Alert 56 – ESM Volume 84, 1 (abstracts)

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An ideology critique of the use-value of mathematics
Alexandre Pais

The idea that mathematics is needed for our mundane everyday activities has raised the question of how people deal with mathematics outside the school walls. Much has been written in mathematics education research about the possibility of transferring knowledge from and into school. Whereas the majority of this literature commends the possibility of transfer, thus assuming both the desirability of transfer and the importance of school mathematics for the professional and mundane lives of individuals, I am interested in developing an ideology critique on the beliefs underpinning the research on this issue. It will be argued that the use-value attributed to school mathematics disavows its value as part of a political and economic structure, which requires school mathematics to perform other roles than the one related with utility. This critique will be illustrated through the exploration of a typical transfer situation between school and workplace

Troubling “understanding mathematics in-depth”: Its role in the identity work of student-teachers in England
Sarmin Hossain, Heather Mendick & Jill Adler
In this paper, we focus on an initiative in England devised to prepare non-mathematics graduates to train as secondary mathematics teachers through a 6-month Mathematics Enhancement Course (MEC) to boost their subject knowledge. The course documentation focuses on the need to develop “understanding mathematics in-depth” in students in order for them to become successful mathematics teachers. We take a poststructural approach, so we are not interested in asking what such an understanding is, about the value of this approach or about the effectiveness of the MECs in developing this understanding in their participants. Instead we explore what positions this discourse of “understanding mathematics in-depth” makes available to MEC students. We do this by looking in detail at the “identity work” of two students, analysing how they use and are used by this discourse to position themselves as future mathematics teachers. In doing so, we show how even benign-looking social practices such as “understanding mathematics in-depth” are implicated in practices of inclusion and exclusion. We show this through detailed readings of interviews with two participants, one of whom fits with the dominant discourses in the MEC and the other who, despite passing the MEC, experiences tensions between her national identity work and MEC discourses. We argue that it is vital to explore “identity work” within teacher education contexts to ensure that becoming a successful mathematics teacher is equally available to all.

How preservice teachers interpret and respond to student errors: ratio and proportion in similar rectangles
Ji-Won Son
Interpreting and responding to student thinking are central tasks of reform-minded mathematics teaching. This study examined preservice teachers’ (PSTs) interpretations of and responses to a student’s error(s) involving finding a missing length in similar rectangles through a teaching scenario task. Fifty-seven PSTs’ responses were analyzed quantitatively and qualitatively. Analysis results revealed that although the student’s errors came from conceptual aspects of similarity, a majority of PSTs identified the errors as stemming from procedural aspects of similarity, subsequently guiding them by invoking procedural knowledge. This study also revealed two different forms of address and teaching actions in PST interventions along with three categories of acts of communication barriers. The broader implications of the study for international communities are discussed in accordance with the findings.

An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking
Marta T. Magiera, Leigh A. van den Kieboom & John C. Moyer
Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers’ ability to use the features of the algebraic thinking (AT) habit of mind “Building Rules to Represent Functions” and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers’ own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed the pre-service teachers’ ability to recognize and interpret students’ facility with the AT habit of mind Building Rules to Represent Functions by analyzing their interpretations of students’ written solutions to algebra-based tasks and their ability to plan and analyze AT interviews of middle school students during a concurrent field-based education course. The data revealed that the pre-service teachers had a limited ability to recognize the full richness of algebra-based tasks’ potential to elicit the features of Building Rules to Represent Functions in students. The pre-service teachers’ own overall AT ability to Build Rules to Represent Functions was related to their ability to recognize the overall ability of students to Build Rules to Represent Functions, as exhibited during one-on-one interviews, but not to their ability to recognize the overall ability to Build Rules to Represent Functions exhibited exclusively in students’ written work. Implications for mathematics teacher education are discussed.

Schematic representations in arithmetical problem solving: Analysis of their impact on grade 4 students
Annick Fagnant & Joëlle Vlassis
While the value of ‘schematic representations’ in problem solving requires no further demonstration, the way in which students should be taught how to construct these representations invariably gives rise to various debates. This study, conducted on 146 grade 4 students in Luxembourg, analyzes the effect of two types of ‘schematic representation’ (diagrams vs. schematic drawings) on the solving of arithmetical problems. The results show that the presence of schematic representations has a clear positive effect on overall student performance and that a non negligible proportion of students manage to reuse the representations encountered in order to solve new problems. While showing an effect slightly in favor of diagrams as opposed to schematic drawings, our results do not really permit us to draw any conclusions about the form that these representations should take, in particular since a differential effect was observed depending on the type of problem.