Student Participation in Collective Problem Solving in an After-School Mathematics Club: Connections to Learning and Identity

Abstract

This study examined the participation of a group of middle school students in an after-school mathematics club as they worked on cryptography problems. The analysis focused on interactions characterized by collective problem-solving activity, when intellectual work was distributed and various students took on active problem-solving roles, paying particular attention to intersections between task structures and positioning moves. We found that all open-ended tasks—those tasks that afforded multiple strategies and had multiple solutions—resulted in at least some collective problem solving, though it was not always sustained (Turner, Gutiérrez, & Sutton, 2009). We also found that the task structures, in combination with interactive positioning moves by facilitators and students, served to sustain or disrupt collective problem-solving activity.

Résumé

Cet article analyse la participation d’un groupe d’étudiants du premier cycle du secondaire aux activités d’un club de mathématiques alors qu’ils tentaient de résoudre des problèmes de cryptographie. l'analyse est centrée sur les interactions caractéristiques des activités collectives de résolution de problèmes, au cours desquelles les tâches intellectuelles sont distribuées et plusieurs étudiants assument un rôle actif de résolution des problèmes. l'analyse se concentre en particulier sur les points d’intersection entre la structure des tâches et les actions de positionnement. Nos résultats indiquent que toutes les tâches ouvertes, c’est-à-dire les tâches qui permettent l’utilisation de stratégies multiples et conduisent à de multiples solutions, ont mené au moins en partie à la résolution collective de problèmes, même si la solution envisagée n’était pas toujours durable (Turner, Gutiérrez et Sutton, 2009). Nous avons également constaté que la structure des tâches, alliée à certaines actions de positionnement interactives de la part des facilitateurs et des étudiants, est en mesure de soutenir ou de perturber les activités collectives de résolution de problèmes.

Acknowledgments

This research was funded in part by a National Science Foundation Center for Learning and Teaching Grant to The Center for the Mathematics Education of Latino/as (CEMELA), grant no. ESI-0424983. Any opinions, findings, and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Notes

1. Each episode was fully transcribed and line numbers used, starting at 1, to indicate shifts in the speaker. Both episodes include various segments of each complete transcript (episode A was 305 lines and episode B was 437 lines) and the corresponding line numbers. Our discussion references lines included in the episode text, as well as additional lines that occurred in between episode segments.

2. Though students use the phrasing “reverse the alphabet,” we discuss this as a reflection for purposes of analyzing the transformations considered by students.

3. A fixed point is any point mapping to itself. In this example, reflecting and then shifting the alphabet results in a linear function in which, depending on the shift, you have either 0 or 2 fixed points. The examples Diana considered (shift by 8, 10) each had two fixed points.