Adults' Understanding of Inversion Concepts: How Does Performance on Addition and Subtraction Inversion Problems Compare to Performance on Multiplication and Division Inversion Problems?

Abstract

Problems of the form a + b -b have been used to assess conceptual understanding of the relationship between addition and subtraction. No study has investigated the same relationship between multiplication and division on problems of the form d x e ÷ e. In both types of inversion problems, no calculation is required if the inverse relationship between the operations is understood. Adult participants solved addition/subtraction and multiplication/division inversion (e.g., 9 x 22 ÷ 22) and standard (e.g., 2 + 27 -28) problems. Participants started to use the inversion strategy earlier and more frequently on addition/subtraction problems. Participants took longer to solve both types of multiplication/division problems. Overall, conceptual understanding of the relationship between multiplication and division was not as strong as that between addition and subtraction. One explanation for this difference in performance is that the operation of division is more weakly represented and understood than the other operations and that this weakness affects performance on problems of the form d x e ÷ e.

How we use what we learn in Math: An integrative account of the development of commutativity.

“…Third graders in tendency also profited from these problems, but for them the effect was not significant. Adults, by comparison, did not show such a benefit, probably due to a floor effect based on the simplicity of the problems (for similar results, see Robinson & Dubé, 2009;Robinson & Ninowski, 2003).…”

“…Therefore, we assume that this finding is not due to a sample artefact. Nevertheless, it does not fit the general claim that with experience, children become faster and more accurate at solving addition problems and also tend to use more sophisticated strategies, such as order-irrelevant, decomposition, and retrieval strategies (Baroody et al, 1983;Canobi et al, 1998Canobi et al, , 20022003;Geary, Brown & Samaranayake, 1991;Goldman, Mertz & Pellegrino, 1989;Resnick, 1992;Rittle-Johnson & Siegler, 1998;Siegler, 1987;but see, McNeil, 2007;Robinson & Dubé, 2009;Robinson & Ninowski, 2003;Torbeyns et al, 2009). It also seems to contradict the results of Baroody et al (1983), which show that approximately 80% of their third graders applied the commutativity-based shortcut to solve arithmetic problems (see also, Canobi et al, 2003).…”

The inverse relation between multiplication and division: Concepts, procedures, and a cognitive framework

“…In the context of three-term problems such as 5 þ 2 À 2, fast and accurate solutions have been interpreted as evidence that solvers use their conceptual knowledge of the inverse property of addition and subtraction to simplify the computational requirements. The inverse property of multiplication and division can be analogously invoked to solve problems such as 5 Â 3 Ä 3 without the need for computation; but in contrast to the case of addition and subtraction, relatively little research has explored use of the inverse principle for multiplication and division (Robinson & Ninowski, 2003). In the present paper, we reviewed accumulated research on children and adults' use of the inverse property of multiplication and division on three-term problems.…”

“…Robinson and Ninowski (2003) and LeFevre and Robinson (2010) explored adults' use of inversion on multiplication and division problems in relation to addition and subtraction. In all of the studies with adults discussed in this review, problems ranged from single-digit operands (e.g., 3 þ 4 À 4 and 6 Â 2 Ä 2) to single-and double-digit operands (e.g., 6 þ 27 À 27 and 9 Â 22 Ä 22).…”

Introduction

“…There is also evidence that school children are much less likely to recognize and use the inverse relation between multiplication and division than the equivalent relation between addition and subtraction to solve arithmetical problems (Robinson, Ninowski, & Gray, 2006). Many adults also appear not to understand the inverse relation between multiplication and division: Even university students often find it hard to solve multiplicative problems with the help of this inverse relation (Robinson & Ninowski, 2003).…”