Modelling relationships between measures

To assess the relationship between different measures and arithmetic, we first used factor analysis on each set of measures to derive factor scores. These analyses were conducted separately on the different theoretically defined sets of measures, because of the small sample size. The factor analyses allowed us to derive factor scores to represent the error free common variance from the different sets of measures. For each set of measures, there was clear evidence of a dominant single factor that captured a high proportion of common variance between the measures.

We chose not to create a factor score for “number knowledge” because our measures of number identification and dot counting are theoretically different constructs; number identification relies on knowledge of Arabic symbols and dot counting relies on the count sequence but no Arabic number knowledge. These two measures were therefore retained as separate measures in the analyses.

The first step of exploratory factor analysis is to compare the correlations between measures of a factor. According to Field (2009) if correlations between measures are within r = .3-.8, it is then possible to assess if these measures load onto one factor. Correlations that are too high indicate a single measure, and correlations that are too low indicate different measures. For all of our measures we found that correlations were within the acceptable range: patterning tasks; rs = .44 - .80 (number, letter, object, shape and colour), arithmetic; r = .66 (addition and subtraction) and executive function measures; rs = .28-.43 (backward span, visual search, HTKS and cognitive flexibility). We therefore derived factor scores for each group of measures.

Pattern factor

As correlations were strong between patterning measures (rs = .56 - .80) we assessed if the tasks loaded onto a single factor using factor analysis (principle axis factoring). A single pattern understanding factor was derived from four measures (number, letter, object and shape patterns) which explained 73% of the variance (factor loadings on the pattern factor are reported in Table 5.3).

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Table 5.2 Correlations among measures.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1. Number patterns _{- .68** .76** .75** .43** .38* .41** .46** .28* .16 .45** .37* .48** .43** .38**} 2. Letter patterns _{.63** - .74** .77** .53** .50** .46** .41* .39** .33* .56** .46** .44** .48** .42**} 3. Object patterns _{.72** .69** - .80** .50** .48** .41** .37* .27* .26* .44** .39** .47** .46** .41**} 4. Shape patterns _{.71** .73** .75** - .49** .33** .47** .43** .34* .26* .44** .42** .44** .51** .44**} 5. Calculation score _{.35* .45** .41** .40** - .94** .88** .55** .50** .45** .46** .58** .40** .54** .60**} 6. Addition _{.30* .43** .40** .34* .93** -- .66** .53** .54** .42** .45** .59** .35* .51** .60**} 7. Subtraction _{.33* .37** .32* .38** .88** .62** -- .47** .34** .40* .36** .45** .38** .46** .47**} 8. Number ID _{.36* .27* .22 .22 .46** .45** .36* - .50** .33** .44** .63** .45** .39** .49**} 9. Dot counting _{.18 .29* .15 .23* .42** .48** .24* .39** - .30* .42** .46** 26* .38* .44**} 10. Visual search _{.03 .21 .13 .11 .35* .33* .31* .16 .18 - .29* .32* .31* .45** .48**} 11. Backward span _{.40** .52** .38** .30* .40** .40** .31* .38** .37** .22 - .43** .28* .41** .38*} 12. HTKS _{.27* .36* .28* .30* .50** .52** .36* .52** .39** .20 .37** - .34* .36* .55**} 13. Shifting Task _{.39** .34* .37** .32* .29* .26* .28* .31* .13 .19 .21 .22 - .35* .48**} 14. Non-verbal IQ _{.34* .40** .37* .42** .46** .44** .38* .23 .27* .36* .34* .23 .24* - .52**} 15. Spatial _{.29* .32* .30* .32* .53** .53** .39* .35* .34* .38** .32* .45** .39** .44** -} 16. Age _{.35* .39** .40** .43** .37** .34* .35* .57** .36* .38** .24* .40** .40** .38* .40**} Notes: Partial correlations controlling for age are below the diagonal and simple correlations above the diagonal.

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These factor loadings are all relatively high and relatively uniform. The first factor extracted had a large eigenvalue (2.93) with other eigenvalues being trivial in size. This analysis therefore gives strong support to the view that the different patterning measures can all be seen as tapping a unitary pattern understanding factor.

Table 5.3 Factor loadings for patterning factor.

Factor Loadings Uniqueness

Number .83 .32

Letter .83 .31

Object .88 .23

Shape .89 .21

Arithmetic factor

The addition and subtraction tasks were well correlated (r = .66, p < .001) and as addition and subtraction are the main components of early arithmetic, we used factor analysis to derive an arithmetic factor. The factor explained 55% of the variance accounted for. These factor loadings are relatively high, and the first factor extracted had a moderate eigenvalue (1.10) with the other eigenvalue negligible in size. The factor loadings are reported in Table 5.4.

Table 5.4 Factor loadings for arithmetic factor.

Factor Loadings Uniqueness

Addition .74 .45

Subtraction .74 .45

Executive function factor

Correlations between the four executive function tasks (backwards span, visual search, cognitive flexibility and HTKS) were moderate (rs = .28 - .43). As it is common in the literature to assess executive function as one measure, we assessed if these tasks load onto a single factor. The executive function factor explained 34% of the variance. The factor loadings on the executive function factor are reported in Table 5.5.

These factor loadings are moderate in size and relatively uniform. The first factor extracted had a moderate eigenvalue (1.37) with other eigenvalues being trivial in size.

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This analysis gives support to the view that the different executive function measure can be seen to tap into an executive function factor, however, the loadings are lower than the pattern and arithmetic factors.

Table 5.5 Factor loadings for executive function.

Factor Loadings Uniqueness

Backwards Span .60 .64

Visual Search .62 .62

HTKS .45 .80

Shift task .66 .57

5.3.4 Predictors of arithmetic

We used a regression model to evaluate which measures were unique predictors of arithmetic. In these models, we used factor scores saved from the factor analyses reported above. Age was retained in all models to control for age effects.

In the first stage, we entered all measures (age, pattern factor, executive function factor, number identification, dot counting, nonverbal IQ and spatial awareness) simultaneously as predictors of arithmetic. We then removed iteratively the least significant predictor from the model continuing until all retained predictors were significant. In the final model, executive function, patterning and spatial awareness were significant predictors of arithmetic, explaining 53% of the variation in arithmetic. Together, executive functioning and patterning explained 17% of the variation in arithmetic once spatial awareness and age were controlled (unique R2 = .17, p < .001). In summary patterning, executive function and spatial awareness explained unique variation in arithmetic scores whereas number identification, dot counting, and nonverbal IQ did not (see Figure 5.8).

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Figure 5.8 Path diagram showing significant and unique predictors of arithmetic (age is not significant but is retained in the model). * p < .05.

5.4 Discussion

In this study, we aimed to assess how different patterning tasks relate to one another, and how patterning relates to arithmetic. Firstly, and in line with our prediction, we found that pattern tasks correlate with arithmetic. Four pattern tasks (numbers, letters, objects and shapes) can be defined by one unitary factor which explains unique variation in arithmetic even after controlling for executive function, spatial skills and number knowledge. Secondly, we compared different types of patterning task (repeating, rotating and increasing) and showed that there is some variability in difficulty across task type and stimuli. Repeating patterns were shown to be the easiest and rotating patterns the most difficult type of task. Moreover, we found that letter tasks were significantly more difficult than number tasks, but no other significant differences were found with other measures (objects and shapes).

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Our first and main finding that pattern tasks correlate with and explain unique variance in arithmetic is in line with previous evidence (e.g. Lee et al., 2011; Lee et al., 2012; Pasnak, 2017; Rittle-Johnson et al., 2017; Schmerold, 2015; VanDerHeyden et al., 2011; Warren & Miller, 2013). Previous studies have typically assessed patterning using a small number of tests and have not investigated the relationship between different types of patterning task and arithmetic. To do this, we tested children on a range of pattern tasks with different items (repeating, rotating, increasing) and different stimuli (numbers, letters, shapes, objects) and compared the relationship between different types of patterning task. We found that the different tasks can be loaded onto one factor and therefore are represented by one unitary factor. Moreover, this factor predicts unique variation in arithmetic.

Secondly, we show that repeating patterns are the easiest type of pattern task, being significantly easier than increasing and rotating items across all stimuli. These findings are in line with previous evidence showing that children are capable of completing repeating pattern tasks before increasing patterns (Rittle-Johnson, Fyfe, McLean, & McEldoon, 2013). This is likely because, to complete a repeating pattern (e.g. 1212, 1121) children do not need to abstract a rule but can use basic perceptual skills to identify the correct response. We found that rotating items were the most difficult task for number and object stimuli whereas increasing items were the most difficult task for letter and shape patterns. We suggest that increasing patterns for numbers are simpler than rotating items because children are used to increasing number patterns (e.g. 1357) but less familiar with rotating patterns. On the other hand, children are less exposed to alphabetic patterns (e.g. aceg) in school which may make the increasing letter patterns more difficult.

Finally, we show that pattern tasks are, alongside executive function and spatial skills, a unique predictor of arithmetic. Some potential reasons for this relationship include a domain-specific role of patterning in arithmetic or domain-general role of patterning in arithmetic. A domain-specific role may relate to patterns which exist in numbers, for instance an understanding that 2+1 is the same as 1+2 or that 2+4 is the same as 2+2+2. A domain-general role of patterning may be that general abstraction and manipulation skills that are required to complete pattern tasks are also important for the development of arithmetic and other developmental skills, such as reading. These suggestions do not need to be mutually exclusive and may well both be viable for

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explaining the association between patterning and arithmetic. Future evidence should use different types of pattern task with more outcome measures (e.g. reading fluency) to examine these possible theories, both of which are discussed in more detail later in this chapter.