5.1.1 Testing the Theory of Formal Discipline
The aim of this thesis is to investigate the Theory of Formal Discipline (TFD) claim that studying mathematics improves reasoning skills. The two main re- search questions are (a) is studying mathematics at advanced levels associated with improvement in reasoning skills and (b) if there is improvement, what might its mechanism be? Some light will be shed upon both of these issues in the current chapter, although that is not to say that both will be conclusively resolved.
The TFD suggests that studying mathematics improves one’s logical reas- oning and critical thinking skills. This belief is held by philosophers (Locke, 1971/1706; Plato, 2003/375B.C), mathematicians (Amitsur, 1998; Oakley, 1949), and policymakers (Smith, 2004; Walport, 2010) alike. As of yet, however, em- pirical evidence that supports the TFD is remarkably sparse. There is mixed evidence as to whether thinking skills can be transferred at all (see Chapter 2 for a review of the evidence), and although there is evidence of better reasoning skills from those who have studied mathematics at advanced levels than those who have not (Inglis & Simpson, 2009a), there is no evidence of reasoning skills developing alongside mathematical study. The study presented here investig- ated whether studying mathematics at AS level (the first year of an A level)
is associated with greater improvement in logical reasoning and critical think- ing skills than studying English literature. Improvement in reasoning skills is defined as reasoning that is closer to the relevant normative model, as discussed in Chapter 2 (Stanovich, 1999).
The measures of logical reasoning and critical thinking that I will use are the Conditional Inference and Belief Bias Syllogisms tasks, respectively. The reasons for this were elaborated in Chapter 4, but briefly, conditional reasoning is considered central to logical reasoning (Anderson & Belnap, 1975; Braine, 1978; Inglis & Simpson, 2008), and the ability to decouple one’s prior beliefs from logical validity is considered to be a part of critical thinking (Facione, 1990; S´a et al., 1999; West et al., 2008).
If it is found that studying mathematics is indeed associated with greater im- provement in reasoning skills than studying English literature, it would beg the question of what the mechanism for such an improvement could be. Stanovich’s (2009a) tripartite model was introduced in the literature review as a starting point for identifying potential mechanisms for improvement in reasoning skills. The tripartite model is an extension of dual-process models of reasoning which propose fast and automatic Type 1 processes and slow and deliberate Type 2 processes (Evans, 2003). In the tripartite model Type 2 processing is said to occur at two levels – the algorithmic and reflective levels. The algorithmic level is the computational element to Type 2 processing – the capacity available for effortful processing and the efficiency with which effortful processing can occur. The reflective level is the dispositional element – the processing that regulates when and to what extent the algorithmic level will be used as opposed to Type 1 processing.
It is possible that improvement in reasoning skills could be brought about by changes to any of these three types of processes. Studying mathematics could alter the Type 1 processes that focus our attention on certain aspects of a prob- lem. Alternatively it could improve algorithmic level capacity for or efficiency of effortful reasoning. Finally, it could be that studying mathematics alters the reflective level making the reasoner more willing to put effort into thinking. Alternatively to domain-general factors, a source of improvement in reasoning could be what Stanovich (2009a) termed “mindware”. Mindware consists of domain-specific knowledge, rules and procedures that can be explicitly recalled from memory to aid in solving specific problems, rather than being useful for reasoning more generally. Perhaps knowledge, rules or procedures are taught in mathematics that assist reasoners when solving certain types of tasks.
Type 1 processing is best studied by reaction time or time-limited accuracy measures due to their speed and automaticity (Gillard, 2009; Evans & Curtis- Holmes, 2005). Such methods allow initial responses to be isolated from later
responses that are based on Type 2 processing, but they require computer- based administration and as such were not suitable for the study reported in this chapter (where measures were all administered on paper in schools). In- stead, the role of Type 1 processing in the differences between mathematicians’ and non-mathematicians’ reasoning behaviour is explored in Chapter 8. Here, the algorithmic and reflective levels of Type 2 processing are investigated as potential mechanisms for improvement in reasoning skills.
The algorithmic level can be assessed via measures of intelligence and exec- utive functions, which reflect cognitive capacity and efficiency. The relationship between executive functions and reasoning skills is investigated separately in Chapter 9, while intelligence is considered here. If studying mathematics is as- sociated with improved reasoning skills due to an increase in the capacity for Type 2 processing then this may be reflected in an increased score on intelligence tests. A subset of items from Raven’s Advanced Progressive Matrices (RAPM, Raven et al., 1998) were included as a measure of intelligence in the main study presented below to allow for such a mechanism to be identified. As noted in Chapter 2, RAPM is a non-verbal pattern completion task that is thought to be the best single measure of general intelligence (Jensen, 1998). A time-limited subset of items selected for a student population has been shown to have an ac- ceptable split-half reliability of .79 (Stanovich & Cunningham, 1992). This also allowed between-groups differences in intelligence at Time 1 to be controlled for (see the discussion on quasi-experimental methods in Chapter 3).
The reflective level of cognition is assessed by measures of thinking dispos- ition, such as the Actively Open-minded Thinking scale (AOT, Stanovich & West, 1997), the Need for Cognition scale (NFC, Cacioppo et al., 1984) and the Cognitive Reflection Test (CRT, Frederick, 2005). The Cognitive Reflection Test, introduced in Chapter 2, poses three questions which prompt intuitive but incorrect responses. Participants need to inhibit these responses in order to an- swer correctly. Recently, Toplak et al. (2011) demonstrated that the CRT was a better predictor of normative reasoning than intelligence, executive functions, or the AOT. This suggests two things: firstly, if studying mathematics is asso- ciated with more normative reasoning, then changes to the reflective level may be a more likely mechanism than changes to the algorithmic level, and secondly, the reflective level may be better tapped by performance measures (e.g. the CRT) than self-report measures (e.g. the AOT and NFC scales). Here, the reflective level will be measured with two tasks: the CRT, and the NFC scale. The NFC scale was not included in Toplak et al.’s (2011) assessment and it may provide additional predictive power.
As of yet, there is no suggestion of what the nature of any mathematical mindware responsible for improvement might be, so a starting point is to simply
look at the effect of subject studied (mathematics or non-mathematics) and whether this predicts improvement independently of domain-general factors. If domain general factors are not found to be predictors, then mathematical mindware is a remaining possibility.
As a side point, if mathematics develops reasoning skills via domain-general mechanisms then we might expect a wide-spread improvement in reasoning per- formance. If, on the other hand, mathematics provides some specific mindware, we might expect improvement on only a small set of reasoning tasks for which the mindware is relevant. There will not be enough reasoning measures to test this hypothesis thoroughly here, but perhaps proponents of the TFD would hope for domain-general changes to be responsible for any improvements so that mathematics could be said to have a more useful, widespread influence on thinking skills.
There are two research questions addressed in this chapter: (a) is studying math- ematics at advanced levels associated with improvement in reasoning skills? and (b) if there is such improvement, what might the mechanisms behind it be? These questions are addressed with a longitudinal study, in which mathemat- ics AS level students are compared to English literature AS level students for development in logical reasoning and critical thinking skills. The participants were tested twice, at the beginning and end of their AS year of study.
The Conditional Inference Task was used as a measure of logical reason- ing skills and the Belief Bias Syllogisms task was used as a measure of one aspect of critical thinking. Measures of intelligence (RAPM) and thinking dis- position (CRT and NFC) were included to indicate whether domain-general factors at the algorithmic or reflective levels of cognition (Stanovich, 2009a) were the mechanisms for any development found. These measures also allowed pre-existing differences between the groups, which are likely to exist due to the quasi-experimental design (see Chapter 3), to be statistically controlled for. Fi- nally, a mathematics test was used as manipulation check (to confirm that the mathematics students did learn more mathematics than the English literature students).
Before describing the longitudinal study in more detail, three pilot studies are presented. The first tested whether the Belief Bias Syllogisms task could be split in half in order to save testing time. The second investigated whether the CRT could be imbedded in mathematical word problems, for the sake of making the ‘trick’ nature of the questions less memorable without altering the way that participants respond to them. The final pilot study assessed the duration and
difficulty of the selected measures to ensure suitability for the study.