Building Mathematical Understanding Through Collective Property Noticing

Abstract

In this article we explore the mechanisms through which one group of preservice teachers engage in Collective Property Noticing—a phenomenon in which group members integrate individual contributions such that the group, as a unit, notices mathematical properties of their collective image. Drawing on improvisational theory to help to illuminate these collaborative processes, we claim that Collective Property Noticing is a capacity that is vital for mathematical sense-making in collaborative groups and we propose several conditions under which it is appropriate for a teacher to intervene in students’ learning in a problem-solving setting in order to provoke Collective Property Noticing.

Résumé

Dans cet article, nous nous penchons sur les mécanismes grâce auxquels un groupe de futurs enseignants participe à des activités d’observation des propriétés collectives, durant lesquelles les membres du groupe intègrent les contributions individuelles de chacun de façon à ce que le groupe en tant qu’équipe puisse observer les propriétés de son image collective. Sur la base d’une théorie de l’improvisation servant à éclairer ces processus de collaboration, nous postulons que l’observation des propriétés collectives constitue une habileté vitale pour la construction du sens mathématique dans les groupes de collaboration, et nous formulons plusieurs conditions dans lesquelles il est approprié que les enseignants interviennent dans l’apprentissage des étudiants, dans un contexte de résolution de problèmes, de façon à stimuler l’observation des propriétés collectives.

Acknowledgments

This article is based on a presentation at the 32nd annual meeting of the International Group for the Psychology of Mathematics Education (see Towers, Rapke, Pascuzzo, & Martin, 2010).

Notes

1. A more complete review of this literature is offered in Martin et al. (2006).

2. We recognize that others (e.g., King, 2001; Remillard, 1997; Sassi & Goldsmith, 1995) have adopted the metaphor of improvisation to describe mathematics teaching. This is not our focus here and we do not seek to characterize the actions of teachers as necessarily improvisational. However, in other papers (e.g., Martin & Towers, 2011; Towers, Martin, & Heater, 2012; Towers & Proulx, 2012) we do consider the specific role of the teacher when working with groups of students and in those writings we have positioned teaching as a complex act of participation in unfolding understandings and the teacher as a full participant in the emerging cognitive structure of the group.

3. All names are pseudonyms.

4. We use an ellipsis to convey continuation of a speech act from one line to another—sometimes this occurs for one speaker and sometimes this occurs between multiple speakers.

5. Inset lines of transcript indicate simultaneous speech.

6. Though we are well aware of the critiques of this kind of funnelling (Bauersfeld, 1988) when executed by a teacher, we note that teaching was not the goal of the interviewer here. Others also note the use of such incremental prompts as a research tool when investigating research participants’ understanding (see, e.g., Zazkis & Campbell, 1996).

7. Elsewhere (Martin & Towers, 2011; Towers et al., 2012), we discuss the various roles that a classroom teacher may take in being more fully integrated into the collective.