Three models have been proposed to account for the relationship between the principle of commutativity and the development of more economical addition strategies, which disregard addend order. In the first and second models, it has been proposed that either discovery or assumption of commutativity is a necessary condition for the invention of advanced addition strategies. A third model suggests that children may invent labor-saving addition strategies without appreciating the commutativity principle. A study tested these three models by evaluating 36 kindergarteners' responses on two types of commutativity tasks. Both tasks involved predicting whether commuted and noncommuted pairs of problems would produce the same or different answers. Over two sessions, children's addition strategies were also assessed. Strategies noted were spontaneous counting-all with concrete supports; counting-all mentally, starting with the first addend; counting-all mentally, starting with the larger addend; counting-on mentally from the first addend; and counting-on mentally from the larger addend. (Counting-all strategies begin at the number 1; counting-on strategies begin at the value of the first addend selected by the child.) Findings indicated that, as proposed by the second model, commutativity was not naturally assumed by children, but appeared to be discovered. Contrary to the first model and consistent with the third, an understanding of commutativity was not evident in all subjects who invented labor-saving addition strategies. (Author/RH)