Results

Descriptive Statistics

Children produced a wide range of scores on all of our measures. Scores on the 21-item Colored Progressive Matrices, were approximately normally distributed (M = 10.47, SD = 2.99, min = 5, max = 18). Numeracy scores on the 24-item PENS-B were slightly right-skewed (M = 16.56, SD = 4.66, min = 5, max = 24), presumably due to the age of our sample being at the upper-limit of eligible ages for this measure.

Responses to the Competing-units Counting Task are shown in Table 1. Children on average counted incorrect units on 3.11 trials (SD = 1.53) and counted inaccurately on 1.75 (SD = 1.43) of the eight trials of the task.

Descriptive Statistics
	cpm	pens	counting_incorrect_units	counting_inaccuracy
Valid	64	64	64	64
Missing	0	0	0	0
Mean	10.469	16.563	3.109	1.750
Std. Deviation	2.987	4.659	1.534	1.425
Minimum	5.000	5.000	0.000	0.000
Maximum	18.000	24.000	6.000	6.000

Distribution Plots

cpm

pens

counting_incorrect_units

counting_inaccuracy

Correlation Matrix

We calculated a Spearman correlation matrix, shown in Table 2. There were strong correlations between children’s numeracy skills and all other measured variables. The number of instances in which children counted the incorrect units was strongly negatively related to children’s general numeracy skills.

Spearman Correlations
		age		cpm		pens		counting_incorrect_units		counting_inaccuracy
age	Spearman's rho	—
age	p-value	—
cpm	Spearman's rho	0.211		—
cpm	p-value	0.094		—
pens	Spearman's rho	0.414	***	0.413	***	—
pens	p-value	< .001		< .001		—
counting_incorrect_units	Spearman's rho	-0.279	*	-0.266	*	-0.576	***	—
counting_incorrect_units	p-value	0.026		0.033		< .001		—
counting_inaccuracy	Spearman's rho	-0.303	*	-0.426	***	-0.564	***	0.373	**	—
counting_inaccuracy	p-value	0.015		< .001		< .001		0.002		—

* p < .05, p < .01, * p < .001

Linear Regression

We calculated a hierarchical linear regression in order to isolate the relationship between counting of incorrect units and numeracy. We predicted children’s numeracy skills, as measured by the Preschool Early Numeracy Scale, in the first step by children’s age, intelligence and frequency of counting inaccuracies. This set of control variables explained 41.7% of variance in numeracy skills, F(3, 60) = 14.31, p <.001.

In the second step we entered counting incorrect units scores as an additional predictor of numeracy. Counting incorrect units explained an additional 9.0% of variance in children’s numeracy, F_change(1, 59) = 10.82, p = .002. Moreover, counting incorrect units was the strongest predictor in the full model

Model Summary
Model	R	R²	Adjusted R²	RMSE	R² Change	F Change	df1	df2	p
0	0.646	0.417	0.388	3.645	0.417	14.311	3	60	< .001
1	0.712	0.507	0.474	3.379	0.090	10.816	1	59	0.002

Note. Null model includes age, cpm, counting_inaccuracy

ANOVA
Model		Sum of Squares	df	Mean Square	F	p
0	Regression	570.475	3	190.158	14.311	< .001
	Residual	797.275	60	13.288
	Total	1367.750	63
1	Regression	693.994	4	173.499	15.193	< .001
	Residual	673.756	59	11.420
	Total	1367.750	63

Note. Null model includes age, cpm, counting_inaccuracy

Coefficients
Model		Unstandardized	Standard Error	Standardized	t	p
0	(Intercept)	2.497	4.591		0.544	0.588
	age	2.709	0.908	0.320	2.985	0.004
	cpm	0.278	0.172	0.178	1.615	0.112
	counting_inaccuracy	-1.156	0.356	-0.354	-3.246	0.002
1	(Intercept)	8.862	4.676		1.895	0.063
	age	2.083	0.863	0.246	2.414	0.019
	cpm	0.214	0.161	0.137	1.335	0.187
	counting_incorrect_units	-1.044	0.317	-0.344	-3.289	0.002
	counting_inaccuracy	-0.819	0.346	-0.250	-2.369	0.021