Elsa Malisani and Filippo Spagnolo have written an article called From arithmetical thought to algebraic thought: The role of the “variable”. This article was published online in Educational Studies in Mathematics last week. Here is the article abstract:

The introduction of the concept of the variable represents a critical point in the arithmetic–algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of “unknown” interferes with the interpretation of the variable “in functional relation” and the kinds of languages used by the students in problem-solving. We also wanted to study the concept of the variable in the process of translation from algebraic language into natural language. We present two experimental studies. In the first one, we administered a questionnaire to 111 students aged 16–19 years. Drawing on the conclusions of this research we carried out the second study with two pairs of students aged 16–17 years.

Jean-Baptiste Lagrange and Emel Ozdemir Erdogan have written an article called Teachers’ emergent goals in spreadsheet-based lessons: analyzing the complexity of technology integration, which was published in Educational Studies in Mathematics on Tuesday. Here is the abstract of the article:

We examine teachers’ classroom activities with the spreadsheet, focusing especially on episodes marked by improvisation and uncertainty. The framework is based on Saxe’s cultural approach to cognitive development. The study considers two teachers, one positively disposed towards classroom use of technology, and the other not, both of them experienced and in a context in which spreadsheet use was compulsory: a new curriculum in France for upper secondary non-scientific classes. The paper presents and contrasts the two teachers in view of Saxe’s parameters, and analyzes their activity in two similar lessons. Goals emerging in these lessons show how teachers deal with instrumented techniques and the milieu under the influence of cultural representations. The conclusion examines the contribution that the approach and the findings can bring to understanding technology integration in other contexts, especially teacher education.

Jason Silverman and Patrick W. Thompson have written an interesting article entitled Toward a framework for the development of mathematical knowledge for teaching. This article was published online in Journal of Mathematics Teacher Education on October 14. In the article, they draw upon the research that has been done in the area of Mathematical Knowledge for Teaching (MKT), and they try to navigate towards a framework for this. Silverman and Thompson present a framework that is “not only informed by the work of mathematics teaching, but also a developmental trajectory for mathematics learning and the learning sciences” (from their concluding comments).

Here is the abstract of their article:

Shulman (1986, 1987) coined the term pedagogical content knowledge (PCK) to address what at that time had become increasingly evident—that content knowledge itself was not sufficient for teachers to be successful. Throughout the past two decades, researchers within the field of mathematics teacher education have been expanding the notion of PCK and developing more fine-grained conceptualizations of this knowledge for teaching mathematics. One such conceptualization that shows promise is mathematical knowledge for teaching—mathematical knowledge that is specifically useful in teaching mathematics. While mathematical knowledge for teaching has started to gain attention as an important concept in the mathematics teacher education research community, there is limited understanding of what it is, how one might recognize it, and how it might develop in the minds of teachers. In this article, we propose a framework for studying the development of mathematical knowledge for teaching that is grounded in research in both mathematics education and the learning sciences.

Michelle T. Chamberlin, Jeff D. Farmer and Jodie D. Novak have written an article called Teachers’ perceptions of assessments of their mathematical knowledge in a professional development course. The article was published online in Journal of Mathematics Teacher Education a couple of days ago. Here is the abstract:

The purpose of the project reported in this article was to evaluate how assessing teachers’ mathematical knowledge within a professional development course impacted from the teachers’ perspective their learning and their experience with the course. The professional development course consisted of a 2-week summer institute and the content focus was geometry. We had decided to assess the mathematical learning of the teachers during this professional development course for various accountability reasons, but were concerned about possible negative by-products of this decision on the teachers and their participation. Thus, we worked to design assessment in ways that we hoped would minimize negative impacts and maintain a supportive learning environment. In addition, we undertook this evaluation to examine the impacts of the assessment, which included homework, quizzes, various projects, and an examination for program evaluation. Seventeen grade 5–9 teachers enrolled in the course participated in the study by completing written reflections and by describing their experiences in interviews. We learned that while our original intent was “to do no harm,” the teachers reported that their learning was enhanced by the assessment. The article concludes by describing the various properties of the assessments that the teachers identified as contributing to their learning of the geometry content, many of which align with current recommendations for assessing and evaluating grade K-16 mathematics students.

Herbert Gerstberger has written an interesting article about the connection between Mathematics learning and aesthetic production. In the article, he introduces several interesting aspects concerning aesthetics, arts, metaphor, semiotics, etc. The article was published online in ZDM, two days ago. Here is the article abstract:

Some teaching projects in which the learning of mathematics was combined with mainly theatrical productions are reported on. They are related and opposed to an approach of drama in education by Pesci and the proposals of Sinclair for mathematics teaching and beauty. The analysis is based on the distinction between aesthetics as related to beauty or as related to sensual perception. The usefulness of concepts of model and metaphor for the understanding of aesthetic representations of mathematical subject matter is examined. It is claimed that the Peircean concept of the interpretant contributes to a concise analytical approach. The pedagogical attitude is committed to a balanced relationship of scientific and aesthetic values.

Guershon Harel has written an article called A DNR perspective on mathematics curriculum and instruction. Part II: with reference to teacher’s knowledge base, which was published online in ZDM on Tuesday this week. In this article, Harel touches upon many interesting issues concerning the teaching and learning of mathematics. Here is the abstract of the article:

Two questions are on the mind of many mathematics educators; namely: What is the mathematics that we should teach in school? and how should we teach it? This is the second in a series of two papers addressing these fundamental questions. The first paper (Harel, 2008a) focuses on the first question and this paper on the second. Collectively, the two papers articulate a pedagogical stance oriented within a theoretical framework called DNR-based instruction in mathematics. The relation of this paper to the topic of this Special Issue is that it defines the concept of teacher’s knowledge base and illustrates with authentic teaching episodes an approach to its development with mathematics teachers. This approach is entailed from DNR’s premises, concepts, and instructional principles, which are also discussed in this paper.

Guy Brousseau, Nadine Brousseau and Virginia Warfield have written an article called Rationals and decimals as required in the school curriculum Part 3. Rationals and decimals as linear functions. The article was published in The Journal of Mathematical Behavior a few days ago. Here is the abstract of the article:

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment, carried out by a team of researchers and teachers that included his wife, Nadine, in classrooms at the École Jules Michelet, was to teach all of the material on rational and decimal numbers required by the national programme with a carefully structured, tightly woven and interdependent sequence of “situations.” This article describes and discusses the third portion of that experiment.

Ronald A. Beghetto and James C. Kaufman have written an article in ZDM that was published on Friday. The article is entitled Do we all have multicreative potential? and it deals with the issue of creativity and multicreativity. Here is the abstract of the article:

Are only certain people destined to be multicreative—capable of unique and meaningful contributions across unrelated domains? In this article, we argue that all students have multicreative potential. We discuss this argument in light of different conceptions of creativity and assert that the likelihood of expressing multicreative potential varies across levels of creativity (most likely at smaller-c levels of creativity; least likely at professional and eminent levels of creativity). We close by offering considerations for how math educators might nurture the multicreative potential of their students.

Stefan Krauss, Jürgen Baumert and Werner Blum have written an article entitled Secondary mathematics teachers’ pedagogical content knowledge and content knowledge: validation of the COACTIV constructs. The article (which is an Open Access article!) was published online in ZDM last week. This is a very interesting article, which gives a nice contribution to the field of research related to teachers’ knowledge. It builds upon the framework of Shulman, and it gives a nice overview of these theories, as well as an overview of some of the other research projects that have been contributing to this field (like the study of Ball, Hill, Schilling et al. in Michigan). Here is the abstract of the article:

Research interest in the professional knowledge of mathematics teachers has grown considerably in recent years. In the COACTIV project, tests of secondary mathematics teachers’ pedagogical content knowledge (PCK) and content knowledge (CK) were developed and implemented in a sample of teachers whose classes participated in the PISA 2003/04 longitudinal assessment in Germany. The present article investigates the validity of the COACTIV constructs of PCK and CK. To this end, the COACTIV tests of PCK and CK were administered to various “contrast populations,” namely, candidate mathematics teachers, mathematics students, teachers of biology and chemistry, and advanced school students. The hypotheses for each population’s performance in the PCK and CK tests were formulated and empirically tested. In addition, the article compares the COACTIV approach with related conceptualizations and findings of two other research groups.