Indigo Esmonde

Ideas and Identities: Supporting Equity in Cooperative Mathematics Learning:

This review considers research related to mathematics education and cooperative learning, and it discusses how teachers might assist students in cooperative groups to provide equitable opportunities to learn. In this context, equity is defined as the fair distribution of opportunities to learn, and the argument is that identity-related processes are just as central to mathematical development as content learning. The link is thus considered between classroom social ecologies, the interactions and positional identities that these social ecologies make available, and student learning. The article closes by considering unresolved questions in the field and proposing directions for future research.

Mathematics Learning in Groups: Analyzing Equity in Two Cooperative Activity Structures.

Many mathematics classrooms use cooperative learning to support equitable learning environments for all students. Past research in the field has focused primarily on increasing achievement rather than on contexts that support equitable interactions. This year-long study in 3 secondary mathematics classes compares 2 activity structures—a group quiz and a presentation—by examining group interaction within the 2 activities. The analysis shows that groups constructed a range of work practices, including a practice focused on collaboration, one focused on individual work, and one focused on “helping.” In addition, students adopted a variety of positions, including expert, novice, in-between, and facilitator. In this data corpus, experts tended to dominate interactions during group quizzes, whereas presentation preparations were more equitable, particularly when a student was positioned as a facilitator. Based on the analysis, suggestions are provided for structuring more equitable mathematics group work.

Revoicing in a Multilingual Classroom

The concept of revoicing has recently received a substantial amount of attention within the mathematics education community. One of the primary purposes of revoicing is to promote a deeper conceptual understanding of mathematics by positioning students in relation to one another, thereby facilitating student debate and mathematical argumentation. Our study reexamines revoicing in a multilingual high school algebra classroom; our findings challenge the assumption that revoicing is necessarily tightly connected with classroom argumentation. We demonstrate that a single discursive form, such as revoicing, can play a wide range of valuable functions within the classroom. More importantly, we investigate systematic differences in the ways that revoicing is used, by a particular teacher, across languages. Implications for policy and practice are discussed.

Explanations in Mathematics Classrooms: A Discourse Analysis

This article focuses on student explanations as a discourse practice central to mathematics teaching and learning. I discuss classrooms as hybrid discourse spaces and focus on how talk is used to accomplish social action. In doing so, I contrast several different social and sociomathematical norms for explanation and suggest that students’ choices of discourse practices position them within the classroom. Further, I caution educators against assuming that complete and detailed explanations are always best to support student learning. I discuss how explanations that are coconstructed by several students can actually support joint engagement in mathematical work and help peers stay “on the same page” while avoiding hierarchical positioning.RésuméCet article se penche sur les explications fournies par les étudiants comme pratique du discours essentielle en enseignement et apprentissage desmathématiques. Je présente la salle de classe comme un espace hybride de discours, et j’analyse les façons dont on se sert du discours comme acte social. Je compare ainsi plusieurs normes sociales et socio-mathématiques dans les explications, et je postule que les pratiques du discours choisies par les étudiants positionnent ces étudiants au sein de la classe. De plus, je suggère aux enseignants de ne pas présumer que les explications complètes et détaillées soient toujours celles qui sont le mieux en mesure de favoriser l’apprentissage des élèves. Au contraire, les explications construites en collaboration avec d’autres élèves semblent favoriser l’engagement en groupe lorsqu’il s’agit du travail mathématique, et aider les pairs à partager les mê;emes connaissances tout en évitant les positionnements hiérarchiques.

Nobody's Rich and Nobody's Poor … It Sounds Good, but It's Actually Not: Affluent Students Learning Mathematics and Social Justice.

This article investigates how affluent students made sense of social justice issues that were embedded in mathematics learning activities. I present 2 case studies of such activities at the intermediate and secondary levels in 2 different schools. The analysis draws on video records and classroom artifacts and applies the theoretical framework of figured worlds to consider how students drew on their past experiences and on the structure of the classroom activities to understand the mathematics and the social justice issues. The analysis demonstrates how the 1st activity provided a familiar figured world to support learning about issues of wealth distribution. In the 2nd activity, because of a lack of what are termed intermediary figured worlds, students were left to draw on only their own experiences and background knowledge, including stereotypes about poor neighborhoods.

Teaching Mathematics for Social Justice in Multicultural, Multilingual Elementary Classrooms

This article describes a set of collaborative inquiry projects that emerged from a research study group involving teachers, university researchers (the authors), and school district staff as they worked to teach mathematics equitably in an urban elementary school. The project is analyzed using Marilyn Cochran-Smith’s (2004) six principles of pedagogy for teaching for social justice. In the study group, teachers were involved in designing research questions to honor their students’ cultural and community knowledge and to develop mathematics teaching with a social justice focus. We offer three examples of teaching mathematics for social justice in diverse classrooms and consider the broader implications of inquiry projects such as these.RésuméCet article présente une série de projets d’enquête réalisés en collaboration par un groupe de recherche comprenant des enseignants, des chercheurs universitaires (les auteurs) et des membres du personnel d’un district scolaire, désireux d’enseigner les mathématiques de façon équitable dans une école élémentaire en milieu urbain. Le projet est analysé au moyen des six principes pédagogiques de Marilyn Cochran-Smith pour l’enseignement de la justice sociale. Au sein du groupe d’étude, les enseignants ont participé à la création de questions de recherche misant sur les connaissances culturelles et locales de leurs étudiants, et au développement d’un enseignement des mathématiques centré sur la justice sociale. Nous proposons trois exemples d’enseignement des mathématiques pour la justice sociale dans différentes classes, et nous nous penchons sur les implications plus générales de projets de recherche de ce type.

Getting Unstuck: Learning and Histories of Engagement in Classrooms.

This article focuses on the role of history in shaping learning interactions in a high school mathematics class, in which we argue that students participate in two key activity systems: Learning mathematics and doing school. Within the context of these two activity systems, we highlight the nature of sociogenesis, the patterns of shift in communities as people build on one anothers accomplishments, jointly solve problems, and disseminate new and old ways of solving problems. Drawing on a yearlong study of group work in a high school mathematics classroom in California, we discuss how mathematical inscriptions in the classroom and the groups mathematical interactions were influenced by and also influenced the groups shared history. With this article we contribute to cultural-historical activity theory by providing insights into the study of history in classroom interactions.

RESPONSE: Genetic Method and Empirical Techniques: Reply to Hatano and Sfard's Commentaries on Cognition in Flux

Some time ago, Geoffrey Saxe (1994) published an article in Mind, Culture, and Activity that invoked a distinction between a method of inquiry and empirical techniques. Techniques are approaches to data gathering like observations, interviews, and surveys, as well as research designs. They are a tool kit for empirical work. Deeply linked to conceptual frames and units of analysis, a method of inquiry is something bigger. A method, like the genetic method outlined in our article, positions the investigator to pursue particular leads while leaving others to remain unexplored, if not unnoticed. It supports an investigator’s efforts to parse nature at its joints, given theoretical assumptions about the character of knowledge and knowledge change. Our method of inquiry targets forms and functions used in collective practices, with a focus on continuities and discontinuities in processes of micro-, onto-, and sociogenetic change. In the case at hand, we set our sights on the lexical expression fu, using it as a microcosm to unravel some complex issues in culture–cognition relations. Sfard (this issue) and Hatano’s (this issue) commentaries spark a host of interesting issues or tensions between our techniques and our genetic method, and in our reply, we comment on three: (a) our genetic method and techniques to document naturally occurring conversation, (b) cross-cohort design techniques and what they afford in relation to our method, and (c) epistemological groundings of our method and techniques. In each case, we make efforts to clarify issues and in so doing point to productive tensions between methods and techniques in fieldwork.

Mathematics Learning in Groups: Analysing Equity Within an Activity Structure

Many mathematics classrooms use cooperative learning to support equitable learning environments for all students. Past research in the field has focused primarily on increasing achievement rather than on contexts that support equitable interactions. This year-long study in 3 secondary mathematics classes examines group interaction within a single type of group activity – preparing to give a class presentation. The analysis shows that groups constructed a range of work practices, including a practice focused on collaboration, one focused on individual work, and one focused on “helping.” In addition, students adopted a variety of positions, including expert, novice, in-between, and facilitator. In this data corpus, presentation preparations were most equitable when a student was informally positioned as a facilitator. Based on the analysis, suggestions are provided for structuring more equitable mathematics group work.

Discursive Authority and Sociocultural Positioning in the Mathematics Classroom: New Directions for Teacher Professional Development

This article discusses findings from a two-day teacher conference focusing on discursive authority and sociocultural positioning in mathematics classrooms. The conference was designed to study how research on classroom discourse could be transformed into effective professional development activities. We describe how the focus on discourse, positioning, and authority was operationalized and made pragmatic for the teachers by focusing very closely on specific language use in classrooms. For each of the six workshops of the conference, we outline the goal of the workshop, the activities that were structured to reach those goals, and the theoretical constructs that were introduced to the teacher participants. We then draw on (a) transcripts of video and audio recordings, (b) material produced by the teachers, and (c) survey data, to discuss how these theoretical constructs were taken up and leveraged by participants as a means of reflecting on their own and other’s practice.RésuméCet article se penche sur les conclusions issues d’un colloque sur l’enseignement, d’une durée de deux jours, centré sur l’autorité discursive et le positionnement socioculturel dans les cours de mathématiques. Le colloque visait à faire le point sur les façons dont la recherche sur le discours en classe pouvait être transformée efficacement en activités de développement professionnel. Nous expliquons comment l’analyse du discours, du positionnement et de l’autorité a été rendue plus pragmatique pour les enseignants en ciblant particulièrement l’utilisation d’un langage spécifique en classe. Pour chacun des six ateliers du colloque, nous donnons le but de l’atelier, les activités structurées conçues pour atteindre ce but et les construits théoriques présentés aux enseignants participants. Nous avons également utilisé les transcriptions d’enregistrements vidéo et audio, le matériel produit par les enseignants et les données provenant d’enquêtes. Les résultats montrent que ce type d’activités de développement professionnel a un impact significatif sur les enseignants, car il augmente leur capacité d’aborder et de maˆıtriser des construits théoriques puissants dans le but d’analyser aussi bien leurs propres pratiques d’enseignement que celles d’autres enseignants.