Does studying mathematics enhance your overall mental prowess?
Abraham Lincoln certainly believed so, embarking on the arduous task of mastering Euclid’s treatises on geometry to increase his cognitive capacities, in particular his linguistic and logical abilities. This idea — that mathematics strengthens your mind much as physical exercise strengthens your body, helping you negotiate a variety of mental challenges — goes all the way back to Plato. Alive and well in today’s world, it is one reason popularly given for why everyone should study mathematics.
So it can come as a surprise to learn that cognitive psychologists have a different take on the issue. Various studies point to the conclusion that subjecting the mind to formal discipline — as when studying geometry or Latin — does not, in general, engender a broad transfer of learning. There is no sweeping increase of a general capacity for tasks like writing a speech or balancing a chequebook.
But surely a narrower claim is true: that mathematics, so systematically built as it is on inference, must develop logical thinking. Right?
By “logical,” I mean the kind of thinking needed to solve the following problem:
Four cards are laid in front of you, each of which, it is explained, has a letter on one side and a number on the other. The sides that you see read E, 2, 5 and F. Your task is to turn over only those cards that could decisively prove the truth or falsity of the following rule: “If there is an E on one side, the number on the other side must be a 5.” Which ones do you turn over?
Clearly, the E should be turned over, since if the other side is not a 5, the rule is untrue. And the only other card that should be flipped is the 2, since an E on the other side would again disprove the rule. Turning over the 5 or the F doesn’t help, since anything on the other side would be consistent with the rule — but not prove it to be true.
This innocuous-looking puzzle, a variation of which was introduced by the British psychologist Peter Wason in 1966, has been called “the single most investigated paradigm in the psychology of reasoning.” If you answered E and 2, congratulations: You are among the roughly 10 per cent of the public able to solve the puzzle. Many reasons have been advanced for this poor showing, including the lack of relevance of such an abstract exercise to people’s daily lives.
Most people reflexively eliminate the cards not explicitly specified in the rule (the F and the 2) and then continue with slower, more analytic processing only for the E and the 5. In this, they rely on an initial snap judgement about superficial similarity, a tendency that some scholars speculate evolved in humans because in most real-world contexts, quickly detecting such similarities is a good strategy for survival.
Interestingly, though, it turns out that if the puzzle’s abstract rule is translated into terms that are logically equivalent but grounded in real-world experience — as in, “If someone is drinking at a bar, she must be at least 21 years of age” — then the success rate jumps to 75 per cent or more.
I learnt about the Wason selection task and its intricacies from a fascinating recent book, Does Mathematical Study Develop Logical Thinking? by the education and cognition researchers Matthew Inglis and Nina Attridge. They conducted experiments that found that university students studying mathematics were just as likely as those studying history to quickly reject the F and the 2 cards. But differences emerged in the slower, more effortful cogitative phase that followed, leading to divergent success rates in the end: 18 per cent for the mathematics students versus 6 per cent for the history students.
Based on results from a slate of such reasoning tasks, Inglis and Attridge show that studying higher mathematics (at the advanced secondary and college levels) does lead to an increase in logical ability. In particular, mathematics students become more sceptical in their reasoning — they begin to think more critically. But these gains, though significant enough to establish a causal relation between mathematics education and logical thinking, are too modest to settle the debate on how much mathematics should be prescribed as part of a general education, and for which students. (An 18 per cent success rate is hardly compelling.) Moreover, there is the possibility of a self-selection effect: Students with the greatest potential to get a benefit in their logical reasoning might be disproportionately attracted to mathematics classes in the first place, so these gains might not apply to the entire population.
In any event, the most crucial finding of such research, in my view, is how much insight the psychological study of learning can contribute to the practical teaching of mathematics — two fields of endeavour that are too often pursued separately. It is sadly telling that while the Wason selection task is well known among psychologists, it is not familiar to most mathematicians and maths teachers.
I propose we start to teach the Wason selection task in mathematics courses at the high-school level and higher. The puzzle captures so much that is essential to mathematics: the nuts and bolts of inference, the difficulty of absorbing abstract concepts when removed from the context of real-world experience, the importance of a slow, deliberative cogitative process and the pitfalls of instant intuitive judgements. I presented the puzzle to a recent college class of mathematics majors and they listened with rapt attention afterward — startled by their lowly 19 per cent success rate.
Logical thinking may be promoted by mathematics, but it is a gradual and complex learning process. Psychological insight into learning, such as that offered by Wason’s puzzle, can give students a head start by educating them on the challenges they will face.
— New York Times News Service
Manil Suri, the author of the novel The City of Devi, is a mathematics professor at the University of Maryland, Baltimore County, and a contributing opinion writer.