So far, the author has suggested the viewpoint of writing positively, focusing on the characteristics of written language.　But he hasn’t considered how to shift “drawn” language to written language, or the viewpoint of the instruction which introduces students to use figures positively.　Therefore, together with examining the preceding researches, the author reconsidered the difficulties again which written language contains, by returning to the original text of the researches.　

　　As a result, the author developed his idea to the degree of“difficulties of drawing”which contains even the difficulties of positive use of figure expressions such as (1) the knowledge of drawing (2) the use and activity of drawing (3) the cost of drawing (4) the viewpoint of learning which drawing has.　

　　And students, through solving problems, by learning together, conquer the difficulties of drawing.　At the first stage, teachers show students drawing as knowledge.　At the second stage, teachers have students gain how to use and utilize drawing through the process of solving problems by both teachers and students.　At the third stage, instructors, helping in a casual manner, have students use and activate drawing and solve problems cooperatively.　

　　On the other hand, the author, at each stage, incorporate the method to have students draw and the viewpoint to overcome the difficulties of drawing.　By this process, together with the understanding of tactics of drawing, the author aims to have students use, utilize and get accustomed to drawing.　In other words, the author, making intentionally the situations for experience, has students gain the utility of drawing and how to activate drawing.　

　　Besides, changing the teachers’ qualities of instructions about drawing and repeating this process, the author shift drawn languages to written language, positive activities of various kinds of expressions.　

　　This article is concerned with “class culture” in mathematics learning based on the widely accepted idea of ‘culture’.　We define the purpose of the article as follows: ・Building on previous studies about the characteristics and effects of culture, we will consider and discuss the status and role of “children’s questions” in mathematics learning.

　　As a result of our discussion, we have gained several insights (from ① to ⑫).　In addition, we have given several examples of children’s questions.

① “Children’s questions” are generated in a “class culture.” Moreover the former in turn contributes to the shaping of the latter.

②　Individual “children’s questions” are often related to something universal or paradigmatic.

③　“Children’s questions” can provide strong motives for cooperative learning in school.

④　“Children’s questions” can help overthrow the stale procedures of learning.

⑤　“Children’s questions” can stimulate the children to surpass themselves. ⑥　“Children’s questions” can enliven activities for collective learning. ⑦ “Children’s questions” can instigate interaction between tangible resources (such as textbooks, reference books, teaching materials, etc.) and intangible ones (value judgments, beliefs, thought processes, behavior, etc.).

⑧　“Questioning” can be one of the effective strategies for mutual nurturing.

⑨ Learning activities in which the people involved appreciate and recommend “questioning” can lay the foundation of the understanding of others.

⑩　“Questioning” can encourage learners to strive for superior values.

⑪　 Learning activities in which “children’s questions” are appreciated and recommended may be valuable not only in terms of efficiency and economy but also in terms of spiritual and moral cultivation.

⑫　“Questioning” can trigger the formation of true identities.

　　New Instrumentalism of Mathematics Education, I compare it with the mathematics education of Models Modeling Perspective which Richard Lesh advocates.　Both are based on the Dewey philosophy.　Constructivism is a learning theory based on constructing activity.　Rather than constructing activity, Models Modeling Perspective prizes a construct.　The construct is modeling which can be successful beyond school.　New Instrumentalism of Mathematics Education prizes the construct as a tool.　New Instrumentalism of Mathematics Education and Models Modeling Perspective have three steps of learning processes.　In the 1st step, each extracts a model or a content of mathematics as a tool from an everyday situations.　In the 2nd step, each polishes a model or the contents of mathematics.　In the 3rd step, each aims at use the model or the content of mathematics as a tool in an everyday situations.　Thus, both study form is very similar.　Therefore, when making a lesson on the New Instrumentalism of Mathematics Education, the lesson of Models Modeling Perspective is consulted very much.

　　The purpose of this paper is to examine the effectiveness of lesson organization based on the Koyama’s (2010) “two-axis process model” of understanding mathematics for developing students’ understanding in high school mathematics classroom.　By using the principles and methods for lesson organization based on “two-axis process model”, the authors clarified levels and embodied learning stages for understanding the theorem of sines and cosines as a topic of “Figure and Measurement” in high school subject Mathematics I.　We did an action research with a pretest, mathematics lessons, and a posttest in order to examine the effectiveness of our lesson organization in terms of improving students’ mathematical understanding of this topic.

　　As a result of qualitative and quantitative analyses, the following two points were found out.

(1) About the understanding level of “general mathematical relations”, all students who understand how to prove the theorem can use it in solving mathematical problems, while the opposite is not the case.

(2) In order to shift their understanding up to the level of “general mathematical relations”, beforehand students need to understand the “relations of mathematical entities” at a certain level.

　　The first suggests that it is useful for teacher to make a distinction between understanding the theorem itself and using it in solving mathematical problems at the level of understanding “general mathematical relations” in teaching and learning of mathematical topic in “Figure and Measurement.” The second suggests that it is important for teacher to aim at students’ understanding of “relations of mathematical entities” in order to shift their understanding up to level of “general mathematical relations”.　These findings verify the effectiveness of our lesson organization based on the two-axis process model in high school mathematics classroom.

　　Zambian Ministry of Education has emphasized learners as the centre of the entire education in their educational policy enacted in 1996. However, there is a gap between the ideal lesson, which is “LearnearCentred” and the real Zambian lesson, which is “Teacher-Centred”.　In this context, “Learner-Centred” is obscure for both the Ministry of Education and the teachers.　Therefore, the objective of this study is to find out the existing conditions and their challenges of the “Learner-Centred” lesson in mathematics at basic education level in Zambia.

　　Three challenges which are to expect pupils’ reactions in every lesson, to give pupils the necessity of learning mathematics and to introduce the opportunity for considering the meaning of the mathematical contents have been mentioned for teachers.　Moreover, the educational policy has two challenges which are to give teachers concrete advises for the “Learner-Centred” lesson implementations and to discuss the principle in the background of “Learner-Centred” in order to clarify the intention of components about “Learner-Centred”.

　　The purpose of this paper is to show whether mathematical activity is really activated by open approach, and to clarify the educational value of the open approach.

　　The value that the author describes refers to the transition of students into higher levels based on “the levels of mathematical ability” defined by Nagasaki (2007).

　　The author developed the open problem in the unit of a “linear indefinite equation with two unknowns” in action research training.　Moreover, the lesson which solves it by group study was tried. The author analyzed the proposals from every group, the videos, the protocol, and the students’ reports. The following things became clear from the result.

(1) Although it was analysis of the specific group, providing an environment where students can speak in comfort makes it easier to improve “the ability to discuss topics in mathematical terms” more effectively for the students who are hesitant to make a presentation or raise questions.

(2) Although there is the necessity for examination of an assessment standard, the open problem method effectively raises “the ability of students to use mathematics by following the rules.”

(3) Many proposals beyond the teacher’s anticipation were made through productive group study.　By devising how to take up various views, “the ability to find mathematical law, a rule, etc.” is raised.

(4) Through the open approach, a teacher can evaluate the students’ activities as well as the lessons carried out previously.

　　However, the educational value of open approach will not be achieved without enough structured lessons using a teacher’s tools, ideas and mathematical activity efficiently.

　　While “literacy = reading/writing ability” has been the basis of the early stages of modern societies as an essential and elementary component, i.e. an entrance, in modern education systems, which was later characterized as the indication of “academic capability,” the author of the present article assumes that “literacy = wisdom” in the contemporary context refers to the basis of the postmodern world and an exit point of education systems, which is standardized as “practical capability.”　The present study discusses the educational practices of Fukuyama Junior And Senior High School Attached To Hiroshima University in the aim of proposing new curriculum with “literacy” positioned as practical capability in secondary education in the age to come. The school was designated as a pilot school for research purposes by MEXT in 2009 and has made efforts in researches for “Development of Secondary Education Curriculum to Cultivate Critical Thinking” for 3 years.　 In this article, efforts in the “Period for Integrated Study” by the math department for the research are reported and specific classroom practices to cultivate critical thinking from the development unit “Random Numbers and Simulation” are discussed. As a result, reality in critical thinking could be seen in reactions of students in the classroom.

　　In our country, ‘in-school teacher training’ have been planned and conducted for the purpose of ‘the formation of teaching competence for the teachers’ in each school, where lesson study, as a method of research, is carried out traditionally centered on some classes open to other teachers. It was also shown in “The Teaching Gap (Stigler & Hiebert 1999)” that lesson study has improved the quality of Japanese mathematics teaching, and then Japanese traditional lesson study is now spreading internationally.　However, the concrete effects and their factors on the formation of mathematics teachers’ competence for teaching through lesson study have not been yet clarified enough.

　　In this paper, we focused on the qualitative change of the teacher K’s ways of teaching which occurred while we conducted the ‘School Support Project’ with lesson study.　The project was done in a seventh grade classroom with the class teacher K for improving mathematics classes over a four-month period from September to December, 2012.　We took two typical teaching episodes in oder to compare the qualitative differences before and after the project.　We analyzed them in detail by using mainly two theoretical perspectives: “the Types of MATOME” and “the Didactical Situation Model”.

　　The analysis of the two episodes revealed the qualitative changes of the teacher K’s ‘ways of teaching’, which are characterized as follows:

　　　･ In some teaching situations, the subject of control in the class interaction changed from the teacher to the students at all levels except the goal level.

　　　･ Not only the type Ⅲ-a of MATOME but also the type Ⅲ-b of MATOME, Student-Guided MATOME, came to be observed.

　　We clarified some important elements causing these changes.　Among these elements, the teacher K’s shift in her listening way seems to be closely related with her new experience, in which she watched her students’ learning processes carefully as an observer several times when the other member of the project team taught mathematics in different ways in her classroom.　This fact suggests the necessity of long-term cooperation with others in Japanese traditional lesson study.

　　This study explores the development of high school mathematics classrooms in terms of the teaching and learning for connecting students and mathematics from a cultural perspective.　We in particular focus on the learning of high school students who are less confident and less motivated through examining what cultural perspectives are, what the bridges are between everyday thinking and thinking in mathematics, and the learning for enhancing students’ self-efficacy.　Culture is in this study conceived as thinking and habits held in common by human beings.　As a result of our analysis and discussion, we got three points for developing high school mathematics classrooms.

　　First, we discussed the learning space of play, the route of learning based on theories of the didactical situation or the semiotic chaining, and the learning environment in which students’ self-efficacy is enhanced.

　　 Second, in order to develop the teaching and learning bridging between students and mathematics, we have to articulate the gap between students’ thinking based on everyday life and thinking in mathematics, and develop the classroom lesson as semiotically-connected practices.

　　Third, students may gain both knowledge and activity at the same time through participating in an activity, and enhance their self-efficacy through the activity in the integrated environments where their ideas connecting between their everyday thinking and thinking in mathematics are activated and encouraged.

　　Wakayama University has a large amount of archives of the Wakayama Normal School, the predecessor of the present university.　Last year, third grade test papers and some other items around the late Meiji era were newly organized.　These papers serves us actual contents of solid geometry instructions at the time before the reform movement of mathematics education.   

　　The test problems especially concerning the relationship between a line and a plane seem extremely difficult from the point of today’s view.　For example, “Find the locus of the foot of the  perpendicular line given to the straight line in this plane passing through the fixed point in this plane from the fixed point besides the plane.” 

　　 It seems that even making a sketch is not so easy.　In this case it is necessary to make reasoning with demonstration using some definitions and theorems, and the reasoning would also refine the sketch.　This is a characteristics of the solid geometry problems at that time.　We can notice that reasoning with demonstration were emphasized in the solid geometry even more than the plane geometry.　We can also realize that normal schools had a important role of the secondary education. 

　　Many difficulties in learning probability have been reported by several researchers from psychological perspectives.　One example is the Law of Small Numbers, which is a well-known misconception in probability that has been described and considered in the literature for several decades.　People who hold this misconception mistakenly assert that the Law of Large Numbers is applied to small samples as well.　The reasons why the Law of Small Numbers happens are usually explained by means of heuristics and biases. However, the question how are these heuristics and biases overcome? do not have yet any clear answer.　The purpose of this paper is to clarify a process of overcoming the Law of Small Numbers from an epistemological perspective of negation.　For this purpose, the Iwasaki’s framework on negation in concept formation is applied for describing overcoming the Law of Small Numbers.　A process of overcoming the Law of Small Numbers consists of three phases from the framework.　A first step is negation of proportion and construction of substance of proportion.　A second step is analytic negation in which substance of proportion is negated by viewpoint of necessity, and the concept of probability is constructed in this phase.　A third step is synthetic negation in which concept of probability is negated by viewpoint of contingency, and overcoming the Law of Small Numbers occurs in this phase (Fig.).

　　The purpose of this research throws new light upon a connection between research of the Jigsaw Method on Mathematics Education and research of mathematical communication based on these comparisons.

　　The greatest feature of Jigsaw Method on Mathematics Education is securing the opportunity of an argumentation to all the children.　This is a foundation of the structural aspect of the Jigsaw Method on Mathematics Education as a group learning method.　The foundation of this structural aspect has the feature of four more points.　They are “a setup of two or more expert subjects”, “a setup of the integrative viewpoint of an argumentation”, “problem solving according to people two or more”, and “serious consideration of justification.”

　　Two points are mentioned as a practical feature for a teacher to actually practice the Jigsaw Method on Mathematics Education.　They are “support in an expert group”, and “ascertaining of group support.” As for these, both are concerned with momentary judgment within a lesson.　This is the contents relevant to teacher education.

　　In my general survey of the previous study of mathematical communication, it was arranged by four views which were aim of education, way of education, development of mathematical communication, and structure of mathematical communication.　

　　The research as aim of education was able to find out the common feature with the structural aspect of the Jigsaw Method on Mathematics Education. The research as way of education was able to point out the common feature with the Jigsaw Method on Mathematics Education about growth of the knowledge by an argumentation from the epistemological feature based on social constructivism.　The research as development of mathematical communication pointed out that two points, a setup of the viewpoint of the integrative argumentation from the structural aspect and the practical feature, had a common feature.　However, two points of the practical feature showed the necessity of making it concerned with a theory of teacher education.

　　The research as the structure of mathematical communication pointed out that the epistemology in mathematics education is related with the Jigsaw Method on Mathematics Education.　

　　The next question of this research was pointed out four things.　First of all, it is constructs the creation principle of the teaching materials that is based on the structural aspect of the Jigsaw Method on Mathematics Education.　Secondly, it is constructs of the creation principle of the lesson that is based on the practical feature of the Jigsaw Method on Mathematics Education.　Thirdly , the research on the relation of the practical feature of the Jigsaw Method on Mathematics Education and teacher education are needed to investigate.　Finally, the new practice of the Jigsaw Method on Mathematics Education which utilized ICT are also needed to investigate.

　　The present study is a survey research targeting Japanese Grade 5 students at elementary school as well as Grade 9 students at junior high school, which used the framework of the international comparative study “The Third Wave” , in which 11 countries are participating.　The questionnaire carried out in this study consists of three types of questions regarding “What I Find Important in Mathematics Education”.　This paper discusses general trends, as well as commonalities and differences between the target groups (HiroshimaMiyazaki-Osaka prefectures, elementary-junior high school levels, or among schools) regarding the 64 items in the Section A of the questionnaire.　

　　As a general trend, many of the answers were positive, and general items such as “Problem solving” and “Explanation by the teacher” are shown to have a strong positivity.　On the other hand, “Appreciating the beauty of mathematics” was shown to have a lower value among the answers, which might be a point from the attitudinal side to be considered.　A significant difference of more than 80% (55 items) could be observed when elementary and junior high school levels’ answers are compared.　Similarly, a significant difference could be observed among the three prefectures ranging from 30% to 60% (22 to 44 items), as well as among the schools in Hiroshima prefecture, ranging from 20% to 50% (14 to 39 items).　Only the items “Problem solving” and “Explanation by the teacher” do not show a significant difference among prefectures and schools.　In this sense, it is necessary to be cautious when discussing the common values.　Regarding the comparison among classrooms in the same elementary school, almost no difference was observed; in that sense, it might be said that the influence of the homeroom teacher system is not strong.

　　The purpose of this paper is to reveal the tasks to structure the performance in the arithmetic education. For the purpose, the first described an idea of PA, considered some concrete instances in Pedagogy and Mathematics education. These view pointed out that there was the need to pay its attention to the original performance in arithmetic education. It was clarified that Mathematization had included position as a core of performance from immutability and trendy point of view.

　　Therefore, the second considered the historic flow in Mathematization of Freudenthal, Treffers, Lange. By these considerations and from the perspective of “mathematical method”, it was pointed out that Mathematization of Lange was required by arithmetic education in the future. Furthermore, based on them, it was clarified that a performance in the arithmetic education was globally structured as a cycle of Mathematization described by OECD / PISA.

　　It is difficult to push forward learning instruction, however, without a viewpoint to catch the performance in a cycle of Mathematization.　As a policy of that purpose, it is thought that a performance is caught as a model.　Furthermore, it is necessary to pay its attention to the transformation of the model. As a solution for the task, some researches of Gravemeijer to be focused on the model-of / model-for transition are very suggestive. According to this activity, there were four levels activity “activity in the task setting”, “referential activity”, “general activity”, “formal activity” in the process of constructing models.　These four levels of activity illustrate that models are initially tied to activity in specific settings and involve situation-specific imagery. These illustrated the transition of from “model-of” to “model-for” in Mthematization.

　　As a result, it was clarified that a rise of the mathematical recognition was appropriate to assess as the formation of the model and the model-of / model-for transition in learning instruction by PA in the arithmetic education.

　　The purpose of this paper is to clarify the mathematical activities as purpose.　For attaining this purpose, firstly it is described the mathematical activities by two points of view of mathematical modelling as process and thought of function as product of learning.　In addition, it is described finally the ideal of the mathematical activities from as purpose these two points of view focusing on proportion.　In the end, it is described future problems to materialize the activities.

　　Focusing on the purpose of learning and teaching, the mathematical activities are divided into two parts: “purpose model” and “method model”.　And for the nature of mathematics activities and in the current discussion, it is necessary to learn mathematical methods.　However, the status of learning and teaching is focused on mathematical contents, so not method so much.　Consequently the problem is that find out the method of learning and teaching for mathematical activities as purpose.

　　Then in the activities of proportion, it is necessary to find two depending quantities.　In such a case of using mathematical model as a solution, it is necessary to make and remake mathematical model.

　　And to materialize the activities, there are four problems: (1) To design lesson and practice; (2) To study another products of learning; (3) To develop teaching methods; and (4) To develop a curriculum centered on methods.

　　The purpose of this paper is to characterizing the teaching and learning aiming to foster today’s mathematical literacy in statistic region, using unit on descriptive statistics as an example.　To achieve this purpose, the teaching and learning which exemplified in this paper were considered from perspectives of mathematical literacy.

　　The results are followings.

(1)　 The process of teaching and learning aiming to foster today’s mathematical literacy was clarified from the viewpoint of social change.　This process was the teaching and learning incorporating structure-oriented into application-oriented mathematical methods.　In this process, it was important that ways of solving (i.e. model) toward a real world’s problem are gradually extended as they become sharable (used by other people) and reusable (in other situation) (cf. Lesh & Zawojewski, 2007).

(2)　 “Variability” was thought as the statistical conception which plays central roles in statistics. And this conception’s way of developing was identified through considerations of prior researches on statistics education (e.g. Reading & Shaughnessy, 2004).

(3)　 The teaching and learning in statistic region which exemplified in the latter part of this paper were characterized from the perspectives on mathematical literacy. The characteristics will play a significant role when we want to embody the teaching and learning aiming to foster today’s mathematical literacy in statistical region and other topic region.

　　The purpose of this research is to clarify the aspects of students’ development of functional concepts during the transitional period from elementary to secondary mathematics based on the findings from the qualitative research on the observations of the elementary fourth and fifth grade classrooms and the teaching experiment research on proportion in the seventh grade.

　　The findings and the points as a result of our analysis and discussion are as follows.

（1） The viewpoints of symbolization and signalization in the van Hiele theory can work for examining the development of students’ functional concepts during the transitional period from elementary to secondary school mathematics.

（2） It is important as the foundations in the learning of function that students have their images of motion of the event, switch them from the analog viewpoint to the digital one, and conceive the dependency relations in which they control the dependent variable based on the independent variable.

（3） Symbolization may be caused when students explain their activities on table, graph and expression using the languages for telling the event. Also, mediating the plural representations with each other through situating the event as a core may enhance their symbolization of each representation, and moreover it can lead to the signalization among the representations.

（4） When students explore an aspect of change in the event （e.g. speed） in the representations and connect them with each other based on the interpretations of the event, they can conceive the mutual relationships among table, graph and expression, and in particular enhance the meanings of rate of change in the functional concepts.

　　It is problematic that middle school students cannot formulate an equation or represent the situation in a graph in their problem solving, while they can solve the given equation and represent the graph of the given equation solely.　Expressions or equations, graphs, and tables should be effective tools for their problem solving. Therefore, it is needed to develop integrated curriculum for Function and Equation.　We have edited experimental textbooks as actualized our intended curriculum.　In this paper, we insist that it is necessary that the notion of mathematical activity is embedded consistently throughout the curriculum for coordinating the intended curriculum in the implemented curriculum.　In many curriculum development studies, the focus devoted to this point is lack or short because of the difficulty of a long-term implementation.　Our research emphasizes this point, and thus is intended to develop curriculum through a long-term lesson studies.