How is subjectivity understood?

Tony Brown has written an article that was recently published online in Educational Studies in Mathematics. The article is entitled "Signifying “students”, “teachers” and “mathematics”: a reading of a special issue", and here is the abstract:

This paper examines a Special Issue of Educational Studies in Mathematics comprising research reports centred on Peircian semiotics in mathematics education, written by some of the major authors in the area. The paper is targeted at inspecting how subjectivity is understood, or implied, in those reports. It seeks to delineate how the conceptions of subjectivity suggested are defined as a result of their being a function of the domain within which the authors reflexively situate themselves. The paper first considers how such understandings shape concepts of mathematics, students and teachers. It then explores how the research domain is understood by the authors as suggested through their implied positioning in relation to teachers, teacher educators, researchers and other potential readers.

ESM, July 2008

Educational Studies in Mathematics has released the July Issue (Volume 68, Number 3). It contains 5 articles:

• What makes a counterexample exemplary? by Rina Zazkis and Egan J. Chernoff
• The roles of punctuation marks while learning about written numbers by Barbara M. Brizuela and Gabrielle A. Cayton
• Lacan, subjectivity and the task of mathematics education research by Tony Brown
• Learning opportunities from group discussions: warrants become the objects of debate by Keith Weber et al.
• An international comparison using a diagnostic testing model: Turkish students’ profile of mathematical skills on TIMSS-R by Enis Dogan and Kikumi Tatsuoka
  

Open-ended problems

The International Journal for Mathematics Teaching and Learning has recently published an article called: "Teaching and Evaluating 'Open-Ended' Problems". The article is written by Rama Klavir and Sarah Hershkovitz, and it is freely available in pdf format. Here is the abstract:

This paper focuses on an open-ended problem. The problem comprises a group of four numbers from which the students are asked to find the one that does not belong. Each of the numbers can be selected as not belonging, each one for different reasons. The problem was given to 164 fifth-grade students. The paper suggests tools for teachers to analyze and evaluate the work of their students when dealing with problems of this kind.

NCTM and the development of mathematics education in the US

NCTM is a huge organization for teachers of mathematics in the US, and it has certainly had a strong impact through the years. Michael Paul Goldenberg - author of the blog: Rational Mathematics Education - has written a very long and interesting article where he criticize this grand organization. For me - a Norwegian researcher with both legs planted firmly in Europe - this provides an interesting insight into the US discourse. I recommend reading the article, whether you agree with his views or not!

The system of coordinates and the concept of dimension

Constantine Skordoulis et al. have written an article called "The system of coordinates as an obstacle in understanding the concept of dimension". This article has recently been published online in International Journal of Science and Mathematics Education. Here is the abstract of the article:

The concept of dimension, one of the most fundamental ideas in mathematics, is firmly rooted in the basis of the school geometry in such a way that mathematics teachers rarely feel the need to mention anything about it. However, the concept of dimension is far from being fully understood by students, even at the college level. In this paper, we examine whether the Cartesian x-y plane is responsible for student difficulty in estimating the value of the dimension of an object, or is it only students misconceptions about dimension that lead them to a false estimation of the value of the dimension of various objects. A second question discussed in this paper examines whether the system of coordinates acts as an epistemological obstacle or whether it has only a didactical character.

Several new articles

International Journal of Mathematical Education in Science and Technology (IJMEST) has published several new articles online. Unfortunately, I have been to busy to cover them all, but you can take a look here!

Teaching Mathematics and its Applications has published a new issue (the June issue of 2008), and you can view these articles and abstracts by following this link.

ESM, June 2008

Educational Studies in Mathematics has published the June issue. This issue is in memory of Jim Kaput, and has a title: Democratizing Access to Mathematics through Technology: Issues of Design, Theory and Implementation— In Memory of Jim Kaput’s Work.

There are several interesting articles in this issue. Here are the headlines:

• Introduction: Building on the vision of Jim Kaput (1942–2005) by Richard Lesh and Stephen Hegedus
  
• Next steps in implementing Kaput’s research programme by Celia Hoyles and Richard Noss
• From static to dynamic mathematics: historical and representational perspectives by Luis Moreno-Armella, Stephen J. Hegedus and James J. Kaput
• A science need: Designing tasks to engage students in modeling complex data by Richard Lesh et al.
• Students’ expression of affect in an inner-city simcalc classroom by Roberta Y. Schorr and Gerald A. Goldin
• The role of scaling up research in designing for and evaluating robustness by J. Roschelle et al.
• Studying new forms of participation and identity in mathematics classrooms with
  integrated communication and representational infrastructures by Stephen J. Hegedus and William R. Penuel
• James J. Kaput (1942–2005) imagineer and futurologist of mathematics education by David Tall
  

Attrition of mathematics teachers

Gillian Hampden-Thompson, William L. Herring and Gregory Kienzl have written a report called Attrition of Public School Mathematics and Science Teachers. A 4-page abstract of the report is available as downloadable PDF. Here is the abstract:

Using data from the Teacher Follow-up Survey (TFS), this Issue Brief reports on trends in the attrition of public school mathematics and science teachers over a 16-year period and examines the reasons given by mathematics and science teachers for leaving teaching employment. Findings from the analysis indicate that the percentage of public school mathematics and science teachers who left teaching employment did not change measurably between 1988–89 and 2004–05. However, the percentage of other public school teachers who left teaching employment did increase over the same period. Differences were found between mathematics and science leavers and other leavers. For example, of those teachers with a regular or standard certification, a smaller percentage of mathematics and science teachers than other teachers left teaching employment. In addition, when asked to rate various reasons for leaving the teaching profession, greater percentages of mathematics and science leavers than other leavers rated better salary or benefits as very important or extremely important.

Everyday Mathematics and 'cognition in practice'

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.

New IJSME articles

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.
  

Mathematical language in early childhood settings

Loretta C. Rudd, Matthew c. Lambert, Macy Satterwhite and Amani Zaier have written an article that has recently been published (online first) in Early Childhood Education Journal. The article is entitled: "Mathematical Language in Early Childhood Settings: What Really Counts?" and it presents a study of what kinds of mathematical language that was used in six kindergarten classrooms (ages 0-6). Here is the abstract of the article.

Previous research indicates that, prior to entering kindergarten, most children are exposed to some type of formal or direct mathematics instruction. However, the type of mathematical language and the frequency of its use vary greatly in terms of its emphasis on academic content. This study investigated the types and frequency of mathematical language used in six classrooms for children ranging in age from birth to five years. The study site was a quality early childhood setting at a state university in Southwest. Results indicated that utterances pertaining to spatial relations exceeded any other type of mathematical concepts by approximately twice the frequency. In addition, there was a paucity of higher level mathematical concepts observed. These data suggest a need for enhanced attention to higher level mathematical concepts explored in early childhood settings.

Two new ZDM articles

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.
  

IJSME - New online articles

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.
  

BSHM Bulletin, Issue 2, 2008

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.

Some interesting reading

Education Week has a couple of interesting articles relating to mathematics education this week:

• Why the Best Math Curriculum Won’t Be a Textbook is an article that takes up discussions about mathematics curriculum standards and textbooks. One of the recommendations from the report of the National Mathematics Advisory Panel was shorter, more focused and more coherent textbooks, and this is discussed in the article.
  
• Math Group Tries to Help Young Teachers Stay the Course takes up the problem of young teachers that quit from the teaching profession, and an effort made by NCTM to help in that respect.
  

Both these articles are unfortunately only available to subscribers of Education Week, but they address interesting issues related to mathematics education.

The third and last reading tips in this connection, is a post from the "Let's play math!" blog. The post is entitled "How to teach math to a struggling student", and it starts off this important discussion with a practical example. If you don't agree with the advice given in the post, you might consider dropping a comment in the blog, because this is an important and interesting discussion!

Mathematics Teaching - pdf archive

The ATM journal Mathematics Teaching has a very nice (and growing) archive where several back issues are available in PDF format. Some articles/issues are only available to ATM members, while others (quite a few, actually!) are available to all, for free. In the archive, you can even take a closer look at the first issue (ever) of the journal, from 1955.

So, if you are interested in mathematics teaching in general (and in the UK in particular), you should definitely take a look! Hopefully, the archive will continue growing, and I wish other journals would follow up and do the same thing (preferably with a large collection of freely available back issues)!

School mathematics - everyday mathematics

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.

Real-world examples and transfer of learning

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.

Fibonacci numbers

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.

Is There a Role for Executive Functions in the Development of Mathematics Ability?

Blackwell Synergy - Mind Brain Education, Volume 2 Issue 2 Page 80-89, June 2008 (Article Abstract):

This article examines the role of working memory, attention shifting, and inhibitory control executive cognitive functions in the development of mathematics knowledge and ability in children. It suggests that an examination of the executive cognitive demand of mathematical thinking can complement procedural and conceptual knowledge-based approaches to understanding the ways in which children become proficient in mathematics. Task analysis indicates that executive cognitive functions likely operate in concert with procedural and conceptual knowledge and in some instances might act as a unique influence on mathematics problem-solving ability. It is concluded that consideration of the executive cognitive demand of mathematics can contribute to research on best practices in mathematics education.

Animating an equation

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.

New ZDM-articles

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.