In this study, I analyzed the mathematical characteristics that contribute to the discrimination and prediction of successful applicants by conducting the following two studies.

　　　(1) Logistic regression analysis was conducted for each group of students according to their academic ability and installer, and the characteristics that contribute to the discrimination and prediction of successful applicants were estimated.

　　　(2) Logistic regression analysis was conducted for all general high school students to supplement the above conjecture.

　　The following five mathematical characteristics were identified as a result of the study.

　　　・Interest in mathematical analysis

　　　・Interest in mathematical problem solving

　　　・Logical thinking ・Visualization and summary of arguments and results

　　　・Understanding of the main points of problems and the methods of their solutions

　　For the top four, the stronger the feature, the more it is expected to contribute to the discrimination and prediction of successful candidates, and for the bottom feature, the stronger the feature, the lower the probability of discrimination and prediction.

　　In Tamura (2018), the“ creativity” factor was extracted in a factor analysis conducted on successful candidates. On the other hand, Sternberg (1988) divided thinking styles into five categories, all of which are necessary for creativity, and it is inferred that creativity has a role in connecting between item attributes. The five characteristics mentioned in this study straddle all four attributes of the questionnaire items: ① interest, motivation, and attitude toward mathematics, ② mathematical thinking, ③ skill and expression in thinking, and ④ knowledge and understanding of mathematics.

　　This result is suggested to complement the claim of Sternberg (1988).

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　　In this research, we consider the “subjects”─the targets of consideration of mathematically active persons ─as “SIGNs”─in Peirce’s semiotics─consisting of representamen, object, and interpretant, and explore the ontological status of this topic’s evolutionary development. The purpose of this paper is to clarify the status of the evolutionary development of the subjects in a lesson on the magnitude relationship between square roots for 9^th graders through an analysis using semiotics and comparing the lesson with a lesson introducing square roots analyzed in previous research.

　　In the beginning of the lesson on the magnitude relationship between square roots, the subject was “a sign that looks like a number,” but with the introduction of a diagram, this was changed to “the length of a side of a square.” During the exploration of the successive approximation method, the initial “　  squared to something close to 2” continued to exist. In this process, the possible range of existence and the method of diminishing intervals were revealed, so by “matome (summing up)” along the way, the subject became “something in between other numbers.” Finally, the existence of “something that is approximated by the successive approximation method” involved the process and product of the final matome.

　　The following hypothesis was thus obtained: In a classroom situation where the developmental process of accidental evolution is shown, a subject that has a “legisign (sign as law)” as its representamen can be changed to a “sinsign (individual sign).” It is suggested that in order for a new subject to become a being for many participants, it needs to become at least a productive object of existence, and it thus changes into a being as a class containing a sinsign within Peirce’s “ten classes of signs.” If the same situation had not changed to a subject containing a sinsign, it would be difficult for the participants to interpret the rules for application, etc., of the subjects, so another subject could emerge that transformed in to a different mode of representamen or the representamen might not change but the object or interpretant could change. We also obtained some suggestions about matome, which has a condensing and integrating function.

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　　Students have ample opportunities to prove theorems, but they do not have much opportunity to realize what a proof is or why a proof is necessary.  In this paper, we take the standpoint that it is impossible to understand the meaning of a proof without trying to understand the theorem.  We want to clarify the process wherein the conception of a proof is fostered through the reciprocal understanding of the theorem and the proof.  When a theorem is proven by dividing it in some cases, we would observe clearly that the conception of the proof is fostered in a reciprocal understanding of the theorem and the proof.  

　　First, the pragmatic view of the fostering of the conception of a proof is outlined through the review of previous work (Hanna, 1990; Hanna & Jahnke, 1993).  

　　Second, some theories concerning the construction and reading comprehension of a proof are examined. The construction of the proof is explained by clarifying the chain of generating to and reflecting on the proof language, referring to the figures and descriptions used for the proof.  Such chains are clarified based on seeing the figure and the description as generic examples (Mason & Pimm, 1984).  In addition, we review previous works concerning the reading comprehension of the proof (Yang & Lin, 2008; Mejia-Ramos et. al., 2012), and derive the features that encompass the entire proof from a holistic viewpoint.  We propose that we can clarify the commonalities and the differences of the proof language in proofs which are done by dividing in some cases.   

　　Third, to concretely catch the reciprocal understanding of the theorem and proof, the proof of the theorem of the inscribed angle is examined.  The proof is done divided, as shown in the figure on the right, into three cases.   

　　Finally, we declare that the process of understanding the divided cases of the theorem and the process of the proof of the  theorem are reciprocally related. Figure: Three cases of the theorem of the inscribed angle

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　　We interviewed four junior high school mathematics teachers about their class views and classroom practices. Analyzing the interview data identified factors supporting and hindering implementing teachers’ class views through their practices. Knowledge of teaching materials and methods supported practicing class views. In contrast, the specific contents of a class view, such as inconsistency with the current school system, hindered its practice. A detailed analysis of knowledge of teaching materials and methods that supported practicing the class view indicated that learning environments must be consistent with teachers’ interests. We also examined the backgrounds to class views and identified factors in acquiring class views that are difficult to achieve in classes and hinders its practice. The results indicated that class views’ backgrounds were affected by factors unrelated to the class, such as personality, personal interests, and significant life events. The nature of the Japanese teaching profession, which is responsible for forming the character and socialization in classes, combined with the class view, might make it difficult to practice. We have provided suggestions to help teachers practically implement their class views based on the above discussion and this study’s findings.

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　　In this paper, we will compare and analyze teaching plans for regular classe s and distance classes by Japanese math teachers. To investigate the “ability to create teaching plans”, which is related to the potential teaching ability of J apanese teachers, reveals the common points and differences between teaching and evaluation in regular classes and distance classes. When comparing and analyzin g the actual lesson practice, it is quite difficult because many elements are in tricately intertwined, but if it is a teaching plan before the lesson, it is pos sible to grasp objectively what plans are conceived by teachers.
　　 In order to make comparisons and analyzes, teachers of public high school mathematics in Hiroshima Prefecture create teaching plans for regular classes and distance classes using preset common teaching materials, student views, learning contents, and teaching materials. At the preparation stage of this paper, the teaching plan was received from 11 teachers, and by comparing and analyzing them, the issues and possibilities of distance learning became clear through discussi on and consideration.

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　　The purpose of this paper is to investigate elementary school children’s mathematical inquiry and its phase about expansion and reflection of the multiplication table. Therefore we carry out the teaching unit “review and expand the multiplication table” for third graders by an 8-hour class.

　　We investigated children’s mathematical inquiry by teaching unit about reviewing and expanding the multiplication table by qualitative methods.

　　As a result of our discussions, we obtained several insights:

(1) Interpretation through expressing the multiplication table in various ways, such as expressing a figure appearing on the circumference by connecting mutually promotes the mathematical inquiry about finding property and regularity of the multiplication table.

(2) The reasoning of the analogical inference that is going to apply the relations between steps of two multiplication promotes the expansion and reflection of the multiplication table and its progressive thinking.

(3) Third grader children have difficulty about the meaning interpretation of multiplication of 0. Children are going to overcome this conflict through applying thinking to lead to a distributive law and extrapolation.

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　　The purpose of this research is the unit of the junior high school mathematics “square root” and is to make the aspect of the learning process of the student who interprets the meaning of the square root clear using the point of view of representation and a way of thinking. Because a way of a study considers a square root by the operation activity that I have synthesis of representation in a mathematical background, expresses the act by a word and becomes a target, the mathematical activity that the meaning of the square root is understood is performed. And because the student’s characteristic activity that this learning process can see and a description thing are analyzed qualitatively, the aspect of the learning process is made clear.
　　As a result, the following 3 points became clear.
(1)　The operation activity that I have synthesis of representation in a mathematical background and the activity that I express the act in a word and become a target respectively deepen their understanding about the meaning of the square root.
(2)　You can replace double meaning understanding with something abundant using the point of view of multiplication as synthesis of representation.
(3)　Foundation formation to the learning which understands the meaning of the multiplication of an insight to learning of“ the Pythagorean theorem” and learning of radical root and a complex number at least is born by the operation activity that I have synthesis of representation in a mathematical background and target-ization of the act.

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　　The purpose of this paper is to clarify the characteristics of teachers’ interventions in inquiry-based lessons through comparison with “problem solving lessons” generally conducted in Japan. In this study, we designed an inquiry-based lesson using the framework of “Study and Research Paths (SRP)” formulated within the “Anthropological Theory of the Didactic” (ATD). In SRP, instead of learning the piece of mathematics knowledge one by one, students engage in activities to acquire the necessary knowledge as needed and use anything which is available to create an answer. In order to achieve the purpose of this paper, we analyze SRP lessons using the perspectives of “triple dimension of teachers’ work” proposed in ATD: mesogenesis, topogenesis, and chronogenesis. Then, based on the results of the analysis, we tried to clarify the difference of the teacher’s interventions between SRP and “problem solving lesson” through the comparison. The differences identified as a result imply teacher’s expertise necessary to carry out inquiry-based lessons like SRP.

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