統計を支配しているくみひも群(braid group)を考察することにより, 特に自明でない位相的構造を持った曲面上における統計粒子系の構造を解説する. くみひも群の生成子の満たす関係式を統計粒子の量子力学的な性質との関係で議論する. その結果, 統計粒子系の統計にある制限がつき, また曲面の穴を貫く磁束に起因するアハラノフ・ボーム効果にある特徴があらわれる. 統計の系の基本的な構造を考察することにより, 量子ホール効果における準粒子に対するミクロな理論のいくつかの結論と統計粒子の位相的構造を反映するくみひも群による考察とが無矛盾であることがわかる. またくみひも群の表現論からの要請を満たすように統計粒子系を格子上で構成し格子上での具体的な数値計算の結果を示す.

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The study of fractions in elementary school begins in the second grade and continues every year through the sixth grade. However, there is an interval of about a year before they learn fractions in the following school year, and there are concerns about entrenching and forgetting what they have learned. We conducted a survey on the recognition of unit fractions among fourth-grade children, about one year after learning fractions in the third grade, in order to identify learning characteristics in the third grade of children who were able to identify some of the fractions based on unit fractions, and to obtain suggestions for learning guidance. Only children who could write following the objectives, children who only added the numerator, and children who were biased toward the division operation were included in the actual analysis. We attempted to document how the children in their third year learned fractions. This implies that beginning in the second semester of third grade, the viewpoint of relying on division begins to shift to the viewpoint of unit fractions, and beginning in the fourth semester, the viewpoint converges to the viewpoint of unit fractions.

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It is well known that many children confuse the quantitative aspect of fractions with the operational aspect of them. In this paper, three instructions for fourth and fifth graders were investigated and analyzed. The purpose of the instructions was to separate the aspects and to (re)construct the quantitative aspect of fractions. In one of the classes of fourth grade, though some children asserted that one half meter was the half of a meter, many children insisted that the half of ten meter was called not only five meter but also one half meter, so one half plus one half was one but one half meter plus one half meter was not always one meter. Since the discussion was continued without reaching an agreement, the teacher instructed the difference between one half and one half meter. In the class of five grade, children could reach smoothly an agreement of what one half meter was. The results of investigation carried on after the first instruction, however, showed that the figurative expression of liquid with using fractions was relatively poor. The confusion of the aspects of fractions is stable and it seems not possible for all children to separate them clearly within one hour instruction. Since four fundamental rules of arithmetic on fractions is taught with using the model of quantity, we must encourage children to separate them at every opportunity.

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第3学年で学習する量

では，その時々に応じて1とみるものが変わるといった経験をすることとなる．これは，それまでの自然数の学習では出会わなかった見方であるため，子どもにとっても容易なことではなく，見方を育成するうえでの手立てが求められるといえる．そこで本稿では，単位量を強調した授業展開が有効であるとの視座から，単位量を強調した実験授業を計画，実施し，その授業でみられた子どもの概念形成過程を詳細に捉えることを試みた．本稿で分析対象とした主な授業は，第4時，第6時，第7時の3時間分である．その分析結果からは，単位量を強調することで何を1とみなければならないのかということや，2m問題であっても1mをもとにしてみなければならないといった見方が働く様子が確認された．また，ポストテストの結果からは，単位量の見方を強調した学級Aが他の二つの学級よりも単位量の見方が促進される様子が明らかとなった．

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階微積分とは，従来の整数階で表された微積分を非整数階へ拡張した作用素であり，階微積分を用いることでリチウムイオン電池のインピーダンス特性を少ないパラメータで正確に表現できることが知られている．本研究では階微積分で表されるシステム（階システム）を対象とした適応観測器をリチウムイオン電池に用いることで，電池の状態推定を行えることを示す．

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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 Japan, fraction is fought of the 2nd grade and more over.  It is one of the important subject in mathematics education research.  However few precedent literature and studies in Japan about this subject develop curriculum based on any theoretical framework.

　　 Hence, I review and critique The Rational Number Project (RNP), especially their theoretical framework for teaching and learning of the rational number.

　　 And I propose a few agenda for the RNP framework as follows;

(1) How the “mathematical variability” can be ordered?;

(2) How the “perceptual variability” can be ordered and what any other “manipulative aids” can be?;

(3) What kind of teaching and learning can be considered in the any cell of “Matrix”?; and (4) How “Translation system” can be used for the teaching and learning process? 

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次期学習指導要領では，

の乗除に関する内容が移行となる．本稿では，÷の算数科教科書における，面積図と数直線の複合図に着目する．複合図は，除数が単位である場合の2つに大別できる．例として，G社とKe社に掲載されている複合図の作成過程について整理する．これにより，複合図の作成過程において働く数学的な見方・考え方の特定に向けた考察を行う．

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It is considered that fraction concepts are difficult to teach/learn in Japan's elementary schools. There are a lot of factors including the following: ・A variety of the meanings, the complexity of the meanings, and the complicated notations for fractions (Ishida, 1985) ・The continual argument about introductory lessons for fractions ・The appearance of new meanings for fractions in the third grader's mathematics textbooks and some theories In addition, children's everyday concepts of fractions have been taken into little consideration in teaching/learning fraction concepts. In other words, the following have not yet been explained. 1) What everyday concepts of fractions do children have, before they begin learning fractions at school? 2) How are children's fraction concepts formed through fraction lessons, based on their everyday concepts of fractions? 3) What teaching/learning are suitable for fraction lessons when we consider children's everyday concepts of fractions? This paper answers the first problem. That is, the purpose of this study was to clarify children's everyday concepts of fractions. Here, everyday concepts, and mathematics concepts corresponding to the scientific concepts are defined by Vygotsky's theory. Three investigations were conducted to clarify children's everyday concepts. In the first investigation, six third grade children were interviewed to collect information for a questionnaire (Yoshida, 2000). In the second one, twenty-one second grade children were interviewed using not only a questionnaire but also some concrete objects such as ribbons, Origami (folding papers), kumquats (small oranges), and sliced cakes and cookies made of cardboards etc. Finally, thirty-nine third grade children were given questionnaires as a test at the same time without any concrete objects. All children have not learned fractions at school at all. The first and second investigations were recorded on videos and tape recorders. The results from the second study, particularly, reflected the children's everyday concepts of fractions of the three investigations, because it was conducted in the most realistic everyday context using concrete objects. Moreover, the children were younger. The questionnaires for the second and third investigations were the same but different from the first one. They were based on the result from the first study, and were classified into six categories: (I) equipartition, (II) a view of unit '1', (III) quantity sense, (IV) equivalence, (V) order, and (VI) comparison between two quantities. Finally, children's everyday concepts of fractions were clarified for each category. And furthermore, the children's everyday concepts of fractions were integrated concisely into the following: EF1) Equipartition concepts that are used when children partition the whole equally by regarding the whole as a unit '1' EF2) Construction concepts that are used when children construct a unit '1' subjectively corresponding to an integer EF3) Identification concepts that are used when children identify a unit '1' subjectively corresponding to an integer according to the existing scale EF4) Identification concepts that are used when children identify the objects which has no scale, severing the connection with the whole EF5) Quantity concepts that are used when children consider the quantities in a real context EF6) Fundamental concepts for equivalent fractions that are used depending on the contexts EF7) Basic concepts for ordering fractions that are used in everyday concepts

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変位の非線形関数と周期係数の積で表される復原力を有する自励振動系では、係数共振や調波共振のほかに2/3次超調波共振も発生することが確認された。2/3次超調波共振域の定常解と安定性は回転座標系への変換と平均法の導入により容易に得られる。超調波共振付近の解はうなり振動であり、うなりの振幅はリミットサイクルを近似することにより求まる。数値計算例によりこれらの近似解は定性的によい精度をもつ。

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階微積分を用いたフィードバック制御を柔軟な機械構造物の振動制御に応用する。まず、機械構造物の階微分応答は直接既存のセンサから獲得できないので、複数のセンサを組み合わせたセンシング法を提案する。つぎに、制御系のモデルを構築するために、階微分応答を系の中に含む状態方程式モデルを構築する。最後に、階微分フィードバックを講じることで、粘弾性体と類似の制御効果を制御対象に与えることができることを明らかにする。

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Fractions of quantity are considered difficult to understand due to confusion with division fractions that were learned before. Therefore, this study focused on group learning when studying fractions where existing mathematical knowledge is superior. The aim was to characterize the mathematical negotiations influencing the selection of an original unit that is necessary to capture a fraction of quantity as a fraction of subordinate units. We conducted a survey on learning fractions of quantity in the third grade of elementary school. Further, we analyzed, both quantitatively and qualitatively, results of the pretests and intentions behind the discussions observed during group learning. Specifically, we extracted three groups from the experimental lesson and examined aspects of the discussions using an analytical framework we developed based on Searle’s speech act theory. The results showed two types of mathematical negotiations—integrated and distributed—which are necessary to select an original unit in a fraction of quantity. We also considered differences from discussions in which mathematical negotiations did not occur, and examined implications to provide learning guidance.

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本稿の目的は，算数科教科書における「もと」という言葉に係る分析を通すことから，算数科における「もと」という語の使われ方の特徴について指摘することである．本研究の目的を遂行するために，算数科教科書における「もと」という言葉の使われ方について，「もと」という言葉の種類と，学年及び領域を観点とした分析を行った．それらの分析を通すことから「もと」という語の使われ方の特徴として，教科書会社ごとの比較を通すことから，①教科書会社によって「もと」という語の使用頻度及び，使われ方に違いがあること，②算数科における「もと」という語の使用は主に第2学年から始まり，特に数と計算の領域において「初めの(original)という意味でのもと」が多用されていること，③算数科において「基準(base)という意味でのもと」が，学年の上昇に伴って多用されるようになり，特に変化と関係の領域において多用されていること，の3点を指摘した．

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本稿では,教室においての意味がどのようにつくられていくのかを明らかにするために,「折る」という子どもの生活経験を用いたの授業を分析している。Vygotsky理論に基づく生活的概念と数学的概念,分割量,教師の役割という観点から授業を分析した結果,次の結論を得た:1)子どもの生活的概念は概念発達の原動力となっている。2)「分割操作」を中心とした分割量Aの意味づくりを基に,そこに暗黙的に存在していた「量」を顕在化する分割量Cの意味づくりが行われた。3)教師の果たした役割は,の意味づくりを行う適切な場の設定,子どもの生活的概念を理解し,適切なZPDを引き出したこと,適切な教具の使用という3つであった。

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The purpose of this paper is to consider the influence of medium of instruction on both cognitive and affective sides in learning mathematics. The index for cognitive side in this paper is "pupil's everyday concepts of fraction" and "the influence of everyday concepts on the formation of concepts of fraction in the school". Here, everyday concepts are defined by Vygotsky's theory. And the index for affective side is "pupil's preference for mathematics" and "How pupils look English and Filipino as a medium of instruction". The survey was conducted for 157 fifth graders in two schools. In one school, medium of instruction for mathematics is English and in the other it is Filipino. Cognitive side is examined in terms of test items and affective side is in terms of questionnaires. As a result, the following 5 points were found out. (i) The pupils of both schools regard "kalahati", the everyday concept of half in Filipino, as the word to represent the half of continuum. (ii) The concept of "kalahati" have an effect on the formation of concepts of "1/2". (iii) Pupils tend to switch English to Filipino when they are talking a conceptual matter. (iv) Most pupils in both schools like mathematics. Even after medium of instruction has been changed to English, this tendency doesn't change. Pupils accepted this change positively. (v) Most pupils in both schools prefer English as a medium of instruction in learning mathematics. The affective side was underpinned by not only the educational reason such that "it is easier to learn numbers in English", but also the social reason such that "English is an international language". Here, we can see the gap between the fact of code-switching in cognitive side and the preference in affective side.

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本発表では小学校算数科授業において子どもの具体物利用がでの等分概念理解にどう影響を与えたか報告する。小学□年□学級を対象に「」導入授業を, 1)単位に依拠せずインフォーマルな知識をもとに表記する実験群(□学級)と2)数量の単位に依拠し,長さ・かさの端の大きさを表記する教科書群(□学級)に分け実施した。授業前後に実施したの等分概念に関するプレテストおよびポストテスト,ワークシートに記録された子どもの方略分析の結果より,等分理解には等分割に関連するインフォーマルな知識を活性化させる具体物利用の活動が有効であることが示唆された。

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