Two experiments examined kindergartners', first graders', and second graders' numerical estimation, the internal representations that gave rise to the estimates, and the general hypothesis that developmental sequences within a domain tend to repeat themselves in new contexts. Development of estimation in this age range on 0-to-100 number lines followed the pattern observed previously with older children on 0-to-1,000 lines. Between kindergarten and second grade (6 and 8 years), patterns of estimates progressed from consistently logarithmic to a mixture of logarithmic and linear to a primarily linear pattern. Individual differences in number-line estimation correlated strongly with math achievement test scores, improved estimation accuracy proved attributable to increased linearity of estimates, and exposure to relevant experience tended to improve estimation accuracy.

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The authors examined developmental and individual differences in pure numerical estimation, the type of estimation that depends solely on knowledge of numbers. Children between kindergarten and 4th grade were asked to solve 4 types of numerical estimation problems: computational, numerosity, measurement, and number line. In Experiment 1, kindergartners and 1st, 2nd, and 3rd graders were presented problems involving the numbers 0-100; in Experiment 2, 2nd and 4th graders were presented problems involving the numbers 0-1,000. Parallel developmental trends, involving increasing reliance on linear representations of numbers and decreasing reliance on logarithmic ones, emerged across different types of estimation. Consistent individual differences across tasks were also apparent, and all types of estimation skill were positively related to math achievement test scores. Implications for understanding of mathematics learning in general are discussed.

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...

Many intelligent strategy choices may be accomplished through relatively low-level cognitive processes. This article describes a detailed model of how such "mindless" processes might lead to intelligent choices of strategies in one common situation: that in which people need to choose between stating a retrieved answer and using a backup strategy. Several experiments testing the model's applicability to children's single-digit multiplication are reported. These include tests of predictions about when different strategies are used and how early experience shapes later performance. Then, the sufficiency of the model to generate both performance at any one time and changes in performance over time is tested through the medium of a running computer simulation of children's multiplication. The simulation acquires a considerable amount of multiplication knowledge, and its learning and performance parallel those of children in a number of ways. Finally, several implications of the model for understanding cognitive self-regulation and cognitive development are discussed.