A study on the representation of the arithmetic facts memory: cognitively speaking, is the commutativity a property of multiplications and additions?

In this thesis we deal with the role of the commutative property in the organization of the memory that encodes the knowledge about two arithmetic operations: multiplication and addition. Despite its relevance in arithmetic the commutative property has been largely neglected. This theoretical underestimation of the order of the operands and of the cognitive processing of commutative property is likely due to the lack of strong empirical evidences. In this thesis we provide new and clear empirical evidences of a operands-order effects. Five experiments were conducted by using various methodologies. Our results indicate that commutative pairs are processed differently for the two operations. For additions, we found an effect of the order of the operands. Namely, the problems in which the first operand is the larger (e.g., 7+4) were solved faster than the commuted problems (e.g., 4+7). For multiplication, we found an extremely surprising result that can not be predicted by any of the current models of arithmetical cognition, that is an interaction between the order of the operands and the size of the problem. Namely, for large multiplications (both operands larger than 5) the problems in which the first operand is the smaller (e.g., 7×8) were solved faster than the commuted problems (e.g., 8×7); for small multiplications (both operands smaller than 5) and medium multiplications (one operand smaller and the other larger than 5) the problems in which the first operands is the larger (e.g., 3×4, 7×3) were solved faster than the commuted problems (e.g., 4×3, 3×7). For both operations the relevant role of the order of the operands (and its interaction with size for multiplication) in the processing of the problems is also confirmed by a ERPs study. This results are interpreted as the evidence that the arithmetic facts memory is organized by the commutative property. More precisely, we propose to interpret the effects we found as due to the interaction of two factors that shape this memory according to the commutativity: (1) the order of acquisition of the commuted pairs and (2) a reorganization of the memory due to the use of non-retrieval procedures that privilege one of the two orders of the operands. Clearly, further researches are needed in order to better understand the relation between arithmetic facts memory and the commutative property. However, our result have showed a new effect that can usefully be used to investigate how the cognitive system encode, represent and process the arithmetical knowledge.