The Role of Non-symbolic and Symbolic Skills in the Development of Early Numerical Cognition from Preschool to Kindergarten Age

Figures & data

Table 1. Descriptive statistics of non-symbolic number line estimation, quantity comparison, enumerating, working memory, symbolic number line estimation and counting

	M	SD
Time 1
Non-symbolic number line estimation (PAE)	32.11	9.23
Quantity comparison	19.67	4.67
Enumerating	9.21	5.64
Working memory	11.06	2.74
Time 2
Symbolic number line (PAE)	17.17	5.95
Counting	9.16	3.27
Working memory	16.58	2.68

Note. PAE = percentage absolute estimation error.

N = 43, except for enumerating N = 42.

Table 2. Posterior summaries of the correlation between non-symbolic number line estimation, quantity comparison, enumerating, working memory, symbolic number line estimation and counting

		Time 1				Time 2
		Non-symbolic number line (PAE)	Quantity comparison	Enumerating	Working memory	Symbolic number line (PAE)
Time 1
Quantity comparison	Pearson’s r	−0.31	–
	95% Credible Interval	−0.55; −0.01	–
	Bayes Factor	1.44*	–
Enumerating	Pearson’s r	−0.10	0.23	–
	95% Credible Interval	−0.38; 0.20	−0.08; 0.49	–
	Bayes Factor	0.23°°	0.55°	–
Working memory	Pearson’s r	−0.40	0.25	0.23	–
	95% Credible Interval	−0.61; −0.11	−0.06; 0.50	−0.08; 0.49	–
	Bayes Factor	6.02**	0.66°	0.54°	–
Time 2
Symbolic number line (PAE)	Pearson’s r	0.42	−0.13	0.03	−0.30	–
	95% Credible Interval	0.13; 0.63	−0.40; 0.17	−0.26; 0.32	−0.54; 0.00	–
	Bayes Factor	9.25**	0.26°°	0.20°°	1.27*	–
Counting	Pearson’s r	−0.33	−0.03	0.44	0.16	−0.25
	95% Credible Interval	−0.56; −0.02	−0.32; 0.27	0.14; 0.64	−0.14; 0.43	−0.50; 0.05
	Bayes Factor	1.70*	0.19°°	10.60***	0.32°°	0.70°

Note. PAE = percentage absolute estimation error.

° Weak/inconclusive evidence H₀; °° Moderate evidence H₀; °°° Strong evidence H_0.

* Weak/inconclusive evidence H_a; ** Moderate evidence H_a; *** Strong evidence H_a.

N = 43, except for correlations concerning enumerating N = 42.

Table 3. Posterior summaries of the regression coefficients of non-symbolic number line estimation, quantity comparison, enumerating, and working memory on symbolic number line estimation and counting (initial models)

		Symbolic number line (PAE)^a	Counting^b
Intercept	Mean (SD)	17.24 (0.87)	9.10 (0.45)
	95% Credible Interval	15.49; 19.01	8.18; 10.01
	Bayes Factor Inclusion	1.00	1.00
Non-symbolic number line (PAE)	Mean (SD)	0.16 (0.12)	−0.07 (0.06)
	95% Credible Interval	0.00; 0.36	−0.18; 0.00
	Bayes Factor Inclusion	2.97*	2.62*
Quantity comparison	Mean (SD)	0.00 (0.10)	−0.07 (0.10)
	95% Credible Interval	−0.26; 0.25	−0.28; 0.05
	Bayes Factor Inclusion	0.46°	1.36*
Enumerating	Mean (SD)	0.03 (0.09)	0.19 (0.10)
	95% Credible Interval	−0.13; 0.28	0.00; 0.34
	Bayes Factor Inclusion	0.49°	9.55**
Working memory	Mean (SD)	−0.14 (0.27)	−0.01 (0.11)
	95% Credible Interval	−0.82; 0.17	−0.30; 0.24
	Bayes Factor Inclusion	0.71°	0.81°

Note. PAE = percentage absolute estimation error.

^aBayes Factor = 0.87, R² =.21.

^bBayes Factor = 6.25, R² =.31.

° Weak/inconclusive evidence H₀; °° Moderate evidence H₀; °°° Strong evidence H_0.

* Weak/inconclusive evidence H_a; ** Moderate evidence H_a; *** Strong evidence H_a.

N = 42.

Table 4. Posterior summaries of the regression coefficients of non-symbolic number line estimation on symbolic number line estimation and non-symbolic number line estimation, quantity comparison and enumerating on counting (final models)

		Symbolic number line (PAE)^a	Counting^b
Intercept	Mean (SD)	17.24 (0.86)	9.10 (0.45)
	95% Credible Interval	15.56; 19.04	8.18; 9.99
	Bayes Factor Inclusion	1.00	1.00
Non-symbolic number line (PAE)	Mean (SD)	0.21 (0.11)	−0.08 (0.06)
	95% Credible Interval	0.00; 0.38	−0.18; 0.00
	Bayes Factor Inclusion	7.55**	3.51**
Quantity comparison	Mean (SD)	-	−0.08 (0.10)
	95% Credible Interval	-	−0.29; 0.05
	Bayes Factor Inclusion	-	1.76*
Enumerating	Mean (SD)	-	0.20 (0.09)
	95% Credible Interval	-	0.00; 0.34
	Bayes Factor Inclusion	-	13.41***

Note. PAE = percentage absolute estimation error.

^aBayes Factor = 7.55, R² =.18.

^bBayes Factor = 15.40, R² =.31.

° Weak/inconclusive evidence H₀; °° Moderate evidence H₀; °°° Strong evidence H_0.

* Weak/inconclusive evidence H_a; ** Moderate evidence H_a; *** Strong evidence H_a.

N = 42.