The Role of the Number-After Rule in the Invention of Computational Shortcuts

How children invent a counting-on from the larger addend (COL) strategy is un-clear. An effort to understand how a child classified as mentally handicapped devised this strategy has suggested a plausible developmental mechanism. The child appeared to master the n + 1 combinations just before the invention of counting-on. This suggests he used a number-after rule for n + 1 combinations as a scaffold for constructing a counting-on strategy. (For example, if the sum of 7 + 1 is the number after seven in the count sequence, then 7 + 2 must be two numbers after seven in the count sequence-i.e., "Eight, nine.") Moreover, the child appeared to master the 1 + n combinations just before he recognized that addend order was irrelevant and started counting-on from the larger of the addends. This suggests that a general number-after rule (the sum of either n + 1 or 1 + n combinations is the number after n) may provide a basis for recognizing that order can be overlooked for any combination. Follow-up analyses of nine other mentally handicapped children and five kindergartners in the normal IQ range confirmed that the number-after rule for n + 1 combinations develops before and may provide a basis for counting-on. The follow-up data indicated that a generalized number-after rule for 1 + n combinations provides one (but not the only) way that children discover the computational shortcut of disregarding addend order.