I’m going to make a somewhat counter-intuitive claim about how to go about preparing for exams (or tests) which is – in order to be really well prepared for exams you need to almost completely forget about preparing for exams (or tests).

This sounds ridiculous doesn’t it? But I’ll try to explain why I think this.

Exams and tests are a completely unnatural thing when it comes to learning. The anticipation of an upcoming test or exam can cause unnecessary stress and, in some cases, outright panic neither of which are any good for learning. The anticipation of exams and tests causes people to learn at a rate that is a fraction of the rate at which they are capable of learning because they will only ‘learn’ what they expect to be examined on – repetitively going over the same few, scanty bits of information trying to cram some poorly-understood and seemingly non-sensical facts into their heads – rather than challenging themselves and delving deeper into a subject to get a much greater understanding.

And of course, why bother to learn things if you won’t be examined on them? Who in their right mind would ‘waste time’ learning things that they won’t be examined on? Well that’s exactly the kind of viewpoint you will probably take if you see education and learning as nothing more than a series of exams and tests. And no wonder people with this attitude hate learning! So would I if that’s all learning meant to me.

But isn’t learning about more than just exams? Yes, I know that exams are an important part of your life if you want to go to university or become a chartered accountant etc. so I’m not saying that exams can be completely ignored – what I AM saying is that preparation for exams should not be the main focus of your learning. If you really want to learn more, learn faster and learn better then you need to be learning because you love learning about what you’re learning about and have a genuine desire to want to go further and acquire more knowledge simply because you can.

Does it sound weird to take such an approach to learning? Of course it does because no-one really encourages people to learn in this way in school. There’s constant testing week-in and week-out and so people only learn the bits of information that they need to know for the test, and as long as they can vomit up these bits of information in the test, regardless of whether they understand (which in many cases people know but don’t understand), then they get a good mark; a couple of days later and they’ve forgotten everything that they learned, rinse and repeat! And this approach is considered to be a good learning approach!!!!

If you are genuinely interested in what you’re learning and you’re learning because you love learning, then learning is easy and effortless. A depth of understanding will come to you that will never come if you see your learning as just learning to pass a constant stream of tests and exams. If you learn because you love to learn then you will pass the tests and exams with flying colours because you will be streets ahead of the rest of the crowd – you will learn at a rate that is unimaginable if you’re learning to just pass an exam and, moreover, what you learn will be much more permanent.

When I was learning GCSE maths I was learning to pass an exam – and it was hard, hard, hard! I hated it. When I was learning A-Level maths I was learning because I was excited about what I was learning – I learned way more than I needed to pass the A-Level exams and I didn’t even think about the exams until a couple of weeks before I sat them, at which point I realised, I didn’t need to do any revision specifically for my exams because I’d been doing it all along without even realising it! The outcome was that the exams were a piece of cake!

I’ve always maintained my position that good resources and, especially good textbooks, are a must for anyone learning maths at any level. I still stand by that but unfortunately, A Level Maths students find themselves in something of a difficult situation nowadays. Why? Well to put it bluntly – the textbooks are terrible!

This is a fairly recent development as in the past there have been some very good (and also some very bad) A Level Maths textbooks. But what makes the latest sets of A Level Maths textbooks, for the most part at least, so bad? After all, when you look at the textbooks they look great; lovely colourful pictures, bright, jazzy etc. Well the thing is, it’s all well and good having a lovely glossy, colourful textbook but if the actual content – the stuff that people are actually supposed to learn – is no good then a few pictures isn’t going to make any difference. It reminds me of the well-known saying that ‘you can’t polish a t…’; well, you get the idea.

The latest sets of A Level maths textbooks are, in my very honest opinion, some of the most uninspiring textbooks I have ever seen. I hate to have to be so negative here, as if there isn’t already enough negativity around, but, sadly, it’s true – they stink! They are awful!

A good textbook, which I loosely define to be a textbook with good, solid content, is interesting and can inspire without the need for any jazzy and colourful pictures. It seems that the more colourful the textbook is, the worse the content will be! I’ll use as an example the textbooks that I used for my A-Level maths (2003-2005); they didn’t contain ANY colour pictures and only a few basic graphs and diagrams. However, they were great! Why? Because they went into sufficient depth, the problems were challenging, and you actually learnt things. If, like me, you were interested in mathematics and WANTED to learn about it then the books were interesting by virtue of the fact that they contained great content on something that you were deeply interested in. If you’re interested enough in your subject then what you need to learn could be written on toilet-paper and it wouldn’t matter.

So, what if you’re not particularly interested in maths but you have to do A Level maths for whatever reason, what do you do to get interested? Well this is a different matter and the solution depends on each individual, but what I can say with 100% certainty is that trying to fob people off by putting fantastic looking pictures in a book with weak content won’t make someone interested in the content – it’s just insulting! Poor content is just an all round lose-lose situation and it’s that simple.

I don’t know who thought that the new A Level Maths textbooks were good enough to be published (oh, and a separate issue – they are very often riddled with errors!) but if you ask me, you would be much better off buying some of the older textbooks to work from. Don’t worry if you think that the old textbooks are out of date – what is inside the books is still very relevant and what’s more, you will probably learn a lot more from them. Some of the older style books that I recommend (off the top of my head) are

  • Heinemann A-Level maths books first published up to around 2004
  • OCR A-Level maths textbooks first published up to around 2004
  • MEI A-Level maths textbooks first published up to around 2005

You can go even further back than these and there are some really amazing A-Level textbooks – bright and jazzy? No! Challenging and interesting? 100% Yes. If you really want to learn mathematics and get a good foundation then these are the places to start and sadly, NOT in the latest sets of textbooks.

A few months ago, I wrote a couple of posts on my website mentioning that I was learning the Polish language. I still am learning Polish and enjoying it so much more than I first imagined that I would.

Even though I don’t subscribe to a particular fixed learning style or methodology I do like to find out how other people go about doing things that I want to be able to do – some people will do things that I really like and that I can incorporate into my learning technique and other people will go about things in a way that is the complete opposite to how I would like to learn and so, even though the technique may work for them, I may choose not to follow their example.

A name that came up during my reading and research was Stephen Krashen – Krashen is (according to Google) a linguist and educational researcher. I decided to find out more about his work and I’m so glad that I did!! In fact, my only regret is that I never came across his work before.

Much of Krashen’s work is focused on language learning and language acquisition (these two concepts are distinct) which, obviously, makes it highly relevant to my quest to learn Polish – but I believe that a lot of what he says regarding how language is acquired is applicable, at least in part, to learning mathematics.

Krashen says that for language acquisition (as opposed to language learning) to take place then the individual must receive large quantities of input that is both comprehensible and interesting (he even goes so far as to say that the input should be compelling and not merely interesting). Language acquisition is viewed as being the more permanent knowledge and makes the greatest contribution to fluency in a language; it may in many cases be a completely unconscious and effortless process. Language learning is stuff like grammar drills and vocab lists which Krashen says is almost completely worthless and unnecessary in the face of language acquisition.

Is it true that large quantities of comprehensible and interesting (if not compelling) input is required for mathematics to be ‘acquired’? looking back over my personal experiences of learning mathematics then I would have to say a huge yes in response to that question. In fact, I think it confirms some things that I often recommend to my students (and which usually get ignored) such as

  • Read around the subject
  • Use a range of different resources
  • Learn because you want to and not because you have to and not just to pass an exam

If you want to read some of Krashen’s work then you can download some of his older books and some of his journal articles from his own website for free at www.sdkrashen.com. Some of his more recent books you will have to buy but I highly recommend even if language learning and acquisition is not your thing then it’s still a real eye-opener.

Some of Krashen’s books that I’ve read over the last couple of months are

  • Principles and Practice in Second Language Acquisition
  • Language Two (written with two other authors)
  • Explorations in Language Acquisition and Use

Obviously, learning a language and learning mathematics are not the same thing and I’m not saying that everything that Krashen says regarding language learning and acquisition is applicable to mathematics but I think there is a lot of stuff in there that is very relevant and can be very useful to mathematics learners.

As I mentioned in my previous post, last year I started to (seriously) learn Polish. So how are things coming along?

Of course the first thing to say is that learning Polish is not easy – but then, I’ve never heard anyone say that any language is easy to learn so it’s really no surprise to learn that Polish is no different. Since the beginning of August 2017 I have spent hundreds of hours on the language – some of these hours are very concentrated and intense, others are a bit more laid back – sometimes I will read quite intensively and other times I will read more extensively but the main thing is that I am enjoying the learning process. I’m not bothered about if I have the most efficient langauge learning method – in fact I want to have the freedom to experiment and to find my own way of learning. That’s not to say that I don’t take an interest in how other people have, and do, learn languages – it’s just that, part of the fun for me comes from the experimentation.

Of course at this point I have to draw a parallel with how I went about learning maths (that thing about drawing a parallel wasn’t meant to be a pun by the way) when I was doing my A Levels etc. I didn’t have a specific study technique (that I knew of), I wasn’t doing it because I felt forced to do it against my will – I learned about it because I loved every second that I spent learning about it; I enjoyed working through the frustrations and every little discovery that I made was thrilling – it was brilliant. And now this is how I feel about learning Polish – it can be frustrating, but I enjoy ploughing through those frustrations.

At present I’m not really in a position to judge the level of my Polish. It could be beginner, it could be intermediate – I’m not really able to make that judgement. Possibly I am low intermediate at a push – I’m planning to take the B1 Level Polish State Certificate Examination in May of this year – B1 Level is considered to be low intermediate and so, I suppose if I pass the exam then that will be objective evidence of my level but I’ll wait until that result comes through.

I spend most of my time reading the language – this is for a few reasons. The first reason is that I can read more often than anything else – if I stop off for a coffee I can just pull a book out of my bag and get on with it. The second reason that I read more is because I like reading – I read a lot in English so it makes sense for me to do this in Polish if that’s what I enjoy doing. The third reason is that I like the theories of Stephen Krashen which says that extensive reading is a very effective way of acquiring a langauge. I don’t understand everything that I read in Polish – sometimes it is very difficult – but of course, as with anything, with practice it gets easier. I try to read authentic material that is intended to be read by native speakers of Polish if I can – obviously right at the very beginning when I started learning this wasn’t really possible but it’s surprising how quickly you can make progress in that direction. It helps, also, to try to read things that you have some familiarity with – for example, the first full book that I read in Polish was the Polish translation of A Short History of Time by Stephen Hawking (Krotka Historia Czasu). Of course this wasn’t an easy book to read but I was able to follow it quite well since I was familiar with a lot of the material. On the tother hand, fiction is much more difficult for me to read because there are so many more literary words whose meanings are not clear for me just yet.

Overall things are going well with my quest to learn Polish – I like that I am making progress but I also like that I have lots to learn. It’s a challenge and it’s a challenge that I’m happy to undertake.

Over the last hal year or so I haven’t really been doing much with my website – no new posts, only a couple of new videos. What’s been going on?

Well the good news is that everything is (and has been) perfectly fine and I’m still doing maths tuition – the reason for my absence has been more to do with getting myself interested in a new (old) hobby. As much as you love doing something, sometimes it’s always nice just to have a bit of a break and not be quite as full-on as you may have been.

So what is this new hobby that I’ve been getting into? Well I’ve been learning a new language – Polish! I started around the end of July/beginning of August last year (2017) and I’ve been really enjoying it. Arguably, this is not the first time that I have dabbled with foreign languages, including Polish; like most people I studied a language at school – French – and also like most people, I hated it. I hated it so much! I couldn’t understand why anyone would put themselves through the torture of learning French in a classroom. It was so dull, so drab and so painful. I hated having to do role-playing where I had to pretend to be a tourist asking ‘ou est la gare’ is or how to say ‘ j’aime aller a la plage’.

After leaving school in 2002 and somehow managing to get a C in my GCSE French (they must have awarded me a few pity-marks, I’m certain of it) I never even thought about learning a language – I had no reason to think about it. In around 2006, though, something changed. I don’t know what it was but I felt compelled to learn French – this time to do it on my own terms. I learned how to read farily well in French, I had regular written conversations (at the time when MSN messanger was popular) with French people, I had Skype conversations in French, I learned the grammar and I loved it – because I was now doing it because I wanted to do it. I never got to a fluent level but I could get by.

A few years later I decided to start learning Polish – why Polish? Why not? This lasted for a few months but I eventually had to set it aside for a while as there were lots of other things going on at the time which meant my time was getting spread a bit to thin. Once again I went back to not even thinking about languages – until last year. I decided that now was the right time for me to do this – to learn a language and to see it done properly. Even though it would probably have been easier for me to start back learning French I felt more drawn to Polish – so I chose Polish.

So now, I want to make my goals for 2018 official – they are

  • Sit (and pass) the Polish State Certificate Exams, B1 Level, in May 2018 (already registered for this!).
  • Sit (and pass) the Polish State Certificate Exams, B2 Level, in November 2018.

For me, I hated learning languages in a classroom. I much prefer to learn the language on my own terms; I enjoy learning grammar when I can do it at my own pace; I enjoy reading authentic material in the language and not textbook-style role-play conversations; I enjoy having authentic conversations in the language (you don’t learn how to talk about which kind of radiator you want in one of the bedrooms in Polish textbooks but these kinds of things come up in real life). As it stands I don’t have any plans to learn other languages just yet – but who knows, once I’ve got to a decent level in Polish I may turn my attention to other languages as well. For the time-being let’s go one step at a time.

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 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.

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.

A really useful skill to develop when learning mathematics is graph sketching; although some of the GCSE and A Level textbooks do have a short chapter covering this somewhere amongst the pages, unfortunately the value of graph sketching is, for the most part, almost invariably overlooked. Students think that this is something that they only need to do when they are asked to do it and even then it’s clear in many cases that they don’t really understand what it is that they are doing.

An equation such as $y=x^2-5x+7$ gives a clear, unambiguous, relationship between $x$ and $y$. So what? Well this means that for each $x$ value that I put into the right hand side of the equation I will get a corresponding y value. So for example if I choose $x=3$ (there is nothing special about this choice of value for x; I could just as easily have chosen $x=5.87$ or $x=-29$), I simply replace any $x$’s on the right hand side with $3$ to get $y=3^2-5×3+7$ and so $y=9-15+7$ and therefore $y=1$.

For many GCSE and A Level students this is fairly straightforward – but they miss the important part. This $x$ value and its corresponding $y$ value form a pair of coordinates, in this case $(3,1)$ – remember that the $x$ coordinate is the first number, $3$, and the $y$ coordinate is the second number, $1$.

Again – SO WHAT?

Well if I do this for several x values then I get a few more corresponding y values and so more pairs of coordinates which I can start to plot on some $xy$-axes. The more $xy$ pairs that I find the more points I can plot; what emerges is the graph of this equation. In many cases you don’t even need to plot a great deal of points to figure out what the shape of the graph is going to look like; many equations can be grouped together into families of equations – equations that have certain similarities – and the graphs of these families of equations (quadratic, exponential, logarithmic) have characteristic shapes and behaviours.

This is important – the equation and the graph represent the same thing but in different ways; the equation determines what the graph will look like and the graph will (at least in theory) determine the equation. If I choose any point on the graph and study the coordinates of that point I will find that the relationship between the $x$ and $y$ coordinates, for all points, is the relationship described by the equation and, moreover, any $x$ and $y$ values that are related via the equation WILL, without exception, be the coordinates of a point on the graph of that equation. So now, rather than seeing an equation as a string of miscellaneous symbols on a page I can start to draw (in some cases of my own volition) the graph of an equation and from this I can see (in the literal sense) the behaviour of the equation at a glance. The equation suddenly becomes more than an equation – it has more character to it. Drawing the graph of an equation might seem like more work – why would you inflict that on yourself? But that isn’t the case. By knowing what the graph of an equation looks like you get a visualisation of the situation and this usually means that overall you have don’t have to work anywhere near as hard to get your solutions.

Sadly, many GCSE and A Level maths students never get to grips with this. Not because they CAN’T draw graphs but because it is never made clear to them that this relationship between equation and graph exists. They go on thinking that in order to draw the graph of an equation they have to have been told beforehand what the graph of that particular equation looks like. This is a disaster for them! I often ask my students to sketch the graph of an equation and to start with they may look at me blankly or say, “I’ve never been told what that looks like”. My response might usually be something like, “Well think about what the equation is telling you is going to happen for different values of $x$”. This might not hit home straight away but eventually it does in most cases and, finally, they understand that the graph (even of unfamiliar equations) can be figured out themselves by thinking about what the equation is doing at various (well chosen) values of $x$.

Once you get your head around this then you’ll find that maths becomes ten times easier.

I just want to talk about something that really holds a lot of people back when it comes to learning maths; something that stops (too) many people from achieving what they can truly achieve – the fear of failure.

The fear of failure and of making mistakes is a very common issue amongst maths students (though, I’m certain that it’s also the case in other subjects as well). Failure (or at least what is perceived to be failure in someones own mind) and mistakes are always seen as a bad thing. Sometimes failures and mistakes are bad things because the consequences may be very severe – but when you’re learning GCSE or A Level maths, the good thing is that no-one dies if you fail or make a mistake; there is no catastrophic nuclear accident; the world continues to turn and the sun continues to rise and set.

Mistakes are a necessary part of learning and doing mathematics at all levels; and the possibility that you might fail to solve a problem is something that you have to learn to accept as this will ALWAYS be the case even if you happen to be called Albert Einstein (OK, he wasn’t a mathematician but you get my drift).

I know that it feels good when you get the correct answer to something and you succeed in writing out a correct solution – but if you only continue to do what you know how to do then you will only ever be able to do what you know how to do; in other words you will stop learning. The possibility of failure is something that I very often have to work very hard to get my students to accept. Sometimes, and this is particularly true of people who have done well with GCSE maths and moved on to A Level maths, students are so used to getting things right first time and without any difficulties, once they’re presented with an actual problem that they have to solve they will give up. The thing is, it’s not that they can’t solve the problem because in many cases they can, but just that they think that if they make a mistake that they have failed miserably and that only ridicule and embarrassment will follow. They are so afraid of making a mistake that they won’t even make an attempt – they just say, “I haven’t been shown how to do that”, or simply “I don’t know what to do”.

As a mathematician (or student of mathematics) you have to be willing to make an attempt at applying your knowledge and accept that you may (and in all likelihood, will) make mistakes; you may even be unsuccessful in finding a solution for some time and you may NEVER find a solution. Have you failed in these cases? No – not at all. You have failed if you make no attempt at a solution; you have failed if you’re not prepared to make the mistakes that you need to make in order to learn.

Sometimes I have to put my students in situations where they HAVE to make an attempt at solving an unfamiliar problem and where they might make many mistakes along the way; I will not give any clues about what to do until a good attempt has been made by themselves to find a solution. Cruel of me? Not really as this is what they’ll have to do in the A Level or new GCSE maths exams so it would be cruel and wrong of me to not do this and only have them do things that they can easily do. By doing this I’m not saying that I expect them to find a full solution and I even say to them that I’m not bothered if they don’t find a solution; what I do expect is that they put themselves out there and try things out; I AM bothered if they don’t make any attempt even if they claim that they don’t have any ideas. I have to get them to understand that mistakes are a natural part of learning and doing mathematics; that no-one is going to laugh at them if they make mistakes; that their otherwise impeccable reputation will remain intact despite their mistakes; and that they will learn more from making these mistakes than by not making them.

I can see their faces contorting when they first write something that they know might be wrong – but this is what doing mathematics is about; trying things out until you find something that works. As a mathematician you’re not expected to be able to blurt out the correct answer or solution immediately upon being presented with a problem. What you are expected to do is to work to find a solution and this means making umpteen false-starts in the process. Once my students accept this fact and free themselves from the fear of failure (whether real or perceived) then I know that they’ve taken a huge step in the right direction.