Stochastic phenomena induce us to construct a probability model and structure our thinking; corresponding models help us to understand and interpret the reality. They in turn equip us with tools to recognize, reconstruct and solve problems. Therefore, various implications in terms of methodology as well as epistemology naturally flow from different adoptions of models for probability. Right from the basic scenarios of different perspectives to explore reality, students are occasionally exposed to misunderstanding and misinterpretations. With realistic examples a multi-faceted image of probability and different interpretation will be considered in mathematical modelling activities. As an exploratory investigation, mathematical modelling activity for probability learning was elaborated through semiotic analysis. Especially, the coherence structure in mathematical modelling discourse was reviewed form a semiotic perspective. The discourses sampled from group activities were analyzed on the basis of semiotic perspectives taxonomical coherence relations.

Division of fractions is always a difficult topic for primary school students. Most of the presentations in teaching the topic in textbooks are procedural, asking students to invert the second fraction and multiply it with the first one, that is, $$\frac{a}{b}{\div}\frac{c}{d}=\frac{a}{b}{\times}\frac{d}{c}$$. Such procedural approach in teaching diminishes both the understanding of structure in mathematics and the interest in learning the subject. This paper discussed the formulation of teaching the division of fractions, which based on research lessons in some primary five classrooms. The formulated lessons started with an analogy to division of integers and working with division of fractions with equal denominators and then extended to division of fractions in general. It is found that the using of analogy helps students to invent their procedure in working the division problem. Some procedures found by students are discussed, with the focus on the development of their invention and mathematical thinking.

Introduction: According the 2011 Taiwanese Government Statistics, the lower secondary school enrollment number of the new-immigrant-children is about 200,000. As known, most of the new immigrants are from the Southeast Asian countries, such as Vietnam, China, Indonesia, Thailand, the Philippines and Cambodia. In order to satisfy the increasing needs and demands on education of the children of new immigrant (CNI, henceforth), Taiwanese government not only develops, but also puts the after-school learning assistance policy into practice from 2006. Therefore, the main purpose of this study is to explore the mathematics achievement of the CNI after the implementation of the after-school learning assistance policy (AsLA policy, henceforth). Purposes: Firstly, to compare the mathematics achievement of the CNI by countries. Secondly, to compare the mathematics performance among the CNI, the children from high-risk family (CHRF, henceforth) and the children of general families. Samples: The 2,452 samples, selected from two junior high schools located in central Taiwan, include 157 CNI, 522 CHRF. Methods: The main method used in this study is interval fuzzy number (IFN, henceforth) in order to compare the mathematics achievement of the children after the implementation of the AsLA policy from different type of families. Results: To reach the two purposes of this study. We can find the effectiveness of mathematics performance from three group's children of new immigrants, high-risk, general family. Therefore, the results provide one of the ways to review the new immigrant's education policy of after-school learning assistance in Taiwan.

This paper aims to explore whether Learning Study improves teachers' subject content knowledge, pedagogical content knowledge, and attitude toward teaching mathematics. A Learning Study was conducted in a Hong Kong primary school for a research lesson on comparing the size of fractions to explore the new teacher roles.

Trigonometric ratios are difficult concepts to teach and learn in middle school. One of the reasons is that the mathematical terms (sine, cosine, tangent) don't convey the idea literally. This paper deals with the understanding of a concept from the learner's standpoint, and searches the orientation of teaching that make students to have full understanding of trigonometric ratios. Such full understanding contains at least five constructs as follows: skill-algorithm, property-proof, use-application, representation-metaphor, history-culture understanding [Usiskin, Z. (2012). What does it mean to understand some mathematics? In: Proceedings of ICME12, COEX, Seoul Korea; July 8-15,2012 (pp. 502-521). Seoul, Korea: ICME-12]. Despite multi-aspects of understanding, especially, the history-culture aspect is not yet a part of the mathematics class on the trigonometric ratios. In this respect this study investigated the effect of history approach on students' understanding when the history approach focused on the mathematical terms is used to teach the concept of trigonometric ratios in Grade 9 mathematics class. As results, the experimental group obtained help in more full understanding on the trigonometric ratios through such teaching than the control group. This implies that the historical derivation of mathematical terms as well as the context of mathematical concepts should be dealt in the math class for the more full understanding of some mathematical concepts.