Associativity and Understanding of the Operation of Addition in Children with Learning Differences.

Abstract. This study sample consisted of children from two learning categories, learning disabled (LD, n = 27, [bar]X = 9.08) and not identified as learning disabled (NLD, n = 42, [bar]X = 7.46), who were individually tested on three different mathematics tasks. The modified nonverbal task and the associativity of length task investigated the quality of students' structures of organizing activity by noting the complexity of grouping relationships abstracted among and between object sets (i.e., composite unit structures). Additionally measured in these two tasks was response accuracy. The flashcard task measured accuracy in response to the same number problems as in the modified nonverbal task as well as strategy type used. However, the strategies scored on the flashcard task were indicative of explicitly taught procedures regardless of children's structures of organizing activity. Significantly more NLD children abstracted composite unit structures suggestive of operational logic on the modified nonverbal and associativity of length tasks, although there were no significant differences in the rate of success on the modified nonverbal task. On the flashcard task, there were no significant differences between the two groups on strategy type used, although the LD children achieved greater success. These results suggest that although children state correct answers on the flashcard and modified nonverbal tasks, they may be reflecting on the tasks using thought structures that are not yet operational. Mathematical knowledge has traditionally been thought of as the "acquisition" of problem-solving strategies that include counting fingers, verbal counting, decomposition, and retrieval. The execution of these computational strategies is thought to result in the development of an association between the problem integer and the stated answer. As memory representations between problems and answers are developed, facts can be directly retrieved (Siegler, 1986, 1988; Siegler & Jenkins, 1989). The inability to compute number facts is due to deficits in: (a) fact retrieval, (b) procedures used, and (c) spatial representation (see Geary, 1993, and Jordan, 1995, for a review of related research). Language difficulties also negatively impact the mapping of conventional mathematical symbols onto a mental model of number and number transformation (Jordan, Levine, & Huttenlocher, 1995). We believe that the above deficits owe their existence to the information-processing perspective of mathematical knowledge. If we change our perspective of what constitutes mathematical knowledge, the source of children's difficulties also changes. From a constructivist perspective, logical-mathematical knowledge consists of biologically based forms and structures of mental activity that evolve their form through the coordinations of specific nervous activity (i.e., schemes) as objects are acted upon (Sinclair, 1990). As specialized organs regulate thinking activity between the organism and the environment while solutions to authentic problems are pondered, these cyclical structures of nervous activity evolve their internal order by their gradual extension and eventual reorganization onto more complex, higher-order levels. In turn, the coordination of more complex grouping relationships within and between object groups is made possible as the environment is acted upon (Piaget, 1971, 1985). The type of relationship coordinated within and between object groups is referred to as a composite unit structure (Behr, Harel, Post, & Lesh, 1994; Lamon, 1993, 1994, 1996; Piaget, 1987). To understand and extend children's logical-mathematical activity, it is therefore necessary to attend to the interactive relationship between students' structures of organizing activity and the type of composite unit structures that this activity coordinates while pondering solutions to problems (Bovet, 1981; Piaget, 1971, 1987). Finally, while language and equations are cultural tools that support reflective activity (e. …