How Best to Teach the Cardinality principle? Veena Paliwal, University of West Georgia Arthur J. BAROODY, UNIVERSITY OF ILLINOIS AT Urbana-Champaign and university of Denver This research was supported by the spencer foundation grant #201600029

The cardinality principle (cp) Specifies that the last number-word used in counting represents the total number of items in a collection. Typically develops between 3 and 5 years of age.

Why learning cp is important? Underlies meaningful one-to-one object counting. It is a developmental milestone that provides a foundation for early numeracy (Bermejo, 1996; Bermejo & Lago, 1990, 1994; Fuson, 1988; Gelman, Meck, & Merkin, 1986; (Sarnecka & Wright, 2013; Slusser & Sarnecka, 2011). Key preschool and kindergarten goal (e.g., various State pre-K standards, Common Core).

Prior Research on How to Best Teach the CP Almost no published intervention research on the CP. Mix et al. (2012) did evaluate four instructional approaches: (a) counting alone without separately specifying the cardinal value/total (count-only) (b) specifying the total alone (c) alternating between counting alone and labeling the total alone (d) labeling a collection with its cardinal value (total) first and then counting (labeling- first). Only labeling-first was significantly efficacious than other conditions at the posttest in Give- me-6 and Give-me-10 tasks.

CONCERNS WITH THE MIX ET AL. (2012) Study Mix et al.’s (2012) results do not actually provide guidance on how to teach the CP. The CP is a basic cardinality concept, which Fuson (1988) called the count-cardinal concept. Mix et al. (2012) used a give-me-n task, which assesses the advanced cardinality concept that Fuson (1988) called the cardinal-count concept. So (many) participants may already have constructed the count-cardinal concept (CP) at pretest.  Posttest gains indicated improvement in the advanced cardinal-count concept, NOT the basic count-cardinal concept (CP). For example, an adult might count a picture of three cookies by saying, “One, two, t-h-r-e-e (in a higher pitch); see three cookies.”

CONCERNS WITH THE MIX ET AL. (2012) Study Most common method of teaching the CP—modeling one-to- one counting and emphasizing/repeating the last number word (count-first method)—was not evaluated, because Mix et al. considered it too confusing. For example, an adult might count a picture of three cookies by saying, “One, two, t-h-r-e-e (in a higher pitch); see three cookies.”

Key issues addressed The purpose of the present study was to use a direct and more precise measure of the CP to examine which of instructional approaches most efficaciously promoted the CP. Compare the label-first (method Mix et al., 2012, found effective); count-first (common method of teaching the CP), and count-only (active control, which Mix et al., 2012, found to be ineffective).

KEY ASPECTS of the Design Table 4 CP Testing and Interventions KEY ASPECTS of the Design 1 week Pretest 3 weeks Random assignment to and implementation of the interventions Delayed Posttest (2-3 weeks after the intervention)

PARTICIPANTS 80 three- and four-year-old children from three prekindergarten schools participated in the preliminary intervention on subitizing. Pretesting revealed that 6 of the 47 three-year-old and 25 of the 33 four-year-old demonstrated understanding of the CP. CP intervention involved 41 3-year-old and 8 4-year-old children (n=49).

PRETEST/POSTTEST Measures used How many—a direct measure for the CP was used as main task. Give-me-n: An indirect measure of the CP that estimates meaningful application of the CP, was used as a transfer task.

interventions Three interventions: Label-first (14 3-year olds; 3 4s): Label collections with its cardinal value and then count the collection. Count-first (14 3-year olds; 2 4s): Count, emphasize last number word, and repeat the last number word. Count-only (13 3-year olds; 3 4s): Count collections only. One-on-one training involved 25-30 minutes sessions twice a week for 3 weeks (6 sessions). Participants practiced counting with 12 examples consisting of counting 1 to 6 items using picture books, concrete objects, and dots on an index card.

RESULTS—MAIN (How Many) TASK ANCOVA showed significant variation among groups, F(2, 45) = 10.25, p < 0.01. Tukey test showed that— Count-first and label-first groups performed significantly (and substantively) better than the counting-only group p < .001 (Hedges’ g = 1.4) and p < .04 (Hedges’ g = 1.1), respectively. Count-first group performed better than the label-first group at a marginally significant (p < 0.06) but substantive level (Hedges’ g = 0.7).

RESULTS-transfer (Give-me-n) TASK ANCOVA showed significant variation among groups, F(2, 45) = 15.22, p < 0.001 Tukey test showed that Count-first and label-first groups performed significantly (and substantively) better than the counting-only group p < .001 (Hedges’ g = 1.6) and p < .03 (Hedges’ g = 0.9), respectively. The count-first group performed significantly and substantively better than the label-first group at p < .023; Hedges’ g = 0.8.

SCIENTIFIC AND SCHOLARLY SIGNIFICANCE CP learning is dependent on how the concept is modeled. Emphasizing the total number of items in a collection (both count-first and label-first training) facilitates learning the CP (count-cardinal concept) and cardinal-count concept, whereas counting only does not. Count-first instruction is more efficacious than label-first training in promoting e CP (count-cardinal concept) and cardinal-count concept.

Table 2 Subitizing Intervention Games Contact information vpaliwal@westga.edu