The purpose of this paper is to clarify curriculum issues concerning the development from descriptive statistics to inferential statistics based on an epistemological perspective of negation, and to refer to perspectives about solving the issues.  For achieving this purpose, three tasks are worked on.  The first task is to analyze history of statistics based on Otaki’s conception model so that the static conception of descriptive and inferential statistics is interpreted.  The second task is to make clear the dynamic conception of descriptive and inferential statistics based on Iwasaki’s framework on the negation theory in concept formation.  A framework about the development to inferential statistics is made by solving these tasks.  The third task is to analyze mathematics textbooks from the perspective of the framework.  As a result, it is found out that the development to inferential statistics is not necessarily intended.  

　　In the development to inferential statistics three steps of negation are required.  The first step is negation of statistic and construction of substance of a statistic, which is not the same of one in descriptive statistics. The next step is analytic negation in which substance of a statistic is negated by viewpoint of decision, and it is regarded as a random variable in inferential statistics.  The final step is synthetic negation in which a random variable is negated by viewpoint of a parameter, and the relation between descriptive and inferential statistics is made clear.  This sequence of negation is shown below (Fig.).

Fig. the development from descriptive statistics to inferential statistic

　　In the Republic of Zambia, Lesson Study Support Project was started by JICA (Japan International Cooperation Agency) technical cooperation in 2005.  The main purpose of this project was to promote implementation of Lesson Study through existing In-service training program.  The Lesson Study contributed to disseminate methodology of Lesson Study to stakeholders.  However, the Zambian Lesson Study was pointed out several qualitative problems (Ministry of Education, 2009).  There were so many teachers who consider simply conducting the group work as the learner-centered lesson.  So, Zambian teachers have easily received superficial idea and they have some difficulty to develop those ideas through lesson practice.

　　Therefore, in this research, author considered the ideal future for the Zambian Lesson Study through grasping and analyzing the current situation of lesson practice by focusing on Teachers’ group discussion during the Lesson Study.  The sources of data were mainly participation observation during the Lesson Study cycle.

　　Through the result of analysis, author clarified the components of the Lesson Study to activate the teachers’ change.  1) Enhancement of research on subject pedagogy (kyozai-kenkyu).  2) Perspective of curriculum development.  3) Reframing of learner-centered lesson.  The further research is necessary for polishing this components.  In developing countries, the research on subject pedagogy is not developed. Therefore there is no opportunity to touch the knowledge to be created from the research.  This situation is more difficult to activate the Lesson Study.

　　Many Zambian teachers believe positively that the Lesson Study is one of the best way to improve teaching competency.  On the other hand, there are things which teachers’ change possible (General teaching method such as group work) and impossible (Subject matter knowledge) in a short period.  They will be also has a significant effect on the development and continuation of the Lesson Study in the future.

　　 The purposes of this study involved in mathematical representation in mind are following two.  The first one is to consider the reality on mathematical education and second one is to suggest refining its reality.

　　 From previous studies about the reality on mathematical education, this study considers that the definitions of reality are to actualize the existence of learners and to win the sympathy of learners with considering both substantiality and humanity.  Then, this study suggests following three statements for refining the reality. 

(1) This statement considers the meaning of the value of reality.  The reality that stands on its own merits, but actually it is the basement that produces the new values.  Therefore, the reality is not only to be actualized  the existence of learners as Kikuchi mentions.  Rather it actualizes the existence of learners.  In short, feeling the reality actualizes learners’ independence. 

(2) This statement considers the features on reality.  The reality on mathematical education is influenced by the context of learning in classrooms, the contents on learning, the existences of others, and sociality.

(3) This statement considers one of the basements of reality.  The reality is supported by the tacitness and it activates mathematical activities, provides broader possibilities, and facilitates the production to new values.

　　 The purpose of this paper is to develop and propose the strategies for theoretical prediction of mathematical conception formation.  We elaborate the notion of hypothetical learning trajectory proposed by Simon (1995).  The paper mainly consists of three parts. First, we formulated the general strategies for predicting a hypothetical learning trajectory through the reviews of previous research. Second, we formulated the conditions of some types of mathematical conceptions, focusing on the operative origin of mathematical conceptions and on the educational implication from constructivism.  Third, in order to show the usefulness of our formulation of the strategies for theoretical prediction of mathematical conception formation, we predicted a hypothetical learning trajectory when a student reads a Japanese high school mathematics textbook, and discussed the implication from our analysis.

　　 In mathematics learning, we expect students’ activity that they create meaningful mathematical knowledge and make use of it by themselves.  In such activities, we need to distinguish the stage of guess and justification and we focus on the stage of guess.  Wada (2009) pointed out the importance role of the reasoning of abduction in the stage of guess.

　　 However, it is not fully clarified what types thinking are the basis of abduction.  Thus, we focus on the Yonemori (2007) because he argues about the nature of abduction.  According to him, there are the stage of insight and inference in the process of abduction, and both of them play essential role for to form a hypothesis. In the stage of inference, there are four criteria for to choose a hypothesis. When subjects form and choose a hypothesis, s/he use the criterion consciously and reflectively.  For this reason, we argue the criteria by to analyze examples and to compare Nakazima (1981).

　　 As a result, we propose three criteria (the criterion of the hypothesis formation) that played primary role for abduction.

　　 The aim of this paper is to find theoretical foundations of teaching mathematical proof for the sake of curriculum development based on the mathematical activities throughout six years of secondary schools in Japan.  To accomplish this aim, we first search for, through the review of related literatures, the principal aspects of proof and proving that should be taken into consideration for the curriculum development.  In particular, we examine the distinction often made in Japan between “proof” (shoumei) and “demonstration” (ronshou), the idea of “local organization” introduced by Freudenthal (1971, 1973), and the idea of “mathematical theorem” proposed by Italian research group (Mariotti et al., 1997).  This literature review shows that proof and proving are often discussed in relation to the propositions to be proven and the system of mathematics within which proof is carried out.  And we consider that three elements―statement, proof, and theorem―that characterize a “mathematical theorem” are principal aspects that evolve throughout the learning in secondary schools.  We then develop and propose a framework that allows us to design a curriculum by studying two questions: what kinds of teaching contents should be included in each aspect?; what kinds of evolution should be envisioned on the nature of each aspect? In this development, we lean on different perspectives such as the viewpoint of mathematical logic to identify different kinds of “statement”, Freudenthal’s idea of “local organization” and “global organization” to characterize different levels or natures of “theory”, etc.  In this paper, we also discuss the relationship between the developed framework and the mathematical activities which is another crucial point to be considered in our curriculum development around mathematical proof. 

　　 The purpose of this paper is to consider of effect and influence which focus on children's questions core in arithmetic lessons. Therefore we focus the characteristic activities, children’s discussion, children’s writing concerned in children’s questions and theme in teaching unit of decimal division. We consider of effect and influence which focus on children’s questions by qualitative research method.

　　 As a result of our discussion, we have gained several insights such as below.

　(1) To promote collaborating learning in arithmetic lesson, children’s questions have effective and central role. At the same time children’s questions generated by children’s discussion take direction of the collaborating learning.

　(2) Children’s questions and discussion based on the childen’s question promote to listen various opinion by other people, to catch and interpret other view, to advance thinking in the class. Children’s questions and discussion based on the childen’s question make and foster foundation of understanding about others. Moreover children’s question form meaning renewal about mathematical concept and thinking which they learned previously. But teacher must grasp, interpert and appreciate mathematically about children’s questions in arithmetic as soon as possible.

　(3) Through arithmetic lesson move toward from beginning to the end of teaching unit of decimal division, childen’s questions proceed to something universal and mathematical.