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

Abstract

One crucial issue in mathematics development is how children come to spontaneously apply arithmetical principles (e.g. commutativity). According to expertise research, well-integrated conceptual and procedural knowledge is required. Here, we report a method composed of two independent tasks that assessed in an unobtrusive manner the spontaneous use of procedural and conceptual knowledge about commutativity. This allowed us to ask (1)

Fostering Formal Commutativity Knowledge with Approximate Arithmetic

“…On the one hand, this result suggests that our approximation task is not suitable to boost the integration of both knowledge types in first graders. On the other hand, this missing correlation additionally supports the findings of Canobi et al [ 19 , 44 ] or Haider et al [ 48 ] who found first measureable integration of procedural and conceptual commutativity knowledge in second or third graders. Therefore, to maximize the chance of finding a possible beneficial influence of our approximation task on integration of conceptual and procedural knowledge in Experiment 3, we tested whether our approximation task might affect conceptual knowledge in third graders.…”

“…Thus, participants of Experiment 2 should not only be trained in solving addition problems, but also should have (at least) some formal conceptual knowledge about commutativity (see for example [ 42 ]). In order to assess conceptual knowledge, we used the above mentioned judgment task ([ 48 ]; Table 1 ) as an additional task. It resembled the computation task as it contained 30 addition problems altogether; 18 of which formed 9 commutative pairs.…”

“…Three-addend problems were used to test whether our approximation task is suited to activate commutative knowledge to entirely unfamiliar commutative problems. To this end, we accept that three-addend problems imply associativity as well as commutativity (see also [ 19 , 48 ]). Problems were spread over five pages, with six problems on each page.…”

“…The judgment task [ 48 ] also consisted of 30 three-addends-addition problems distributed over three pages. Among the 10 problems per page, three problems were commutative to their respective precursor problem (same addends in different order).…”

“…They were simply asked to mark those problems which they believed required no calculation in order to get to the result. If children possess conceptual knowledge, they should be able to figure out that this only applies to commutative problems (see [ 48 ]). We assume that the judgment task taps children’s metacognitive knowledge of the commutativity principle which according to Flavell [ 49 ] is an essential component of conceptual understanding.…”