The purpose of this study is to suggest the contact between drawing mathematically and its authenticity for considering the cue of expressive changing which will be expanded in the future.

　　 According to previous studies, the results of examining the relationship among the mind, the understanding, and the representation suggested that learners’ representations are the results of thinking and learners rethink with them.  Through such trial and error processes, “the mutual-beneficial relationship” which means thinking representations support mutually is suggested.  Moreover, better representations are as the results of trial and error processes.  This study suggests the powers of broad mathematical representation which can practice and utilize linguistic and symbolic expressions actively, properly, and flexibly with using graphical representations as the present question of mathematical education.  With considerations of previous studies, this study proposes the following three suggestions about the contact between drawing mathematically and its authenticity.  The first one is the teaching about teachers’ conscious drawing.  The second one is that the transformation of learners’ schema and cognitive aspect through trial and error processes is the contact.  The third one is that learners’ motivation of drawing actively is fostered by experiencing its authenticity through trial and error processes.  

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　　 The purpose of this article is to construct a framework for analyzing various types of representations used in mathematics classes, which is based on Nakahara’s study on the representational system (Nakahara, 1995). In our framework, “manipulative representations” in Nakahara’s representational system is reconceputualized as “manipulative/embodied representations” to include “verbal natural language” and “gesture”. And, the framework is constructed by combining Nakahara’s five extended modes of representations (symbolic, linguistic, illustrative, manipulative/embodied, realistic) with three types of representations (formal, preformal, informal) in iceberg model (Webb et al., 2008) in two-dimensional array.

　　 With our framework, various types of informal representations used in mathematics classes are extensively exemplified, and an excellent classroom practice that informal representations seem to be effectively used (Masaki, 2009) is also analyzed.

　　 As a result, using the framework, some informal representations corresponding to each cell of the framework are identified, and the potentiality to describe mathematics classes using informal representation is illustrated.

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In learning of mathematics, writing mathematics and speaking mathematics are very important in the process of solving the problems. Because students ought to comprehend of the learning contents through activities of writing mathematics and speaking mathematics. This study deals with the following: 1) The development of a new index to measure the change between the power of writing and the power of speaking in the process of solving the mathematics problems. 2) The relation between the power of writing and the power of speaking in the process of solving mathematics problems. We found a new index which was divided the value of the slope of a regression line by using the cumulative marks of the tests by allotment of marks of the tests. We call this new index the coefficient of variation of the cumulative marks. We experimented on the power of writing mathematics and the power of speaking mathematics for the junior high shcool students grade 2. As a result, the following became clear: 1) The value of the correlation coefficient between the marks of writing mathematics and the marks of speaking mathematics is very large. 2) Generally, the power of writing mathematics is higher than the power of speaking mathematics.

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物理がわからない原因として，学習者の基本的なのスキル不足を挙げる教員は 少なくない。それではにおける数式に習熟していれば，物理で学習する数式が理解で きるかと言われれば，必ずしもそうではない。それは，物理における数式とにおける 数式とは異なるからである。物理はを工夫し，これを 活用して自然現象の観察・実験から物理理論を構築してきた。しかしながら多くの教員 は，それゆえに初学者が直面する概念的困難に対して無自覚である。この小論では，物理 独特の数式に焦点を当て，初等物理教育における対応について考える。

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教育がその指導対象としている知識(的知識)の本性を明らかにすることは,教育学の根本問題の一つである。筆者は,的知識の特性を一つずつ拾い集め,それを精査していくという作業を積み重ねるにことで,的知識の本性の理解に近づいて行きたいと考えている。本稿では,知識内容とその方法の関係に焦点を当てることで,的知識の重要な特性として,以下の2点を指摘した。(ア)知識内容とその方法は不可分の関係にあり,の変更は不可避的に知識内容の変容を伴う。したがって,方法の置き換えによる指導教材の平易化は,いつでも行えるわけではない。(イ)的知識の発展を促す。この特性は,学習の系統性,階層性に反映されることになるので,教育の観点からも重要となる特性である。

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は一つの「言語」である。とりわけ数字や文字を用いた式(数式・文字式)には,日本語と同じように,特有の文法がある。重要なことは,文法的に正しい式の的意味を理解することであり,具体的な状況に応じてその式を適切に用いることができるということである。こうした認識のもとに,数字や文字を用いた式(数式・文字式)に図,表,グラフ及び用語を含めたものを「言語」ととらえ,することのできる力としての「言語力」の育成について考察した。

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　　As for the study on power of expression, many studies has been made in mathematics education in former studies.　If these previous works are taken into consideration, the power of thinking, transmitting and reading a person’s thought should be realized to be“power of expression”in the future of the mathematics education. However, the framework which can grasp correctly about children’s“power of mathematical expression”has hardly been studied from rationale about the training method of“power of expression” .

　　Moreover, the way children’s“power of mathematical expression”is correctly grasped by researcher is one of the important issues.

　　Therefore, in this paper, the framework which can grasp correctly about children’s“power of mathematical expression”and can interpret a type of“power of mathematical expression”from children’s activity should be built from a theoretical aspect.　Moreover, I presume to explore about type of“mathematical power of expression”which children are holding by using the theoretical framework built in this paper.

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本論はキリスト教や仏教といった世界宗教をし解析する試みである。宗教は、その哲学的なである存在論あるいは形而上学と同様に、科学と同じ意味における検証可能性は持ちえないが、論理的一貫性を問うことは出来る。もまた検証可能性は持ちえないが論理的一貫性は問われねばならない。そこに宗教分けても存在論として哲学的にし解析する可能性が開けて来る。本論はまず第一節において、キリスト教や仏教といった世界宗教を哲学的にしうる存在論を提示し、続いて第二節において、その存在論によるキリスト教と仏教の哲学的なを試みる。これを受け第三節において、宗教をである集合論の基本的な結果を整理し、最後に第四節において、その集合論による宗教のと解析を試みる。

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算数・

において，多様なを活用することは，古くからのストラテジーとして問題解決の糸口になることが広く知られている．そのためには，あるに表し変えることが有用である．すなわち，中原(1995)の方法の変換が，何をきっかけとして起きているのかを検討することは価値がある．そこで，今後，実際の授業場面のデータを集め，どのようなときにあるに表し変え，どのようなときに表し変えていないのか，データの傾向から検討したい．本稿は，先行研究をもとに，どのようにしてをみとり，どのような方法でデータを集めることができるか,その方法を検討した．

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離散の一分野であり，情報科学の基礎理論として最も重要な位置を占めている．地質学では連続な空間や時間を地質体や地質年代のように離散化して取り扱うことが多い．空間や時間を離散的に把握し，それらの相互の関係を考えるための的道具として，離散の考え方が有効である．本講座では，野外調査データから地質構造を推定する3次元地質モデリングに深く関連する部分に限定して，離散の有効性と必要性を解説する．主な考察の対象は，(1)地層の分布域と境界面の関係（地質構造の論理モデル），(2) 地質構造の論理モデルから地層の分布を定める関数，(3)地層の対比や区分を同値関係として・解析する手続き，(4)地層の形成順序を半順序による順序づけとして・解析する手続き，(5)野外調査データから地下の地質構造を3次元モデルとして確定するまでの手続きである．本講座を通じて，3次元地質モデリングにおける離散の重要性についての認識が深まることを期待する．

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はじめに

電気関連分野では，「

」は物理現象をするための重要な道具である。実際，大学や高等専門学校において，専門科目のより深い理解を得るためには，は必要不可欠

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　　 The purposes of this study involved in mathematical representation in mind are following two.  The first one is to consider the reality on mathematical education and second one is to suggest refining its reality.

　　 From previous studies about the reality on mathematical education, this study considers that the definitions of reality are to actualize the existence of learners and to win the sympathy of learners with considering both substantiality and humanity.  Then, this study suggests following three statements for refining the reality. 

(1) This statement considers the meaning of the value of reality.  The reality that stands on its own merits, but actually it is the basement that produces the new values.  Therefore, the reality is not only to be actualized  the existence of learners as Kikuchi mentions.  Rather it actualizes the existence of learners.  In short, feeling the reality actualizes learners’ independence. 

(2) This statement considers the features on reality.  The reality on mathematical education is influenced by the context of learning in classrooms, the contents on learning, the existences of others, and sociality.

(3) This statement considers one of the basements of reality.  The reality is supported by the tacitness and it activates mathematical activities, provides broader possibilities, and facilitates the production to new values.

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The main purpose of this study is to reflect upon the teaching of geometry from a viewpoint of visualization. If the meanings and the roles of visualization in the teaching of geometry would be recognized, then the characteristics of the teaching of geometry in elementary mathematics are also to be clarified. There have been many researchers who investigated the characteristics of visual recognition on the concept of geometry: Pimm, Bishop, van Hiele, Hoffer, Triadafillidis, Vinner, Fischbein and so on. But their works were restricted within narrow specific regions that were to regard the visual recognition as an initial stage in the developmental process of the concept of geometry or as a nuisance in the teaching of geometry. I insist on the need to recognize the important roles of visual recognition and to make the best use of it positively. From reference to the researches about visualization, the functions of visualization in the teaching of geometry can be prescribed as follows. Functions of Visualization ◎Visual Interpreting Function: Understanding, reading and interpreting the visual representations and spatial vocabulary used in geometrical works, graphs, diagrams of all types. (It also involves meanings of figural representation) ◎Visual Processing Function: Visualizing and translating abstract relationships and nonfigural information into visual terms. It also includes the manipulation and transformation of visual representation and visual imagery. ○Translating Function ・translation between different types of representations ・dimensional change from 3D to 2D, from 2D to 3D and so on ○Mental manipulating Function ・transformation (geometrical transformations) ・figural change (partition and composition) A keen interest has been shown only into the visual interpreting function because of the progress of information processing researches about a way of looking at figures and developmental process of the concept of geometry. But the visual processing function has not so interested in. Therefore I investigated the characteristics of the visual processing function both in problem solving situation and in reasoning process. In problem solving situation, the translating function from linguistic representation to figural representation and the mental manipulating function were recognized. For example in "rectangle problem", mental manipulation of partial figure fills crucial role. From the result of questionnaire I identified the weakness of visual strategies and the necessity of visualization in problem solving situation. In reasoning process, there exist three types of cognitive stages: a purely configural stage (visualization), a natural discursive stage, and a theoretical discursive stage. Visualization embodies an initial stage of reasoning process that is independent from other stages and indispensable for next discursive stages. Therefore it cannot to be lacked in reasoning process. What is significant on the teaching of geometry in elementary mathematics from a viewpoint of visualization is to become conscious of the mental manipulating function which would be a key function as a strategy of problem solving and as a basis of reasoning process.

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船体の運動状態を重視するミクロな航行シミュレーションを実施する場合の船体運動のモデルを概説する。モデルとしては,船体に働く流体力を直接的にする流力モデルと,船体運動を操作量に対する応答方程式としてする応答モデルがある。

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"Journal Writing" is one of the methods to use writing in mathematics classes, and it has two functions; "(1) Expressive function", and "(2) Transactional function". In this paper, "(2) Transactional function" is focused on, and the case described in Ninomiya (1996 (c)) is investigated. First of all, "the framework for the investigation of Transactional Writing", which has four stages; impression, summary(1), summary(2), consideration, is established from the framework of Nakamura (1989). Since the writing activities are taken place during the second term of 1996, the difference of the paper tests' results between the first and the second terms should be considered as to be caused by the writing activities. Every student is over the national average during the second term while more than one-fourth of the students were below national average during the first term. Furthermore, the students who are not good at math improves their score a lot, which means the advantages of writing activities work very effectively. It also means that writing activities makes "the education for every student" policy possible. Writings which have written by students are analysed in two ways; by the class teacher and from "the framework for the investigation of Transactional Writing". The correlation of these two analysis' results is 0.846. Since the analysis by class teacher is reliable, the reliability of the framework is suggested. The correlation between writing and mathematical ability is also high. This means that mathematical writing is defferent from writings in language classes. Moreover, mathematical writing ability is correlated with higher-order abilities such as an attitude or way of thinking, a lot. From such outcome, mathematical writing ability should be considered as one of the higher-order abilities, which is stressed in the current curriculum for school mathematics.

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

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　　So far, the author has suggested the viewpoint of writing positively, focusing on the characteristics of written language.　But he hasn’t considered how to shift “drawn” language to written language, or the viewpoint of the instruction which introduces students to use figures positively.　Therefore, together with examining the preceding researches, the author reconsidered the difficulties again which written language contains, by returning to the original text of the researches.　

　　As a result, the author developed his idea to the degree of“difficulties of drawing”which contains even the difficulties of positive use of figure expressions such as (1) the knowledge of drawing (2) the use and activity of drawing (3) the cost of drawing (4) the viewpoint of learning which drawing has.　

　　And students, through solving problems, by learning together, conquer the difficulties of drawing.　At the first stage, teachers show students drawing as knowledge.　At the second stage, teachers have students gain how to use and utilize drawing through the process of solving problems by both teachers and students.　At the third stage, instructors, helping in a casual manner, have students use and activate drawing and solve problems cooperatively.　

　　On the other hand, the author, at each stage, incorporate the method to have students draw and the viewpoint to overcome the difficulties of drawing.　By this process, together with the understanding of tactics of drawing, the author aims to have students use, utilize and get accustomed to drawing.　In other words, the author, making intentionally the situations for experience, has students gain the utility of drawing and how to activate drawing.　

　　Besides, changing the teachers’ qualities of instructions about drawing and repeating this process, the author shift drawn languages to written language, positive activities of various kinds of expressions.　

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