Children’s understanding of addition and subtraction concepts

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

“…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).…”

“…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).…”

“…However, as already discussed concerning Baroody et al's (1983) findings, Canobi (2009Canobi et al, 1998Canobi et al, , 2002 assessed conceptual knowledge by asking participants to explain their strategies after they had solved an addition problem. Thus, even though Canobi et al used different tasks for assessing procedural and conceptual knowledge, the knowledge assessed by their addition task might have resulted from a mixture of procedural and conceptual competencies (see also, Robinson & Dubé, 2009). This might have led to a higher correlation between both tests compared to a variant were spontaneous application of procedural and conceptual knowledge is independently assessed.…”

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

“…Nevertheless, researchers have typically assumed that when solvers apply an inversion shortcut (i.e., faster or more accurate solutions of inversion problems such as 4 þ 9 À 9 compared to equivalent standard problems such as 4 þ 9 À 6), this performance is a useful, albeit indirect measure of children's understanding of the inverse principle (Gilmore & Papadatou-Pastou, 2009). Similarly, we assume that use of the associativity shortcut reflects children's understanding of the relation between operations while also including and acknowledging that additional measures of conceptual knowledge are necessary in this field (Prather & Alibali, 2009;Robinson & Dubé, 2009a).…”

“…In Robinson and Dubé (2009a), children in grades 2, 3, and 4 solved both inversion and control addition and subtraction problems. The control problems were designed such that subtracting the third number from the second number would always yield a positive number (e.g., 6 þ 5 À 2).…”