Forget speed: why the tortoise will always prevail in mathematics
Maths is an endurance event
Nothing snaps me out of my morning daze quite like a maths problem. Here’s one I caught on social media a while back:
In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?
Brushing my teeth is usually two minutes of dead time, but this problem got my neural juices flowing. Two minutes is about all it took as I reasoned my way to the answer (see the footnote to this post). As I read the original article in full, my delight turned to dismay. It turns out that my solution was sound, my speed not so much. The problem had appeared in a national mathematics competition, in which thirteen-year old Luke Robitaille solved the problem within a second to clinch the title. One second: that revelation was enough to spoil my breakfast.
My colleagues, some of them maths graduates, fared even worse when I shared the problem at work. Two minutes remains the company record, and this is a company that thinks a great deal about mathematics and education. Should we give up our day jobs?
Just how important is speed in mathematics?
Mathematical talent — often labelled ‘genius’ — is often associated with speed. Think of that kid at school who always came up trumps in maths class. Think Arthur Benjamin, the self-declared mathemagician who performs impressive feats with numbers at frightening pace.
Speed is the fuel of competition in maths. Luke and his opponents had an upper limit of 45 seconds to tackle each problem. Similar time constraints govern the format of other contests such as the UK’s Child Genius, an annual televised showcase of numerical wizardry (and highly strung parents). We are all party to the games; what are exams if not timed competition?
Competition has its place, within context. That context is usually premised on closed problems that occupy a tiny space within the vastness of the mathematical universe. Taken to an extreme, competition cheapens the enterprise of problem solving. The Kaplans put it best when they implore educators to focus on the battle of man versus mystery rather than man versus man. There should be room for all everyone at the rendesvous of learning.
The brand of speedy mathematics that dominates public perception and excludes so many is but one representation of the subject. For the majority of folks who do not excel in the rapid-fire format of mathematics, there is good news: maths is an endurance event.
Mathematicians have long known this. Says Timothy Gowers:
“The most profound contributions to mathematics are often made by tortoises rather than hares.”
Andrew Wiles contributed something quite profound when he ended the 358-year search for the proof of Fermat’s Last Theorem. Wiles points to scale and novelty as the two aspects of mathematics that elude competitions. Let’s hear it straight from the tortoise’s mouth:
“Let me stress that creating new mathematics is a quite different occupation from solving problems in a contest. Why is this? Because you don’t know for sure what you are trying to prove or indeed whether it is true.”
When I encountered the chicks, I was certain a solution existed because the problem had appeared in a maths competition. Just knowing that a solution exists changes the shape of a problem. My mind instinctively searched through available strategies, rather than confronting the unknown.
I was even confident that a simple solution was within grasp, which narrowed my range of approaches. Had the unpecked chicks been presented as an open problem, I may not have gravitated to such an elegant solution, so quickly. Contests shorten the arc from problem to solution, which is why exam preparation is often focused on how to deal with time pressures rather than the subject matter.
The mathematician is conditioned through uncertainty; speedy maths only satiates the need for immediate resolution. The sprint mentality of school mathematics needs to make way for an endurance mindset.
In my formal study of mathematics, speed became less prevalent as I delved deeper into the subject. I hammered through my GCSE exams at a pace of around 5 minutes a question, increasing to 15 minutes at A Level. My undergraduate exams typically required four solutions in three hours. And my doctoral research imposed no time constraints — the blunt calculation is that I solved two significant problems in the space of four years. I recall weekly meetings with my supervisor that lasted well over 2 hours, and others that ended within minutes. My supervisor informed me early on that our collaboration would not be time-bound; it’s just not how mathematicians work.
There is nothing innate about the mathematics of schooling that should demand speed. The usual justification is that speed builds fluency, and that students need to call on facts readily in order to free up their limited working memory when tackling tougher problems. That’s why you faced all those drills.
The problem arises when we fixate on speed as the only way for students to demonstrate their knowledge and skills. An excessive focus on speed reinforces a binary view towards mathematics — you’re good if you’re fast, and tough luck otherwise. But fluency has to encompass more than mere fact recall; it must be fused with understanding and reasoning. Speed isn’t the differentiator it once was for humans; machines have put paid to that.
The stress that is so often induced by speed requirements only incapacitates working memory. Imagine, as William Thurston did, if we treated writing the same way:
“If there was a popular confusion between good spelling and good writing, many potential writers would be unnecessarily discouraged.”
Speed is just one facet of mathematical thinking. Mathematics is an endurance event, open to all, and unbounded by the constraints of time. It should be treated as such.
My solution to the chicks problem: A chick will remain unpecked when the two chicks either side peck the other way, which they each do with a probability of a half. Since those two pecks are independent, the probability that they will both occur is just the product: ½ * ½ = ¼. So of the 100 chicks sitting in a circle, we would expect a quarter of them to remain unpecked, or 25.
I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.
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