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Number-recognition tasks, such as the how-many task, involve set-to-word mapping, and number-creation tasks, such as the give-n task, entail word-to-set mapping. The present study involved comparing sixty 3-year-olds’ performance on the two tasks with collections of one to three items over three time points about 3 weeks apart. Inconsistent with the sparse evidence indicating equivalent task performance, an omnibus test indicated that success differed significantly by task (and set size but not by time). A follow-up analysis indicated that the hypothesis that success emerges first on the how-many task was, in general, significantly superior to the hypothesis of simultaneous development. It further indicated the how-many-first hypothesis was superior to a give-n- first hypothesis for sets of three. A theoretical implication is that set-to-word mapping appears to develop before word-to-set mapping, especially in the case of three. A methodological implication is that the give-n task may underestimate a key aspect of children’s cardinal understanding of small numbers. Another is that the traditional give-n task, which requires checking an initial response by one-to-one counting, confounds pre-counting and counting competencies. 

Very few questions have cast such an enduring effect in cognitive science as the question of “symbol-grounding”: Do human-invented symbol systems have to be grounded to physical objects to gain meanings? This question has strongly influenced research and practice in education involving the use of physical models and manipulatives. However, the evidence on the effectiveness of physical models is mixed. We suggest that rethinking physical models in terms of analogies, rather than groundings, offers useful insights. Three experiments with 4- to 6-year-old children showed that they can learn about how written multi-digit numbers are named and how they are used to represent relative magnitudes based on exposure to either a few pairs of written multi-digit numbers and their corresponding names, or exposure to multi-digit number names and their corresponding physical models made up by simple shapes (e.g., big-medium-small discs); but they failed to learn with traditional mathematical manipulatives (i.e., base-10 blocks, abacus) that provide a more complete grounding of the base-10 principles. These findings have implications for place value instruction in schools and for the determination of principles to guide the use of physical models.