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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. 

Efforts to improve instruction frequently focus on fostering meaningful learning—learning based on conceptual understanding—as opposed to knowledge memorized by rote. Consistent with Dewey’s (1963) principle of interaction, fostering meaningful learning entails identifying what children already know and do not know and building on the former to learn (moderately) new knowledge (Claessens & Engel, 2013; Fyfe et al., 2012; Piaget, 1964; Vygotsky, 1978). A learning trajectory (LT) approach to instruction—which includes conceptually and research-based and goals, a research-based learning progression of successive developmental levels, and research-based teaching activities to promote each level—epitomizes such an effort (Clements & Sarama, 2008; Confrey et al., 2012). Formative, classroom-based assessment—ongoing assessment to guide and monitor student learning (Black et al., 2003; Cizek, 2010; Author, 2018a)—is an integral aspect of the LT approach (Daro et al., 2011). In contrast to more commonly used summative assessment strategy (e.g., a unit test given at the end of an instruction unit to assess whether unit content has been mastered and grade progress), formative assessment serves to identify what developmental level a child has already achieved and the next developmentally appropriate level on which instruction should begin (Author, 2018a). Moreover, children are regularly assessed during instruction to gauge whether they–individually or collectively–have mastered a developmental level before instruction proceeds with the next higher level. In sum, “the LT approach involves using formative assessment (National Mathematics Advisory Panel, 2008; Shepard et al., 2018) to provide instructional activities aligned with empirically validated developmental progressions (Fantuzzo, Gadsden, & McDermott, 2011). Although research has shown that LT-based instruction is more efficacious, research is needed to evaluate the add-on value of the formative assessment components of LT-based instruction on student outcomes and the professional development of teachers. This presentation will highlight future lines of research that would provide insight into underlying theory and more productive strategies. Because LTs “need to be supplemented with consideration of obstacles that the student must overcome,” much needs to be learned about the obstacles posed by the content itself, instructional materials, and teachers (Ginsburg, 2009).