This study examined the generality of the logarithmic to linear transition in children's representations of numerical magnitudes and the role of subjective categorization of numbers in the acquisition of more advanced understanding. Experiment 1 (49 girls and 41 boys, ages 5-8 years) suggested parallel transitions from kindergarten to second grade in the representations used to perform number line estimation, numerical categorization, and numerical magnitude comparison tasks. Individual differences within each grade in proficiency for the three tasks were strongly related. Experiment 2 (27 girls and 13 boys, ages 5-6 years) replicated results from Experiment 1 and demonstrated a causal role of changes in categorization in eliciting changes in number line estimation. Reasons were proposed for the parallel developmental changes across tasks, the consistent individual differences, and the relation between improved categorization of numbers and increasingly linear representations.

We tested the hypothesis that encoding the numerical-spatial relations in a number board game is a key process in promoting learning from playing such games. Experiment 1 used a microgenetic design to examine the effects on learning of the type of counting procedure that children use. As predicted, having kindergartners count-on from their current number on the board while playing a 0 -100 number board game facilitated their encoding of the numerical-spatial relations on the game board and improved their number line estimates, numeral identification, and count-on skill. Playing the same game using the standard count-from-1 procedure led to considerably less learning. Experiment 2 demonstrated that comparable improvement in number line estimation does not occur with practice encoding the numerals 1-100 outside of the context of a number board game. The general importance of aligning learning activities and physical materials with desired mental representations is discussed.

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