Although it may seem impressive from the outside and make for good movie storyline (think ‘Good Will Hunting’ and ‘A Beautiful Mind’), performative mental mathematics is not at the heart of what we do as mathematicians
My son was recently given a Rubik’s cube for his birthday. The friends who gave it to him assured him, in the card that accompanied the gift, that “your dad will be able to teach you how to solve it”.
They had assumed that, as a mathematician, I would naturally be able to solve the puzzle. It is a mere children’s toy (和 43,252,003,274,489,856,000 —— 43 quintillion – possible configurations), 毕竟. As it happens I can’t, or at least I couldn’t.
It’s been something I had felt uncomfortable about for years, especially given everyone’s expectations. So I decided to sit down and learn how.
It’s not uncommon, as a professional mathematician, for people to have overinflated expectations of your abilities and your desire to use them. When I eat out with friends, 例如, it’s not unusual for the bill to be passed my way to split. Everyone looks surprised when I get out my phone and use the calculator to work out what each person needs to chip in. My mental arithmetic is fine, but when given the choice I’d prefer to do a fast computationally aided job rather than running the risk of making a mental mistake, especially after a few drinks.
Perhaps this misconception of mathematicians as calculating machines stems from the fact that many people’s recollections of maths at school are of difficult sums and times tables. I genuinely believe many of my friends think that I sit in my office all day, multiplying extremely large numbers together or doing really, really long division. The portrayal of mathematicians in popular culture (思考 Good Will Hunting 或者 A Beautiful Mind) also fuels the impression that to be a mathematician is to be a socially inept genius able to do complicated sums rapidly in your head.
Although it may seem impressive from the outside and make for good movie storylines, performative mental mathematics is not at the heart of what we do. In reality, the job of a mathematician is much more creative than many of the lessons we are taught early on in school.
Generically, mathematics is the search for and the understanding of pattern. It’s fair to say, 然而, that mathematics as a subject is extremely diverse. Saying someone is a mathematician is like saying someone is a sportsperson. Sportspeople share common traits such as fitness and athleticism which make them good at their jobs. 相似地, mathematicians share a common language. But someone at the pure end of the mathematical spectrum can be as different to someone, 像我这样的, at the applied end of the spectrum, as a professional rugby player is to a synchronised swimmer.
The mathematics underlying the Rubik’s cube is group theory. This is about as far from my specialism in mathematical biology – studying the choreography of pigment cells which give animals their distinctive patterns or the emergent coherence of locusts in their devastating swarms – as Formula 1 is from figure skating.
Learning to solve the cube was certainly not something I could just figure out, or something which came naturally to me. I didn’t study the group theory behind the Rubik’s cube and consequently I have no deep insights into the toy. I am happy to share that it took me several hours to solve my first cube with the help of a YouTube video. I am still quite slow at solving it (my best time is more than two minutes) and I have not reached the stage where I have learned how to correct more than the most basic mistakes I might make when following my algorithmic route to a solution.
During the process, this is what I discovered: there are strong parallels, 我相信, between my feelings in going out of my comfort zone to learn the Rubik’s cube and most people’s experience with maths. 特别是, you don’t need to be a world-record breaker to successfully solve the cube (the fastest time is well under five seconds).
In the same way, you don’t need to be a professional mathematician in order to use maths in your everyday life. A little mathematical knowledge in our increasingly quantitative society can help you to harness the power of numbers for yourself. Simple rules allow us to make the best choices and avoid the worst mistakes.
For percentage calculations, 例如, it’s sometimes useful to use the commutative property of multiplication. When using technical jargon, that sounds complicated, but all “commutative” really means in this context is that we can swap the order in which we multiply numbers together. When faced with a calculation like 16 的百分比 25, it can be easier to flip the problem and instead find 25 的百分比 16 —— 4.
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I didn’t study the group theory underlying the algorithm that I learned to solve the cube. 相似地, you don’t need to understand all of the underlying theory in order to use mathematical tools.
I will admit I felt intimidated by the weight of mathematical expectation on me – feeling I should first master the relevant group theory before I learned how to solve the Rubik’s cube. I think this is the same feeling many people have about the subject – that you have to be some sort of savant just to set foot in the house of maths.
I’m not saying that everyone can or should try to be Einstein, but rather that you don’t need to be a professional mathematician to use maths to your advantage or even, dare I say it, to enjoy doing maths for fun.