It’s been a very long while since I’ve been actively blogging, which I’m going to put down to the intensity of second year and a heavy rehearsal schedule, but in reality is probably due to crippling laziness and my Dad giving me the Netflix password. I’ve been thinking about this post for a while, but I haven’t ever got round to actually putting proverbial pen to paper (fingers to cracked phone screen), and since I’m currently on a very long tube journey from Fulham Broadway to Camden Town I figure now is as good a time as any to start.
“What is this thing?!”
As most scientists and mathematicians will discover, at some point you reach a stage in your academic career where showing your work to someone outside your chosen discipline prompts a despairing cry of “What the fuck is that?! I have literally no idea what any of this means. What is this thing?! Jesus Christ, I could never do physics/maths/chemistry/engineering”. It would be lying to pretend that this sort of reaction isn’t, deep down, quite welcome – it gives you a sort of smug, superior “my subject is hard” glow – but, less superficially, it reveals a common misconception about mathematical subjects which, in my opinion, is a key factor in preventing kids from doing STEM subjects at a higher level.
“Imagine you’re reading a book in Cyrillic…”
Think about it this way. If you skim through a difficult history book, not being a historian yourself, your reaction might be a despairing one – “this is so dry and the vocabulary is overcomplicated, I could never finish this!” – but you would probably be able to power through it, maybe with the aid of a dictionary. Similarly, with a book in a foreign language, you accept that you’ll need google translate (or a dictionary, if you’re a purist or you don’t have the internet). Now imagine that you’re reading a book in Cyrillic, except it’s not regular Cyrillic, it’s just a meaningless string of Cyrillic symbols which are typeset seemingly at random with a couple of incongruous English words tossed in there like confusing croutons on a salad of misery. This is what maths looks like.
“People are told that mathematics is ‘the science of numbers’”
The problem lies in the fact that people are told that mathematics is ‘the science of numbers’ or something, when, in fact, it is a mix of language, inductive thinking and formal logic, and it’s not a science at all. To carry on with the maths/foreign language analogy, we can think of the symbols in a formal statement as the ‘words’ of the language, while being able to put them together in a way that makes logical sense is the ‘fluency’ that you get when you’ve been studying that language for a really long time. The problem is only in the fact that you only realise this much later on in your mathematical life – nobody tells you at school that it’s ok to be mystified by equations, because you’re not good enough at the language to understand them yet.
The easiest way to show my point is with a couple of examples – my favourite by far for illustrating the complexity of formal language is the Intermediate Value Theorem. Have a look at this. (I was going to latex this stuff, but I can’t be bothered and I prefer good old fashioned pen and paper anyway)
This is the Intermediate Value Theorem in its most condensed form (including the definition of continuity, for completeness purposes). Nope? How about this:
“If a continuous function f with an interval [a, b] as its domain takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval”
That was poached directly from the Wikipedia page on the IVT. Let’s simplify this some more:
“If you draw a wiggly line that doesn’t cross itself from the bottom left of a rectangle to the to the top right corner of a rectangle, without taking your pen off the paper, then any horizontal line you draw on the rectangle will cross the wiggly line at least once”
And finally, here is a picture:
There. That wasn’t so hard. What we’ve done here is essentially put the maths through a three-step translation process – taking the formal symbols and turning them into the wordy statement, then interpreting that in layman’s terms, and then thinking about it pictorially – and it’s this sort of visualisation which is absolutely crucial to the understanding of mathematics as a whole. It’s not universally possible to visualise something in pictorial terms in higher level maths, and it’s then where you have to revert to using axioms and theorems rather than trying to think intuitively (topology, for example) but as a general rule, it always helps to try and think what the statement is actually trying to say before you work with it.
What I’m trying to say here is that kids are taught implicitly that maths is hard because it looks hard, and that only serves to further the thought that maths is like marmite – you either ‘get it’ or you ‘don’t get it’ – which is one that stops so, so many kids from doing A-Level maths. Of course, this isn’t helped at all by the fact that “good at maths” in kid-world (and most of the adult world, as I’ve discovered too many times in bars: “you study maths? What’s 347*892?” GO AWAY) directly translates to “being able to do lots of big sums in your head very quickly” which is not the same thing at all. “Lightning calculator” and “good at maths” are not tautological statements, far from it – plenty of people who can do lots of sums very fast couldn’t recognise an epsilon/delta proof if it punched them in the face. Teach maths it as what it is – a formal language and logical reasoning – not as ‘magical number science’ and I guarantee that the bright, well-reasoned kids who would end up being lawyers or historians might well end up being mathematicians instead. Think about it, anyway.
Edit: I may spend the rest of my academic life compiling a giant “Maths to Easy English” dictionary