Nezahat Cetin has written an article called The performance of undergraduate students in the limit concept. The article was published in the last issue of International Journal of Mathematical Education in Science and Technology. Here is the article abstract:

In this work, we investigated first-year university students’ skills in using the limit concept. They were expected to understand the relationship between the limit-value of a function at a point and the values of the function at nearby points. To this end, first-year students of a Turkish university were given two tests. The results showed that the students were able to compute the limit values by applying standard procedures but were unable to use the limit concept in solving related problems.

Dumma C. Mapolelo from University of Botswana has written an article that was recently published in the International Journal of Mathematical Education in Science and Technology. The article is entitled Students’ experiences with mathematics teaching and learning: listening to unheard voices. Here is the abstract of the article:

This study documents students’ views about the nature of mathematics, the mathematics learning process and factors within the classroom that are perceived to impact upon the learning of mathematics. The participants were senior secondary school students. Qualitative and quantitative methods were used to understand the students’ views about their experiences with mathematics learning and mathematics classroom environment. Interviews of students and mathematics lesson observations were analysed to understand how students view their mathematics classes. A questionnaire was used to solicit students’ views with regards to teaching approaches in mathematics classes. The results suggest that students consider learning and understanding mathematics to mean being successful in getting the correct answers. Students reported that in the majority of cases, the teaching of mathematics was lecture-oriented. Mathematics language was considered a barrier in learning some topics in mathematics. The use of informal language was also evident during mathematics class lessons.

Nikolaos Metaxas and Andromachi Karagiannidou have written an article called When Two Circles Determine a Triangle. Discovering and Proving a Geometrical Condition in a Computer Environment. This article was published online in the International Journal of Computers for Mathematical Learning on Sunday. Here is the abstract of their article:

Visualization of mathematical relationships enables students to formulate conjectures as well as to search for mathematical arguments to support these conjectures. In this project students are asked to discover the sufficient and necessary condition so that two circles form the circumscribed and inscribed circle of a triangle and investigate how this condition effects the type of triangle in general and its perimeter in particular. Its open-ended form of the task is a departure from the usual phrasing of textbook’s exercises “show that…”.

The April issue of Educational Studies in Mathematics has been published, and it contains five articles (including a book review):

• The array representation and primary children’s understanding and reasoning in multiplication, by Patrick Barmby, Tony Harries, Steve Higgins and Jennifer Suggate’. Abstract:  We examine whether the array representation can support children’s understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a ‘representational-reasoning’ model of understanding, where understanding is seen as connections being made between mental representations of concepts, with reasoning linking together the different parts of the understanding. We examine in detail the implications of this model, drawing upon the wider literature on assessing understanding, multiple representations, self explanations and key developmental understandings. Having also established theoretically why the array representation might support children’s understanding and reasoning, we describe the results of a study which looked at children using the array for multiplication calculations. Children worked in pairs on laptop computers, using Flash Macromedia programs with the array representation to carry out multiplication calculations. In using this approach, we were able to record all the actions carried out by children on the computer, using a recording program called Camtasia. The analysis of the obtained audiovisual data identified ways in which the array representation helped children, and also problems that children had with using the array. Based on these results, implications for using the array in the classroom are considered.
• Social constructivism and the Believing Game : a mathematics teacher’s practice and its implications, by Shelly Sheats Harkness. Abstract:  The study reported here is the third in a series of research articles (Harkness, S. S., D’Ambrosio, B., & Morrone, A. S.,in Educational Studies in Mathematics 65:235–254, 2007; Morrone, A. S., Harkness, S. S., D’Ambrosio, B., & Caulfield, R. in Educational Studies in Mathematics 56:19–38, 2004) about the teaching practices of the same university professor and the mathematics course, Problem Solving, she taught for preservice elementary teachers. The preservice teachers in Problem Solving reported that they were motivated and that Sheila made learning goals salient. For the present study, additional data were collected and analyzed within a qualitative methodology and emergent conceptual framework, not within a motivation goal theory framework as in the two previous studies. This paper explores how Sheila’s “trying to believe,” rather than a focus on “doubting” (Elbow, P., Embracing contraries, Oxford University Press, New York, 1986), played out in her practice and the implications it had for both classroom conversations about mathematics and her own mathematical thinking.
• Investigating imagination as a cognitive space for learning mathematics, by Donna Kotsopoulos and Michelle Cordy. Abstract:  Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University mathematics professor Barry Mazur (Imagining numbers (particularly the square root of minus fifteen), Farrar, Straus and Giroux, New York, 2003). The work of Mazur led us to question whether the features and steps of Mazur’s re-enactment of the imaginative work of mathematicians could be appropriated pedagogically in a middle-school setting. Our research objectives were to develop the framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging in an application of it in middle school setting. Findings from our application of the model suggest that the framework presents a novel and important approach to developing mathematical understanding. The model demonstrates in particular the importance of shared visualizations and problem-posing in learning mathematics, as well as imagination as a cognitive space for learning.
• Teachers’ perspectives on “authentic mathematics” and the two-column proof form, by Michael Weiss, Patricio Herbst and Chialing Chen. Abstract:  We investigate experienced high school geometry teachers’ perspectives on “authentic mathematics” and the much-criticized two-column proof form. A videotaped episode was shown to 26 teachers gathered in five focus groups. In the episode, a teacher allows a student doing a proof to assume a statement is true without immediately justifying it, provided he return to complete the argument later. Prompted by this episode, the teachers in our focus groups revealed two apparently contradictory dispositions regarding the use of the two-column proof form in the classroom. For some, the two-column form is understood to prohibit a move like that shown in the video. But for others, the form is seen as a resource enabling such a move. These contradictory responses are warranted in competing but complementary notions, grounded on the corpus of teacher responses, that teachers hold about the nature of authentic mathematical activity when proving.
• Book Review: The beautiful Monster by Mark Ronan (2006), Symmetry and the Monster, one of the greatest quests of mathematics. New York: Oxford University Press, 255 pp. ISBN 978-0-19-280723-6 £8.99 RRP

Kaye Stacey and Jill Vincent has written an article about Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. This article was published online in Educational Studies in Mathematics a few days ago. Here is the abstract of their article:

Understanding that mathematics is founded on reasoning and is not just a collection of rules to apply is an important message to convey to students. Here we examined the reasoning presented in seven topics in nine Australian eighth-grade textbooks. Focusing on explanatory text that introduced new mathematical rules or relationships, we classified explanations according to the mode of reasoning used. Seven modes were identified, making a classification scheme which both refined and extended previous schemes. Most textbooks provided explanations for most topics rather than presenting “rules without reasons” but the main purpose appeared to be rule derivation or justification in preparation for practise exercises, rather than using explanations as thinking tools. Textbooks generally did not distinguish between the legitimacies of deductive and other modes of reasoning.

Marshall Lassak has written an article about Using dynamic graphs to reveal student reasoning. This article was published earlier this month in International Journal of Mathematical Education in Science and Technology. Here is the (rather short) abstract of the article:

Using dynamic graphs, future secondary mathematics teachers were able to represent and communicate their understanding of a brief mathematical investigation in a way that a symbolic proof of the problem could not. Four different student work samples are discussed.

A. Bruno and M.C. Espinel have written an article called Construction and evaluation of histograms in teacher training. The article was published in International Journal of Mathematical Education in Science and Technology a couple of days ago. Their study shows, among other things, that students confuse histograms with bar diagrams. Here is their abstract:

This article details the results of a written test designed to reveal how education majors construct and evaluate histograms and frequency polygons. Included is a description of the mistakes made by the students which shows how they tend to confuse histograms with bar diagrams, incorrectly assign data along the Cartesian axes and experience difficulties in constructing the frequency polygon.

Three articles have been published in The Journal of Mathematical Behavior recently that are all related to “the catwalk task”.

1. Steven Case: The catwalk task: Reflections and synthesis: Part 1
Abstract: In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional.

2. Emiliano Vega and Shawn Hicks: The catwalk task: Reflections and synthesis: Part 2
Abstract: In this article we recount our experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, we individually reflect on our own mathematical work on the catwalk task. Second, we reflect on our experiences working with a group of community college students on the catwalk task and our interpretations of their mathematical thinking. In so doing we also detail a number of innovative and novel student-generated representations of the catwalk photos. Finally, we each individually reflect on the entire experience with the catwalk problem, as mathematics learners, as teachers, and as professionals.

3. Chris Rasmussen: Multipurpose Professional Growth Sequence: The catwalk problem as a paradigmatic example
Abstract: An important concern in mathematics teacher education is how to create learning opportunities for prospective and practicing teachers that make a difference in their professional growth as educators. The first purpose of this article is to describe one way of working with prospective and practicing teachers in a graduate mathematics education course that holds promise for positively influencing the way teachers think about mathematics, about student learning, and about mathematics teaching. Specifically, I use the “catwalk” task as an example of how a single problem can serve as the basis for a coherent sequence of professional learning experiences. A second purpose of this article is to provide background information that contextualizes the subsequent two articles, each of which details the positive influence of the catwalk task sequence on the authors’ professional growth.

So, you may ask, what is this catwalk problem really about then? The problem is originated in a set of 24 time-lapse photographs of a running cat. The question is simply: How fast is the cat moving at frame 10? Frame 20? (See this pdf for a presentation of the problem!)

The December issue of Nordic Studies in Mathematics Education (NOMAD) has already reached the subscribers (in the paper format). Now, it has also appeared online – or at least the abstracts. Here is the list of contents:

• Morten Blomhøj and Paola Valero: Bringing focus to mathematics education in multicultural and multilingual settings (Editorial)
• Kay Owens: Culturality in mathematics education: a comparative study
• Eva Norén: Bilingual students’ mother tongue: a resource for teaching and learning mathematics
• Troels Lange: Homework and minority students in difficulty with learning mathematics: the influence of public discourse
• Paola Valero, Tamsin Meaney, Helle Alrø, Uenuku Fairhall, Ole Skovsmose and Tony Trinick: School mathematical discourse in a learning landscape: understanding mathematics education in multicultural settings
• Barbro Grevholm: Activities for 2009 in the Nordic Graduate School in Mathematics Education