An often overlooked aspect of mathematics at GCSE and A Level is presentation. How to present solutions doesn’t usually get much attention – there will be those who naturally present their solutions in a neat and organised way, there are those who will eventually figure out that they need to present their solutions in a neat and organised way and then there are those who don’t realise the importance.

I think good presentation of solutions is one of the most important aspects of mathematics – in some ways even more important than the content. Think about it this way – if someone wrote a book and they didn’t pay any attention to their word order, their grammar and spelling was completely wrong or they didn’t indicate when they were moving from one subject to another in a clear way – then even though they might have some brilliant ideas in their head they have not been able to present their story, arguments and ideas in a coherent way. Would you be at fault for not understanding what they intended to say? Would it be your job to unscramble their words? In this case is the intended content (however brilliant) as important to you as the presentation? Probably not. Yet many GCSE and A Level maths students will pay little or no attention to their presentation. My advice to my students is not to present a page of indecipherable heiroglyphics and symbols as a solution…ever! It is up to you as the author of your solution to make sure that it is presented well and can be understood.

But why is the presentation of solutions in mathematics so overlooked at these levels? Well for a start off it could be because ‘How to present your solutions’ is not a chapter in any of the maths textbooks and a lot of people will only learn what is in a chapter of their set textbook so if it’s not there then it doesn’t get learned. But I think mainly it’s because it is difficult to get to grips with in some cases; it’s difficult to articulate exactly how to present solutions in a concise way. There are many conventions and unwritten rules that mathematicians will obey when writing out their solutions. Some of these conventions can seem a bit arbitrary, and in some cases contradictory, and so this can be a bit of an issue for people who like a fixed set of written rules to abide by; for example it is much more usual to see $2x$ rather than $x2$ or $x^{2}+2x+3$ rather than $2x+3+x^{2}$ even though there is nothing, strictly speaking, wrong with either expression in both cases. It’s easy to make a mistake when learning the conventions and to feel silly when you realise what you’ve done – but unfortunately this can’t be used as an excuse for not trying.

There isn’t any standardisation with how to present your work and solutions and in some ways everyone has their own little idiosyncracies and preferences when it comes to writing out their solutions – I know I do. So how do you learn how to present your work well? The answer for me was to observe other people; look at how other people set out their working, particularly your teachers or lecturers; analyse the layout of model solutions in textbooks; find what you like and what you don’t like. There is a lot of trial and error involved and your presentation is something that will change over time. Your presentation may be quite rudimentary to start with but you need to analyse the layout of your own solutions; check whether you think your solutions would be comprehensible to someone else; write and re-write solutions to problems until you’re satisfied with what you see – you’re not necessarily focusing on the content of the solution but on its layout.

Something that can really raise the standard of your presentation is using words to explain what you’re doing throughout your solution. For some reason, people feel that once they get into a maths lesson or exam that words are forbidden. This is simply not the case. I’m not saying that you have to write paragraphs explaining in minute detail each and every thing that you do – but a couple of short sentences here and there really help someone reading your work to understand what’s going on. Using words is a good way to give a short conclusion to your solution; even though you might know exactly what you’ve done in your solution and what the implications are, you still need to point these out using words in some cases.

There is no perfect way to present your solutions; there is no set of rules to just follow. Sometimes you will receive some criticism for your presentation style but as long as the criticism is constructive then you can take it on board and either change your style or not. Just because someone criticises your presentation doesn’t mean that it’s wrong – just that it’s not to their liking. But is your work is clear and can be understood then you have a good style; it can always be improved in one way or another, but all the same it is a good style. It takes conscious effort of your own volition to learn how to present your work but whatever you do, don’t overlook your presentation.

 

There’s an old saying that practice makes perfect; it’s very easy to trot out this old saying every now and again without really thinking about it. Does practice really make perfect? I don’t think so. Of course there is the argument that you would need to define what you meant by perfection and whether perfection is something that can ever be attained; I don’t really want to get into that right now – I want to look at this from a more practical perspective.

I completely agree that practice is important when learning a new skill; mathematics being no exception. You have to practise mathematics by doing mathematics and it takes a lot of hard work and enough time to get good. When you first start learning a skill, whether that’s playing the piano, playing tennis or driving a car, your technique will probably be fairly bad. For some things you will, over time, automatically gravitate towards better and more efficient techniques – from my own experience it’s almost like your brain is sifting through all of the possible ways that you’ve done something and choosing the optimal way on your behalf. I remember this happening when I used to train for the long jump all those years ago – some aspects of my technique just came about automatically and I didn’t have to think about them too much such as the leg shoot at the end of my jump; I didn’t need to consciously learn this – just practise it.

But what about when your brain doesn’t do this automatically or if your brain chooses (on your behalf) a poor technique? Simply rehearsing and practising a skill with a poor technique, no matter how many hours you toil away at it, will only reinforce that poor technique and your progress will quickly grind to a halt. If you don’t realise that you have a poor technique then it may not come to your attention until a later time when someone points it out to you – but it’s surprising how many people will practise something in complete knowledge that they are using a poor technique. In this case practice will not make perfect…no matter how much you practise.

Mathematics is a skill like anything else and as such, to really become confident and competent at mathematics (of any kind) you have to practise – but you have to make the effort to use good techniques and not just fall back on ‘comfortable’ but poor techniques.

Mental arithmetic is a good example of this – particularly in relation to the mantal arithmetic part of QTS Numeracy test. I have provided tuition for many people preparing for the QTS Numeracy test and I think the reason that the majority of them felt that they needed tuition was because of their poor mental arithmetic (MA) skills. The reason that their MA skills are seemingly so poor is because of their technique; it isn’t a good idea to try to do long multiplication or long division in your head using methods that are primarily written methods. I point this out to people and introduce them to more ‘mental arithmetic friendly’ methods of doing calculations – after all they’re wanting to improve their MA skills. This means that they sometimes have to re-condition their brains to use this new technique as a first option when doing mental arithmetic rather than what they have been using until now. But many will choose to ignore what I tell them and continue to practise using their ‘comfortable’ yet poor techniques. The result – well…unfortunately they don’t make much progress. The ones who take on board what I say, are willing to work through the initial discomfort of using the new technique but ultimately make it part of their general approach have much more success.

It’s not just mental arithmetic where I see this – I see it with people learning to solve algebraic equations, trigonometry and problem solving in general. But it also extends into anywhere that a skill of some description is being learned – in all of these cases, just practice alone does not necessarily make perfect. At some point you have to make a conscious change to your technique and approach by learning (or even re-learning) a skill using good (or better) techniques and being motivated to practise these good techniques perfectly each and every time if you want to progress beyond a certain level. I’m certainly not the first person to say this but the saying shouldn’t be “Practice makes perfect” but more “Perfect practice makes perfect”.

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.

 

The wait is over – here’s the Leeds Maths Tuition monthly review video for January 2017. That was a joke by the way.

In this video I spend a bit of time talking about the new GCSE and A Level maths specifications and exams – I’ve had a few discussions about this topic with some of my students and some of their parents over the last month. Many people have had their first taste of the new GCSE maths exams by now and I think it’s taken a lot of people by surprise just how different it may be compared to previous years exams. Are the changes good or bad? I think you would be hard pushed to find any GCSE maths students (and possibly teachers for that matter) who approve of the changes; well how about maths tutors? Well you can find out what this maths tutor thinks in the monthly review video.

I also talk about some books that I’ve read over the last month which were:

  • In the Key of Genius : The Extraordinary Life of Derek Paravicini – Adam Ockelford
  • Extraordinary People – Darrold A. Treffert
  • Moonwalking With Einstein : The Art and Science of Remembering Everything – Joshua Foer

These were some interesting books; the Darrold A. Treffert book was maybe a bit too in depth for me at this stage – I enjoyed reading it but I think I jumped in the deep end a bit with that one. The other two books were much more easy going. Anyway, watch the video and you’ll find out a bit more about them; though, of course the best thing to do is read them for yourself.

So I’ll be back next month, all being well, with the next monthly update; hopefully see you then. In the meantime, if you really can’t wait until next month (yeah right, whatever John) I’ll be writing my regular posts for my website so you can keep an eye out for those. TTFN.