succeed. Example 5 is an example of the cognitive load-obstacle arising from cause that it is not able to notice that disjunctive generalization occur necessarily from expanding a set of the objects of consideration. Example 6 is an example of the cognitive load-obstacle arising from cause that generalization is impeded by specializing a figure I show some similar examples for each example. In Section 4, I point out about the problems of the actual teaching-leaning as follows. Children who can't overcome the cognitive load-obstacle, as described in Section 3, can't gain the mathematical concepts generalized. However, in no little daily teaching-learning, before children overcome the cognitive load-obstacle, many teacher give the general formula and the solution of problem for children or give many exercises to make children avoid these cognitive load-obstacle. I make these educational problems clear by the flow chart with two different learning route to gain a problem solving.

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The purpose of this paper is to investigate the actual conditions of teacher's listening on KIKANSHIDO. There are three forms of listening of teacher; the evaluative, interpretive and transformative listening of teachers, which I adapt for analysing the listening of teachers. (1) Evaluative Listening: If a teacher is listening in an evaluative manner they will characteristically have an evaluative stance. For a teacher, student's contributions are judged as either right or wrong. (2) Interpretive Listening: Interpretive listening is characterised by an awareness of the fallibility of the sense being made. If a teacher is listening in an interpretive manner they will characteristically have an active interpretive stance. (3) Transformative Listening: There is an attempt to interpret and make sense of what the speaker says, but always from the point of view of the listener. If a teacher is listening in a transformative manner they will characteristically have an open stance to the interrogation of assumptions they are making. There are three forms of mislistening of teacher; fallible, Unsuccessful and selective listening of teachers, which I adapt for analysing the mislistening of teachers. There are three roles of teacher's listening on KIKANSHIDO. (1) Teacher confirms the actual conditions of probrem posing. Teaher confirms if children grasp the meaning of problem. When they misunderstand the problem teacher explains the meaning of problem again. (2) Teacher evaluates children's responses. Teacher confirms how they solve the problem. When they have errors teacher instructs them. (3) Teacher makes a plan for the next instruction. Teacher attends to multiplicity, confrontation, fallibility of children's contributions. Teacher takes up the ideas to be confronted with and differentiate. There are three roles of teacher's listening on KIKANSHIDO: confirming the actual condition of problem posing, evaluating children's responses, making a plan for the next instruction. Teacher's listening on KIKANSHIDO play an inportant roles on mathematics instruction.

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In this paper, the theoretical frameworks of semiotic chaining and meta-representation are examined. Taking account of a Peircean model with three components, a representation of nested chaining, which consists of object, representamen, and interpretant, is identified. From Hirabayashi's (1987) implication, it is found that meta-representation can be either a method of learning mathematics or an explanation of the content of mathematics. Referring to the triadic model of chaining, both object and representamen are object-representation in Hirabayachi's terms, whereas the interpretant is a meta-representation. In this paper, a framework of reflexive writing is examined in terms of these constructs, and an example is analyzed using the nested (triadic) model of chaining. From this model, three types of note-taking are identified, and the importance of expressing an interpretant is emphasized.

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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 studies (VII), we investigated "looking-back" activity after problem solving to examine the nature of variable "UEPS/AEPS" in problem solving. In each study, we have focused on one variable in our framework to elaborate each variable or to clarify the arguments. In this article, two self-referential activities during a problem-solving process were analyzed using multiple variables as analytical points of view in order to verify the effectiveness and validity of the framework and to discuss pedagogical suggestion to problem solving instruction. By combination of variables, the descriptive power increased, and it became possible to capture some self-referential activities in detail. Especially, in the analysis, a certain pattern of occurrence of self-referential activity connected to success control was suggested. In regard to problem solving instruction, it was suggested that "Good-Judge" for the current/ongoing problem situation and going back to past successful problem-solving process/product (a kind of Previous/Current Goal Self-Referential Activity), and therefore such an intentional instruction that the above-mentioned self-referential activity was internalized were important.

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The inquiring similarity lesson is the school lesson which students construct mathematics by recognizing similarity. In this lesson, the teacher presents a mathematical entity in the first, then he (or she) presents the two entities similar and non-similar to it. At this stage, the teacher makes students inquire the similarity between the mathematical entity presented in the first and the entity presented as similar to it. After then, the teachcr presents a entity and makes students judge it to be similar or non-similar to the entity presented in the first. The teacher repeats this action. It can be said that the student constructed mathematics when he (or she) was able to recognize the similarity among the entities presented as similar in these process. In the first half of this paper, it has been clarified that it is effective in the inquiring similarity lesson to present the entities having the aliginable difference between the entities presented as similar and as non-similar. In this case, it must be taken into consideration that the significannt features for the construction of mathematics are not hidden under the ground. In the latter half of this paper, we divided the similarity required of students to recognize into the following four types and gave some means to facilitate the recognition of each type of similarity. ・objective similarity with perception ・objective similarity with no perception ・relational similarity with perception ・relational similarity with no perception

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This research is related to the group discussion method to promote the formation of the mathematical concept. Especially, this research applies the focus to the analysis and consideration of the lesson by the model of a mathematical concept as an empirical research. And the characteristics and validity of the lesson that it is conscious of the relations are examined. The main contents of this research are as follows. (1) About the group discussion method to promote the formation of the mathematical concept, it is examined from the point of view of the sociality and the relations, so the role of the teacher in the concept formation and the meaning of the group discussion method were discussed. (2) The frame of evaluation problem of the relations of the mathematical concept is built, we enforced the analysis of the relations of the mathematical concept of the child. (3) We enforced the experimental lesson by the model of the formation of the mathematical concept, we investigated an influence on the child by this lesson. (4) We examined the matter that this lesson promoted a solution of a problem of not only the contents being learned at present but also the contents which will be learned in the future. (5) We examined the matter that this lesson improved a solution of a problem of the child of the group of middle and low ranks.

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Vygotsky categorized concepts into two types: everyday concepts and scientific concepts (the latter is called mathematical concepts in this paper). Although Vygotsky pointed out that these concepts are interrelated in concept formations, he did not mention how these concepts themselves are formed. Yoshida (2000, 2004), therefore, posed a concept formation process using the idea of "sublation" through the following three stages: everyday and mathematical concepts 1) conflict with each other, 2) are lifted to higher levels, and 3) are preserved as a unified concept, or sublated concepts. Based on this model of process, this paper aims at clarifying how children's everyday concepts prepare the ground for fraction concepts before they encounter fraction concepts in systematized lessons at school. Following the interview survey for 21 second graders and the questionnaire survey for 39 third graders in Japan in 2001, this paper compares the questionnaire survey for 23 fourth graders in the U.S. in 2004 with the Japanese ones. The main findings in this paper are as follows. Broadly speaking, the questionnaire data on the everyday concepts of fractions from American children are almost identical to those of Japanese children, although there are small differences in detail (e.g. the variety of answers by American children). The use of "everyday concepts of fractions" is readily visible in these data. For instance, responses such as "half is something less than a whole," or "half is something divided evenly into some parts (not in two)" show that the concept of half remaining ambiguous. The "structure of fractions" composes of (A) fractions as quantity (the object of the fractions is quantity), (B) fractions as ratio and (C) fractions as operation (the object of these fractions is the relation between quantity and quantity) , and (D) fractions as number (the object of the fractions is number) (Yoshida, 2002a). This "structure of fractions" was derived theoretically, and is shown in Figure I. Finally, relating the results of the surveys with the "structure of fractions," two fundamental principles (P1 and P2) emerge, which run through the "structure of fractions" (cf. Figure2). P1 is a principle of equality, in which equality of size in fractions is in common to all kinds of fractions. P2 is a principle of comparing and relating two quantities or two numbers, or more specifically, a principle of relating with ONE-whole. For example, you can describe the "quantity" of juice in a cup as "1/3 of a cup" (fraction (A)) relating the amount of juice as a part with the cup as ONE-whole. You can compare and describe the "relation" between Sylvia's and Daniel's oranges as Sylvia's orange is "1/4 of Daniel's" (fraction (B)). While children have to regard the quantity of Daniel's oranges as ONE-whole, some children describe that "Daniel likes oranges more than Sylvia." Furthermore, a "number 1/5" is positioned in a number line relating with ONE-whole, or 1. Although the principles reflect essential aspects of fractions as mathematical concepts, children do not become aware of these in their everyday life as demonstrated in the surveys.

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　　In this paper, I have analyzed three examinations of procedural knowledge on "Velocity Context", "Duration Context", and "Distance Context". "Velocity Context" is judged by duration and distance, "Duration Context" is judged by distance and velocity, and "Distance Context" is judged by velocity and duration. The subjects are 227 pupils of 5 elementary schools in Kobe and their grades are from four to six.
　　I have conducted Guttman's scalogram analysis (White, B. W. & Saltz, E., 1957, pp.81-87) to three examinations, and validity of them is statistically confirmed by CR, MMR, and PPR. Also reliability of them is evidenced in terms of coefficient of reliability by KR-20.
　　As results of other analyses, I have clarified the following contents.
(1) There is a weak correlation between "Velocity Context" and "Duration Context", but not much between "Duration Context" and "Distance Context", or "Distance Context" and "Velocity Context".
(2) The result of F-test yielded significant value. : F(2, 225)=5856.883, p < .05.
(3) The result of multiple comparison tests of Ryan's procedure is the following for Table 1.
　(1) For (Z_i-Y_i), the critical value is t_<1.67,226>=2.406, so t≒7.158 (p < .05).
　(2) For (X_i-Y_i) and (Z_i-X_i), the critical value is t_<3.33,226>=2.134, so t=2.913 (p < .05).
　　　So each absolute value on Table 1 has yielded significant value.
　　　According to this result, it is said that chilren are easy to understand in order of "Distance Context", "Velocity Context", and "Duration Context".
(4) According to order of (3), Children have started from X_0Y_0 Z_0, passed through X_1Y_1Z_1, and got to X_2Y_2Z_2 as Figure 1 shows. In Figure 1, X, Y, and Z are "Contexts" like Table 1, and numbers of 0, 1, and 2 on X_1Y_0Z_2 et cetera show levels in respective "Contexts". Also numbers within round brackets are persons.

Table 1 Absolute Values of Subtraction among Mean Values on three "Contexts"

Figure 1 Deveropment of Procedural Knowledge on Velocity

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　　In "Ratio," I have investigated aspects of emotions, conceputual knowledge, procedual knowledge, and school lessons. I make some investigations of "Ratio" and clarify processes for conceptualizatin of "Ratio" in children, I hope.
　　In this paper, I have mentioned the investigation for comparison of two objects from the viewpoint of consideration to aspects of conceputual knowledge in "Ratio." The investigation was carried out in the third term. The subjects are 205 pupils of two elementary schools in Kobe and their grades are five and six.
　　Reliability of the investigation was guaranteed by stability of it, Cronbach's a coefficient, GP analysis, and x^2-test.
　　Validity of the investigation was guaranteed by D index and differentiation of statistical category.
　　By the way, I have set up the following 4 hypotheses and examined to verify them.
Hypothesis (1)
　　Maybe the highlier subjects of Control Group (N=167) score, the highlier they compare in some standards.
　　By the right table 1, statistical categories of subjects and their answer patterns verified "Hypothesis (1)"

Table 1. Answer patterns of Question 1〜6 for CG of each Category

　　Hypothesis (2)
　　Maybe there don't exist meaningful differences between 5th grade pupils and 6th grade pupils (i.e. Control Group (N=167)) who don't take lessons in "Ratio" which I have proposed.
Hypothesis (3)
　　Maybe there exist meaningful differences between 5th grade pupils (i.e. Experimental Group (N=38)) who take lessons in "Ratio" which I have proposed and 5th grade pupils and 6th grade pupils (i.e. Control Group (N=167)) who don't take ones.
　　As the result of the investigation on Control Group (N=167), F-test and t-test verified "Hypothesis (2)" (i.e. F (100,67)≒2.005 > F (100,65), t≒1.896 < 1.98 (p≦.05)).
　　As the result of the investigation on Experimental Group (N=38) and Control Group (N=167), F-test and t-test verified "Hypothesis (3)" (i.e. F (37,166)≒2.170 > F (30,160), t≒5.635 * (* p≦.05)).
Hypothesis (4)
　　Maybe there exist obviously differences for comparison by "Ratio" between Experimental Group (N=38) and Control Group (N=167).
　　As the result of the investigation, the following table 2 verified "Hypothesis (3)."

Table 2. The percentage of comparison by "Ratio" between Experimental Group (N=38) and Control Group (N=167)

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The continuous research on mathematical attainment is a part of the International Project on Mathematical Attainment (IPMA) in which such countries as Brazil, Czech Republic, England, Hungary, The Netherlands, Ireland, Japan, Poland, Russia, Singapore and USA are participated. The aim of this project is to monitor the mathematical progress of children from the first year of compulsory schooling throughout primary school and to study the various factors which affect that progress, with the ultimate aim of making recommendations at an international level for good practice in the teaching and learning of mathematics. In Japan, the total of eight different public primary schools have agreed to participate in the project. We asked all two-cohort children and their classroom teachers from these schools to be involved and to take mathematical attainment tests for six years. At the present we have carried out six tests, i.e. Test 1, Test 2, Test 3, Test 4, Test 5 and Test 6 to about 300 children of first cohort for five years. The purpose of this paper is to analyze the indices for evaluating the progress of mathematical attainment of first cohort children at middle grades. As a result of analysis in terms of some indices, we found out the following: ・Some indices including Value-added Scores can be adopted for evaluating the progress of mathematical attainment between successive tests. ・We could grasp the evidence of the progress by introducing the indices for 329 children in thirteen classes and comparing the average scores of the classes. ・It is highly possible that we can point out the true improvement of children, if we have some indices from the continuous research on mathematical attainment such as IPMA.

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In this paper we describe the development of the creativity power and the creativity attitude in mathematics learning for junior high school student's grade 1 to grade 3. For the creativity power measurement, we used a creativity test of "The Number and Formula" and "The Figure and Congruence" used the test score. For creativity attitude measurement, we used a Creativity Attitude Scale (CAS) that was developed by Saito N. We examined the development of the creativity test score and the creativity attitude scale score, and the causal relation between the creativity test score and the creativity attitude scale score for "The Number and Formula" and "The Figure and Congruence". The results are as follows: - The students who get higher score of the fluency get higher score of the diffusion for junior high school student's grade 1 to grade 3. - The students who get higher score of the flexibility get higher score of the originality for junior high school student's grade 1 to grade 3. - The creativity attitude scale score hardly develops for junior high school student's grade 1 to grade 3.

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This paper proposes the development of a teaching-evaluation system to deepen students' understanding of the basic contents in mathematics. This system was tested with two teachers; T_1 and T_2. Group A is the class which is introduced to the teaching-evaluation system and is instructed by teacher T_1 and T_2. Group B is the ordinary class which is not introduced that system and is taught by teacher T_1. Teacher T_1 gives lesson instruction and homework to students in group A. Another teacher T_2 gathers homework of group A class and marks, and gives teacher T_1 feedback on the results of the homework. Teacher T_2 also guides each students in group A who got low marks. Teacher T_1 infuses the feedback information in his teaching. We examined differences of the results between group A and group B. The results for the unit "The Linear Function" which students learn in junior high school grade 2 are as follows; - The average mark of the test at the end of the unit, group A got 14.7% higher than that of group B. - The degree of connection among learning items, group A got 39.3% and group B got 20.9%. The average mark of group A is about twice of that of group B. - For the connection among the whole learning items of the unit, group A students were able to make knowledge network in their mind. But group B students could not. These results shows that the teaching-evaluation system is effective in deepening students' understanding of the basic contents in mathematics. Moreover this system has following advantages; - The class evaluation can be performed objectively by the teacher T_2. - The teacher can guide timely those students who need assistance. Therefore the students' achievement will be restored.

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The purpose of this research is to clarify the way of the treatment of Sakumon in "life-centered arithmetic" in the early part of the Showa era. We pay attention to the arithmetic education on which Iwashita, Fujiwara and Inatsugu insist. They play a central role in the practice of "life-centered arithmetic". Through consideration, we find the following facts. (1) The chief aim of arithmetic education by Iwashita is to develop the qualitative thinking. To realize this aim, he introduces the practice to develop the qualitative thinking in daily life. For instance, gathering the qualitative facts, measurement are examples of this activity. Children pose the problem by the use of these facts or the result of measurement. Sakumon is a part of the practice to develop childeren's qualitative attitude in daily life. (2) Fujiwara restricts the position of Sakumon in his arithmetic education with the reflections that arithmetic education based on Sakumon is not able to preparete the curriculum. He changes Sakumon's position into one of methods of teaching arithmetic, namely the representation of the qualitative life. (3) Inatsugu attemptes to accord logicism with psychologism in arithmetic education. Generalization and specialization of mathematical thinking are the scaffold to accord them. Sakumon becomes the teaching and learning method to cultivate specialization. Simizu's study of curriculum development based on Sakumon has an attraction for him.

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Before the promulgation of "Ren Yin education system" (1902) and "Gui Mao education system" (1904) in China, mathematics was taught by traditional methodology of "teaching and memorizing scriptures" in private schools in most area. During the implementation of new education system for modernization, "five stages teaching", which originated from J. F. Herbart, was introduced and applied into primary arithmetic education in China through Japan. This paper examines influence of Japan upon mathematics education in modern China, especially about teaching methodology. The following points are clarified through this study. (i) The introduction of teaching methodology from Japan was done through translations from Japanese books which was carried on the first academic journal about education in China, 'Educational World", and some others volumes which was compiled by Chinese students studying in Japan, such as 'Koubun Academy'. (ii) The introduction of arithmetic education in Japan, which was represented by 'Points and Teaching Methodology of Arithmetic' of Rikitaro Fujisawa, brought formal discipline to develop one's thinking in objectives, as well as practical aim, in arithmetic education in China. (iii) The teaching methodology of the Herbart, which was shown in Japanese books such as 'Teaching Methodology for Primary Subjects', was introduced and diffused as a modern teaching methodology instead of traditional teaching of "teaching and memorizing scriptures", which was conducted in private schools before establishment of modern formal education.

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This research aims to compare and analyze the educational activities of Mathematics in primary schools between Paraguay and Japan in order to assess the Paraguayan educational activities. For this purpose, the following methodology was employed: (a) comparing objectives and contents of the curriculum of Mathematics and the textbooks, (b) field work in some schools of both countries, (c) questionnaire to teachers of both countries, and (d) recording and analysis of videotapes of lessons in both countries. The relevant conclusions are as follows: 1. In the Paraguayan case, there are discrepancies between the objectives of the educative reform and the intended curriculum contrary to the Japanese case where they are well-related. 2. In Paraguay, there are differences between teachers' objectives and the curriculum developed, since they try to promote the logical thinking and its application in their surroundings by means of passive and mechanical learning of students. Contrary to Japan where teachers promote active learning and encourage discovery. 3. The evaluation criteria of lessons in Paraguay are based on the interest and the discipline of students, being frequently a positive evaluation and thus, the necessity for improvement is not observed. On the contrary, evaluation in Japan is not only based on students' achievement but also on evaluations of the lessons' progress and the teachers' observation on each student. Frequently, the evaluation is negative and teachers pick up the points that require improvement.

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　本研究は,ガーナにおける現職教員夜間研修プログラムが基礎学校教員の教科知識に与えた影響について,事例研究を行うことを目的としている。また同時に,教師がこの研修への参加動機を,自らを高めることとしていることについても確認しようというものである。この研修プログラムは,教員の質やその不足の問題を解決することが目的で,1998年に始められた。本プログラムの良い点は,研修を受講する教師が教壇に立ちながら,同時に研修も受けることができるということである。したがって教師は研修の中で新しく学習したことを,授業の中ですぐさま活用する事ができるのである。ところがこの研修プログラムの効果について,未だ研究がなされておらず,より良い研修を求めていく上で,まだ為すべきことは多数存在する。
　そこで本研究では,ガーナ国中央州における現地調査を実施した。質問紙調査において,研修受講者58名,研修未受講者54名,校長40名,指導主事20名の有効回答を得た。さらに研修受講者の内6名が選出され,さらに6名の未受講者と合わせて計12名の教師の授業が観察され,さらにその内の受講者1名と未受講者3名より授業についてコメントを得た。このようにして収集されたデータに対して,平均や標準偏差などの記述統計による量的分析と,コメントの内容分析を行った。これらを通じて本事例では,この研修プログラムにおいて基礎学校教師の数学教科知識が高まったが,他方で未だに分数を指導する上での問題が存在していることが分かった。

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In this paper, we discuss the way of calculations of Σ^^n__<k=1>k^p in Wasan from the view point of mathematical activities. Wasan is the mathematics which was studied until the Edo period in Japan. In Wasan, calculations were based on the number of bamboos in a bunch, as described in "kuchizusami in Japanese" which is one of the oldest mathematical books in Japan. We have considered the calculation of Σ^^n__<k=1>k^p which is called "houda in Japanese". We introduce "shyousahou in Japanese", elementary methods by Yasumoto Okayu (1794〜1862) who was a mathematician in the Edo period, and the method using diagrams and "suida in Japanese" which is a very important formula. And from the view point of mathematical activities we introduce the way of teaching calculations of Σ^^n__<k=1>k^p in the high school mathematics.

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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 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 Ecology. And another feature is to give students enough time to make conjectures on results and get the numerical calculation by using computer to solve the mathematical models. Actually, in the practice, some students had difficulty in making and solving mathematical models by using computer, but gradually felt interested in considering about some phenomenon, which might not be made without using computer. 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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The following is in the exercises of mathematics for high-school students. Ex.1. As the real number t changes, illustrate the range which contains the straight lines y=tx+2t^2-1. Ex.2. As the real number t changes in the region -1≦t≦1, illustrate the range which contains the straight lines y=tx+2t^2-1. Ex.3. As the real number t changes in the region 0≦t≦1, illustrate the range which contains the straight lines y=3(t^2-1)x-2t^3. These exercises are difficult for high-school students. In order to let them understand a graphical meaning, it is necessary to consider the concept of the envelope for a family of straight lines. In this paper, we will try to persuade high-school students that each line of a family of straight lines is the tangent of the boundary of the domain obtained by the discriminant D≧0. First, we explain that the straight lines y=tx+2t^2-1 in Ex.1 have the contact point of the boundary of the domain obtained by the discriminant D≧0. Next, by using this idea, we explain how to find the domain which contains the straight lines in Ex.2. As for Ex.3, they can not use the discriminant D, so we need to explain the method by using the envelope and another method. Finally, we report the result that we actually made our students solve these exercises in mathematics classes of Kure College of Technology.

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