ESM, April 2009

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