Konstantinos Tatsis, Sonia Kafoussi and Chrysanthi Skoumpourdi have written an article called Kindergarten Children Discussing the Fairness of Probabilistic Games: The Creation of a Primary Discursive Community. The article was recently published in Early Childhood Education Journal. Here is the abstract of the article:

In this paper we analyse the language used by kindergarten children and their teacher while they discuss the fairness of two games that involved the concept of chance. Their discussions show that the children are able to overcome their primary intuitions concerning the fairness of a game and to comprehend the important role of materials. The children mostly used counting strategies in order to justify their opinion; this reveals the establishment of a primary discursive community based on the premise that each opinion should be justified in order to be accepted by the other children and the teacher.

Pessia Tsamir and Dina Tirosh have written an article about Combining theories in research in mathematics teacher education. This article was published in ZDM two days ago. In this interesting article, they examine how the combination of the theories of Shulman and Fischbein “may contribute to the evaluation of mathematics teachers’ (prospective and inservice) knowledge”. Here is the article abstract:

In this paper, we describe how the combination of two theories, each embedded in a different realm, may contribute to evaluating teachers’ knowledge. One is Shulman’s theory, embedded in general, teacher education, and the other is Fischbein’s theory, addressing learners’ mathematical conceptions and misconceptions. We first briefly describe each of the two theories and our suggestions for combining them, formulating the Shulman–Fischbein framework. Then, we present two research segments that illustrate the potential of the implementation of the Shulman–Fischbein framework to the study of mathematics teachers’ ways of thinking. We conclude with general comments on possible contributions of combining theories that were developed in mathematics education and in other domains to mathematics teacher education.

Ngai-Ying Wong from The Chinese University of Hong Kong, has written an article with the interesting title: Confucian heritage culture learner’s phenomenon: from “exploring the middle zone” to “constructing a bridge”. The article was published online in ZDM on Tuesday. The article gives some interesting insight into aspects of the Chinese culture, and it did represent several new issues and aspects to me. Besides, it is the first scientific article that I have ever seen (within our field, at least) that includes martial-art pictures. In the article, Wong also draws upon variation theory (which derives from the work of Swedish scholar Ference Marton and colleagues). Here is the abstract of the article:

In the past decades, the CHC (Confucian heritage culture) learner’s phenomenon has spawned one of the most fruitful fields in educational research. Despite the impression that CHC learners are brought up in an environment not conducive to learning, their academic performances have been excelling their Western counterparts (Fan et al. in How Chinese learn mathematics: perspectives from insiders, 2004). Numerous explanations were offered to reveal the paradox (Morrison in Educ J, 2006), and there were challenges of whether there is “over-Confucianisation” in all these discussions (Chang in J Psychol Chin Soc, 2000; Wong and Wong in Asian Psychol, 2002). It has been suggested that the East and the West should come and discuss at the “middle zone” so that one can get the best from the two worlds. On the other hand, at the turn of the new millennium, discussions on mathematics curriculum reform proliferate in many places. One of the foci of the debate is the basic skills—higher-order thinking “dichotomy”. Viewing from the perspective of the process of mathematisation, teaching mathematics is more than striking a balance between the two, but to bridge basic skills to higher-order thinking competences. Such an attempt was explored in recent years and the ideas behind will be shared in this paper.

Daniel Chazan, Michael Yerushalmy and Roza Leikin have written an article that was published online in The Journal of Mathematical Behavior yesterday. The article is entitled An analytic conception of equation and teachers’ views of school algebra, and here is the abstract:

This interview study takes place in the context of a single small district in the United States. In the algebra curriculum of this district, there was a shift in the conception of equation, from a statement about unknown numbers to a question about the comparison of two functions over the domain of the real numbers. Using two of Shulman’s [Shulman, L. S. (1986). Paradigms and research programs in the study of teaching: A contemporary perspective. In Wittrock, M. C. (Ed.), Handbook of research in teaching (3rd ed., pp. 3–36). New York: Macmillan] categories of teachers’ knowledge – pedagogical content knowledge and curricular content knowledge – we explore whether in this context teachers’ content knowledge give signs of being reorganized. Our findings suggest that the teachers see this conception of equation as useful for equations in one variable. They struggle with its ramifications for equations in two variables. Nonetheless, this conception of equation leads them to reflect on the algebra curriculum in substantial ways; two of the three teachers explicitly spoke about their curricular ideas as being associated with this conception of an equation or with their earlier views. The third teacher seems so taken with these curricular ideas that he explored their ramifications throughout the interview. We argue that the consideration of this new conception of equation was an important resource that the teachers used to construct their understandings of alternative curricular approaches to school algebra. As they work with this new conception of an equation, we find an analogy to their situation in Kuhn’s description of the individual scientist in the process of adopting a new paradigm.

Ajay Ramful and John Olive wrote an article entitled Reversibility of thought: An instance in multiplicative tasks, which was published online in The Journal of Mathematical Behavior yesterday. Here is the abstract of the article:

In line with current efforts to understand the piece-by-piece structure and articulation of children’s mathematical concepts, this case study compares the reversibility schemes of two eighth-grade students. The aim of the study was to identify the mechanism through which students reverse their thought processes in a multiplicative situation. Data collected through clinical interviews depict the precise strategies that the participants used to work back to find the missing values in an inverse proportional task. This study also illustrates how a conceptual template generated by one of the participants afforded him considerable flexibility in the multiplicative task. Another outcome of the study is that it shows how the numerical characteristics of the parameters in the problem affected the students’ ability to reverse their thought processes. We infer that there is a need for further research on how students might represent their reversibility schemes in the form of algebraic equations.

Florence M. Singer and Cristian Voica wrote an interesting article that was recently published in The Journal of Mathematical Behaviour: Between perception and intuition: Learning about infinity. Here is the article abstract:

Based on an empirical study, we explore children’s primary and secondary perceptions on infinity. When discussing infinity, children seem to highlight three categories of primary perceptions: processional, topological, and spiritual. Based on their processional perception, children see the set of natural numbers as being infinite and endow Q with a discrete structure by making transfers from N to Q. In a continuous context, children are more likely to mobilize a topological perception. Evidence for a secondary perception of arises from students’ propensities to develop infinite sequences of natural numbers, and from their ability to prove that N is infinite. Children’s perceptions on infinity change along the school years. In general, the perceptual dominance moves from sequential (processional) to topological across development. However, we found that around 11–13 years old, processional and topological perceptions interfere with each other, while before and after this age they seem to coexist and collaborate, one or the other being specifically activated by the nature of different tasks.

Shelly Sheats Harkness and Jonathan Thomas have written an article that is entitled: Reflections on “Multiplication as Original Sin”: The implications of using a case to help preservice teachers understand invented algorithms. This article takes a case report called “Multiplication as original sin” as point of departure. The article was published online yesterday in The Journal of Mathematical Behavior. Here is the abstract of the article:

This article describes the use of a case report, Multiplication as original sin (Corwin, R. B. (1989). Multiplication as original sin. Journal of Mathematical Behavior, 8, 223–225), as an assignment in a mathematics course for preservice elementary teachers. In this case study, Corwin described her experience as a 6th grader when she revealed an invented algorithm. Preservice teachers were asked to write reflections and describe why Corwin’s invented algorithm worked. The research purpose was: to learn about the preservice teachers’ understanding of Corwin’s invented multiplication algorithm (its validity); and, to identify thought-provoking issues raised by the preservice teachers. Rather than using mathematical properties to describe the validity of Corwin’s invented algorithm, a majority of them relied on procedural and memorized explanations. About 31% of the preservice teachers demonstrated some degree of conceptual understanding of mathematical properties. Preservice teachers also made personal connections to the case report, described Corwin using superlative adjectives, and were critical of her teacher.

Alayne C. Armstrong has written an article that was published online in The Journal of Mathematical Behavior yesterday. The article is entitled The fragility of group flow: The experiences of two small groups in a middle school mathematics classroom. Here is the abstract of the article:

This article considers two small groups of students in the same Grade 8 mathematics classroom whose approaches to the same mathematical problem result in very different experiences. Using videotapes and written transcripts, an analysis of the groups’ working processes was undertaken using Sawyer’s pre-existing structures required for the presence of group flow, and Davis and Simmt’s conditions for complex systems. It is suggested that although both groups had the prerequisite structures in place to experience group flow, the second group was not decentralized enough to enable all members to establish a working collaborative proximal zone of development in which they could develop their ideas as a collective, while the first group was sufficiently decentralized and appeared to demonstrate episodes of experiencing group flow. If teachers are aware of conditions that encourage the experience of group flow, this may help them in forming productive small groups within the classroom and developing successful group-oriented learning tasks.