Whereas Piaget established conservation task to asess children's understanding of extensive quantity, this study aimed at establishing another kind of conservation task to asess children's understanding of intensive quantity, e.g. the concept of density. The following study examined children's misconception of density by using our conservation task. Subjects were sixth graders. A typical question was asked in the conservation task: Which density is greater, a big or a small aluminium lump? Although subjects were taught in advance that density of substance was given by its weight per unit volume (1 cm3), commonly explained in school education, half of the subjects failed to answer the question. They answered that the density was greater for the big lump, suggesting they did not understand the concept of density by a commonly given definition. Results were discussed from the viewpoint of the formation process of intensive quantity concept. A teaching method to lead children to a better understanding of the nature of intensive quantity was proposed.

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Whereas Piaget established a conservation task to assess children's understanding of extensive quantity, this study aimed at establishing another kind of conservation task to assess children's understanding of the concept of intensive quantity, e.g. the thickness of paint on a board. The following study examined children's misconception by using our conservation task. Subjects were sixth graders and undergraduates. A typical question was asked in the conservation task : Which is thicker, the large part of a board or the small part, when it is painted uniformly? Although Subjects were taught in advance that the thickness of paint was given by its volume per unit area (1 cm^2) , commonly explained in school education, two-thirds of sixth graders and one-fourth of undergraduates failed to answer the question, suggesting they did not understand the concept of intensive quantity through the commonly given explanation. Results were discussed from the viewpoint of the method employed to teach the concept of intensive quantity.

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The purpose of this paper is to show the previous research results by the author, and to construct the principles for mathematics class to promote students' number sense. In this research, I was able to bring the activities together which became the references of number sense by using the research results in this area, but was not able to answer what the number sense was. Then, I clarified the inner nature of number sense by using phenomenology which searched for the nature of thing. Being based on these suggestions, the principles for mathematics class to promote students' number sense were constructed. By using these principles, I analyzed an actual arithmetic class. As a result, the following points became clear. First of all, in the class where the number sense is promoted, the teacher always does pay attention to make students' consideration turned to the problem solving process and their outlooks on value. And, the teacher makes the discussion among students' active by using students' surprise opinions. Simultaneously, it was also suggested that others and the class environment were important to promote number sense.

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　　 The first purpose of this paper is to resolve the mixed unit by analytical geometry and function and to clarify the difference between them, and the second one is to clarify the learning content of function and analytical geometry, and the third one is to explore how to instruct analytical geometry in junior high school.

　　 As a result, it can be considered that common representation of analytical geometry and function is the graph in the side of each shape, but it is found that the meaning of shape, which is derived from the side of each concept, is different.

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1959年1月、日本初の教育専門局として開局したNHK東京教育テレビジョンは2019年で60年を迎える。当初は学校放送番組や語学番組などの講座番組が中心であったが、1980年代には社会全体が生涯学習を志向するようになったこともあり、子ども向け番組や、社会人そして高齢者を対象とした番組が増え、90年代からは「ゾーン編成」が取り入れられる。2000年代になると、さらに多様な番組を展開するとともに学校放送番組や趣味実用番組を中心に、インターネットへの展開を積極的に進める。2009年、教育テレビ開局50年には「ETV50 学ぶ冒険」をテーマに特別番組やイベントを開催。2011年には「Eテレ」という愛称を採用、「未来を志向するチャンネル」として番組を改変、2017年からは「みつかるEテレ」をキャッチコピーとして、さまざまなキャンペーンやイベントを展開している。本稿では教育テレビ60年の歴史を、各年度の「国内放送番組編集の基本計画」と放送番組時刻表を基に、NHK放送文化研究所の調査や論考、教育テレビ開局周年記念番組の内容などを参照しながらみていく。

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　　In order to realize the lessons where students think autonomously and developmentally, the teacher’s consciousness about how to grasp the problem solving and thinking that the students do will influence.  With reference to Ernest’s（2015）  Fallibilism and Absolutism, we developed a questionnaire on teaching students to think autonomously and developmentally, and from the case studies of two teachers, we caught the trend of “classroom practice emphasizes teachers’ planned developmental guidance and short-term academic improvement, fixing instruction and support to control learners’ thinking and attitudes”.

　　These findings are findings within the scope of case studies, and are not findings that sufficiently demonstrate versatility.  In this study, we conducted a questionnaire survey on lessons where students think autonomously and developmentally for elementary, middle school and high school teachers nationwide.  The purpose of this paper is to analyze Akita prefecture data and obtain suggestions for guidance.

　　As a result of analyzing the data of 124 elementary school teachers, 71 middle school teachers, and 35 high school mathematics teachers in Akita prefecture, the following 2 points were clarified.

　　1. As  for teachers’ consciousness, there is a tendency for Awareness of Developmental Mathematics to be seen when the lesson is planned, while to increase absolutely Awareness of Mastery Mathematics  when the lesson is practiced.

　　2. In  the composition of each factor, “G.  Numerical values, conditions, content, and sequence of problems” was not included, indicating that teachers have an absolute, fixed view of mathematics and its teaching  as the subject of instruction, which may be considered a manifestation of their Awareness of Mastery Mathematics.

　　In the future, following the Akita prefecture data, we will proceed with the analysis of national wide data and clarify the trends of students and university faculty members.  In addition, we will proceed with the analysis of whether there is a difference depending on the number of years of teaching profession, or whether there is a relationship with the good results of the national academic ability / learning situation survey.  Furthermore, we will consider the training method using this survey.

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The author examines the features of pedagogical content knowledge (pck) and its implications. Pck was first contrived by L. S. Shulman. He was indicated pck as very important knowledge among teacher knowledge base. Pck is combined knowledge, including knowledge of content, pedagogy and knowledge of the learner's conceptions and preconceptions, including misconceptions. And pck is invented by teacher who will teach some topics. Pck is producted through the process of pedagogical reasoning (pr). Shulman especially emphasized transformation in that process. Transformation is constituted by four process, preparation, representation, selection and adaptation & tailoring to student characteristics. The features of pck is as follows. (1) pck is amalgam of some knowledge (2) pck is producted through pr which is the problem-solved process (3) pck is special forms of teacher knowledge which indicates the features of professional aspects of teaching The implication of pck from point of teacher education, is the next two points. (1) pck points out the importance of the courses of content and method (2) pck indicates the needs of means for 'apprenticeship of observation' formed before entering the teacher education program The contents are as follows. 1. Introduction 2 . Shulman's pck and its formation process 3. The evolution of the study of pck 4. The features of pck 5. Conclusion

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本研究では，GIGAスクール構想下における，1人1台の情報端末のICTを活用した授業実践を把握し，小学校の授業実践に関するDXの現段階を検討することを目的として，2021年1月から2021年11月までに出版された「GIGAスクール構想」「1人1台端末」に関する書籍21冊に掲載された小学校の1人1台端末を活用した授業実践を，SAMRモデルを用いて分類した．その結果，①中学年になるとM・R（変換）の授業実践が増加すること，②教科全体ではS・A（強化）の授業実践が多いが，総合的な学習の時間ではM・R（変換）の実践が多いことが確認できた．

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