Does Mathematical Study Develop Logical Thinking?

The following sections are included:

• Dedication
• Acknowledgements
• Contents
• List of Figures
• List of Tables
• List of Abbreviations

Why should students study mathematics? Over the last few years, more British 16 year olds have taken mathematics examinations than any other subject. Figure 1.1 paints a clear picture: only English gets close to having the same number of students as mathematics. Students rate mathematics as being the most important school subject, and in most countries they are required to study it for more hours per week and until an older age than other disciplines. But what justifies the subject's privileged place in the eyes of students and on the school curriculum?

In the first chapter, we discussed some of the background to the TFD, arguing that although the theory is widely accepted by mathematicians and policymakers, most psychologists regard it with scepticism. Our goal in this chapter is to discuss how the veracity of the TFD could be empirically investigated. This discussion requires us to focus on two main topics. First, we discuss how reasoning can be assessed. If we are to see whether the study of mathematics changes a person's reasoning behaviour, we need a way of measuring that behaviour. Second, we discuss how research studies that address the TFD can be designed. Unfortunately the best research design available — a true experiment where students are randomly allocated to study either mathematics or non-mathematics — would be frowned upon by modern ethics committees, so we are forced to make do with designs that have threats to their validity. In the latter part of this chapter, we discuss these threats and what steps can be taken to reduce them.

The first step in our TFD-testing strategy is simply to determine whether or not there are group differences in reasoning behaviour. Do people who have studied advanced mathematics perform differently on our three identified tasks — syllogisms, conditional inference and the Wason selection task — to people who have not studied advanced mathematics? If the answer were no, then it would cast substantial doubt upon the TFD.

In Chapter 3, we presented some evidence that groups of students who have studied advanced mathematics reason differently to groups of students who have not. We used the three reasoning tasks highlighted by our stakeholders in Chapter 2 as being most likely to be facilitated by advanced mathematical study. In most cases, we found the pattern of results that the TFD would predict: the mathematics group's responses were closer to the norms of classical logic than the comparison group's. This was particularly the case with the tasks that involved abstract content, and especially the abstract conditional inference task.

In Chapter 4, we saw that our group of post-compulsory A level mathematics students became more likely to reject the valid modus tollens inference over time. Our goal in this chapter is to explore this finding further, in two directions. First, we ask whether this is an educational problem: is it important that mathematics students be able to make MT deductions? Second, we ask how the propensity to make the MT inference changes with more advanced mathematical study. In particular, we report a longitudinal study of undergraduate mathematics students, similar in form to those reported in Chapter 4.

In the last three chapters, we have presented converging evidence from a number of studies that studying advanced mathematics is associated with changes in conditional reasoning development. Although consistent results were found with various different reasoning tasks, our primary method of investigating these issues was to use the abstract conditional inference task, developed by Jonathan Evans.

Let us suppose that the evidence presented earlier in the book is sufficiently convincing for us to conclude that studying advanced mathematics probably does cause students to change the ways in which they interact with conditional ‘if … then’ statements. Why would this be?

The following sections are included:

• Evidence Presented in the Book
• Does Studying Mathematics Actually Improve Logical Thinking?
• The ‘Real World’ Objection
• Concluding Remarks

The following sections are included:

• Appendix A Reasoning Tasks
• Appendix B Abstract Conditional Inference Task
• Appendix C Thematic Conditional Inference Task
• Appendix D Abstract Syllogism Task
• Appendix E Thematic Syllogism Task
• Appendix F Abstract Truth Table Task
• Appendix G The Cypriot Curriculum
  • Low-Intensity Curriculum: Year 1
  • Low-Intensity Curriculum: Year 2
  • High-Intensity Curriculum: Year 1
  • High-Intensity Curriculum: Year 2
• Appendix H Proof Comprehension
  • The Proof
  • Proof Comprehension Items

The following sections are included:

• Bibliography
• Index