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!)

• 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

We start by visiting the maths section of the web site answers.yahoo.com. Here, anybody can ask a question from anywhere in the world at every possible level. Answers are given by anyone who wants to contribute and then askers/readers rate the responses. A brief look here and it is starkly clear that our young people are struggling and their ability to think logically—that is understand a problem, organize data into knowns and unknowns, explore possibilities and assess solutions is definitely on the decline. In our opinion, this is more insidious than the actual decline in their overall mathematics skills. Further, one is struck by the fact that technology seems to be contributing to this decline when in fact it should be the opposite. We then examine two question/answer cycles in detail and show how the freeware GeoGebra (www.geogebra.org GeoGebraWiki: http://www.geogebra.org/wiki GeoGebraForum: http://www.geogebra.org/forum)—which gives the freedom to explore and learn to everyone, everywhere and at any time—can be of tremendous value to pupils and students in their understanding of mathematics from the smallest ages on up.

Lectures are a familiar component in the delivery of mathematical content. Lecturers are often challenged with presenting material in a manner that aligns with the various learning styles and abilities within a large class. Students complain that the old-fashioned lecture style of copying notes from a board hinders the learning process, as they simply concentrate on writing. In recent times, distributing elaborate lecture notes has become a widespread alternative, but has its own problems, alienating the audience with lack of participation. The authors have developed a system of lecture notes, we call partially populated lecture notes, that have enjoyed success with students and addressed these difficulties.

We introduce the sociomathematical norm of speaking with meaning and describe its emergence in a professional learning community (PLC) of secondary mathematics and science teachers. We use speaking with meaning to reference specific attributes of individual communication that have been revealed to improve the quality of discourse among individuals engaged in discourse in a PLC. An individual who is speaking with meaning provides conceptually based descriptions when communicating with others about solution approaches. The quantities and relationships between quantities in the problem context are described rather than only stating procedures or numerical calculations used to obtain an answer to a problem. Solution approaches are justified with logical and coherent arguments that have a conceptual rather than procedural basis. The data for this research was collected during a year-long study that investigated a PLC whose members were secondary mathematics and science teachers. Analysis of the data revealed that after one semester of participating in a PLC where speaking with meaning was emphasized, the PLC members began to establish their own criteria for an acceptable mathematical argument and what constituted speaking with meaning. The group also emerged with common expectations that answers be accompanied by explanations and mathematical operations be explained conceptually (not just procedurally). The course and PLC design that supported the emergence of speaking with meaning by individuals participating in a PLC are described.

Working for meaningful mathematical change in the schools isn’t easy. There are issues of politics, turf, and sometimes unreasonable expectations on the part of the school district and the volunteers who work with it. But with good intentions, goodwill, and tenacity, there are ways to make a difference. This paper describes some of the ups, the downs, and the ultimate progress in a collaboration between U.C. Berkeley and the Berkeley Unified School district. It offers lessons to mathematicians who want to understand and/or work with their local schools.