1. 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 #

2. 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.

3. 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).

4. 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.

5. 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.”

6. 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.”

7. 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).

8. 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)

9. 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).

10. 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.

11. 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 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.

12. 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 < (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).

13. 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.

14. 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.

15. Table 2
Subitizing Intervention Games
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