The area of the Pythagoras triangle is the sum of area of the Pythagoras triangle that is smaller than it except some exceptions.　The exception is the case of M=2N (M,N is an independent variable of the solutions of Euclid).

　　Furthermore, these relations are expressed as the sequence and constructed in the Fibonacci series Next, the Pythagoras number is distributed on various parabolas group on the coordinate which assume two axes into two sides sandwiching the right angle.　The degree of leaning of the axis of symmetry of the parabola group is 0 in case of the basic formula (Euclid solution) of the Pythagoras number. In addition, it is 0 and ∞ in case of “the unit formula”of sum of area.　Furthermore, the axial degree of leaning converges to 2 at an early stage in case of “the general formula”.

　　The theory of“Transcending Recursive Model”(Pirie, S. & Kieren, T., 1994) is originally used to describe students’understanding in mathematics learning.

　　In this study, I applied this theory normatively in the lesson for problem solving learning, and I considered how to organize mathematical activities in the lesson.　Mathematical activities play the central role on the step of understanding the problem or devicing the plan (Polya, G., 1980).　Moreover, they were refered by students depending on the need at the step of carring out the plan (Polya, G., 1980).

　　To use“Transcending Recursive Model”in the lesson for problem solving learning enables to discuss understanding and problem solving on the same base, although they were told in the different context.

　　The purpose of this study is to consider the unit about multiplication and factorization of polynomial in lower secondary school, and to develop a learning environment about this unit.

　　Especially, to achieve this purpose, we focus attention on a calculation format that is called“Multiplication Cross”.　This calculation format has been introduced in“Das Zahlenbuch”by E. Ch. Wittmann.　And this concretizes trial-and-error method in student’s learning activities.　So we used this calculation format as a problem for designing learning environment of multiplication and factorization of polynomial.

　　As empirical study, we designed learning environments of 6 unit times, and practiced experimental lessons in grade 9 class.　In consequence, we got following 4 suggestions for this design of learning environment.

1）“Multiplication cross”can be a scaffolding for students to solve problems of multiplication and factorization of polynomial.

2）“Multiplication cross” induce more inquiry learning in students learning activities.

3）“Multiplication cross” give students the ability to solve evolution problems of multiplication and factorization of polynomial.

4） We have to pay attention to how the formulas are introduced in learning of multiplication and factorization of polynomial.

　　In accord with these suggestions, we would like to continue to investigate how this learning environment in lower secondary school can be redesigned.

　　Recently， the importance of mathematical-activity has been increasing and， in high school mathematics，the task-based learnings as mathematical-activities are obligated.　As its contents， the application-oriented mathematics， in which the functional value is emphasized， tends to be taken up. However， as it is pointed out by Wittmann， the balance between application-oriented and structure-oriented mathematics is important．In this paper， we propose the “structure-oriented mathematical-activity”that can induce the learner’s proactive and dynamic considerations， and in which， the learner can value the contents and the considerations for themselves， not for their functional values．

　　Quoting the theory of learning motivation by S. Ichikawa，the practical research on SLE by M. Suzuki，Shigematsu & K. Hino，and the study on introducing experiments in mathematical teaching by Y. Iijima，we propose the cyclic model of structure-oriented mathematical-activity， which is also a miniature of the researching activities of real mathematicians．

　　Also we summarize the view points to develop the materials for structure-oriented mathematicalactivities， and offer an example of such materials， relevant to the geometry of polygons and polyhedra．

　　In most of the developing countries， it is recognized that the connection between mathematics and science should be emphasized in education in order for them to industrialize.　However， pupils often fail to connect these two subjects by themselves.　This phenomenon is known as context­dependency.　The objective of this study is to clarify context dependency between mathematics and science in the case of function at high school level in Zambia.　

　　One hundred sixty five pupils in Grade 12 at a girls’ high school in Zambia were chosen for this research. The research was conducted by using the same two types of tests about functions， providing different contexts between physics and mathematics.　Context­dependency was studied through pupils’ used solving methods and interview answers, as well as by using chi­square test.　

　　The results showed that more pupils got correct answers in the question to find pattern from the table in the physics context.　Many pupils noticed horizontal relations in the physics context and vertical relations in the mathematics context.　When the interviewer explained the situation by using the physics context， they noticed more relations.　On the other hand， more pupils answered correctly the question to find the unknown value from the table in the mathematics context.　Most of the pupils noticed similarity between the physics test and the mathematics test.　With instruction from the interviewer， pupils were able to use mathematics formula in the physics context.　

　　It was found that these pupils had difficulty finding patterns in the mathematical context while the physics context helped them understand.　They had difficulty finding unknown values in the physics context and the mathematics formula helped them to find the answer.　However, challenge for clarifying the relations between mathematical connection with other subjects and characteristic of subjects and the relations between context­ dependency and mathematical connection with other subjects were also pointed out.　

　　In mathematics education the process of generalization is one of most important issue.　We define generalization as knowing process that has epistemological direction from particular to general.　It is distinguished from the process of extension.　Then, we focus on the process of validation in generalization. Because requisites for the validity of generalization and extension are completely different.　In this article, our interest is peculiar validation of generalization; that is, what particulars are involved in the general?

　　Proof and/or proving are necessary conditions for the validation, but they’re not sufficient conditions.　We focus on the“abstract from some things”and“abstract something”(Kant, 1781) in the process of abstraction．The former intends to leave from one’s perception; for this reason, previous studies have paid attention to the former.　By contrast, the latter is intended to stay one’s perception and to lead abstracted concepts to one’s perception; that is, how to use concepts.　So, the latter rejects the notion of“general concept”; generality of any concepts are derived from only how to use concepts．Every“general”contains infinite particulars, therefore “general concept”will be self-existent and limitless generality.　On the contrary, any generality derived from “how to use concepts”must relate to concrete somethings. Thus, we can clarify“ what particulars are involved in the general”．

　　As a result of this study, the following didactical suggestions are concluded: we must be seeing students’s abstraction as two different knowing, and focus on each of them in a didactical situation; when students ask about“what particulars are involved in the general?”in a specific situation, their generalization will be improved．

　　The purpose of this research is to clarify the aspect of symbolization of tables, formulas,and graphs,and signalization in development of a functional concept on the basis of the theory of levels of thinking by van Hiele (1986).　In development of the functional concept, I think that symbolization is that it is to recognize the situation in tables, formulas, and graphs, and when the situation that a symbol works on other expression appeared, I consider that a symbol functions as a signal.

　　In order to clarify the aspect of symbolization and signalization, I observed and analyzed the lessons of a fourth grader “how to change” and the fifth grader “formula using ○ and △”, and　investigated the aspect of development of a children’s functional concept.

　　As a result, the following knowledge was acquired :

・ In order to symbolize graphs, it is important that children consider motions of hands (gestures) as the concept of the function and the change.

・ In order to symbolize formulas, it is important that children focus on the structure of the situation by unitizing and norming (Lamon, 1994).

・ By following a change of the phases of thinking (van Hiele, 1986), children found that it was difficult to exceed “free orientation (phase 4)”.

　　Faculty of Education and Culture and Graduate School of Education, University of Miyazaki, have been implementing a research project for the support of continuity from elementary through junior secondary levels since 2011.　This project aims at conducting comprehensive research on the theory and practice of continuity from elementary through junior secondary levels, developing a teacher education programme (pre-service and in-service) based on the research, and pronouncing research outcomes.

　　For the purpose of contributing the research project, this paper intended to identify the trend of previous researches on the continuity from elementary through junior secondary levels in Mathematics Education and to gain suggestions for the programme by reviewing the 63 selected previous research papers in terms of research questions, research methods and outcomes.　As a result, the following trends were identified:

　　a) In terms of “teaching contents”, the previous researches have proposed a lot of perspectives and concrete ideas on development of teaching contents and lesson designs especially for the domains of “Numbers and calculation/algebra” and “geometry”.　In addition, they have also developed and practiced some curriculum for continuity from elementary through junior secondary levels.

　　b) In terms of “students”, they have identified the characteristics of students’ understanding and learning during the transitional period (from 10 to 13 years old) at a certain level. However, it is necessary to research more about their affective dimension.

　　c) In terms of “teachers”, although they have proposed some desirable teaching activities, there is a need to implement more comprehensive research.　For instance, the points of view on the gap of school/teacher cultures between at elementary and junior secondary levels, the difference of teachers’ academic backgrounds should be considered.

　　d) In terms of “context”, there is also a need to conduct more researches, for instance, focusing on the influence of examination, the expectation of parents, the perspectives of learning in community etc.

　　In this paper, the trend of problem solving is first analyzed. The research issue is identified that values, which are related to students’ sense making, are rarely taken up in problem solving activities. Therefore we take up the values aspect of problem solving, and especially focus on social values.

　　The results are followings.

　(1) Three types of values in mathematics education are identified based on reviewing the previous studies. The first values are directly related to the characteristics of mathematics and the second ones correspond with the societal events and values. The third ones are rather individual preference, but also under the influence of social values. And through this review, it is also found that there are only few researches which deal with student’s values in mathematics education.

　(2) It summarizes aspects and examples of students’ values emerging in problem solving activities. There are three aspects of individual, small group such as family, friends, class, and large group that is society at large.

　(3) In addition to the above summarization of values in (2), four types of socially open-ended problems, such as “distributing”, “making a rule”, “selecting”, “foreseeing” are identified to prompt students to come up with social and individual values.

　　The purpose of this research is to show the efficiency of the developed curriculum of geometric figures in compulsory level based on the theoretical framework in order to mediate between elementary and secondary school mathematics.   Especially, we focus on the transformation of the students’ logical recognition of geometric figures through a four-year prospective practice.   During this four-year practice, teaching units that aimed to progress students’ awareness toward the relationship between properties of a geometric figure were developed and practiced.

　　In this paper, we are interested in transformation of the students’ logical recognition who have been taking part in through out the four-year prospective practice.   After the lessons which were designed to initiate into the learning/teaching of proof at the 8^th grade, the transformation of the students’ logical recognition of geometric figure were evaluated by the use of the same performance task developed four year ago.   The result of the evaluation shows that the developed curriculum is effective to promote the transformation of the logical recognition of geometric figures.