9.3.4 Executive functions and Conditional Reasoning
The relationships between the three executive function measures and four in- terpretation indices were investigated with Pearson correlations and are sum- marised in Table 9.2. Hypothesis 2 stated that working memory scores would be positively correlated with the MCI, and this was supported by the data, r(94) = .34, p = .001. Those with better working memory had higher MCI scores (Figure 9.5).
Hypothesis 3 stated that inhibition scores would not be related to the inter- pretation indices due to the abstract nature. Contrary to this, inhibition scores were significantly positively correlated with the BCI, r(93) = .21, p = .041, and marginally negatively correlated with the DCI, r(93) = −.18, p = .083, suggest- ing that better inhibition (represented by a lower score) was associated with a lower BCI and a marginally higher DCI. These two correlation coefficients were marginally different, t(91) = 1.93, p = .056. This could be because in non-mathematicians the biconditional interpretation stems from the heuristic level of cognition, and when inhibited it is replaced with a defective conditional interpretation. In previous studies there was no indication of a relationship between inhibition and abstract conditional reasoning behaviour, but this may be because previous studies only looked at consistency with the material inter- pretation and didn’t consider a BCI or DCI. It may be the case that taking a material interpretation depends on other factors (e.g. working memory) whereas the BCI and DCI are related to inhibition in the manner suggested above.
There were no firm predictions about the relationship between shifting and conditional reasoning, but shifting was found to be significantly positively cor- related with the BCI, r(94) = .22, p = .036, and marginally negatively correlated with the DCI, r(94) = −.18, p = .077. This suggests that better shifting ability (represented by a lower score) was associated with a less biconditional inter- pretation of the conditional, and a marginally more defective interpretation, mirroring the relationships found for inhibition.
W orking memo ry sco re (0-1) 0.6 0.7 0.8 0.9 1.0
Conformity to Material Conditional Index (0-32)
12 14 16 18 20 22 24 26 28 30
Figure 9.5: Correlation between MCI and working memory scores.
Biases in Conditional Reasoning
Lastly, the relationships between participants’ three executive functions and their NCI and API were examined, again with Pearson correlations. Hypothesis 4 stated that NCB would be related to inhibition or working memory scores, and Hypothesis 5 stated that APB would be related to inhibition scores.
NCIs were significantly negatively correlated with shifting scores, r(93) = −.26, p = .013, indicating, rather counterintuitively, that those with better shift- ing ability had higher NCIs. An intriguing possibility is that the quadratic re- lationship between DCI and NCI found in Chapter 5 was driven in some part by shifting skills, i.e. better shifting ability is associated with higher NCIs and a more defective interpretation of the conditional (marginally significant here) meaning that NCIs tend to be higher in people with a more defective interpret- ation of the conditional, at least up to a point.
NCIs were not correlated with working memory scores, r(93) = .08, p = .472, nor with inhibition scores, r(92) = −.07, p = .493, contrary to Hypothesis 4. APIs were significantly negatively correlated with shifting ability, r(94) = −.21, p = .043, where those with better shifting ability showed higher APIs, similarly to NCIs. APIs were marginally negatively correlated with inhibition, r(92) = .19, p = .066, suggesting that those with better inhibition ability tend to show a greater API, contrary to prediction. Lastly, APIs were not correlated with working memory scores, r(93) = .07, p = .485.
The aim of this chapter was to investigate the potential for the algorithmic level of cognition to be a source of change in conditional reasoning behaviour. It was suggested that the algorithmic level might be a mechanism via which the TFD could operate, but for this to be the case it would be necessary for there to be a relationship between measures of the algorithmic level and measures of reason- ing behaviour. The study presented in Chapter 5 suggested that intelligence, at the algorithmic level, was not a mechanism for the changes found in the math- ematics students’ reasoning behaviour. Here, the three executive functions of working memory, inhibition and shifting were used as further measures of the algorithmic level. The Conditional Inference task used in Chapters 5 and 6, which mathematics students were found to significantly change on during AS level, was used as a measure of reasoning ability. Performance on the executive function tasks was as expected: the distributions of scores were in line with pre- vious studies and the three skills were found to be clearly separable (Hypothesis 1).
Hypothesis 2 predicted that working memory would be positively correlated with the MCI, and this was the case. Better working memory was associated with a more material interpretation of the conditional, confirming the findings of previous studies (e.g. De Neys et al., 2005; Verschueren et al., 2005). The other interpretations did not appear to be related to working memory and MCI was not correlated with inhibition or shifting skills. This could reflect that, rather than relying on the inhibition of intuitive responses in favour of an avail- able alternative interpretation, a material interpretation relies on having the necessary working memory capacity to apply or calculate the alternative.
Inhibition and shifting were found to be positively correlated with the BCI and marginally negatively correlated with the DCI. This could indicate that the BCI is the intuitive interpretation of the conditional statement, and that those with better inhibition and shifting ability are more able to inhibit this response in favour of a defective interpretation. This has implications for the TFD: it is possible that when mathematics students become less biconditional and more defective in their reasoning patterns (as found in the AS level students in Chapter 5), it is because their inhibition and shifting skills have improved. This is not incompatible with the hypothesis put forward in Chapter 5 which suggested that exposure to forward (‘if then’) inferences encourages students to consider not-p cases irrelevant. Perhaps it is this exposure which trains mathematics students to inhibit a biconditional interpretation.
The results related to the NCI and API scores were surprising. For NCI it was hypothesised that either inhibition (if NCB is a Type 1 bias) or working
memory scores (if NCB is a Type 2 bias) would be correlated, yet neither was. For API it was hypothesised that inhibition scores would be correlated, and they were not. However, it may be the case that the type of inhibition measured here is not the same type of inhibition required to avoid NCB and APB. In executive function tasks participants are told how to perform optimally and the measure reflects how well they can do so (Stanovich, 2009a). In the case of the inhibition task used here, participants were told to inhibit the interference of font colour on their responses and the measure was the RT cost of doing so. The spontaneous inhibition of Type 1 errors by Type 2 processing intervention when solving a reasoning task may have an altogether different nature. This likely depends on the thinking dispositions of the reasoner at the reflective level of cognition rather than the efficiency of implementing inhibition at the algorithmic level (Stanovich, 2009a). A more effective way to test for the relationship between inhibition and NCB and APB would be to use a measure of reflective-level inhibition such as the Cognitive Reflection Test (Toplak et al., 2011).
Such an analysis is possible in the data presented in Chapter 6 where another set of undergraduate participants completed the CRT and the Conditional Infer- ence Task. A reanalysis of this data showed that reversed intuitive scores on the CRT were significantly negatively correlated with APB, r(55) = −.30, p = .028, indicating that those with better inhibition of intuitive responses to the CRT had lower APB scores. Conversely, there was not a significant correlation between reversed intuitive scores to the CRT and NCB scores, r(55) = −.18, p = .187. These additional analyses suggest that APB is indeed a heuristic bias, while there is no evidence of this being the case for NCB.
It is important to note that the results of this chapter do not support the idea that executive functions are responsible for changes to mathematics students’ reasoning behaviour, simply that they could be. The longitudinal study reported in Chapter 5 suggested that thinking disposition (measured by CRT scores) was not a factor in changes to reasoning behaviour, but that it was a predictor of starting reasoning ability. It may be the case that shifting and inhibition play a similar role, influencing a person’s baseline reasoning ability but not playing a role in development alongside mathematical study. In order to investigate this further, another longitudinal study would be required in which mathematics and non-mathematics students’ executive functions are measured at the start and end of a period of study. Nevertheless, the results of this chapter do put forth some evidence that executive functions could be an interesting place to look for the mechanisms of reasoning change.
Summary of novel findings
1. Inhibition and shifting were positively correlated with conformity to a biconditional interpretation and marginally negatively correlated with con- formity to the defective conditional interpretation of conditional state- ments.
2. Affirmative premise bias was not related to an executive function measure of inhibition, but it was related to a reflective-level measure of inhibition, the CRT, in a reanalysis of a previous dataset.
For millenia it has been assumed that studying mathematics improves general reasoning skills (Plato, 2003/375B.C; Locke, 1971/1706; Smith, 2004; Walport, 2010; Oakley, 1949). This idea is known as the Theory of Formal Discipline (TFD). Previous research suggested that there may be a relationship between advanced mathematics and conditional reasoning skills (Lehman & Nisbett, 1990; Inglis & Simpson, 2008), although it remained unclear whether the re- lationship was developmental (Inglis & Simpson, 2009a). This thesis aimed to do two things: establish whether studying mathematics at advanced levels is associated with changes to reasoning skills, and if so, to investigate some po- tential mechanisms for the changes found. The findings of my studies suggest that studying mathematics is associated with a less biconditional and a more material, and in particular, more defective, interpretation of conditional state- ments. This appeared to be limited to abstract ‘if then’ conditional reasoning. The source of the change is discussed further below, after a summary of each study’s findings.