Does Mathematical Study Develop Logical Thinking? Testing the Theory of Formal Discipline

Are advanced mathematics students more prepared for logical thinking, careers, and life in general? Are they just naturally “better” at logic and drawn to it, leading them to study a field that is supremely logical? This book attempts to address questions such as these.

For many years I have heard that majoring, or getting a doctorate, in mathematics is considered desirable by employers. When students ask me what career options are open to them through a mathematics major I respond in kind, along the lines of “Studying math trains your brain to think critically and logically. You could get lots of jobs!” As a pure mathematician, I have been told that the National Security Agency loves to hire people like me, not because of our particular field of study but because of how we are trained to think. When talking to undergraduates, I tout that math students are known to perform well on logical tests like the LSAT, making mathematics majors prime candidates for law schools. (This is actually true, as physics/math majors score at the top out of 29 disciplines, seen in Nieswiadomy, Michael, “LSAT Scores of Economics Majors: The 2008–2009 Class Update.” Available at SSRN.)

Despite my willingness to believe that my chosen field developed my general thinking skills, I had no real evidence to back up this claim. For that reason, I was excited to read this book and learn whether mathematicians’ hubris is well-founded. Inglis and Attridge begin with a bit of history, describing how the belief in the usefulness and transferability of mathematics skills is grounded in classical philosophy and current policy. The Theory of Formal Discipline (TFD) is at one point explained through the words of Plato: “Those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull … become much quicker than they would otherwise have been” (p. 3). I found it especially interesting how the TFD influenced the formation of early schools in countries like the US and England. Even now students are generally required to learn mathematics, often leading to jokes about how people never actually use algebra in real life. (Math, the only place where people can buy 60 watermelons and no one wonders why…).

Apparently the TFD is not as universally accepted today, despite the prominence that mathematics maintains as a compulsory subject. Inglis and Attridge strive to determine whether there is truth to the theory by collecting data about the conditional reasoning of advanced mathematics students in the UK and Cyprus. In chapter two, the researchers select tasks based on the reasoning gains expected by proponents of the TFD. These tasks revolve around conditional reasoning and students’ understanding of “if…, then…” statements. At this point in the book I actually found myself in suspense about what the results would show, as if it were a novel. Although I had some concerns about basing general thinking skills on investigations of conditional reasoning, I understood the authors’ basis for choosing this direction and felt that it allowed for a decent foundation of evaluating broader logical thinking.

A series of studies were conducted. The first couple focus on comparing mathematics and literature students in the UK. Following that came a longitudinal study of high-level and low-level mathematics students in Cyprus. More research followed in the UK, some involving psychology students. I was impressed by the authors’ research methodology and frequent consideration of alternative explanations of results. On the other hand, I was surprised at what was considered to be “advanced” mathematics. I always assumed that most of the benefits to studying mathematics, when it comes to logical thinking, derived from exposure to proof-based mathematics. The research participants in this book were not often in proof-based courses, with the exception of some Cyprus students who took geometry. It would be interesting to extend the research to include students who have seen advanced algebra or analysis, or perhaps even doctoral mathematics students.

The conclusions of the various studies in this book were interesting but not always satisfying. I won’t spoil future readers’ suspense by describing all of the results. I will mention, however, that there is convincing evidence that studying mathematics helps students reject invalid inferences. This outcome holds through longitudinal studies where there was no initial difference between the reasoning skills of different groups of students, or where those differences were controlled. When I first mentioned to a colleague that I was reading this book their response was to point out that mathematics students probably go into the subject because they are naturally more inclined to logical thinking. Inglis and Attridge are mostly successful in contradicting such a “filtering” theory through longitudinal studies. To summarize in the words of the authors, “we found that mathematicians and non-mathematicians do appear to reason differently, and that this cannot easily be explained by group differences in intelligence” (p. 102).

There are two less satisfying results. One is that the benefits of mathematical study extend mainly to abstract tasks and not to thematic thinking. Another is that mathematics students did not become better at accepting valid modus tollens inferences (from \(p\Longrightarrow q\) and not-\(q\), infer not-\(p\)). In fact, the students sometimes became worse at judging these statements. This negative finding was followed-up and reiterated through another study; the authors spend considerable time explaining why mathematics students persistently made this mistake.

In all, I felt that this book was interesting and well-written. Their research methods were explained clearly and conclusions were summarized nicely. It is a relatively quick read at only 130 pages (before appendices). There were no definitive conclusions about the TFD, which is to be expected with such a difficult research project, but there were convincing results. I think that anyone who has been told, or who has told others, that mathematicians make better thinkers should read this book.

Mindy Capaldi is an associate professor at Valparaiso University. Her current research area is mathematics education, but she enjoys the Scholarship of Teaching and Learning realm. Her favorite area of mathematics is Abstract Algebra. She is a fan of reading fiction and doing math, and spends much of her time on these two activities.