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        
p-value        
cpm Spearman's rho 0.211      
p-value 0.094      
pens Spearman's rho 0.414 *** 0.413 ***    
p-value < .001 < .001    
counting_incorrect_units Spearman's rho -0.279 * -0.266 * -0.576 ***  
p-value 0.026 0.033 < .001  
counting_inaccuracy Spearman's rho -0.303 * -0.426 *** -0.564 *** 0.373 **
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, Fchange(1, 59) = 10.82, p = .002. Moreover, counting incorrect units was the strongest predictor in the full model    

Model Summary
Model R 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

Residuals vs. Predicted

Q-Q Plot Standardized Residuals