Prospective Elementary Teachers’ Understanding of Division Involving Rational Numbers in a Teaching Through Problem-Solving Context

The purpose of this qualitative study was to explore prospective elementary teachers’ (PTs’) understanding of why, when dividing numbers, we multiply the dividend by the reciprocal of the divisor when the divisor is a rational number. We analyzed a sample of 44 PTs’ written responses on pre- and post-unit assessments, class homework, end-of-semester exam questions that involved division by rational numbers, and task-based interviews. We coded data deductively and inductively to investigate PTs’ procedural and conceptual understanding, and their errors and misconceptions about rational number ideas. We found that (a) a large number of PTs demonstrated procedural knowledge of fraction division across the semester, (b) the number of PTs who demonstrated conceptual knowledge and partial conceptual knowledge of fraction division remained considerably low in data collected earlier in the study, and almost equal to the number of PTs who demonstrated procedural knowledge in data collected later in the study, and (c) a small number of PTs showed misconceptions and errors about fraction division across all data sources. We also discuss the implications of this research in mathematics teaching and mathematics teacher education programs, in mathematics education research, and some of the limitations underlying our study.