What’s to enjoy about maths?

I think it’s fair to say that maths gets quite a lot of stick; for many, school maths lessons were lessons where you catch up on a bit of sleep or spend some time looking out of the window while the teacher was droning on about algebra and trigonometry. How could anyone derive any kind of pleasure from this form of legalised torture? It’s a good question and to be honest with you I don’t know the answer as to how people do enjoy it – but some people do.

The fact is – and I’m sorry to have to say this but I’m sure you’ll agree – that a lot (not all) of the maths that you may have learned while at school and preparing for your GCSE maths exam (or O Level maths exam) or even your SATS exams was boring. Fractions are boring; percentages are boring; ratios are boring – I don’t even try to hide the fact that these topics are boring. I know that as a maths tutor I’m supposed to be all enthusiastic about this stuff and trying to inspire you – but, frankly, this stuff is completely uninspiring. This is why I don’t tutor primary maths – I don’t feel able to make fractions fun and interesting and quite honestly it makes me cringe to think about trying to teach ‘funky fractions’ so I leave it to those who want to do it and let them get on with it however they see fit.

BUT….I am enthusiastic about maths and I do try to inspire my students when it comes to maths but not through fractions or percentages or ratios. You see, these boring topics are necessary to know about if you want to get to the interesting maths. Just like when you learn a musical instrument you have to learn all of the boring scales and chords and all that lot, and if you don’t know the boring stuff then you’ll never get to the interesting stuff – well the same applies with maths. And also, just like when you learn a musical instrument there most people will drop off before they get to the interesting stuff but there are some who, for whatever reason, persevere – then again, the same applies with maths. For those that drop out early on, maths will always remain dull and boring and it will always be bewildering to them why anyone would enjoy it – but for those who persevere then they get their reward eventually.

The reality is that it can, and does, take years to get to the really interesting stuff in maths. When you’re in the primary stages of learning maths at school then you might be told that all of this stuff that you’re learning about leads on to this or that; and that might sound interesting but, sadly, it’s a long way off. You’ll have to invest many years of learning before you get to it – some do but most don’t.

So what do I find interesting about maths? Well certainly not fractions, times-tables, reverse percentages (I still don’t know what the difference is between a percentage and a reverse percentage!) ratios or converting top-heavy fractions to mixed numbers – are you still awake? What I find interesting is learning about Algebraic Topology, Group Theory, Mathematical Logic, Number Theory…I could go on; I enjoy reading works by some of the greatest mathematicians (and philosophers) who have ever lived such as Bertrand Russell, Euclid, Georg Cantor, Hermann Weyl – it’s like seeing inside their minds; I like the fact that there are still a lot of unsolved problems in mathematics and the philosophy of mathematics; I like that there is always something to challenge myself with and that I might have to spend days, weeks or months learning to understand something but then being able to see the beauty of the subject first-hand.

Maths is a fascinating subject – whether you believe me or not is up to you – but I feel in my element when I’m reading about some of the mathematical theories that have been developed. Mathematicians aren’t creative? Pull the other one! If mathematician’s weren’t creative there wouldn’t be anything like Non-Euclidean geometry; Georg Cantor would never have been able to develop his theories on Transfinite Numbers if he was creatively barren. If you want to see these things for yourself then you have to have the drive at the outset to get through the boring stuff – believe me if you get through it then you will see for yourself why maths is so interesting! But I also say (and I don’t mean to sound sneering or that I’m trying to belittle people when I say this) that maths is not for everyone – it may be that if you find maths boring then maybe your interests and talents lie elsewhere and you absolutely should be investing your time elsewhere.

How am I supposed to know to do that?

As I go through problems with my students and show them possible solutions, it might be that from time to time I solve a problem that someone isn’t familiar with, though it uses theory that they do know; at this point a common response is “How am I supposed to know to do that if I get a question like that in the exam?” This question really shows the misunderstanding that people have about what goes into producing a solution to a problem.

Unfortunately, maths education within the UK school system has become very sterile over the years – though steps have and are being taken to change this with the introduction of the new GCSE maths specification and the proposed new A Level maths specification to begin in the not too distant future. I’m speaking, here, from a point of view of having observed many GCSE and A Level maths students over the last half a decade or so and a big problem for them has been that they don’t understand that maths is about problem solving; many of them have never really had the opportunity in school to solve a problem.

But what about all of the textbooks that they have the and worksheets that they get? What about all the homework that they do? Well when you look at the homework sheets that are given out, or if you look at the homework ‘problems’ that students are set they often are very routine and uninspiring. What happens (in many cases) is this: the teacher introduces a topic and gives a few worked examples; the teacher then sets a bunch of questions that effectively follow the exact pattern of the wroked examples; then move on to the next topic Sadly, this is not problem solving by any stretch of the imagination. This is what I call ‘pub-quiz maths’ – either you know the answer and can parrot it out or you don’t know the answer and you hit a brick wall.

The thing is with maths is that, even though you might not know the final answer to a problem or the method of solution, you can figure it out. You can use what you DO know to find out answers to what you DON’T know. You can’t do this in a pub-quiz: either you know what the highest mountain in Europe is or you don’t. With maths you can use what you know to come up with original solutions to problems that are not necessarily anything like you’ve come across before; and many schools will stop short of this. Many teachers will try to analyse past exam papers and distill out what the most common ‘types’ of problem are and try to show their students model solutions for each ‘type’ of problem. But then when these students are faced with an unfamiliar situation (and it might only be very slightly different to what they are familiar with) they haven’t got the problem solving experience to modify and apply their knowledge and they say, “I’ve never been shown how to do this.” Well that’s the point – you have to find a way of solving it yourself.

My tutoring style concentrates more on this problem-solving aspect of mathematics. It’s hard work for the students and for me but I can’t, and don’t, shy away from putting my students in situations where they have to apply their knowledge and solve problems themselves. This is what maths is about and this is what makes maths interesting and exciting. For me to take away that opportunity from my students would be disgraceful and a complete disservice to my students.

So the answer to the question, “How am I supposed to know to do that if I get a question like that in the exam?” is this; get the problem-solving experience that you need to solve problems confidently and independently. I can teach the knowledge and encourage problem-solving skill development but I can’t teach experience – you have to get that yourself and you’ll get it through practice. You have to solve problems to learn to solve problems and then eventually you become flexible enough with your thinking to apply knowledge very naturally in quite creative ways. When you learn to do this THEN you are doing maths.

What the Rubik’s Cube taught me about learning maths

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.