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.

As with anything, sometimes you need a break from maths.

For me, maths is something that I have a real drive to learn more about. I don’t want to stop learning more about the subject and seeing which interesting and unexpected directions it can lead me. Maths is by far one of the most important aspects of my life – I feel that maths forms a substantial part of my identity. I don’t want to just accept what I know and accept what I don’t know and leave it at that – I want to keep pushing to know more and more.

But this comes at a price. Every now and again and without really realising it I can feel like a kind of exhaustion hits me and I realise I’m doing too much. I don’t realise it at first because I’m caught up in the whole learning process; the enthusiasm is there; I feel good when I’m learning and thinking about things in minute detail; dissecting proofs and theorems and trying to squeeze every last drop of knowledge that I can out of what I read and learn. But this can’t go on forever; it is very draining and energy intensive. It can be difficult to stop because I feel that I will forget things or that I’ll end up squandering my time when I could be doing something useful like doing learning more. At this point it’s got to the point of obsession and I’ve been here many times before; It’s neurotic John again.

As much as I don’t want to, I know that I have to take a break. I need to do something else for a few weeks or even a couple of months. And I think now might be a good time for me to do that. Maybe I won’t take a complete break from maths – after all, I’m a maths tutor so I can’t exactly not do ANY maths at all. But maybe I can turn to other things for a while like spending some time in my garden, spending time with my daughter or getting a bit more exercise.

I always find this part difficult because I feel that much of what I’ve been learning will be forgotten and that I’ll be taking a step backwards. This might be true; I might forget some things – I’m human after all, right? But by taking a break I give my brain a chance to have a bit of a change; in fact some things I will remember with even more clarity after having a break. And even though I might forget some things (which I might have forgotten anyway), in the long run I’ll be refreshed and my enthusiasm will return (it always does when it comes to maths) and then I can throw myself into things again and let the cycle play out all over again.

I think I’m writing this post, more than anything else, to convince myself that I need a break – but there is something that I want to make clear: sometimes you just have to take a break from what you’re doing. This is really true if you’re like me and you tend to get completely absorbed in things and become oblivious to what your body is telling you – I know there are, and have been, lots of my A Level maths students who remind me of myself when I was doing my A Level maths; they just want to learn more and more about the subject and nothing will stop them. But you do eventually have to draw the line somewhere; admit that you need a break and that you deserve a break from time to time. Though, you have to be honest with yourself – do you genuinely need a break or are you just looking for yet another reason to not do any maths for a while because if that’s the case then a break really won’t do you any good.


It always feels good when you work through a maths problem and you get to the right answer, doesn’t it? How about when you work through a problem and you get to an answer that’s wrong? It doesn’t feel so good then. If it’s a homework sheet that you need to hand in and you can’t see where you went wrong then you might be tempted to copy someone else’s work; at least you’ll get some nice ticks all over your work telling you what a great job you’ve done instead of those horrible crosses. If it’s a past exam paper that you’re working through you might just look at the mark scheme to see how to solve a problem instead of bothering to spend time figuring it out and before you know it you feel like you can solve anything – the solution is always obvious when you know how it’s done.

But is the answer what you really need right now? I agree it feels nice to get a right answer but a wrong answer is telling you something; like it or not you still have some learning to do.

When I set homeworks for my students, or if they’re working through past papers (shockingly, some schools seem to try and teach the whole GCSE or A Level maths courses through past papers) it’s all too tempting for them to just look in the back of the textbook at the answers or just siphon off a model answer from the mark scheme and expect that I’ll be happy with that. The thing is – I know the ability levels of each of my students and I can recognise what is their work and what isn’t fairly well. If I see something that I know isn’t their work or their answer I will challenge them about it; I will ask them how they arrived at their answer or how they came up with their solution. The silence can be deafening.

So I have before me a page of beautifully correct answers and solutions – yet my student has learned nothing. Is that a good result for the student? As far as I’m concerned, every one of these answers is incorrect because if there is no evidence to support the answer, if the student can’t even begin to explain the solution then how can it be believed to be correct? Obviously if I let this continue then my student will get the feeling that all they need to do is write the correct answer and their job is done. It doesn’t seem to dawn on some, though, that lifting the answers from the back of the book or from the mark scheme is not something that they will be able to do in an exam – but more importantly, what if there is currently no answer to the question that they’ve been asked and it is up to them to provide an answer. After all, at some point in their lives they will probably have a job where they have sole responsibility for certain decisions and the solutions to certain problems – the answers can only come from them; they won’t be able to consult the back of the book and nor will they be able to just give up because they didn’t know where to start otherwise their competency will be seriously questioned.

There is a reason that maths problems are called problems and that’s because you have to look for solutions. It takes practice to find solutions; it takes practice to understand concepts and piece bits of knowledge together and it may take some time before you get the hang of things and consistently produce correct answers. But not going through this process, and indeed denying yourself the opportunity to go through this learning process, means that your problem solving skills will remain shrivelled and weak. You won’t be able to be decisive about what to do; you will not have any degree of confidence in you solutions or results and you will be entirely dependent on external sources to validate your solutions and answers, you might not even be able to make your own mind up about where to make a start.

I put much more emphasis on the solution to a problem than I do on the answer and I make this clear to my students so most of them (if not all of them) learn quite quickly that whatever antics they might get up to with their school homeworks won’t wash with me. I only tutor for ages 14 and up so I feel that I can treat my students as young adults; so I make it as clear as I possibly can to them that if they continue to copy their answers then they only damage their own chances and that they just give the appearance of understanding. They now have to take some responsibility for their own learning; I would rather see several pages of unsuccessful attempts at a problem then either a page of correct answers without solutions (big red flag that one) or simply no attempt at all because they “couldn’t think where to start”. These unsuccessful attempts are the starting point for understanding and there is no way to bypass this stage. Of course you would prefer to get the right answer straight away, who wouldn’t? And sometimes that will happen. But you have to be prepared to think through problems and make a lot of false starts – it is the solution that matters at this stage and not necessarily the final answer.


What is the purpose of mental arithmetic in the modern world? In what way does it benefit someone to be able to do mental calculations considering that a calculator can be bought for about the same price as a bar of chocolate – if not less.

In the past when schoolchildren have questioned the use of mental arithmetic, teachers were at least able to say “Well you won’t always have a calculator handy”. Well I think that’s just not true at all now – everyone has a calculator practically all of the time in their pocket, although it’s usually known now as a phone rather than a calculator. So what do you say to schoolchildren now?

Unfortunately, at least for mental arithmetic, there isn’t really anything that’s going to convince schoolchildren now of the use of mental arithmetic. Some will happily learn mental arithmetic but the vast majority won’t. That’s not to say that being able to do mental arithmetic isn’t useful, it’s just not as obvious how it’s useful. Considering how little time is spent on mental arithmetic in schools and the little mental arithmetic abilities that many students have now it certainly seems that teachers have just about given up on this one.

Maybe they’re right. Maybe there is no need for mental arithmetic. Pocket calculators are faster than humans, more efficient, less prone to error – why bother learning the times tables? Isn’t it just a complete waste of time? Well…yes and no.

When you consider how abundant electronic calculators are in the modern day then mental arithmetic really is mostly a waste of time for doing calculations. With a calculator it’s a cinch to multiply huge numbers together; I can tap into a calculator a list of numbers and it will do all kinds of statistical analysis for me on those numbers – whereas once upon a time it would have taken a good five minutes to manually calculate the variance of a list of numbers, now my calculator does it in less than a second. Why should anyone even bother trying to compete with that? Maybe at some time in past decades computers (of the human kind) were important – people who could do calculations quickly were essential before pocket calculators came along. I don’t know whether being a ‘calculator’ was a particularly lucrative job to have but it was a job. Which company or organisation would be crazy enough to employ someone to sit there doing manual calculations nowadays?

However, letting an electronic calculator do the work for you all the time has some downsides. Just like using a car to always get you from A to B is far easier and quicker than walking or running, you end up getting lazy and out of shape (at least physically). Similarly, using an electronic calculator all the time means that you get lazy and out of shape (at least mentally). I come across many maths students right up to A Level who struggle with mental arithmetic – and it shows. It’s not because they’re incapable it’s just that they’ve never been discouraged from using a calculator – it saved classroom time in the past but now it comes and bites them on the backside.

As it turns out – not being able to do simple mental arithmetic such as addition and subtraction of two-digit numbers and not knowing your times tables means that you lack, to a certain degree, understanding of how numbers work so you don’t have any real idea how to start extending those ideas to algebraic problems where an electronic calculator might not be able to help (though, you can buy calculators that deal with algebra now but they’re a bit more pricey).You also end up using all of your energies trying to figure out really simple things like 12×15 in the middle of a complex algebraic problem and then you don’t have any energy left to do the more complicated parts of a problem; the result is a very bumpy stop-start, stop-start solution to a problem which can be compared to driving a car and slamming on the brakes every fifty yards – you might get where you want to be but with much more wasted effort.

I don’t think it’s necessary to be able to do huge mental calculations; there are people around today who can carry out huge mental calculations. They do it either because it’s something that they can just do or because mental arithmetic is something that they enjoy doing and have spent many hours learning how numbers work in great detail. However, I can’t see that mental arithmetic will be making a comeback in the classroom any time soon – I think that maths students are impoverished because of it’s absence and lack of mental arithmetic skills makes their lives more difficult further down the line if they decide to go on to do higher level maths; but even if they don’t want to do higher level maths they will still lack the confidence of dealing with numbers in a world where numbers and statistics are everywhere you look. Sadly, I think that mental arithmetic is a dead skill.

I’m going to let you in on a big secret about how to get good at mathematics. You might not learn this in school but there IS a quick way of learning mathematics; it’s been known for hundreds of years but seldom talked about and it’s this – hard work.

If that’s not really what you wanted to hear and feel disgusted that I would say such a thing then please do not continue reading this post as I don’t want to waste any more of your time; if you would like to know what my reasons are for believing that hard work is THE QUICKEST way of learning mathematics then please read on…

Unfortunately many of my students get conned during their school maths classes; they’re told “quick and easy” or “cheaty” methods of doing everything from simple multiplications and percentages to trigonometry and integration. It has to be remembered that these cheaty methods were devised by people that understood the theory in the first place – a good example that springs to mind is the CAST diagrams method for solving trigonometric equations (which I don’t encourage using). If these methods are then shown to people that don’t understand the theory from which the cheaty method comes then, sooner or later, you end up with big problems – yet this is done year in and year out in many schools by many teachers and tutors alike in an effort to try and circumvent the hard work aspect of learning mathematics. Teaching these cheaty methods from the outset eventually leads to spectacular failure and lack of understanding as it shows students how to do something in a very limited and narrow range of cases and doesn’t usually provide any kind of flexibility or ability to adapt to unfamiliar situations. Sadly, students are duped into believing that they don’t need to know the theory and develop an unhealthy expectation that maths can always be reduced to cheaty methods.

Here’s the thing – if you understand the theory you can adapt very easily to new sitiuations, solve a wider range of problems and generally enjoy the learning process more because you get more out of it. If you rely on cheaty methods you have to learn a new method of solution for each and every “type” of question that you encounter. To start with this might not be too much of a problem – at GCSE for example you will only really encounter a fairly limited range of possible questions – but further down the line at A Level the doors are flung wide open and if you don’t have some understanding you’re up the proverbial creek without a paddle.

By teaching along these short-sighted lines you encourage an expectation within a student that everything can be reduced to a cheaty method. Which it can’t. From my personal experiences as a maths tutor the largest category of people that fall victim to this way of thinking are those wanting to do the QTS Numeracy test. I’ve lost count of the number of times that I have been asked to provide some tuition for the QTS Numeracy test but insisting that I just tell them all of the “cheaty short-cut methods” to do the questions which they’ll get on the test (by the way; I don’t know beforehand exactly which questions you will get asked on the test; and even if I did I wouldn’t tell you). I’m happy to show people how to take short-cuts provided that they have a sufficiently high level of understanding in the first place. If they don’t understand the basics then we are both wasting our time and I may as well go and talk to a brick wall for a while because they will not understand when or how to apply such short-cuts.

I understand what I’m doing when I do maths but it isn’t because I learned all of the short-cut ways of doing everything. Quite the opposite – I learned to understand what I was doing by working hard and then the cheaty methods become trivial facts; in fact they almost become redundant. By understanding what I’m doing I see where these cheaty methods come from and how they work – better still I can make them even more “cheaty” if I want to in some cases. There is NO WAY to cut out the understanding when learning mathematics and the understanding can only come about through hard work. You have to be prepared to use your own brain to solve problems and not leave yourself at the mercy of some miscellaneous method that you don’t understand but which you keep your fingers crossed that you’re using it right and that it will give you the right answer. Is that really a good way to learn?

IMPORTANT!!! By trying to avoid the hard work of learning to do mathematics properly you will end up spending (wasting) more time, energy and effort trying to get to grips with loads of shot-cut, cheaty techniques that you don’t have any understanding of and will most probably forget every couple of weeks and have to keep re-learning. So hard work really is the quickest way to learn mathematics – not really what you want to hear is it? But that’s how it is.

There is a place for short-cut methods when it comes to mathematics; they can sometimes take the pain out of an otherwise lengthy and tedious calculation but they should NOT under any circumstances be a complete substitute for learning through hard work to acquire the necessary level of competence. You wouldn’t expect to become a world-champion 100m sprinter without hard work would you? And nor would any sprint coach who knew what they were talking about tell you that you could become world champion without a lot of hard work. Leading people to believe that all mathematics can be simplified to such a point where you just need to follow a nice cheaty method is cruel and if you do it and encourage it then shame on you!

When I say to people that I’m a mathematician, in a lot of cases people misunderstand what I do. Let me explain…

Mathematics, maths or math if you live in the U.S of A. is often confused with something called ‘numeracy’. When I say to people that I am a mathematician they think that I spend my time adding columns of numbers together, doing long multiplications, busying myself with percentages and getting very excited about pie charts. It’s got to the stage now where sometimes I just don’t even bother elaborating on what I, as a mathematician, am really interested in.

What I’ve just mentioned above, things like addition, multiplication, division, percentages, reading bar-charts and all the rest, are indeed part of mathematics, but really a part of mathematics that mathematicians are not particularly interested in – at least not nowadays. Yes, as I mathematician I can do all of these things quite comfortably but I’m not necessarily any better at adding up numbers than the next person. Yes, I know a few tricks that can speed up basic calculations but not necessarily any tricks that the proverbial Joe Bloggs wouldn’t know. Yes, I have to be able to do these things but not necessarily because I find them deeply interesting and they’re not the kinds of things that get me out of bed in the morning.

Just being numerate and able to handle numbers well doesn’t make you a mathematician just like the ability to wield a spanner doesn’t necessarily make you a mechanic. Obviously you need to know how to use a spanner to be a competent mechanic but just knowing how to tighten and loosen bolts is hardly something that is going to give someone the confidence to let you try and fix their car engine. There are plenty of people out there who know maths but who would still find mental arithmetic difficult beyond a certain point. This might seem crazy because surely as a mathematician they should find mental arithmetic a doddle…right? And that right there is where the confusion is coming in – equating mathematics with numeracy. These two things have an overlap but they are a million miles apart from each other.

Unfortunately many people don’t make any distinction between the two; maths is numeracy and numeracy is maths. I guess this is not helped by the fact that at school, most of the time spent in ‘maths’ lessons is spent doing numeracy; so unsurprisingly when people think maths they think times-tables, long multiplication, columns of numbers and boring stuff like that. But mathematics is about logical deduction, studying abstract concepts, precision, analysis. As a mathematician I’m more interested in studying algebraic structures such as groups, rings, fields, modules and Lie algebras; I’m more interesting in things like homology groups, transfinite numbers, set theory, mathematical logic and representation theory. None of these topics really require me to have anything more than an average level of numeracy – it helps to be comfortable with numbers but I can’t multiply six digit numbers in my head any better than the aforementioned Joe Bloggs. And I don’t need to be able to.

Most of the time I don’t bother telling people what I did during my maths degree – it’s not particularly interesting unless you have an interest in mathematics and (I really don’t want to sound like I’m being elitist or patronising when I say this) most people won’t know what you’re talking about because of the language that’s used. Does Random Man on the Street know what I’m talking about if I told hime that I did a course on Algebraic Topology or a course on Partial Differential Equations, or Fourier Analysis – I doubt it. I let people believe what they want to believe about what my degree entailed. If they believe that I spent most of my time looking at pages of numbers and doing big sums then I don’t usually bother to try and correct them – it’s just not worth it (I know that from personal experience by the way).

Just to be clear – no I can’t add a page of numbers up at a glance; no I can’t multiply two twenty digit numbers together effortlessly; and no I can’t interpret any old random statistic that is just given to me. I suppose this is why maths gets a bad reputation for being too boring and that mathematicians are just weird because they’re interested in percentages things like that – when you see maths and numeracy as one and the same thing then it’s like seeing literature and the alphabet as the same thing. There’s not really anything I can do to change this. Which is just too bad as many people will never experience mathematics and how deep the subject goes and how beautiful some of the theories are. I’ll just keep on keeping on…I think that’s the best way forward.

I’ve been asked countless times in the past – “what’s the point of maths?” Faced with this question many mathematicians will probably give a list of reasons about why maths is so important and how it can be used in society for this and that in order to convince the questioner of the point in the existence of mathematics and of the virtues of studying mathematics or numeracy.

I feel differently; I don’t feel at all inclined to do that. But surely as a maths tutor isn’t it my duty to inspire and recruit to the ranks of mathematicians? Yes it is my duty to inspire whenever I can and inform people about mathematics but if someone doesn’t want to be convinced then they won’t be. Some people call football ‘the beautiful game’ – try and convince me of that and I can tell you right now that you’re really wasting your time. I don’t want to be convinced of it and therefore, I won’t be convinced no matter what anyone says and how passionately they might say it.

Why exactly does there need to be a reason for studying, or a point to, maths? Would it be sensible to ask someone who plays tennis what the point of tennis is? Would it be sensible to ask a musician what the point in playing the piano is? How about a Formula 1 driver? A chef? Not really.

Why would anyone choose to play football? Well I suppose if they play Premiership football then the money might have something to do with it but primarily, I would imagine, because they enjoy it I imagine. It doesn’t solve any of the world’s problems as far as I can see, though.

Why would anyone choose to play the piano? Maybe simply because they enjoy it. Music and the piano don’t really solve any major problems facing humanity even when Bob Geldof and Bono get involved.

Why would anyone choose to learn about mathematics? Because they enjoy it? It doesn’t necessarily solve any ….oh wait actually it does solve some big problems facing humanity.

Isn’t it strange that the thing that clearly has applications and benefits that surround us every second of the day is the thing that is constantly challenged about its point. The things that really don’t make that much difference in the big picture are never even questioned.

I don’t do mathematics because it necessarily changes anything; I do it because I enjoy learning about it and for the sheer pleasure of doing it. The applications that there are of mathematics are not my primary reason for doing it; I’m pleased that there are applications and I sometimes take an interest in them but I wouldn’t know half of what I know about the subject if I was motivated only by it’s actual applications in real life.

No-one says that you have to be interested in mathematics. But nevertheless, some people are. But If you really hate something so much, whether maths or anything else for that matter, then don’t learn about it, don’t do it…simple. You don’t have to know about numbers and mathematics in life any more than you need to know about football or how to speak Latin or how to recognise an original Picasso. But for a few exceptional cases, you would probably find life much more difficult if you were completely innumerate than if you didn’t know a single rule of the game of football or a single Latin word or who Picasso was.

There are parts of mathematics that have applications in real life situations and you don’t have to look very far at all for them – in fact you’re probably reading this on one of those applications right now. Then there are parts of maths that don’t have applications. Should we only limit ourselves to those aspects of the subject that have applications right here, right now? Non-Euclidean geometry came about as a theoretical pursuit and it wasn’t until decades later that it found an application in Einstein’s theory of relativity – one of the single most important theories in the history of science. Some of the theory may never find an application other than within mathematics to produce more mathematics – but how do we know which theory will lead to something and which to nothing?

If we are to limit ourselves to stuff that only has applications here and now then should we abandon much of history, linguistics, physics, literature, music or sport even if people love doing these things for the sake of doing them? Why should mathematics, or anything else for that matter, have a point? Why does it need a purpose other than it makes people happy to do it? Mathematics is one of the most beautiful, and in some instances one of the most pointless, of human creations. Why shouldn’t it be enjoyed?

Well that’s my rant just about over – I’m glad to get it off my chest. I hope that you don’t think that all this means that I’m not enthusiastic about maths or that I don’t want to pass on my enthusiasm to other people, though! I continue to learn about mathematics on an almost daily basis and enjoy doing so; I’m proud of the fact that I can usually answer the deep, probing questions that I’m asked by my students about maths and I get excited when I get asked something that I don’t know the answer to because it means I have more to learn. Mathematics is one of the most important parts of my life; I just don’t feel that its existence needs justification or that it has to have a point. I’ll continue to show people the beauty of mathematics and either they will be convinced by its beauty or they won’t. But no hard feelings, though, if not.