Past studies have documented students' and teachers' persistent difficulties in determining whether 2 quantities covary in a direct proportion, especially when presented missing-value word problems. In the current study, we combine a mathematical analysis with a psychological perspective to offer a new explanation for such difficulties. The mathematical analysis highlights numbers of equal-sized groups and places reasoning about proportional relationships in the context of reasoning about multiplicative relationships more generally. The psychological perspective is rooted in diSessa's (diSessa, 1993, 2006; diSessa, Sherin, & Levin, 2016) knowledge-in-pieces epistemology and highlights diverse, fine-grained knowledge resources that can support inferring and reasoning with equal-sized groups. We illustrate how the combination of mathematical analysis and psychological perspective may be applied to data using empirical examples drawn from interviews during which preservice middlegrades teachers reasoned with varying degrees of success about relationships presented in word problems that were and were not proportional.
Andrew Izsák, Department of Mathematics and Science Education, University of Georgia, Aderhold Hall, Athens, GA 30602; firstname.lastname@example.org
Erik Jacobson, Department of Curriculum and Instruction, Indiana University, 201 N. Rose Avenue, Bloomington, IN 47405; email@example.com