The result of National Assessment of Academic Ability in 2018 indicates that Japanese secondary students have difficulty in identifying the type of a function which is inherent in a given situation, in particular in the case of “linear function”.  Against this issue, we’d like to stress the importance of the deep developed conception of “linear function”, rather than the reinforcement of memorizing what “linear function” is and/or how to identify them. 

　　First, from the view point of example space, we make a consideration on the actual conditions of students’ learning process of linear function in lower secondary schools.  In fact, we point out that, though the intensional definition is given explicitly, students’ learning could advance extensionally using examples.

　　Thus, to obtain deep developed conception of linear function, it is required to step into the intensional conception based on the definition.  To describe students’ changing process from extensional conception of linear function to intensional conception of that, we propose applying the framework of Schema development in APOS theory.  Finally, using this framework, we present a desirable conception of linear function for students.

　　The purpose of this study is to develop probability curriculum for formation of interpretation of probability. To accomplish a purpose, we set lower purpose (a) and (b).

　(a) To Indicate theoretically what process should be used to form a learner’s interpretation of probability.

　(b) To consider as case what kind of teaching materials can support its formation.

　　Firstly, lower purpose (a) was considered based on “Psychogenese et histoire des sciences” (Piaget & Garcia, 1996). As a reason for this is that historical development of interpretation of probability and formation process of interpretation of probability can be considered as mutually negotiable. Secondly, for lower purpose (b), we considered the teaching materials that could become problem situations that could not be solved by existing interpretation of probability based on studies on formation of probability conception. As a reason for this is that interpretation of probability are an aspect of the probability concept.

　　The results were as follows.

　・ We constructed formation process of interpretation of probability. It takes the following steps. The first is “Intuitive subjectivist interpretation of probability” which is formed in daily life before learning probability, the second is “Frequentist interpretation of probability”, the third is “Subjectivist interpretation of probability”, and the fourth is “Common understanding viewpoint between frequentist and subjectivist interpretation of probability”.

　　・ We have developed teaching materials that can be problem situations to which existing interpretation of probability cannot be applied as teaching materials that can support formation of interpretation of probability.

　　・ We discussed the relationship between formation of interpretation of probability and formation of mathematical aspect of probability, and developed probability curriculum for formation of interpretation of probability.

　　It is pointed out that statistics education has a challenge that statistical knowledge that is closely related to a real context cannot be used for solving statistical problem practically.  Inferentialism, which R. Brandom  originally proposed in pure philosophy, brings new light to statistics education research, and it has the potential to be a solution to the traditional challenge to avoid inert statistical knowledge.  In the paper, statistical problems in mathematics textbooks at the first grade of junior high school were analyzed from the perspective of Inferentialism to investigate a part of the cause of the challenge.  Intentionality given by the problem and its abstraction level were used as the framework of the analysis.  Specifically, all statistical problems in seven mathematics textbooks were categorized by giving conceptual or concrete intentionality.  As a result, it is found that most of statistical problems in the textbooks give conceptual intentionality.  Certainly, they appear to be strongly related to real contexts, but in fact they just wear realistic dresses.  Based on the result, the traditional problem of acquiring inert knowledge in statistics education can be understood as an obvious phenomenon derived from typical statistics teaching and learning.  In other words, it can ironically explain that inert statistical knowledge has been acquired as an outcome of statistics teaching and learning.  In order to avoid inert statistical knowledge, it is necessary to let learner tackle statistical problems that give concrete intentionality, and master knowing-how to consider which known statistical knowledge can be applied from the viewpoint of contextual knowledge.

　　The purpose of this paper is to examine the aspects of sixth graders’ logical explanation for spatial figures by proposing the theoretical framework and by analyzing the interviews for the students in terms of the framework.

　　We first set the theoretical framework which consists of the relationships among three dimensional objects, two dimensional representations, and the languages. Moreover, we distinguished two-dimensional representation into sketch, net, and projection in order to analyze the students’ explanation in more detail and reconstructed the framework as the tetrahedron model. The model constitutes the inner and inter relationships among the three-dimensional object and three two-dimensional representations.

　　Next, we qualitatively analyzed the interview data conducted for sixth graders based on our theoretical framework.

　　The main findings of this paper are as follows.

　　1. Sixth graders’ logical explanation can be better examined through our theoretical framework, particularly along the arrows showed in the framework.

　　2. About half of six graders have their abilities to explore the challenging problem of spatial figures and to explain them logically under the conditions where they can image three-dimensional object dynamically. It suggests a possibility to reconstruct secondary curriculum in spatial geometry towards more exploratory learning which can evoke students’ logical explanation.

　　In this research, we consider the subject in mathematics as a “sign”―in Peirce’s semiotics―that directs our attention to different things, and we clarify its ontological status.  According to this perspective, since a sign consists of representamen, object, and interpretant, the existence of the subject could be determined by the relationship between the representamen and the object.  In this study, we semiotically analyzed subjects through Euler’s activity in “Analysin Infinitorum.” The results highlighted the following points.  

　　First, we should consider not only universals but also individuals as subjects in mathematics, because the subjects in mathematical activities evolve from individuals to universals.  Second, the signs as subjects are initially the signs themselves (i.e., icon).  Thereafter, these signs indirectly indicate a tacit class (i.e., index), and finally, they conventionally indicate mathematical objects (i.e., symbol).  Peirce’s semiotics, therefore, gives a suggestion to be effective among frameworks for the subjects, because it covers various ontological positions and explains the evolution between them.

　　The aim of this paper is to clarify that mathematics is the most successful subject for collaborative learning. Abstraction and generalization are the essential characteristics of mathematics.  The separation model of generalization (Iwasaki 2007) based on Dörfler’s generalization theory and model is employed to analyze social and collective process of learning mathematics.  In that process, interactions develop effectively constructive abstraction toward generalization.  In other words, collaborative learning provides students with the opportunity for constructive abstraction in mathematics class.  Here they regard a symbol as object which contains meaning. Finally this paper identifies teacher’s roles in this construction: selection of tasks, designing of process and habituation of collaborative learning mode.

　　The purpose of this paper involves two points: One is to clarify more essential elements for realizing “Group-Competition-Style Mathematics Classes”.  In addition, in light of such essential elements, to clarify the concrete method to realize “Group-Competition-Style Mathematics Classes” and important teachers’ moves in realizing it.  The second one is to obtain suggestions on classes to encourage understanding of “deductive reasoning” at elementary school, especially to clarify the mechanism and function of “Group-Competition-Style Mathematics Classes” in particular what type of understanding about “deductive reasoning” are invited to the students.  

　　For that purpose, teaching experiment was conducted in the “school support project”, which was conducted at public schools in Joetsu City for about 4 months from September 2018 to December 2018, including 4 lessons with a 5th grade class on December 3, 4, and 5 in 2018.  We planned and conducted a competition style of group-based lessons in the 5th grade class.  The mathematical topic of the class was to investigate whether a certain geometric relation was established.  The students were required to form their own assertions and their mathematical explanations together in groups to discuss them with each other.  The lessons were recorded with video cameras and IC recorders and detailed transcripts were prepared for the analysis.  

　　The analysis results are summarized in the following two points:

1. The  most essential element for realizing “Group-Competition-Style Mathematics Classes” is a “gap” in students’ ways of thinking occurring in the lesson process, rather than teachers’ preparing in advance  what will be conflicting.  Therefore, the important thing in realizing “Group-Competition-Style Mathematics Classes” is that the teacher will be able to assess a “gap” between students’ ideas or ways of thinking occurring in the course process.  That “gap” in itself is what teachers can not predict in advance. Therefore, in addition, teachers must be prepared to change the lesson plan according to the “gap”. However, by rethinking “gap” as a more essential element in this way, we could expand the scope of application of “Group-Competition-Style Mathematics Classes”.

2. T he “Group-Competition-Style Mathematics Classes” in our lesson study, more precisely, the lesson that changed the plan with the “gap” that occurred between the two guesses as the core after the stage of  inductive activities, encourages students to understand the effectiveness of “deductive reasoning”.  More specifically, such classes encourage students to understand deductive reasoning as a way to overcome the limitations of inductive reasoning.  In other words, such classes encourage students to understand the generality of deductive reasoning.  The conflict of arguments between inductive reasoning and deductive inference that occurs under the situation of “group competition” has the function of making the students realize the theoretical character which is a feature of deductive reasoning.

　　The overall objectives of this multidisciplinary research are to make mathematics curriculum principles involving designing a learning environment and propose a multifunctional, practical framework for a formative assessment by the participants and observers.  The program of this research consists of three parts: the core theory, designing methodology, and curriculum assessment.  This article deals with the core theory including generation and construction of objects, as well as bringing forth, maintaining and changing a world of mathematical significance.  The theory finally proposes the nested model to describe macro / mezzo / micro structure of worlds of significance in a mathematics classroom.  

　　This research especially focuses on the mathematical working spaces theory developed by Kuzniak et al. to describe the way to generate (mathematical) objects and intention theory to explain the participants’ attitudes and emotions during mathematics lessons.  Behaviors and local interactions between individual students at a micro level will evolve toward actions by a pair or small group of them at a mezzo level.  The participants’ activities ultimately lead to systemic ones at a macro level by harmonizing some mezzo level actions with each other, which seem to have a certain holistic system from a view point of the observers.  This research suggests these findings through theoretical and practical discussions and summarizes them as the nested model.

　　In international educational discourses, economic and political driving forces are introducing computational thinking, which was originally advocated by an information scientist J. M. Wing, for mathematics education. Introduction of programming education in Japanese primary education in 2020 can be considered as a step toward this movement.  However, existing studies on mathematics education have put less emphasis on the idea of computational thinking as a subject of research, although it has the potential to radically change the national curriculum of mathematics education.  The purpose of this study is to argue why we need to consider the idea of computational thinking in mathematics education research, and to elaborate on its potential impact on educational practices and curriculum development of mathematics.  

　　The structure of this paper is as follows: 1) an overview of international research trend on computational thinking; 2) theoretical consideration of the reciprocal relationship between developmental paths of mathematical and computational thinking; 3) illustration of concrete mathematical contents to exemplify the reciprocal relationship; and 4) further consideration on a distinction between mathematical and computational thinking in mathematical activities.  

　　The results reveal the reciprocal relationship, that is, in mathematical activities, a kind of computational thinking can be developed using mathematical thinking, and conversely, a different kind of mathematical thinking can be developed using computational thinking.  In addition, three new roots of mathematical topics through mathematical activities with computational thinking were identified: A) reification of mathematical procedures by creating new mathematical objects; B) methodological consideration of mathematical methods as theoretical ways of problem solving in the idealized mathematical world; and C) methodological consideration of mathematical methods as practical ways of problem solving in the real world.  Based on these findings, future research directions are discussed.