Sharon S. N. Ng and Nirmala Rao have written an article for the journal Early Years. The article concerns teaching of mathematics in Hong Kong, and it is entitled: Mathematics teaching during the early years in Hong Kong: a reflection of constructivism with Chinese characteristics? Here is the abstract:

This paper characterizes early mathematics instruction in Hong Kong. The teaching of addition in three pre-primary and three lower primary schools was observed and nine teachers were interviewed about their beliefs about early mathematics teaching. A child-centered, play-based approach was evident but teachers emphasized discipline, diligence and academic success. Observations also revealed practices reflective of both constructivist and instructivist pedagogies. Results from interviews suggest that teachers’ traditional cultural beliefs about instruction were challenged by western ideologies introduced in continuing professional development courses and by notions promulgated by the educational reforms. Both consistencies and inconsistencies between teachers’ beliefs and practices were identified. Implications of the findings are discussed.

Tracey Muir, Kim Beswick and John Williamson have written an article that was recently published in the Journal of Mathematical Behavior. The article is entitled “I’m not very good at solving problems”: An exploration of students’ problem solving behaviours.

Abstract:

This paper reports one aspect of a larger study which looked at the strategies used by a selection of grade 6 students to solve six non-routine mathematical problems. The data revealed that the students exhibited many of the behaviours identified in the literature as being associated with novice and expert problem solvers. However, the categories of ‘novice’ and ‘expert’ were not fully adequate to describe the range of behaviours observed and instead three categories that were characteristic of behaviours associated with ‘naïve’, ‘routine’ and ‘sophisticated’ approaches to solving problems were identified. Furthermore, examination of individual cases revealed that each student’s problem solving performance was consistent across a range of problems, indicating a particular orientation towards naïve, routine or sophisticated problem solving behaviours. This paper describes common problem solving behaviours and details three individual cases involving naïve, routine and sophisticated problem solvers.

The Montana Mathematics Enthusiast has just published a combined number 2&3. This large issue is filled with content. Here is the list of featured articles:

1. Jeff Babb & James Currie(Canada)
The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context

2. Michael Fried (Israel)
History of Mathematics in Mathematics Education: a Saussurean Perspective

3. Spyros Glenis (Greece)
Comparison of Geometric Figures

4. Giorgio T. Bagni (Italy)
“Obeying a rule”: Ludwig Wittgenstein and the foundations of Set Theory

5. Arnaud Mayrargue (France)
How can science history contribute to the development of new proposals in the teaching of the notion of derivatives?

6. Antti Viholainen (Finland)
Incoherence of a concept image and erroneous conclusions in the case of differentiability

7. Dores Ferreira & Pedro Palhares (Portugal)
Chess and problem solving involving patterns

8. Friðrik Diego & Kristín Halla Jónsdóttir (Iceland)
Associative Operations on a Three-Element Set

9. Jon Warwick (UK)
A Case Study Using Soft Systems Methodology in the Evolution of a Mathematics Module

10. Barbara Garii & Lillian Okumu (New York, USA)
Mathematics and the World: What do Teachers Recognize as Mathematics in Real World Practice?

11. Linda Martin & Kristin Umland (New Mexico, USA)
Mathematics for Middle School Teachers: Choices, Successes, and Challenges

12. Woong Lim (Texas, USA)
Inverses – why we teach and why we need talk more about it more often!

13. Steve Humble (UK)
Magic Math Cards

The issue also contains a couple of articles on logarithms in a historical perspective, a large section of articles with reactions on the report of the National Mathematics Advisory Panel, etc.

Peter Riley Bahr has written an article that was published in the August issue of Research in Higher Education. The article is entitled: Does Mathematics Remediation Work?: A Comparative Analysis of Academic Attainment among Community College Students. Here is the abstract:

Postsecondary remediation is a controversial topic. On one hand, it fills an important and sizeable niche in higher education. On the other hand, critics argue that it wastes tax dollars, diminishes academic standards, and demoralizes faculty. Yet, despite the ongoing debate, few comprehensive, large-scale, multi-institutional evaluations of remedial programs have been published in recent memory. The study presented here constitutes a step forward in rectifying this deficit in the literature, with particular attention to testing the efficacy of remedial math programs. In this study, I use hierarchical multinomial logistic regression to analyze data that address a population of 85,894 freshmen, enrolled in 107 community colleges, for the purpose of comparing the long-term academic outcomes of students who remediate successfully (achieve college-level math skill) with those of students who achieve college-level math skill without remedial assistance. I find that these two groups of students experience comparable outcomes, which indicates that remedial math programs are highly effective at resolving skill deficiencies.

The June issue of International Journal of Mathematical Education in Science and Technology is published (Volume 39, Issue 4, 2008). It has the following original articles in the list of contents:

• Integrating supplementary application-based tutorials in the multivariable calculus course by I. M. Verner;
  S. Aroshas; A. Berman
• If not, what yes? by Boris Koichu
• Mathematical e-learning: state of the art and experiences at the Open University of Catalonia by A. Juan; A. Huertas; C. Steegmann; C. Corcoles; C. Serrat
• Unique factorization in cyclotomic integers of degree seven by W. Ethan Duckworth
• A college lesson study in calculus, preliminary report by Joy Becker; Petre Ghenciu; Matt Horak; Helen Schroeder

Sandra Crespo and Nathalie Sinclair poses this very interesting question in an article that has recently been published in Journal of Mathematics Teacher Education. The entire title of the article is: What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems.

Mathematical problems are an integral part of mathematical learning, and although most pupils encounter mathematical problems as they are posed in textbooks, the teachers have an important role in assigning appropriate problems for the students to solve. Prospective teachers have had few opportunities to focus on problem posing in their studies, and their experience with mathematical problems are mostly in connection with the solving of problems that are posed by the teacher or a textbook. The authors of this article “consider the practice of problem posing to be especially important for prospective teachers because a great deal of the work of teaching entails the posing and generation of what the mathematics education community often refers to as “good” questions—questions that aim to support students’ mathematical work”.

The main research questions in the study described in this article are:

1. What is the role of exploration in the problem-posing process? (What happens when prospective teachers pose problems with and without first exploring the situation that could motivate their questions? What kinds of questions do they pose in each of these two kinds of structured problem-posing setting?)
2. How do prospective elementary teachers decide on the quality of the questions they pose? (What rationale do they provide when asked to justify what makes their questions mathematically interesting? What is the effect of making explicit some of the qualities that make mathematics problems interesting and worth solving?)

The questions were investigated in a course that Sandra Crespo taught herself, and the course was offered in the fourth year of a 5-year teacher preparation program. A central theme in the course was a “pedagogy of inquiry” rather than one of presentation, and the students were given the opportunity to investigate different forms of mathematics teaching. There were 22 students in the course, and the researchers used four tasks and two classroom interventions in the study. The data consisted of written work from the students as well as field notes from observations of the students’ work with the given tasks, and from discussions in class.

Here is the abstract:

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’

Kelli Nipper and Paola Sztajn have written an article that was recently published in Journal of Mathematics Teacher Education. The article is entitled: Expanding the instructional triangle: conceptualizing mathematics teacher development.

Abstract As mathematics educators think about teaching that
promotes students’ opportunities to learn, attention must be given to
the conceptualization of the professional development of teachers and
those who teach teachers. In this article, we generalize and expand the
instructional triangle to consider different interactions in a variety
of teacher development contexts. We have done so by addressing issues
of language for models of teachers’ professional development at
different levels and by providing examples of situations in which these
models can be applied. Through the expansion of our understanding and
use of the instructional triangle we can further develop the concept of
mathematics teacher development.

Teachers are professionals with a rich knowledge that is both content specific and general. They shape instruction by the way they interpret and respond to students and materials (p. 2). The notion of “the instructional triangle” is based on the definition of instruction as (they refer to Cohen and Ball, 1999, p. 5 here): the interaction between teachers and students around educational material. These ideas are also shared by other researchers. One of them, Barbara Jaworski, created the teaching triad, consisting of:

• management of student learning
• sensitivity to students
• engagement in challenging mathematics

Nipper and Sztajn describe how they have tried to expand this instructional triangle to teacher education, and as a response to language issues, they suggest to replace the ordinary triangle: teacher – student – mathematics with the more general: organizer – participants – content. For a further elaboration of their analysis and theoretical suggestions, you should dig deeper into the article!

Lyndon C. Martin has written an article called Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie–Kieren Theory. The article is going to be published in The Journal of Mathematical Behavior, and it was available online yesterday at ScienceDirect. Here is the abstract:

The study reported here extends the work of Pirie and Kieren on the nature and growth of mathematical understanding. The research examines in detail a key aspect of their theory, the process of ‘folding back’, and develops a theoretical framework of categories and sub-categories that more fully describe the phenomenon. This paper presents an overview of this ‘framework for folding back’, illustrates it with extracts of video data and elaborates on its key features. The paper also considers the implications of the study for the teaching and learning of mathematics, and for future research in the field.

For another article discussing the Pirie-Kieren theory and related theories, you might want to take a look at this article by Droujkova et al. from PME29.