Sergei Abramovich and Gennady A. Leonov have written an article called “Fibonacci numbers revisited: technology-motivated inquiry into a two-parametric difference equation“, which was recently published in International Journal of Mathematical Education in Science and Technology. Here is the abstract:

This article demonstrates how within an educational context, supported by the notion of hidden mathematics curriculum and enhanced by the use of technology, new mathematical knowledge can be discovered. More specifically, proceeding from the well-known representation of Fibonacci numbers through a second-order difference equation, this article explores its two-parametric generalization using computer algebra software and a spreadsheet. Combined with the use of calculus, matrix theory and continued fractions, this technology-motivated approach allows for the comprehensive investigation of the qualitative behaviour of the orbits produced by the so generalized difference equation. In particular, loci in the plane of parameters where different types of behaviour of the cycles of arbitrary integer period formed by generalized Golden Ratios realize have been constructed. Unexpected connections among the analytical properties of the loci, Fibonacci numbers and binomial coefficients have been revealed. Pedagogical, mathematical and epistemological issues associated with the proposed approach to the teaching of mathematics are discussed.

A new article called “Animating an equation: a guide to using FLASH in mathematics education” has recently been published in International Journal of Mathematical Education in Science and Technology. The article is written by Ezzat G. Bakhoum. Here is the abstract of the article:

Macromedia’s FLASH development system can be a great tool for mathematics education. This article presents detailed Flash tutorials that were developed and taught by the author to a group of mathematics professors in a summer course in 2005. The objective was to educate the teachers in the techniques of animating equations and mathematical concepts in Flash. The course was followed by a 2-year study to assess the acceptance of the technology by the teachers and to gauge its effectiveness in improving the quality of mathematics education. The results of that 2-year study are also reported here.

Four new articles has been published online in ZDM recently:

• Strategies to foster students’ competencies in constructing multi-steps geometric proofs: teaching experiments in Taiwan and Germany by Aiso Heinze, Ying-Hao Cheng, Stefan Ufer, Fou-Lai Lin and Kristina Reiss. Abstract: In this article, we discuss the complexity of geometric proofs with respect to a theoretical analysis and empirical results from studies in Taiwan and Germany. Based on these findings in both countries, specific teachings experiments with junior high school students were developed, conducted, and evaluated. According to the different classroom and learning culture in East Asia and Western Europe, the interventions differed in their way of organizing the learning activities during regular mathematics lessons. The statistical analysis of the pre–post-test data indicated that both interventions were successful in fostering students’ proof competence.
• Connecting theories in mathematics education: challenges and possibilities by Luis Radford. Abstract: This paper is a commentary on the problem of networking theories. My commentary draws on the papers contained in this ZDM issue and is divided into three parts. In the first part, following semiotician Yuri Lotman, I suggest that a network of theories can be conceived of as a semiosphere, i.e., a space of encounter of various languages and intellectual traditions. I argue that such a networking space revolves around two different and complementary “themes”—integration and differentiation. In the second part, I advocate conceptualizing theories in mathematics education as triplets formed by a system of theoretical principles, a methodology, and templates of research questions, and attempt to show that this tripartite view of theories provides us with a morphology of theories for investigating differences and potential connections. In the third part of the article, I discuss some examples of networking theories. The investigation of limits of connectivity leads me to talk about the boundary of a theory, which I suggest defining as the “limit” of what a theory can legitimately predicate about its objects of discourse; beyond such an edge, the theory conflicts with its own principles. I conclude with some implications of networking theories for the advancement of mathematics education.
• A networking method to compare theories: metacognition in problem solving reformulated within the Anthropological Theory of the Didactic by Esther Rodríguez, Marianna Bosch and Josep Gascón. Abstract: An important role of theory in research is to provide new ways of conceptualizing practical questions, essentially by transforming them into scientific problems that can be more easily delimited, typified and approached. In mathematics education, theoretical developments around ‘metacognition’ initially appeared in the research domain of Problem Solving closely related to the practical question of how to learn (and teach) to solve non-routine problems. This paper presents a networking method to approach a notion as ‘metacognition’ within a different theoretical perspective, as the one provided by the Anthropological Theory of the Didactic. Instead of trying to directly ‘translate’ this notion from one perspective to another, the strategy used consists in going back to the practical question that is at the origin of ‘metacognition’ and show how the new perspective relates this initial question to a very different kind of phenomena. The analysis is supported by an empirical study focused on a teaching proposal in grade 10 concerning the problem of comparing mobile phone tariffs.
• Comparing, combining, coordinating-networking strategies for connecting theoretical approaches by Susanne Prediger, Ferdinando Arzarello, Marianna Bosch and Agnès Lenfant. This is the editorial for the next issue, and it does not have an abstract.

The May issue of Mathematics Teacher has also arrived. The list of contents presents the following articles, whereas the last one is a free preview article:

Deep Thoughts on the River Crossing Game
Dan Canada and Dave Goering

The Power of Investigative Calculus Projects
John Robert Perrin and Robert J. Quinn

Why Aren’t They Called Probability Intervals?
Thomas F. Devlin

The May issue of Teaching Children Mathematics has also appeared, and it contains the following articles:

Instructional Strategies for Teaching Algebra in Elementary School: Findings from a Research-Practice Collaboration
Darrell Earnest and Aadina A. Balti

Insights into Our Understandings of Large Numbers
Signe E. Kastberg and Vicki Walker

The first article is a free preview article.

The May issue of Mathematics Teaching in the Middle School has arrived, and it contains the following articles:

Teaching and Learning Mathematics through Hurricane Tracking
Maria L. Fernandez and Robert C. Schoen

The Importance of Equal Sign Understanding in the Middle Grades
Eric J. Knuth, Martha W. Alibali, Shanta Hattikudur, Nicole M. McNeil and Ana C. Stephens

Exploring Segment Lengths on the Geoboard
Mark W. Ellis and David Pagni

What Do Students Need to Learn about Division of Fractions?
Yeping Li

The May issue of Educational Studies in Mathematics has appeared, with the following articles:

• Deep intuition as a level in the development of the concept image
  
• Concept image revisited
• The power of Colombian mathematics teachers’ conceptions of social/institutional factors of teaching
• Analyses de séances en classe et stabilité des pratiques d’enseignants de mathématiques expérimentés du second degré

Kirsti Hemmi from Stockholm University has written an article that was recently published (online first) in ZDM. The article is entitled: “Students’ encounter with proof: the condition of transparency“. Here is the abstract of the article:

The condition of transparency refers to the intricate dilemma in the teaching of mathematics about how and how much to focus on various aspects of proof and how and how much to work with proof without a focus on it. This dilemma is illuminated from a theoretical point of view as well as from teacher and student perspectives. The data consist of university students’ survey responses, transcripts of interviews with mathematicians and students as well as protocols of the observations of lectures, textbooks and other instructional material. The article shows that the combination of a socio-cultural perspective, Lave and Wenger’s and Wenger’s social practice theories and theories about proof offers a fresh framework for studies concerning the teaching and learning of proof.