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.

Nathalie Sinclair has written an article with the interesting title: Aesthetics as a liberating force in mathematics education? The article was published in ZDM a couple of days ago. Here is the article abstract:

This article investigates different meanings associated with contemporary scholarship on the aesthetic dimension of inquiry and experience, and uses them to suggest possibilities for challenging widely held beliefs about the elitist and/or frivolous nature of aesthetic concerns in mathematics education. By relating aesthetics to emerging areas of interest in mathematics education such as affect, embodiment and enculturation, as well as to issues of power and discourse, this article argues for aesthetic awareness as a liberating, and also connective force in mathematics education.

Ricardo Nemirovsky and Francesca Ferrara have written an article called “Mathematical imagination and embodied cognition” that was published online in Educational Studies in Mathematics on Friday. Here is the abstract of their article:

The goal of this paper is to explore qualities of mathematical imagination in light of a classroom episode. It is based on the analysis of a classroom interaction in a high school Algebra class. We examine a sequence of nine utterances enacted by one of the students whom we call Carlene. Through these utterances Carlene illustrates, in our view, two phenomena: (1) juxtaposing displacements, and (2) articulating necessary cases. The discussion elaborates on the significance of these phenomena and draws relationships with the perspectives of embodied cognition and intersubjectivity.

Máire Ní Ríordáin and John O’Donoghue have written an article about The relationship between performance on mathematical word problems and language proficiency for students learning through the medium of Irish. The article was published in Educational Studies in Mathematics two days ago. Here is the abstract of their article:

Ireland has two official languages—Gaeilge (Irish) and English. Similarly, primary- and second-level education can be mediated through the medium of Gaeilge or through the medium of English. This research is primarily focused on students (Gaeilgeoirí) in the transition from Gaeilge-medium mathematics education to English-medium mathematics education. Language is an essential element of learning, of thinking, of understanding and of communicating and is essential for mathematics learning. The content of mathematics is not taught without language and educational objectives advocate the development of fluency in the mathematics register. The theoretical framework underpinning the research design is Cummins’ (1976). Thresholds Hypothesis. This hypothesis infers that there might be a threshold level of language proficiency that bilingual students must achieve both in order to avoid cognitive deficits and to allow the potential benefits of being bilingual to come to the fore. The findings emerging from this study provide strong support for Cummins’ Thresholds Hypothesis at the key transitions—primary- to second-level and second-level to third-level mathematics education—in Ireland. Some implications and applications for mathematics teaching and learning are presented.

Helen J. Forgasz and David Mittelberg have written an article called Israeli Jewish and Arab students’ gendering of mathematics. The article was recently published online in ZDM. Here is a copy of their article abstract:

In English-speaking, Western countries, mathematics has traditionally been viewed as a “male domain”, a discipline more suited to males than to females. Recent data from Australian and American students who had been administered two instruments [Leder & Forgasz, in Two new instruments to probe attitudes about gender and mathematics. ERIC, Resources in Education (RIE), ERIC document number: ED463312, 2002] tapping their beliefs about the gendering of mathematics appeared to challenge this traditional, gender-stereotyped view of the discipline. The two instruments were translated into Hebrew and Arabic and administered to large samples of grade 9 students attending Jewish and Arab schools in northern Israel. The aims of this study were to determine if the views of these two culturally different groups of students differed and whether within group gender differences were apparent. The quantitative data alone could not provide explanations for any differences found. However, in conjunction with other sociological data on the differences between the two groups in Israeli society more generally, possible explanations for any differences found were explored. The findings for the Jewish Israeli students were generally consistent with prevailing Western gendered views on mathematics; the Arab Israeli students held different views that appeared to parallel cultural beliefs and the realities of life for this cultural group.

Allan Leslie White and Chap Sam Lim have written an article about the use of the Japanese Lesson Study model in Australian and Malaysian classrooms. The article is entitle Lesson study in Asia Pacific classrooms: local responses to a global movement, and it was published online in ZDM on Wednesday.

If you are interested in the topic, this article gives a nice overview of the history and theoretical background of the Japanese Lesson Study approach, and there is also a nice list of references to dig into. In the conclusions of the article, they claim:

However, the significant features of Japanese Lesson Study, such as the use of collaborative work, working on common goals, sharing of ideas, team teaching, lesson observation and cooperation among peers seemed to exert similar impacts on all groups of participants. Participants from all glocal programs reported an improvement in their lesson planning, better pedagogical content knowledge and closer collegial relationship as a result of experiencing the Lesson Study process.

Here is the abstract of the article:

Japanese Lesson Study is a model for teacher professional learning that has recently attracted world attention particularly within the mathematics education community. It is a highly structured process of teacher collaboration, observation, reflection and practice. The world focus has been mainly due to the work of American researchers such as Stigler and Hiebert (Am Educ Winter:1–10, 1998; The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. Free Press, New York 1999), Lewis and Tsuchida (Am Educ Winter:14–17; 50–52, 1998) and Fernandez [J Teach Educ 53(5):395–405, 2002]. These researchers have documented Lesson Study from the perspective of their social, cultural and educational contexts. In order to develop a deeper understanding of Lesson Study in a post-modern global world, there is a need to seek views beyond those presented from an American perspective. This paper will provide further additional perspectives from an Australian state view and a Malaysian state district view and a university view. The aim is to develop an understanding of how the different contexts have influenced the structure and implementation of the Japanese Lesson Study model.

Lisa B. Warner has written an article that was published online in The Journal of Mathematical Behavior yesterday. The article is entitled How do students’ behaviors relate to the growth of their mathematical ideas? Here is the article abstract:

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie–Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school problem-solving sessions. The results suggest that certain types of student behaviors appear to be associated with the growth of ideas and emerge in specific patterns. More specifically, as understanding grows, there is a general shift from behaviors such as students questioning each other, explaining and using their own and others’ ideas toward behaviors involving the setting up of hypothetical situations, linking of representations and connecting of contexts. Recognizing that certain types of student behaviors tend to emerge in specific layers of the Pirie–Kieren model can be important in helping us to understand the development of mathematical ideas in children.

Warner focus a lot on the Pirie-Kieren model in her theoretical framework (see the article of Susan Pirie and Thomas Kieren from 1994). The main focus of Warner’s article is to address the following questions:

Are different types of student behaviors associated with the growth of mathematical ideas in specific ways? If so, how?

In her conclusions, Lisa Warner suggests that for the students in her study, “certain types of behaviors appeared to be associated with the growth of mathematical ideas in certain ways”. She also suggests that further research is needed in order to investigate whether these findings correspond with findings in similar studies of other students, different types of tasks, etc.