Examining aspects of elementary grades pre-service teachers’ mathematical reasoning

ABSTRACT

This mixed methods study investigated undergraduate elementary grades pre-service teachers’ (PSTs) mathematical reasoning. In the context of combinatorics, we examined two aspects of mathematical reasoning, namely, the processes of enumerating and generalizing, and sought to understand the relationship between them through analyses of 150 PSTs’ solutions to a set of combinatorics tasks. A subset of eight PSTs participated in follow-up interviews to further understand these aspects of reasoning when completing similar tasks. Findings suggest that the processes of reflecting and recalling prior knowledge mediated the relation between these two processes of PSTs’ reasoning. An analysis of PSTs’ written work suggested that many did not draw on their process of enumeration to generalize. Subsequent opportunities for reflection during interviews allowed PSTs to refine their enumeration process. For some, recalling prior knowledge assisted them in connecting their observations with an underlying mathematical structure, enabling them to generalize in ways that extended their understanding to explain contextual features of the problem. For others, this recollection shifted their focus and prevented their use of structure to generalize. For instructors of mathematics courses in teacher preparation programs, these results suggest that PSTs benefit from opportunities to reflect on their reasoning and that a careful consideration of how PSTs use their prior knowledge is essential in developing mathematics knowledge for teaching.

KEYWORDS:

• Combinatorics
• mathematical generalization
• mathematical reasoning
• pre-service teachers

Acknowledgments

We would like to acknowledge the contributions of Dr. Sarah Berenson, without whom this study would not have been conducted. We are profoundly grateful for her encouragement, guidance, and feedback. Her assistance was invaluable.