Solution Spaces of Multiple-Solution Connecting Tasks as a Mirror of the Development of Mathematics Teachers’ Knowledge

Abstract

This study aims to deepen our understanding of the development of teacher knowledge in systematic (through learning) and craft (through teaching) modes. The main research tools of this study were multiple-solution connecting tasks. We used the notion of solution spaces to analyze the data and demonstrated that modifications of the teachers’ solution spaces were situated in their practices of varying types. We found that the implementation of multiple-solution connecting tasks in systematic mode meaningfully developed teachers’ problem-solving performance due not only to the reproduction of solutions offered during the course but to the production of new solutions. Furthermore, in creating opportunities for students to solve tasks with multiple solutions, teachers expanded their personal solution spaces. We conclude that the combining of systematic and craft modes is an optimal condition for the development of teachers’ knowledge.

Résumé

Cette étude a pour but d’approfondir notre compréhension du développement des connaissances chez les enseignants en matière de modes systématiques (grâce à l’apprentissage et à l’enseignement). Les principaux outils de recherche de cette étude sont les tâches de connexion à solutions multiples. Nous avons exploité la notion d’espaces de solution pour analyser les données, et nous avons démontré que les modifications des espaces de solutions proposées par les enseignants sont présentes dans différents types de pratiques. Nos résultats montrent que la réalisation systématique de tâches de connexion à solutions multiples améliore de façon significative la performance des enseignants en termes de solution de problèmes, non seulement grâce à la reproduction de solutions offertes durant la formation, mais également grâce à la production de solutions nouvelles. De plus, en créant pour les étudiants des occasions de résoudre certaines tâches au moyen de solutions multiples, les enseignants ont élargi leurs espaces de solutions personnelles. En conclusion, nous estimons que la combinaison des modes systématique et artisanal est une condition optimale pour le développement des connaissances chez les enseignants.

ACKNOWLEDGMENT

This research was made possible by grant #891/03 from the Israel Science Foundation. We thank Irena Gurevich for her assistance in data collection. We are also indebted to the teachers who participated in the study for their collaboration and goodwill.

Notes

1. Note that Figure 5 provides only partial information on the multiple-solution tasks that teachers incorporated in their classes during the second year of experiment: it does not include information on the CTs that teachers incorporated in their teaching spontaneously, without recording it for the purpose of the study. Because of space constraints, the detailed analysis of teaching CTs is reported in another paper (Leikin, in preparation).