Due to social reform concerns, such as rapid aging and low birth rate in recent years, the need for critical thinking abilities to be developed in children, who will be major players in the 21st century, is becoming increasingly important.  This paper explores learning activities called Researcher-Like Activity (RLA) in an effort to foster critical thinking skills when teaching mathematics.  The objective of the research was to verify the effectiveness of incorporating RLA and to clarify its achievements and problems.  As the result of this research, it was found that RLA had a fixed effect not only on the development of critical thinking-related rationality (logicality), reflective thinking (contemplative), and critical thinking (skepticism), but also on that of deep thinking including reflective thinking (contemplative) and critical thinking (skepticism).  On the other hand, an issue that needs to be further examined is how to help students who have difficulties with mathematics because of the importance teachers place on reflective thinking.  Thus, further research is necessary regarding planned, effective intervention that does not inhibit students’ critical thinking, in order to promote their learning. 

　　 Our consideration about ‘the problem on the sum of angles’ began in the following ‘Problem e’.  Nishikawa et al. (2014) insisted that the ‘Problem e’ would be valuable teaching materials in junior and senior high school mathematics classes.

　　　 Problem e:  Find the sum of angle α and β in the figure in which three  congruent squares are placed side by side.  

　　 This article deepened Nishikawa et al. (2014) and its purposes are as follows:

　　　 (1)  To make the similar problems of the ‘Problem e’ and to raise the value as the teaching materials of the problem more.

　　　 (2)  To consider the ‘Problem e’ from the viewpoint of inversion and to explore the possibility as teaching materials.

　　 In Section 2, we made the similar problems that had a different condition from the ‘Problem e’, and gave three kinds of elementary geometrical solutions in different ways of viewing sums of angles i.e. neighboring two angles, an external angle and expression of the product.  As a result, it was revealed that these similar problems could become superior teaching materials.

　　 Through the consideration in Section 2, we reached an idea that it was a general way to view the problem on sum of angles from the viewpoint of inversion.  The idea is reflected in theorem 3.3 in Section 3 as the relationship of sum of angles with respect to a point on circle of inversion and a pair of points of the inversion. It would be said that the figure of theorem 3.3 is ‘the general figure of the sum of angles’.  The result of theorem 3.3 is also viewed as a kind of extension of ‘The relationships between inscribed angles and its central angle in a circle’.

　　 In Section 4, we considered a property in ‘the general figure of the sum of angles’.  In Section 4.1, we made it clear that the figure was closely related to Apollonius’s circle and we came to the conclusion that these facts would be interesting teaching materials in high school mathematics classes.  In Section 4.2, we obtained the proposition 4.4--4.7 by using the properties of inversion.  The proposition 4.7 can be used in constructing one point of the inversion from the other.

　　 There are many contents in this paper which give us a clear image of making students acquire some kind of mathematical ability.  In this sense, we conducted content-study of ‘the problem on sum of angles’ for the development of a teaching material.  In addition, the consideration of this paper clarified the core of the problem that everyone had overlooked by the elegant solution, by various answers to the similar problems and the generalization based on it.  Therefore, we insist that the contents of this paper have values as a teaching material of teacher’s training for both pre-service and in-service teachers.

　　 The purpose of this study is to develop elementary mathematics lessons which associate with moral education.

　　 We developed a teaching material based on series of studies on building of social values in elementary mathematics proposed by Shimada & Baba (2013a, 2013b, 2014).  We adopted five characteristics of socially  open-ended problem in elementary mathematics introduced by Shimada & Baba (2014) in order to cultivate moral values which are a part of composition of social values in elementary mathematics.  The teaching material developed in this research is “the  problem of organizing teams”.

　　 We conducted a lesson of this teaching material for third graders at the elementary school in order to investigate its effectiveness.  As a result of analysis, it was confirmed that students expressed the values of fairness through solving this problem and the lesson had a possibility of building them.

　　 A purpose of this report is to pay the attention to counterexample, and to clarify the way of critical thinking foster in the elementary mathematics education.  Therefore at first, the author considered fostering of the critical thinking in the elementary mathematics education and the problem based on three components. Then it derived that “a thought of reflecting (critical manner)” included a problem.

　　 Next, as one point of view to solve the problem, it paid my attention to the counterexample from others. Contradiction, the thing that it is planning to be opposed and face is because it is at a strong opportunity arresting own thought for reflecting each other again.  It is important that I catch own thought for reflecting again in the cognitive process of the critical thinking.  It was necessary for conviction to be included in the counterexample and thereby clarified that it gave “an opportunity to reconsider the rationality of both”.

　　 In addition, it give you an opportunity to show “a thought of reflecting (critical manner)” that was a problem by the presentation of the counterexample.  It is an opportunity to make a round trip to “the evaluation of the solution” and “the search of the solution” in the conception diagram of the critical thinking.  Thus, it spoke that it could show three components.  In other words, by the presentation of the counterexample, it got the conclusion that fostering of the critical thinking was available for.

　　 It will be a problem in future to consider an evaluation based on cognitive process of the critical thinking.

　　 The purpose of this study is to investigate how to realize explorative proving in schools by using Wittmann’s operative proofs.  In this paper we examine how elementary school children engage in exploratory proof activities with number pattern problems.

　　 In order to investigate dialectic relations between learners and the artifacts they use in operative proofs, we used ‘instrumental genesis’ as our main framework of analysis.  Data taken from 4 experimental lessons in  Japan and the UK suggest children in elementary schools can engage in productive proving activities with appropriate manipulatives.

　　 From an ‘instrumental  genesis’ point of view, we could observe ‘instrumentalized’ manipulatives offered ways for children to construct proofs and explain the patterns as well as produce new statements.  It was also observed that constructing proofs and producing new statements can occur simultaneously and this is a distinctive feature of proving activities in elementary school contexts.

　　 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.  

　　 This study is undertaken to better understand the relationship between mathematics and reading achievement focusing on PISA2003 and PISA2012.  In PISA, the mathematics and reading literacy are examined continuously every three years from the year 2000.  Therefore, the relationship between them can be analysed including its secular changes.  The research on the relationship between mathematics and reading ability has currently focused by researchers of mathematics education.

　　 In this study, the approach of international comparison will be adopted.  The countries which are performing high mathematical literacy tend to also have higher performance in reading literacy.  However, the relationship between them within the country is not always cleared.  In order to make this point clear, we will focus on not only the indexes of reflecting the level of students’ performance of mathematics and reading literacy such as the mean score, but also the indexes of reflecting the relationship between two variables have to be considered to capture the domestic feature of countries, for instance, such indexes are correlation coefficient and regression coefficient and so on.

　　 In this study, there are two points of view.  One is to focus on test score.  By using Hierarchical Linear Model (HLM), the international comparison of mathematics literacy test score will be conducted through controlling reading literacy test score.  Another one is to clarify the answer pattern of each country by focusing on item difficulties based on Items Response Theory (IRT).

　　 As a result, in the countries have gotten higher level mathematics literacy, the students’ mathematics and reading performance are more related domestically than in lower performed countries.  In addition, the differences of the level of reading literacy test score in higher performed mathematics literacy countries come to a head markedly on a particular item.  On the other hand, regarding lower performed countries, we could not make mention the same situation.  The results imply that the relationship between mathematics and reading achievement is different between higher and lower performed mathematical literacy countries.

　　 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.

　　 The objective of this study is developing a theoretical framework for curriculum development of connecting science and mathematics.  First, connection between mathematics and science were classified into 4 ways, thematic relation of content, thematic relation of thinking, structural relation of content and structural relation of thinking.  Next purposes of connecting science and mathematics in each way were discussed.  In thematic relation of content, deeper understanding and ability to assemble the learning content is promoted.  In thematic relation of content, understanding outside school subject topic and problem solving skills by using thinking in a comprehensive manner is developed.  In structural relation of content, conceptual formulation and ability to transfer is advanced.  In structural relation of thinking, understanding from the different view point and generalizing the thinking is fostered. 

　　 The purpose of this paper is to elucidate the way of the concept formation that is inherent to the statistics domain, in other words, the process forming statistical concepts as methodological knowledge.  For achieving this purpose, three tasks are worked on.  At first, the characteristic of statistics is made clear.  Then, the difficulty in forming concepts is revealed. Finally, the formation process of statistical concepts is explained.

　　 One of the characteristic of statistics is that it makes uncertainty targeted for consideration.  But, It characterizes statistics plainly further that statistics are methodological knowledge.  Based on this characteristic of statistics, statistical concepts must be formed as the solution while solving a problem with the uncertainty statistically.  The difficulty of the statistical concept formation is not abstraction but generalization.  In this paper, as approach for this difficulty, the negation theory is paid the attention to.  It is three stages in a process of the concept formation based on the negation theory.  At first, the extension of concept A, which can solve problem P, is limited by an appearance of problem Q.  This is the stage of limiting extension.  Then, concept B which can solve problem Q is constructed.  Concept B overcomes a shortcoming of denied concept A.  It is the stage of revealing intension. In the last stage, two concepts are reconstructed.  The relation between concept A and concept B are made clear.

　　 The objective of this research is to make students’ mathematical noticing related to mathematical thinking clear, and then understand mathematical thinking as activities in a classroom.  For this objective, this article aims to grasp a dynamics of noticing in mathematics classrooms by referring to the results from investigations in Kageyama et al. (2014).

　　 In this research, we defined mathematical noticing as a series of cognitive actions such as selecting, recognizing particular features and rationally interpreting them.  Plus, we propose 4 viewpoints to analyze students’ complex, cognitive actions in a classroom - individual cognitive dispositions, interactions, material resources, and a problematic situation.  We implemented mathematical lessons for the first and fourth graders, obserbed and analysed them by making use of the 4 points framework.  As a result, we identified that students’ mathematical noticing was sometimes persistent, while changing dramatically depended on the situations. Finally, we suggested that there were 3 possible factors to change noticing as below:

・aims of activities by students in action,

・a renovation of a problematic situation,

・an invention of new objects depended on the above both.

　　 Students in action often say that there is a rule.  This might be a key term related to the above, therefore the future task is to grasp students’ mathematical noticing deeply by focusing on a literal objective such as a rule.

　　 The purpose of this paper is to develop a strategy for designing mathematical tasks to motivate the shift to the mathematical problem solving phase, that is, the phase of understanding mathematical problems.  This paper mainly consists of five parts.  First, we review radical constructivism as a theoretical background.  In order to focus on whether each student can understand mathematical problems, we develop the strategy from the radical constructivist point of view.  Second, we formalize the central concepts of radical constructivism. This formalization makes us easily use those concepts for designing mathematical tasks.  Third, we formalize the meaning of mathematical problems for students.  From the radical constructivist point of view, there can be a gap between the students’ interpretations of mathematical problems and those of the teacher.  We discuss what conditions make mathematical tasks easily understandable for students.  Forth, through a consideration on “a problem of determining the minimum value of quadratic functions, where the equations of their vertical  lines include constants” within a Japanese high school mathematics textbook, we propose a strategy for designing mathematical tasks to motivate the shift to the mathematical problem solving phase.  Concretely, the strategy consists of the following procedures for improving the way of posing mathematical tasks: [1] To make students confirming two important conceptual frameworks about the search range of the answer and about the criterion of validity of the answer; [2] To make students experience perturbation when they apply their alreadyknown knowledge-how; [3] To make students experience appropriate accommodation of the conceptual framework for assimilating mathematical problems.  This proposal is based on the application of the conditions to make mathematical tasks understandable for students.  Finally, we discuss the implication from our proposal to future teaching and researching practices.

　　 The purpose of this paper is to investigate senior high school students’ mathematical inquiry and its phase in factorizing the polynomial xⁿ-1, which focus on producing the irreducible polynomial, discovering and proving mathematical character about the irreducible polynomial xⁿ-1. We investigated students’ mathematical inquiry by teaching units about factorizing the polynomial xⁿ-1 by qualitative methods.

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

(1) By its behavior and its product about factorizing the polynomial xⁿ-1 inductively, a student’s mathematical inquiry is encouraged to connect the properties of the polynomial xⁿ-1 with the properties of the integer n.

(2) Various interpretations about factorizing the polynomial x⁶-1 become a foundation of factorizing the polynomial xⁿ-1. For example, students can connect factorizing the polynomial x⁶-1 with established facts about factorizing when they read the exponential number 6 of x⁶-1 to 6=2×3 or 6=3×2. Similarly, students advance their own thinking by changing the role of polynomial xⁿ-1 as the base.

(3) Algebraic matters that students want to prove are generated in inquiry about factorizing the polynomial xⁿ-1 with using a Computer Algebra System (CAS). However, various students’ activities about proving these matters exist.

　　 The purposes of this study are to reveal characteristics and problems of learning contents in terms of Spatial thinking in geometry curriculum in Japan through comparing with NCTM Standards and CCSSM in the  U.S.A.

　　 The method to compare among curriculums has three steps; (1) three learning contents, “location and coordinates”, “coordinates and transformations” and “2D representations of 3D objects” are selected from the aspects of former curriculum reforms, (2) All learning contents in geometry curriculum are analyzed   using the idea “Making activity an object of another activity” by Hirabayashi (1978), (3) three learning contents selected in first step are compared from the viewpoints, objects and methods.

　　 The results shows that two curriculums in the U.S.A emphasis on coordinates in order to represent the location in the space and describe the transformation of geometric figures, and also especially NCTM standards has a coherence in raising spatial visualization.  On the other hand, Japanese geometry curriculum is taken along van Hiele’s level of thinking.  However there are two problems, lack of opportunity to learn dynamic geometry and inadequate specific viewpoints of raising spatial thinking abilities.

　　 Wakayama University has a large amount of archives of the Wakayama Normal School, the predecessor of the present university.  The author had showed some charectaristics of geometry and other subjects in the Wakayama Normal School, mainly concerning on the “boys’ course”.  The archives contain a certain amount of documents of the “girls’ course”, and the “second course” which was established for the graduates from other secondary schools.  In this paper these documents, such as school syllabi, was analyzed and some characteristics are shown as follows; First, “girls’ course” flexibly changed their mathematics curriculum and textbooks, and kept as high level mathematics education as the “boys’ course” including, for example, solid geometry.  It was much higher than girls’ high schools at that time.  Second, although the “second course” students had high basic knowledge of junior high school, the range of educational content are basically limited to arithmetic, which might be learned in the previous schools.  It might be thought that arithmetic was enough to teach at elementary schools.  The “second course” of the normal school is said to have become a source of upgrades from secondary school to university.  Educational content of the “second course” and its transition to higher education need some more research from the point of view of elementary school teacher training.

　　 As nature of mathematics activity, there are two aspects as “adopting mathematics onto daily life situation” and “creating mathematics from daily life situation”.  Even though many people believe that such connections are important, it is still difficult to make them in good use, because of the NOISES in real life situation.  In this paper, the “Mathematics Problem in Real Life Situation” was examined from the view point of “Mathematical Situation Theory” by Prof. Hirabayashi.

　　 First of all, “Social Open-ended Problem”, which has been presented by Shimada & Baba (2013), is examined as the typical example of “Mathematics Problem in Real Life Situation”.  Finding the major characteristics of it, “Open-Closed Continuum” theory by Dienes, Mathematical Modeling theory, explicit and implicit mathematics theory by Chevallard, are also examined to establish the framework of Mathematics activities with Open-Closed Continuum situation.

　　 By using this framework, a proposal for the lesson is presented, with the procedure of ①Socially Closed situation, ②Socially Open situation, ③Mathematically Open-Closed situation, and ④Socially Closed situation. (Fig.15)  Finally, some advantages for “Mathematics Problem in Real Life Situation” were concluded as follows:  1) Smooth connection between real world and mathematics world, 2) Supporting for students’ motivations for finding rules, 3) Making good use of the value of problem solving, and 4) Getting deeper understanding of mathematics contents.