Elementary Preservice Teachers’ Transitional Conceptions of Partitive Division with Proper-Fraction Divisors

ABSTRACT

An understanding of partitive division is foundational for numerous other mathematics topics, including unit rate, slope, and probability. However, research has shown that learners tend to have a limited understanding of partitive division when the divisor is a proper fraction. To extend research on conceptions of partitive division in this study, we used Moschkovich’s (1999) transitional conceptions perspective to examine how conceptions of partitive division evolve. This article reports on preservice teachers’ (PSTs) conceptions of partitive division with proper-fraction divisors before and after they explored partitive division in a mathematics content course for elementary teachers (N = 17). Our analysis of pre- and post-interviews revealed an initial transitional conception and two levels of refinement of their conceptions. Furthermore, we identified two perturbations that PSTs experienced during refinement of their conceptions. By identifying ordered levels of refinement and associated perturbations, this exploratory study extends the transitional conceptions perspective. Furthermore, this study adds new insights into the conceptual complexities that the partitive model for division of fractions presents to PSTs (and to learners in general) and suggests new hypotheses about ways that conceptions of partitive division undergo refinement.

Acknowledgments

The authors would like to thank Judit Moschkovich for her valuable feedback on a prior version of this manuscript, as well as the helpful suggestions from the reviewers and editor.

Notes

¹ A more detailed definition of partitive division is presented later in the article.

² Note that we included units in all number sentences to necessitate the partitive division model over the quotative model (e.g., 24 oz ÷ 1/3 h). In our intervention, the interviews, and this article, we referred to statements like this with units as number sentences.

³ While we recognize that PSTs’ conceptions were transitioning during the intervention, we did not track conceptions during the intervention but chose to limit our comparisons to the pre- to post-interviews. This allowed PSTs a range of experiences with partitive division before the comparisons and increased the likelihood we would observe refinements.

⁴ Note that the minor drop on Task 2 from pre- to post-interview as seen in Table 3 involved a single student. We were unable to arrive at a satisfactory interpretation for this anomaly.

⁵ All names are gender-preserving pseudonyms.