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The general aim of the research was to conduct a rare test of the efficacy of hypothetical learning progressions (HLPs) and a basic assumption of basing instruction on HLPs, namely teaching each successive level is more efficacious than skipping lower levels and teaching the target level directly. The specific aim was evaluating whether counting-based cardinality concepts unfold in a stepwise manner. The research involved a pretest—delayed-posttest design with random assignment of 14 preschoolers to two conditions. The experimental intervention was based on an HLP for cardinality development (first promoting levels that presumably support and are necessary for the target level and then the target knowledge). The active-control treatment entailed a Teach-to-Target approach (first promoting irrelevant cardinality knowledge about recognizing written numbers and then directly teaching the same target-level goals with the same explicit instruction and similar games). A mix of quantitative and qualitative analyses indicated HLP participants performed significantly and substantially better than Teach-to-Target participants on target-level concept and skill measures. Moreover, the former tended to make sensible errors, whereas the latter generally responded cluelessly. 

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.