David W. Carraher has written an article that was recently published (online first) in Educational Studies in Mathematics. The article is entitled: Beyond ‘blaming the victim’ and ‘standing in awe of noble savages’: a response to “Revisiting Lave’s ‘cognition in practice’”. Here is the abstract:

Everyday Mathematics has contributed in important ways to long-standing debates about mathematical concepts, symbolic representation, and the role of contexts in thinking—the latter topic reaching back at least as far as Kant’s notion of scheme. The descriptive work plays a role, of course. But it is only by making sense of the observations that science moves forward. If over time the expression Everyday Mathematics drops from usage, I would be neither surprised nor disappointed. Eventually the field needs to become absorbed into the mainstream traditions of research in mathematics education. However it would be disappointing if it is remembered only for its descriptive and proscriptive aspects, without recognizing the contributions to research, theory, and the cultural context of learning and thinking.

Two more articles have been published online in the International Journal of Science and Mathematics Education:

• The Factors Related to Preschool Children and Their Mothers on Children’s Intuitional Mathematics Abilities is written by Yildiz Güven. Abstract: The aim of this study is to assess the factors that are related to preschool children and their mothers on children’s’ intuitional mathematics abilities. Results of the study showed that there were significant differences in children’s intuitional mathematics abilities when children are given the opportunity to think intuitionally and to make estimations, and when their mothers believe in the importance of providing such opportunities in the home setting. Children who tended to think fast and to examine details of objects had significantly higher scores. Also, the working mothers aimed to give opportunities to their children more often than non-working mothers. The mothers whose children received preschool education tended to give more opportunities to their children to think intuitionally and to make estimations. When incorrect intuitional answers or estimations were made by children, lower-educated mothers tended to scold their children much more than higher educated mothers. Mothers having at least a university degree explained more often to the children why they were in error than did the less-educated mothers.
• The Power of Learning Goal Orientation in Predicting Student Mathematics Achievement is written by Chuan-Ju Lin et al. Abstract: The teaching and learning of mathematics in schools has drawn tremendous attention since the education reform in Taiwan. In addition to assessing cognitive abilities, Taiwan Assessment of Student Achievement in Mathematics (TASA-MAT) collects background information to help depict average student achievement in schools in an educational context. The purpose of this study was to investigate the relationships between student achievement in mathematics and student background characteristics. The data for this study was derived from the sample for the 2005 TASA-MAT Sixth-Grade Main Survey in Taiwan. The average age of the sixth-grade students in Taiwan is 11 years old, as was the sample for the 2005 TASA-MAT. Student socioeconomic status (SES) and student learning-goal orientation were specified as predictor variables of student performance in mathematics. The results indicate that the better performance in mathematics tended to be associated with a higher SES and stronger mastery goal orientation. The SES factor accounted for 4.98% of the variance, and student learning-goal orientation accounted for an additional 10.61% of the variance. The major implication obtained from this study was that goal orientation was much more significant than SES in predicting student performance in mathematics. In addition, the Rasch model treatment of the ordinal response-category data is a novel approach to scoring the goal-orientation items, with the corresponding results in this study being satisfactory.

Last week, two new articles were published online in ZDM as well:

• Building a local conceptual framework for epistemic actions in a modelling environment with experiments by Stefan Halverscheid. Abstract: A local conceptual framework for the construction of mathematical knowledge in learning environments with experiments is developed. For this purpose, the mathematical modelling framework and the epistemic action model for abstraction in context are used simultaneously. In a case study, experiments of pre-service teachers with the motion of a ball on a circular billiard table are analysed within the local conceptual framework. The role of the experiments for epistemic actions of mathematical abstractions is described. In the case study, two different types of students’ approaches to the role of experiments can be distinguished.
• Indirect proof: what is specific to this way of proving? by Samuele Antonini and Maria Alessandra Mariotti. Abstract: The study presented in this paper is part of a wide research project concerning indirect proofs. Starting from the notion of mathematical theorem as the unity of a statement, a proof and a theory, a structural analysis of indirect proofs has been carried out. Such analysis leads to the production of a model to be used in the observation, analysis and interpretation of cognitive and didactical issues related to indirect proofs and indirect argumentations. Through the analysis of exemplar protocols, the paper discusses cognitive processes, outlining cognitive and didactical aspects of students’ difficulties with this way of proving.

Two interesting new articles have recently been published online in International Journal of Science and Mathematics Education:

• HISTORY AS A PLATFORM FOR DEVELOPING COLLEGE STUDENTS’ EPISTEMOLOGICAL BELIEFS OF MATHEMATICS by Po-Hung Liu combines two of my own main research interests: use of the history of mathematics and (epistemological) beliefs. Abstract: The present study observed how Taiwanese college students’ epistemological beliefs about mathematics evolved during a year-long historical approach calculus course. On the basis of the characteristics of initial accounts, seven students were invited to participate in this study and were divided into two groups. An open-ended questionnaire, mathematics biographies, in-class reports, and follow-up semi-structured interviews served as instruments for identifying their epistemological beliefs. Furthermore, four randomly selected students from another calculus class constituted the control group. Results indicated that most of the students receiving this course exhibited relatively significant changes in their epistemological beliefs of mathematics, but trends and extents in their epistemological development varied across groups as well as individuals. This study identifies the potential relationships among the course features, initial beliefs, and the tendency of belief development, followed by a discussion of the mechanism of belief change and an afterthought on HPM approach.
• METASYNTHESES OF QUALITATIVE RESEARCH STUDIES IN MATHEMATICS AND SCIENCE EDUCATION by Larry D. Yore and Stephen Lerman. This article is without abstract.

Journal of the British Society for the History of Mathematics has published the second issue of this year. It contains several interesting features, and an original article called “Mistakes concerning a chance encounter between Francis Galton and John Venn“. Here is the abstract of this article:

A chance encounter at Bournemouth between Francis Galton and John Venn has lain in some obscurity because of a slip by Galton himself and a second mistake by Karl Pearson. The contact with Venn provides insight into the development of Galton’s perception of statistical dispersion, his disenchantment with the notion of ‘probable error’ and adoption of population variability.

Christian Greiffenhagen and Wes Sharrock have written an article that was published in Educational Studies in Mathematics on Friday. The article is entitled “School mathematics and its everyday other? Revisiting Lave’s ‘Cognition in Practice’“. Here is the abstract of the article:

In the last three decades there have been a variety of studies of what is often referred to as ‘everyday’ or ‘street’ mathematics. These studies have documented a rich variety of arithmetic practices involved in activities such as tailoring, carpet laying, dieting, or grocery shopping. More importantly, these studies have helped to rectify outmoded models of rationality, cognition, and (school) instruction. Despite these important achievements, doubts can be raised about the ways in which theoretical conclusions have been drawn from empirical materials. Furthermore, while these studies rightly criticised prevalent theories of rationality and cognition as too simplistic to account for everyday activities, it seems that some of the proposed alternatives suffer from similar flaws (i.e., are straightforward inversions of the to-be-opposed theories, rather than more nuanced views on complicated issues). In this article we illustrate our sceptical view by discussing four case studies in Jean Lave’s pioneering and influential ‘Cognition in Practice’ (1988). By looking at the case studies in detail, we investigate how Lave’s conclusions relate to the empirical materials and offer alternative characterisations. In particular, we question whether the empirical studies demonstrate the existence of two different kinds of mathematics (‘everyday’ and ‘school,’ or ‘formal’ and ‘informal’) and whether school instruction tries to replace the former with the latter.

Jennifer A. Kaminski, Vladimir M. Sloutsky and Andrew F. Heckler wrote an article that was published in Science last week. The article is called: “Learning theory: The advantage of abstract examples in learning math“. A main issue discussed in the article is whether students who learn mathematics through real-world examples are able to apply this knowledge to other situations or not (the old problem about transfer of knowledge from one context to another). The article claims that their findings:

(…) cast doubt on a long-standing belief in education. The belief in using concrete examples is very deeply ingrained, and hasn’t been questioned or tested.

They also discuss the issue of word problems, and they claim that:

[Word] problems could be an incredible instrument for testing what was learned. But they are bad instruments for teaching.

If, like me, you don’t have full access to the articles in Science magazine, you could read a nice summary of the article with comments on Nobel Intent.

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.