Over the last few weeks I’ve been spending some time learning some speed-cubing techniques on the Rubik’s Cube. I’ve been able to solve the cube for about ten years now using the Petrus method and I’ve never really bothered to go any further than just a basic solution – I’ve never been inclined to do so until now.

Anyway – as I learn about speed-cubing there are some similarities that I notice between how to go about learning to speed-cube and how to learn mathematics.

When learning to speed-cube you have to start very slowly – this seems completely counter-intuitive to what you’re supposed to be doing; going fast. You have to have a solid foundation to build on and there are some basics that need to be etched on to your brain so much so that you don’t have to consciously think about them; things like learning the colour-scheme of your cube and the relative positions of the colours (very important). To start with these are a pain in the arse; you mess things up all the time; you struggle to visualise things from different angles and you struggle even more to think ahead. But with lots of practice you eventually start to account for these things without even thinking about them; with practice the basics become automatic and you can direct more of your energy planning ahead with you solution of the cube. It takes time but slowly you make progress – if you don’t get sufficient practice you don’t develop the automaticity that you really need and all of your energy is wasted on doing the basic things leaving nothing in the tank for more advanced things.

And then I realised – this is exactly the problem that many GCSE and A Level maths students have; they have very weak foundations that they’re trying to build on. They might be trying to solve some integration problem for their A Level maths homework but they’re wasting all of their energy trying to remember how to multiply fractions together! Why? Because they haven’t had sufficient practice.

I admit that fractions are not the most interesting thing to learn about; HOWEVER, if you’re not able to automatically add or multiply fractions correctly then you’ve got a problem. I think that this comes about because it is very unfashionable in schools to drill children in the basics nowadays (and has been for a long time) and this is very detrimental to progress further down the line. I know that it’s boring and a very Victorian way of going about things but do you know what – it works! It simply is not enough to show children the idea behind adding and multiplying fractions (or whatever basic skill they’re learning) and expecting that that’s all that is needed – they need drilling until they can just do it automatically otherwise they will never be able to confidently and independently tackle more complicated problems.

I see this in many GCSE and A Level maths students – their knowledge of the foundational skills is so shaky that they can’t get very far into the solution of a problem before they run out of energy. I have to be very firm with my students and make it clear to them that they will struggle to solve problems unless they get completely familiar with the basics; of course I help them here as much as I can but there does come a point where I have to insist on them taking charge of their own destiny – I can teach them how to do basic skills but they themselves have to get the experience and practice because you can’t teach experience.

When I’m solving A Level maths problems (and even back when I was an A Level maths student myself) I was able to think ahead when solving problems and almost see the whole form of the solution in my head before I’d even put pen to paper. Why? Because I wasn’t having to waste energy faffing around trying to remember whether $x \times x$ is $2x$ or $x^{2}$ or whether I need a common denominator or not when multiplying fractions. I know that some of my A Level students think I’m being funny when I say that they need to work on their fractions but I’m serious – if you don’t go slow to start with and learn the basics well then you can almost forget about speeding things up or moving on to more advanced things because you will always be several steps behind those who have solid foundations.