The Development of the Commutativity Principle and Economical Addition Strategies

Abstract

What is the relationship between the principle of commutativity and the development of addition strategies that disregard addend order? It has been proposed that the assumption (Conjecture 1) or discovery (Conjecture 2) of commutativity is a necessary condition for the invention of such advanced addition strategies. A third hypothesis suggests that children may invent labor-saving addition strategies without necessarily appreciating the commutativity principle. This study tested the three conjectures by evaluating 36 kindergartners on two types of commutativity tasks. Both tasks involved predicting whether commuted and noncommuted pairs of problems would produce the same or different answers. Commutativity was not naturally assumed by children (as proposed by Conjecture 1), but appeared to be discovered. However, contrary to Conjecture 2 and consistent with Conjecture 3, an understanding of commutativity was not evident in all those who invented labor-saving addition strategies. This study also confirmed that counting-all starting with the larger addend-a mental strategy recently discovered in a case study-was not an uncommon labor-saving device among young children.