The purpose of this paper is to identify a structure of probabilistic misconception. Many people mistakenly assert that the Law of Large Numbers might be applied to small samples as well. Traditionally, in terms of misconceptions, this problematic situation is investigated through the Law of Small Numbers. This paper gives a new interpretation to this phenomenon from the perspective of commognitive conflicts. Consequently, it is shown that the Law of Small Numbers results from two metarules (namely the compensating rule and the generalizing one).

　　In this paper, a Triangle of Discourse (Figure 1) is constructed and used as a commognitivist framework for interpreting discourses. The triangle of discourse is roughly a commognized version of semiotic triangles on meaning. Then, the simplest commognitive conflicts are interpreted by two Triangles because commognitive conflicts arise between at least two discourses. Thus, a Dual Triangle of Commognitive Conflict (Figure 2) is constructed and used as a framework for interpreting commognitive conflicts. S_T, R_T, and M_T are a learned discourse in a context (the subindex T indicates teachers), whereas S_L, R_L, and M_L are a learners’ discourse in the context (the subindex L indicates learners).

　　Two types of misconceptions are distinguishable within the Law of Small Numbers when we take a detailed look at it. In the first type (LSN 1), “population / probability” as signifier is associated with “small sample” as realization. On the contrary, in the second type of misconception distinguishable within the Law of Small Numbers (LSN 2), “small sample” as signifier is associated with “population / probability” as realization. These two types of the Law of Small Numbers could be described by means of the dual triangles of commognitive conflict, as shown in Table 1. That is, the compensating and the generalizing rules are causes for the law of small number. The compensating rule is a metarule of routine procedure that we must compensate objects. The generalizing rule is a metarule of routine procedure that we must generalize objects. In learning probability, students should learn the new metarule, the swamping rule, which means that we must swamp objects.

Figure 1: Triangle of discourse

Figure 2: Dual triangles of commognitive conflict

Table 1: The two types of the Law of Small Numbers

LSN 1

S_T R_T M_T

Population / Probability Large Sample Swamping Rule

S_L R_L M_L

Population / Probability Small Sample Compensating Rule

LSN 2

S_T R_T M_T

Large Sample Population / Probability Swamping Rule

S_L R_L M_L

Small Sample Population / Probability Generalizing Rule

 　　The purpose of this paper is to enhance the elaboration of the principle of lesson organization (Table 1), summarized in this research (Ⅱ) (Hisadomi, 2014), through an action research of “event and probability” in high school subject Mathematics A.

Table 1. Principle of lesson organization in high school mathematics

Level of Activity＼Learning Stage ◇Intuitive ☆Reflective □Analytical

Activity in the task setting Referential activity Students’ learning activity General activity Formal mathematical reasoning

Students’ learning activity 

　　The result of qualitative and quantitative analyses of the lesson based on the principle of lesson organization showed the following two points:

　(1)  After expressing the task setting as experimental results (model-of), each of the learning stages: intuition, reflection, and analysis work effectively when the level of activity is raised;

　(2)  When Learning and Teaching of probability is conducted based on the principle of lesson organization, there is a tendency for students’ consciousness of “equally likely” to increase. 

　　 Learning and Teaching based on the principle of lesson organization, specifically for “quadratic function” in high school subject Mathematics Ⅰ (Hisadomi, 2014) and “event and probability” in high school subject Mathematics A, demonstrated the effectiveness of the principle of lesson organization.  From these two case studies, the principle of lesson organization was considered to be effective in students’ understanding of a new concept, especially in the introduction of a unit.

　　For children to increase their autonomy as well as to promote the construction of such a system towards the children’s autonomous learning, we intended to clarify how they can develop their autonomy by focusing on the stage of the goal of the learning cycle of RPDCA. In particular, this study tried to examine the four levels of the children’s goal setting which was set in our previous study.  

　　As a result of analysis, we found it difficult to objectively judge between the levels P2 and P3. Therefore, we set the level 2.5 as an intermediate level between P2 and P3, which permitted us to examine the qualitative differences in the children’s own learning and to clarify their development of goal setting activities. Also, it leads to the new hypothesis for developing the levels of goal setting activities. As another results, we found that there is an interrelation between the levels of goal setting and the achievement tests of their learning, and conversely that there are not any large differences between the achievements of mid-layer and low-layer children. Thus, it is possible for children to develop their achievements if they learn elementary mathematics based on the RPDCA cycle, especially with their proactive attitudes for the goal setting.  

　　As our future tasks, we will examine whether the stage of“plan”  (the goal setting) is interrelated with the other stages of the RPDCA cycle and how these are actually related with each other.

　　Generalization is very important knowing process in mathematics learning. However, in daily life, knowledge about the particular is easier than knowledge about general and enough for most of our purpose, and such knowledge sometimes may be more useful than knowledge about the general. Thus, on the one hand students have ability and tendency to generalize something from very young age (Vinner, 2011), but on the other hand they are not intending to generalize something (cf. Tatsis and Tatsis, 2012). Hence, the purpose of this study is to clarify for what do students generalize something in learning mathematics. In this study, we make a distinction between generalization and extension, and focus on the function of generalization in terms of its meaning, purpose, and usefulness for children.

　　Through reviewing literature on generalization (e.g., Harel & Tall, 1991) and philosophical considerations, six functions with their examples are identified; variablization, purification, unification, discovery, association, and socialization. Then we implied three didactical suggestions for teaching and learning mathematics in classroom. First, teacher should design didactical situations where students can discern meaning of the six functions of generalization. Second, the implied order and/or structure of the function has possibility of a principle of didactics for designing mathematics classes. Third, the function may become a guideline in forming mental habit for students through their experiencing the functions of generalization in mathematics classes.

　　Since 1993, the authors have conducted a series of theoretical and empirical studies on Mathematics Learning with a focus on children’s “Question”. In this paper, the author makes a variety of observations to highly deepen understanding of functions as well as attributes of “Question” in Mathematics Learning, by taking up O. F. Bollnow’s humanistic pedagogy as one of the research sources, on the basis of what he treated in his thesis on “Question” and existentialism.

　　The major findings obtained are as follows:

● There are rationale for giving children Mathematics Learning in the way that children as “Questioning Beings”are esteemed and encouraged as “Questioning Beings”. The reason is that “Questioning” is one of the intrinsic actions to human beings.

● Children’s action of “Questioning” manifests a life that seeks to grow freely and step ahead toward creation, resisting an irresponsible order.

● Children’s “Question”, notably, the “Question” that may yield discontinuous processes in Mathematics Learning comes to have the functions of encouraging children to advance toward the new level of Learning as well as returning Mathematics Learning to the basics when a teacher brings children the “Encounter” with worthwhile materials.

● What is demanded of a teacher about dealing with children’s “Question” is making the most of the “Question” as the “Encounter” with worthwhile materials by rousing children into recognizing the importance of mathematical facts that are inherent in the “Question”.

● Actualizing Mathematics Learning in which children as “Questioning Beings” are esteemed and encouraged should presuppose the fact that children themselves as well as a teacher and children have relationships of mutual trust and share hopes to be generated from such relevant relations.

　　 The objective of this study is to clarify the significance of connecting science and mathematics by focusing on methodological difference between science and mathematics.  First, methodological difference between pure science and pure mathematics was identified based on Popper’s three world theory.  In pure science temporal theory is developed through the interaction between World One and World Three.  On the other hand, in pure mathematics temporal theory is developed through the interaction within World Three.  Next, the difference of functional thinking in science and mathematics education was discussed. In science education functional thinking is used in concrete world and in mathematics education it is used in not only concrete world but also abstract world.  Finally, significance of connecting science and mathematics was considered.  The discussion concludes that connecting science and mathematics enable to discuss based on free interaction between World One and World Three.

　　 There is a considerable gap between developing and developed countries not only in the students’ cognitive aspect, but also in the students’ affective aspect in mathematical literacy of PISA2003.  In the case of Japan, the students’ cognitive aspect is in the highest level in this survey, although their affective aspect is ranked in the lowest level.  On the other hand, most of the developing countries have opposite circumstances from Japanese case.  For this background, this paper examines the relationship between cognitive and affective aspect through an international comparison by making a secondary analysis of PISA2003.  In particular, the data of mathematical literacy test and questionnaire for students were analyzed in this study.  The results indicate that there are some common points among 39 participant countries regardless of developing or developed countries.  One of the affective aspects which is termed “self-efficacy” brings about the positive effect to cognitive aspect.  Particularly the different levels of “self-efficacy” can be actualized in the items of openconstructed response.  Generally, the education in developing countries is in a very different circumstance from developed countries.  Thus, identifying a common trend between developing and developed countries could yield a contribution for international cooperation in mathematics education.

　　 This research is to approach the problem of connection of arithmetic and algebra from a viewpoint of algebraic reasoning.  Therefore, in this paper, we focus on the transition from pragmatic recognition to semantic recognition from a point of view of linguistics research.  The purpose of this paper is to practically examine that we constituted classes of mutual relationships between addition and subtraction from a viewpoint of algebraic reasoning and to clarify aspects of the transition based on aspects of algebraic reasoning.  

　　 As a result, the following things become clear. 

(1) The classes are effective to students’ understanding of mutual relationships between addition and subtraction.

(2)  Aspects of algebraic reasoning in the classes are firstly cognition of antonym by agreement of rules in diagrams and objectification of them, then cognition of synonym by interaction between inverse operation of  diagram, inverse calculation, and part-whole diagrams.

(3) Aspects of the transition from pragmatic recognition to semantic recognition are firstly cognition of antonym, secondly ambiguity, finally synonym.  The cognition of ambiguity, however, means not only meanings of  expression but also ideas of expression.  

　　 The purpose of this study is to develop the prescriptive model for designing social interactions in an elementary mathematics class which is effective and applicable to teaching practices at elementary school level. In this paper we verify the effectiveness of this model through a teaching experiment of ‘fractions’ conducted for two fourth-grade classrooms in a school.  The teaching experiment had four characteristics: a teaching material focusing on meanings of fractions, recall of the definition of fractions learned in third grade, illustrative representations of tapes, and an applied problem which promotes mathematical generalization.  As a result of analysis, we found out the following four results.

　　 Firstly the fundamental process of being conscious, solving by the individuals, solving by small group, being reflective, and then making agreement, in particular the small group activity, contributed to solving the problem.  Also, it was suggested that the children’s solving activities progressed from their ‘individual’ solution to ‘quasi-general’.

　　 Secondly the intentional support of illustrative representations by a teacher was quite effective.  Namely some children in one classroom were able to solve the problem of fractions for themselves and explain the  reason why their answer was correct clearly by using two kinds of illustrative representations of tapes.  In addition, the children negotiated that the length of one-third of a tape whose length was two meters was twothirds meter, as the solution of a problem, by differentiating two kinds of meanings of fractions.  Furthermore, it was confirmed that the children also explained such reason by translating illustrative representations into symbolic representations and generalized their solution of the original problem when they solved an applied one.

　　 Thirdly three kinds of social interactions such as social interaction with others, social interaction with the self and social interaction with representations, were activated by setting small groups. These kinds of social interactions contributed to developing the children’s deeper understanding of fractions.

　　 Lastly we could have two concrete suggestions for the improvement of teaching fractions by a comparative analysis of children’s activities in two classrooms.  It was so crucial for the children to have an additive view of the definition of fractions in order to solve the problem.  It was important for the children to realize two kinds of the structure which were embedded in illustrative representations of tapes.

　　 These four findings demonstrate the effectiveness of the prescriptive model of social interactions for teaching practices in elementary school mathematics.

　　 At the beginning of learning geometry at school, it is said that students have some difficulties to form mathematical concept of geometric figures because of keeping their own experienced based concepts. The purpose of this study is to figure out the difficulties of concept formation of geometric figures related to generalization.

　　 The author conducted a questionnaire for Filipino students in grade 1(n=40), grade 2(n=41), grade 3(n=40) and grade 4(n=41) at two schools in Metro Manila. The questionnaire consists of twenty shapes, triangles, quadrilaterals, sectors, unclosed shape and others. These shapes include prototype example and non-prototype examples, stable and unstable position and orientation. And also three choices, “No”, “Maybe Yes”, “Yes” as an answer for “Is this shape triangles?” are set. Through focusing on extension and analyzing data using three choices, three problems are indicated; (1) some grade 1 students have difficulty related to upside-down prototype example, (2) many of the students in grade 2 and 3 undertake the influence of visual features in a figure, for example “pointy” “curved side”, and their decisions in the case of identification are often vague, (3) most of students in each grade do not have mathematical view point relating to formal definition, therefore they recognize three types of shapes, sector, unclosed shape and rounded triangle as a triangle.

　　 In conclusion, in the process of generalization related to prototype, visual features in a figure and orientation, Filipino students have difficulties to use some essential attributes. There are three types of difficulties; (1) influence of horizontal triangle as prototype example, (2) vague and unstable judgment influenced by visual features, (3) scarcity of changing to mathematical viewpoint.

　　 In our country, theorem in the triangle that says the three interior angles of a triangle add up to two right angles is often illustrated as a mathematical relationship, and many junior high school students do not  understand why this needs to demonstrate. However, in a mathematics lesson of the second grade of a certain junior high school which we describe in this paper, a student whose name is Kan expressed the opinion which suggests the necessity of demonstrating theorem in the triangle.

　　 The research purpose for us is to show clearly the meaning of this student’s opinion, and how this opinion was realized. We also aim to suggest the composition of the mathematics lesson which enables students to  understand the meaning of demonstrating.

　　 We analyzed the lesson using‘Didactical  Situation Model (DSM)’ developed by Iguchi, Kuwano, and Iwasaki (2011), in order to achieve the above-mentioned research purpose. DSM is a framework based on mainly two ideas. One is a model developed by Mellin-Olsen (1991) identifying the level of knowledge control in a didactical situation. The other is the types of“MATOME”  identified by Iwasaki and Steinbring (2009). And we made the viewpoint‘the  functions of proof in mathematics’ identified by de Villiers (1990), especially‘proof  as means of systematization’ when considering the lesson design which touches off the necessity of demonstrating theorem in the triangle.

　　 The results of the analysis were summarized as follows:

　(1)  Kan’s opinion states that understanding the meaning of a student demonstrating theorem in the triangle, and it is characterized as a meaning of‘proof  as means of systematization’.

　(2)  The following two are the important elements in the lesson for a student to express the opinion that Kan did: One is designing a lesson so that‘mathematics  constituted locally’ is realized. Moreover, another is  organizing a lesson so that‘an  imperfect situation’ may occur.

　　 Teachers who participate in their “Kounai-kenshu” are necessary to understand the qualitative changes of children’s thinking in the teaching and learning process.  So, they need their viewpoints to grasp the process objectively.

　　 We have two purposes of this paper: one is to develop the viewpoints to grasp the qualitative changes in the children’s thinking by focusing on inductive activities in mathematics classes by an elementary school teacher in her 2^nd graders’ class, which was conducted in the “School Support Project” improving the “Didactical Situation Model;” the other is to answer the following research questions: How can we grasp the development of children’s thinking in the teaching and learning process?; How do the teacher and the children make their qualitative development of the inductive activities?; What makes the qualitative changes in the interaction process?  

　　 To achieve the two purposes, we constructed the “Variable-detailed Version of Didactical Situation Model” based on the “Didactical Situation Model” and the “Type of MATOME” to grasp the qualitative changes of the children’s thinking more precisely.  We have made protocol analyses of 20 class-hours conducted by the teacher from the viewpoints of the new model and the “Mathematics as the Science of Patterns.”   

　　 As the results of these analyses, it was revealed what the qualitative changes of the children’s thinking is and what makes these changes.  It was also suggested that the viewpoints of the suggestive and supporting contacts and the quality of what they are looking for in their inductive activities could be the practical viewpoints in which to grasp the qualitative development of children’s thinking in the inductive activities.

 The purpose of this paper is to reveal the ideal statistics learning and teaching process.  It is not the same process as that of mathematics, such as algebra or geometry, but inherent in statistics.  The knowledge and competence of statistics is required in many daily situations.  It is, however, pointed out that statistics is taught so deterministically like mathematics that statistics is not learned appropriately.   

　　 For achieving this purpose, two problems are worked on.  The first is to reveal what should be learned in statistics domain based on characteristics of statistics.  The second is to reveal it based on analysis from the Shimada’s model of mathematical activities how learning should be carried out in statistics domain.  As a result, it was revealed that it was necessary for statistics concepts to be learned as methodological knowledge and that statistics concepts were learned throughout the statistical problem solving process.  Finally, the learning instruction that is inherent in the statistics domain is illustrated in the Fig. below.

　　 The purpose of this paper is to construct a framework of statistical thinking and to develop teaching unit about the statistics education.

　　 For this purpose, the author clarified a way of statistical thinking in order to decide a view on a development of teaching unit about statistics education, and considered on phenomena caught statistically and critical thinking for catching them.

　　 As a result, the author classified characteristics of the phenomena caught statistically into three, concreteness, operativity and variable tendency, and critical thinking into three, aspirant thinking, reflective thinking and analytic thinking.  Through placing three characteristics on each apex of a triangle and linking their apices with each critical thinking, the author proposed the framework on a way of statistical thinking which attach great importance to critical thinking.  Finally, the author developed teaching unit “Estimate of the Number” about sample survey, made development of classes and analyzed each development through the framework on a way of statistical thinking.

　　 In Japan, fraction is fought of the 2nd grade and more over.  It is one of the important subject in mathematics education research.  However few precedent literature and studies in Japan about this subject develop curriculum based on any theoretical framework.

　　 Hence, I review and critique The Rational Number Project (RNP), especially their theoretical framework for teaching and learning of the rational number.

　　 And I propose a few agenda for the RNP framework as follows;

(1) How the “mathematical variability” can be ordered?;

(2) How the “perceptual variability” can be ordered and what any other “manipulative aids” can be?;

(3) What kind of teaching and learning can be considered in the any cell of “Matrix”?; and (4) How “Translation system” can be used for the teaching and learning process? 

　　 This study tries to clarify students’ states and processes in connecting the representations and the situation of function through analyzing the interviews with two eight grade’s students in terms of the semiotic perspective, in particular in view of the gestures.

　　 First, we examined the meaning of signs as well as the difficulties in dealing with the signs.  Then, we set our research frameworks focused on two roles of gesture: generalization and conceptual blending.  As a result of analysis, we got the following findings regarding the students’ difficulties and the roles of the students’ gestures in reading the meaning of the situation in terms of the representations of function.

◦ The students had their difficulties in transforming from one representation to the other one with their  understanding of the situation.  In particular, their difficulties were related to the understanding of quantity-per-unit which was showed as the gradient of the graph of the linear function and the coefficient of x in the expression.

◦ It was suggested that there are several roles of gesture in connecting the students’ recognitions of the representations and the situation of function: Making the referent explicit, focusing the correspondence  between two components of the situation, enriching the image of the motion and change of the situation, giving their conceptions of function a sense of reality, finding the regularity with their rhythmic actions, facilitating their conceptual blending through making one representation realistic, and becoming the further tool for understanding the related phenomenon.

　　 The first purpose of this paper is to resolve the mixed unit by analytical geometry and function and to clarify the difference between them, and the second one is to clarify the learning content of function and analytical geometry, and the third one is to explore how to instruct analytical geometry in junior high school.

　　 As a result, it can be considered that common representation of analytical geometry and function is the graph in the side of each shape, but it is found that the meaning of shape, which is derived from the side of each concept, is different.