The perils of averaging data over strategies: An example from children's addition.

Abstract

If a person uses different strategies on different trials, averaging data generated by those strategies can distort conclusions about numerous aspects of performance. The present study illustrates both the dangers of averaging data generated by different strategies and the gains that can be realized by examining performance generated by each strategy separately.The context for investigating these issues was young elementary-school children's addition. Previous models have depicted young children as always solving simple addition problems by using the min strategy, in which they count up from the larger number being added. For example, they would solve 3 + 6 by starting at 6 and counting from there to 9. The conclusion that children consistently use this approach to solve simple addition problems has been based primarily on the results of chronometric analyses that have indicated that both individuals and groups of children show the solution time pattern predicted by the min model.The present study involved examining children's verbal reports of their strategy on each problem as well as their solution-time patterns. When solution times on all trials were analyzed together, as in earlier studies, the results were entirely consistent with the view that children always use the min strategy. However, the verbal reports suggested a quite different picture. The min strategy was but one of five approaches that children reported using. This use of diverse strategies characterized individual as well as group performance; most children reported using at least three approaches. Neither the min strategy nor any other approach was used on as many as 40% of trials.Considerable converging evidence supported the validity of the children's verbal reports. Most important, on trials where they reported using the min strategy, the min model was an even better predictor of solution times than in past studies or in the present data set as a whole. In contrast, on trials in which they reported using one of the other strategies, the min model was never a good predictor of performance, either in absolute terms or relative to other predictors.Three factors that can lead to incorrect conclusions about data averaged over strategies were identified: relative frequency of each strategy, relative variability of performance generated by each strategy, and independent-dependent variable relations across and within strategies. The influence of these factors was illustrated both with regard to the present data and, for the more general case, through analyses of synthetic data in which each factor's contribution could be independently examined.There is no reason to think that variability of strategy use within a single person is limited to children or to arithmetic. Previous reports suggest that it is characteristic of adults as well as children and of tasks as diverse as spelling words, telling time, mentally rotating objects, and solving series completion problems. Among the other issues discussed are implications of the findings for when verbal s...

A componential analysis of an early learning deficit in mathematics

“…Table 3 presents the group-level characteristics of addition strategies. Inspection of Table 3 reveals that verbal counting and memory retrieval were the primary strategy choices for each of the three groups (Siegler, 1987). Indeed, initial analyses indicated no significant group differences in the overall distribution of strategy choices, F(8, 88) = 0.75, p > JO, but overall differences in error rates were significant, F(8, 88) = 2.96, p < .Ol.…”

“…If no answer exceeds the value of the confidence criterion and the value of the search length parameter is exceeded, then the child will resort to the use of a backup strategy. Here, the child will typically use either the counting fingers strategy or the verbal counting strategy to complete problem solving; although some children will occasionally use the fingers strategy (Baroody, 1987;Geary & Burlingham-Dubree, 1989;Siegler, 1987).…”

“…Another common computational strategy involves counting, from zero, in a unit-by-unit fashion a number of times equal to the cardinal value of both the augend and the addend. When problem solving requires counting, the majority of problems will be solved with the use of either the former min (or counting-on) procedure or the latter sum (or counting-all) procedure (e.g., Carpenter & Moser, 1984;Siegler, 1987).…”

“…or a general distribution of strategies (e.g., 50% counting and 50% memory retrieval) was inferred based upon patterns of performance on reaction time (RT) measures (Goldman et al, 1988). However, young children clearly use a variety of strategies to solve any given set of arithmetic problems and may in fact use a different strategy to solve the same problem on two different occasions (Carpenter & Moser, 1984;Geary & Burlingham-Dubree, 1989;Siegler, 1986Siegler, , 1987Siegler & Shrager, 1984). Thus, the above described performance characteristics of LD children may be inaccurate due to the fact that the methodologies employed in previous studies did not allow for the precise measurement of either problem-solving strategies or of the duration of the problem-solving process (see Siegler, 1987).…”

Cognitive addition: Strategy choice and speed-of-processing differences in gifted, normal, and mathematically disabled children.

“…For simple addition, two backup strategies are likely to be used by adults if the correct answer is not readily retrievable from long-term memory. The first of these involves the decomposition of the problem into more simple problems (Siegler, 1987). For example, the problem "6 + 9" might be solved by subtracting 1 from the 6, then adding the I to the 9, and finally adding 5 and 10.…”