Although video cases are increasingly being used in teacher education as a means of situating learning and developing habits of reflection, there has been little evidence of the outcomes of such use. This study investigates the effects of using a coherent video-case curriculum in a university mathematics methods course by addressing two issues: (1) how the use of a video-case curriculum affects the reflective stance of prospective teachers (PTs); and (2) the extent to which a reflective stance developed while reflecting on other teachers’ practice transfers for reflecting on one’s own practice. Data sources include videotapes of course sessions and PTs’ written work from a middle school mathematics methods course that used a video-case curriculum as a major instructional tool. Both qualitative and quantitative analytical methods were used, including comparative and chi-square contingency table analyses. The PTs in this study showed changes in their level of reflection, their tendency to ground their analyses in evidence, and their focus on student thinking. In particular, they began to analyze teaching in terms of how it affects student thinking, to consider multiple interpretations of student thinking, and to develop a more tentative stance of inquiry. More significantly, the reflective stance developed via the video curriculum transferred to the PTs’ self-reflection in a course field experience. The results of this study speak to the power of using a video-case curriculum as a means of developing a reflective stance in prospective mathematics teachers.

Keith Weber, Carolyn Maher, Arthur Powell and Hollylynne Stohl Lee has written an article called “Learning opportunities from group discussions: warrants become the objects of debate” that has recently been published online by Educational Studies in Mathematics. The article deals with the interesting issues concerning discourse and learning opportunities in group discussions. Here is the abstract of the article:

In the mathematics education literature, there is currently a debate about the mechanisms by which group discussion can contribute to mathematical learning and under what conditions this learning is likely to occur. In this paper, we contribute to this debate by illustrating three learning opportunities that group discussions can create. In analyzing a videotaped episode of eight middle school students discussing a statistical problem, we observed that these students frequently challenged the arguments that their colleagues presented. These challenges invited students to be explicit about what mathematical principles, or warrants, they were implicitly using as a basis for their mathematical claims, in some cases recognize the modes of reasoning they were using were invalid and reject these modes of reasoning, and in other cases, attempt to provide deductive support to justify why their modes of reasoning were appropriate. We then describe what social and environmental conditions allowed the discussion analyzed in this paper to occur.

Interestingly enough, they use Toulmin‘s model of argumentation as a part of the theoretical framework for their analyses. The research that they report and discuss in this article occurred in the context of a research project called “Informal Mathematics Learning”, which is a project supported by the NSF.

Two new articles has recently been published (online first) by ZDM. The first article is written by Man-Keung Siu, and it is entitled “Proof as a practice of mathematical pursuit in a cultural, socio-political and intellectual context“. Here is the abstract of the article:

Through examples we explore the practice of mathematical pursuit, in particular on the notion of proof, in a cultural, socio-political and intellectual context. One objective of the discussion is to show how mathematics constitutes a part of human endeavour rather than standing on its own as a technical subject, as it is commonly taught in the classroom. As a “bonus”, we also look at the pedagogical aspect on ways to enhance understanding of specific topics in the classroom.

The other article is called “Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework“, and it is written by Susanne Prediger, Angelika Bikner-Ahsbahs and Ferdinando Arzarello. The article has a focus on the diversity of theories in mathematics education research, and how we can deal with that. Here is the abstract:

The article contributes to the ongoing discussion on ways to deal with the diversity of theories in mathematics education research. It introduces and systematizes a collection of case studies using different strategies and methods for networking theoretical approaches which all frame (qualitative) empirical research. The term ‘networking strategies’ is used to conceptualize those connecting strategies, which aim at reducing the number of unconnected theoretical approaches while respecting their specificity. The article starts with some clarifications on the character and role of theories in general, before proposing first steps towards a conceptual framework for networking strategies. Their application by different methods as well as their contribution to the development of theories in mathematics education are discussed with respect to the case studies in the ZDM-issue.

B. Pedemonte has written an article that has recently been published (online first) in ZDM. The article has a focus on a “core activity” in mathematics, and it is called: “Argumentation and algebraic proof“. Here is the abstract of the article:

This paper concerns a study analysing cognitive continuities and distances between argumentation supporting a conjecture and its algebraic proof, when solving open problems involving properties of numbers. The aim of this paper is to show that, unlike the geometrical case, the structural distance between argumentation and proof (from an abductive argumentation to a deductive proof) is not one of the possible difficulties met by students in solving such problems. On the contrary, since algebraic proof is characterized by a strong deductive structure, abductive steps in the argumentation activity can be useful in linking the meaning of the letters used in the algebraic proof with numbers used in the argumentation. The analysis of continuities and distances between argumentation and proof is based on the use of Toulmin’s model combined with ck¢ model.

Algebra is used in several different domains in mathematics, but this article has a focus on the algebra that is taught and learned in secondary school (Grade 12 and 13). After having elaborated and presented a theoretical framework for her analysis of proofs, Pedemonte presents some data that has been collected from prospective primary school teachers. These students were attending a course at the University, and their solutions to two open problems were analyzed according to the theoretical framework (the solutions of 7 students’ solutions to each of the two problems were analyzed).

The May issue of Journal for Research in Mathematics Education (JRME) has already arrived, and it contains the following articles:

ZPC and ZPD: Zones of Teaching and Learning
Anderson Norton and Beatriz S. D’Ambrosio

The Impact of Middle-Grades Mathematics Curricula and the Classroom Learning Environment on Student Achievement
James E. Tarr, Robert E. Reys, Barbara J. Reys, Óscar Chávez, Jeffrey Shih and Steven J. Osterlind

Learning to Use Fractions: Examining Middle School Students’ Emerging Fraction Literacy
Debra I. Johanning

The Linear Imperative: An Inventory and Conceptual Analysis of Students’ Overuse of Linearity
Wim Van Dooren, Dirk De Bock, Dirk Janssens and Lieven Verschaffel

Teaching With Games of Chance: A Review of The Mathematics of Games and Gambling
Laurie Rubel

The first issue of NOMAD this year has finally arrived, at least the web page has finally been updated to indicate that. Unfortunately, the articles are not available online, but you can read the abstracts (and the editorial in its entirety). The issue contains the following articles:

• The International Commission on Mathematical Instruction: a centenary of history and a future to construct by M. Blomhøj and P. Valero (editorial)
• Word problems in upper secondary algebra in Sweden over the years 1960–2000 by T. Jakobsson-Åhl
• Växelverkan mellan intuitiva idéer och formella resonemang – en fallstudie av universitetsstudenters arbete med en analysuppgift by K. Pettersson
• Classroom settings, self-regulated learning skills and grades in mathematics by J. Samuelsson
• The fifth year of the Nordic Graduate School by B. Grevholm (available in its entirety)

Stephen J. Hegedus and William R. Penuel wrote an article that was recently published online in Educational Studies in Mathematics. The article is called “Studying new forms of participation and identity in mathematics classrooms with integrated communication and representational infrastructures“, and here is the abstract of the article:

Wireless networks are fast becoming ubiquitous in all aspects of
society and the world economy. We describe a method for studying the
impacts of combining such technology with dynamic,
representationally-rich mathematics software, particularly on
participation, expression and projection of identity from a local to a
public, shared workspace. We describe the types of mathematical
activities that can utilize such unique combinations of technologies.
We outline specific discourse analytic methods for measuring
participation and methodologies for incorporating measures of identity
and participation into impact studies.

International Journal of Computers for Mathematical Learning has a column called “Computational Diversions”. Michael Eisenberg recently wrote an article/entry in this column called “Rounded Fractals“. The article is both practical and interesting, and it provides several examples concerning the generation of fractal designs. In the beginning of the article, he mentions turtle geometry (Logo), but the examples are made by making use of the method of iterated function systems. The article also contains a challenge, so anyone interested in fractals might want to take a look.

M. Pariès, A. Robert and J. Rogalski recently published an article called “Analyses de séances en classe et stabilité des pratiques d’enseignants de mathématiques expérimentés du second degré” in Educational Studies in Mathematics. The article is in French, but here is the abstract in English:

In this paper we tackle the issue of an eventual stability of teachers’
activity in the classroom. First we explain what kind of stability is
searched and how we look for the chosen characteristics: we analyse the
mathematical activity the teacher organises for students during
classroom sessions and the way he manages the relationship between
students and mathematical tasks. We analyse three one-hour sessions for
different groups of 11 year old students on the same content and with
the same teacher, and two other sessions for 14 year old and 15 year
old students, on analogous contents, with the same teacher (another
one). Actually it appears in these two examples that the main
stabilities are tied with the precise management of the tasks, at a
scale of some minutes, and with some subtle characteristic touches of
the teacher’s discourse. We present then a discussion and suggest some
inferences of these results.