The problem with real-world problem solving
Mathematical thinking spans multiple worlds
My work keeps me in touch with maths educators from across the world. Recently, I have noticed a renewed emphasis on problem solving in the classroom. It is refreshing to see problem solving embraced as a core tenet of mathematical thinking, rather than simply an enrichment activity that resides at the edges of the curriculum.
One problem still lingers, however: the insistence that problem solving must be situated within real world contexts. Apparently, it is only in the real world that students can find mathematics useful and relatable. This world can indeed provide wonderful hooks for mathematics: my social media feeds are occupied with world cup-themed problems which, at the present time, must seem an awfully tempting way to engage students in maths class (those into Football, at any rate).
The problem is this: the real world is just one of many possible worlds. It is bound to physical realities, which, while setting the stage for a wide range of problems, is unable to accommodate some of the most interesting ones.
Problem solving is the most experiential, and therefore the most essential, form of mathematics. A good maths problem is a proxy for helping us develop thinking strategies — that is where it derives its usefulness from (and a more convincing selling point for mathematics than the vague promise that you might one day call on a particular method in an as-yet determined career). It stands to reason then, that we should seek out the widest possible pool from which to source problems. Why settle for the real world when so many others are available to us?
Some of the most alluring mathematical worlds are also the most imagined ones. The worlds of video games, as well as being among the most relatable to students, provide extraordinary opportunities for mathematical study. Video games give freedom to create new worlds, to create a set of rules and explore the consequences. What could be more mathematical than that?
These worlds often capture our imagination by defying the rules of our ordinarily lives. They are worlds where, for example, the laws of gravity and motion are upended as a deliberate design choice. To the naked eye, it makes for enthralling gameplay. To the educator, it offers no shortage of teachable moments. The optimum strategy for shooting red shells on Mario Kart (a matter close to my heart these days as I seek to maintain my status as the champion Karter at home) is a novel exercise in plane geometry; just one of the plethora of problems resolved in Power-Up, Matthew Lane’s a glorious coming together of mathematics and gaming. Mathematics is also the underpinning of gamification systems and social media dynamics. We find ourselves helplessly drawn to these worlds in our everyday lives; we may as well probe the mathematical properties that seduce us thus.
Some mathematical worlds are less tangible, but no less potent for mathematical thinking. I have found myself lost in Martin Gardner’s world of recreational maths more times than I can count. Gardner had a gift for designing puzzles that would stretch your mind without breaking it. Take the following Gardner classic:
Make one cut that will divide this figure into two identical parts:
Easy to play, difficult to master — the mark of any good maths activity. What world does this problem slot into? Certainly not the ‘real’ world — if anything, these seemingly arbitrary problems appeal to our most innate puzzling instincts. We are intrigued by the problem because it sings to our desire for simplicity and symmetry. As we absorb ourselves in solution attempts, we find ourselves in a realm far removed from the heady concerns of our everyday lives. Problem solving is escapism — it satiates our need to occasionally get away from the real world.
I adore prime numbers for the same reason. Their perplexing properties, and the problems they give rise to, transport my mind to a distant galaxy. If you insist on keeping me in the real world, I’m reduced to the familiar tropes of cryptography and cicada hibernation schedules. Nice by-products though they are, we mustn’t forget that these real-world uses of prime numbers were uncovered long after primes themselves. Primes caught the attention of mathematics for their mischievous behaviours. The endless quest to tame the unpredicatibility of primes illuminates our thinking more than any real-world application.
Perhaps there is a way to combine the best of all worlds. Take the World Cup Arithmetic problem from Alex Bellos. As a hook for Football enthusiasts, the problem succeeds right away. Yet the solution requires us to suspend our real-world sensibilities. We would rarely, if ever, be faced with the informational constraints imposed by Bellos. If getting the answer was our only aim, a cursory look on BBC Sport would suffice in retrieving the missing results. The World Cup Arithmetic problem is an invitation to engage in mathematical thinking, where we embrace the constraints, even if they are slightly implausible, because they set up a mind-stretching problem. The problem belongs as much to the world of Martin Gardner as it does the Footballing world.
Problem solving stretches as far as our imagination. It pays to escape the familiarity of the real world from time to time.