7.5 Alternative Analysis
7.5.3 Auditory latent models.
Entering all auditory executive functioning and working memory variables into a single model, also resulted in a non-converging model. However, once again each combination of three variables did converge (see Figures 7.17, 7.18, 7.19 and 7.20.
Visual EF VSSP Visual inhibition Visual switching Arithmetic ability .99 .98 .98 .86 .95
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Figure 7.17. Auditory executive function latent variable as predictor of arithmetic
competence. EF = executive function.
p < .001 for all paths.
Figure 7.18. Auditory executive function latent variable as predictor of arithmetic
competence. EF = executive function; PL = phonological loop.
p < .001 for all paths.
Figure 7.19. Auditory executive function latent variable as predictor of arithmetic
competence. EF = executive function; PL = phonological loop.
p < .001 for all paths.
Auditory EF Auditory updating Auditory inhibition Auditory switching Arithmetic ability .99 .99 .98 .86 Auditory EF Auditory updating PL Auditory inhibition Arithmetic ability 1.00 .99 .98 .85 1.00 Auditory EF Auditory updating PL Auditory switching Arithmetic ability .99 .99 .98 .86
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Figure 7.20. Auditory executive function latent variable as predictor of arithmetic
competence. EF = executive function; PL = phonological loop.
p < .001 for all paths.
Model fit statistics for each of these models can be found in Table 7.15. Whilst each model indicates a good fit, the one containing the phonological loop, auditory inhibition and auditory switching was arguable to best fit closely followed by auditory updating, auditory inhibition and auditory switching.
Table 7.15
Comparison of Fit Indices for Auditory Working Memory and Executive Functioning Models
Figure c2 p c2/df BIC RMSEA CFI TLI SRMR
UP, INH, CF UP, PL, INH UP, PL, CF 1.38 1.96 2.87 .501 .161 .239 0.69 1.96 1.43 2282 1395 2819 .00 [.00, .13] .07 [.00, .23] .05 [.00, .16] 1.00 1.00 .999 1.00 .998 .998 .002 .005 .003 PL, INH, CF 0.02 .898 0.02 1644 .00 [.00, .09] 1.00 1.00 .00 Note: UP = auditory updating; INH = auditory inhibition; CF = auditory switching; PL = phonological loop; df = degrees of freedom; RMSEA = root mean square error of approximation; CFI = comparative fit index; TLI = Tucker-Lewis index; SRMR = standardised root mean square residual.
In this analysis having a single executive function and working memory latent variable did not produce a good fit, however the modality specific latent variables did, with fit statistics for auditory and visual models being similar (see Tables 7.14 and 7.15). However, these models only contained three predictors, and putting an
Auditory EF PL Auditory inhibition Auditory switching Arithmetic ability .99 .98 .99 .86 .98
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auditory and visual latent variable into the same model did not converge, a finding in keeping with the poor fit for the model where the single executive functioning and working memory latent variable predicts arithmetic ability.
7.5 Summary
The first phase of this research looked to address the following research question:
1) Which modality specific cognitive abilities predict arithmetic competence in the general population?
The cognitive abilities that explained the greatest proportion of the variance in arithmetic ability was impacted by the statistical technique utilised to analyse data. Regression analysis highlighted verbal and non-verbal intelligence, and visual and auditory updating as the constructs which explain additional
variance when all other predictors were accounted for. The intelligence
predictors showed the strongest associations. Interestingly coefficients for the intelligence predictors were similar, which was also the case for the updating ones, hence it is unclear if modality is an important consideration.
Analysing data via structural equation modelling raised questions about the best model to utilise, with models incorporating updating, a single working memory and executive function latent variable, and two modality specific executive functioning and working memory latent variables all displaying good fit statistics. However, whilst the updating model is the most parsimonious, given both the inter-relatedness of executive function and working memory constructs and the difficulty in finding tasks that measure single executive function or working memory constructs, if may be more appropriate to utilise either the one or two latent executive function and
working memory variables. Whichever model is utilised it is clear that the modality of task presentation needs to be considered in future research as both visual and auditory predictors impact arithmetic ability.
In the alternative analysis, the single latent executive function and working memory variable did not display a good fit, and a model containing two modality specific executive function and working memory latent variables in the same model did not converge. However, the models that displayed a good fit again highlighted
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the importance of a range of modality specific executive function and working memory constructs for arithmetic ability. The results in this alternative analysis may be impacted by a lack of power, and hence future research should address this.
This concludes the analysis for the typically developing population. An
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Chapter 8: Discussion for Research Question One
Phase one of this study looked to address the following research question:
Which modality specific cognitive abilities predict arithmetic competence in the general population?
To my knowledge this is the first study to investigate concurrently auditory and visual measures of executive functioning (updating, inhibition, switching),
intelligence, and numerical acuity, in addition to measure of the visuospatial sketchpad and phonological loop, and the impact they have on arithmetic ability. Representing executive functions and working memory abilities as modality specific latent variables, facilitated an investigation into how the modality in which stimuli are presented impacts links between cognitive abilities and arithmetic competence. Although the creation of latent variables containing updating, switching and inhibition is not new, including the phonological loop and visuospatial sketchpad is an
important innovation, as they are not always considered executive functions. However, factor loadings for each were good, which is perhaps unsurprising given the proposed structure of working memory which highlight strong links between each and the central executive (Baddeley, 1996). Hence given these abilities are inter- related and both have previously been identified as predictors of maths ability, it would seem appropriate to include them in the same latent variable.
Both visual and auditory modality specific executive function and working memory latent variables displayed direct paths to arithmetic ability, with the visual latent variable explaining a greater proportion of the variance. Despite intelligence being the strongest predictor in the regression analyses, paths within the structural equation model were indirect, through each latent executive function and working memory variable.
The next section contains a more in-depth discussion of the impact that the choice of statistical technique can have on results. The three structural equation models will then be considered; the first utilises a latent updating variable, the
second a single executive function and working memory latent variable, and the third investigated the impact of modality, via two modality specific executive function and working memory variables.
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8.1 The Impact of Statistical Techniques
It has been suggested that inconsistencies in research findings surrounding the cognitive underpinnings of mathematical abilities may be influenced by the statistical techniques utilised. This phenomenon could be particularly pertinent when observed variables (i.e., regression analysis) and latent variables (i.e., structural equation modelling) are compared and contrasted (Filippetti & Richaud, 2017; Lee et al., 2009).
This study provides further evidence for this phenomenon as data analysed via regression analyses highlighted intelligence and updating as significant predictors, with intelligence variables (non-verbal and verbal) explaining more of the variance. However, entering data into a structural equation model resulted in latent updating (visual and auditory) emerging as the strongest predictor, with only an indirect path between non-verbal intelligence and arithmetic ability. This is potentially an important finding as the majority of studies in this field utilise correlation or regression
techniques. However, given the acknowledged impurity problem for measures of working memory and executive function (Van der Ven et al., 2012), representing executive functions as latent variables may be optimal (Miyake & Friedman, 2012). This is because measurement errors in regression analyses can be confounded with true common variance, whilst structural equation modelling uses multiple indicators whose common variance is extracted, and whose measurement errors are modelled explicitly, thereby reducing the confounding effect of tasks which inevitably are not measuring a single construct. Arguably therefore structural equation modelling facilitates a more precise investigation of the relationships between conceptual constructs (Lee et al., 2009), which is particularly pertinent in studies involving executive functions.
However, this study did not set out to provide evidence for or against the unity and diversity debate, and the range of executive functioning tasks preclude
conclusions being drawn.
The statistical technique utilised may explain inconsistencies in the literature investigating the cognitive underpinnings of mathematical ability, and also why intervention studies looking to train a single construct (e.g., working memory, numerical acuity) have to date had limited success.
Most of the intervention studies aimed at improving working memory have focussed on the visual domain, specifically visual updating and the visuospatial
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sketchpad. While some studies have found gains following adaptive working memory training (e.g., Holmes et al. 2009; Kroesbergen, Noordende, & Kolkman, 2014; St Clair-Thompson, Stevens, Hunt, & Bolder, 2010), others including a meta-analysis and randomised control trial reported near but nor far transfer effects (Dunning, Holmes, & Gathercole, 2013; Melby-Lervåg & Hulme, 2013)
The majority of training programs aimed at developing specific mathematics related abilities involve preschool children and include a range of activities designed to improve early numeracy abilities including counting, recognising and writing numbers, one-to-one correspondence, comparisons of symbolic numerals, change operations and, understanding numbers. Once again stimuli is typically presented visually (e.g., Park & Brannon, 2013; Ramani & Siegler, 2008; Whyte & Bull, 2008; Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006). Whilst there is evidence for training improving both symbolic and non-symbolic abilities, the evidence for far transfer to improved mathematical performance is more limited. Indeed it has been suggested that to effect far transfer, interventions should train both specific and general abilities (Passolunghi & Costa; 2016).
This study provides evidence to suggest that rather than focussing solely on visual updating and the visuospatial sketchpad, examining multiple distinct but interrelated general abilities (updating, visuospatial sketchpad, phonological loop, inhibition and switching) affords a better understanding of the cognitive abilities that predict mathematical ability. Hence, forthwith this discussion will focus on findings pertaining to the structural equation modelling. However before doing so it is important to note that whilst this study did not find numerical acuity to be an important predictor of arithmetic ability, this may have been impacted by the task utilised which measured non-symbolic rather than symbolic numerical abilities.
Hence future research should examine whether results are congruent when symbolic numerical acuity is investigated concurrently with bimodal executive function and working memory abilities.