After the article that I wrote recently on presenting your solutions and work, I thought it would be a good idea to give some examples of what I consider bad presentation and what I consider good presentation. I will give a brief analysis of the examples that I give to explain my thoughts. The first example is a farily straightforward example and the second example is more in depth; but just because a problem is straightforward doesn’t mean that it doesn’t deserve a well presented and clear solution.

These examples give an idea of some of the presentational mistakes that maths students often make – some of them minor and some of them more substantial, but still questionable all the same.

Example A – A bad solution and a good solution

Example A – The bad solution

Why do I think the bad solution is bad? Well it isn’t clear what’s going on; I can’t clearly see the rearrangement of the original equation to arrive at the solution. Another problem with this which is a really common problem amongst maths students right up to A Level is linking everything together with equals signs. Would you agree that $14.5=2.5$? Probably not – and yet this is exactly what is suggested here; there is a continuous string of equals signs connecting $3x+7$ with the final answer of $2.5$ (note that if $x=2.5$ then $3x+7=14.5$). I find that the equals sign becomes, for many people, similar to using “erm…” when speaking. It becomes something to fill the in-between spaces that often don’t need to be filled anyway.

Remember that the equals sign $=$ indicates equality of two things. If two things are not equal then you should not put an equals sign between them.

Example A – The good solution

Why do I think this solutin sis good? It’s clear at each stage what I have done to rearrange the original equation even though I haven’t explicitly said what I have done it isn’t difficult to see. I haven’t just linked everything with equals signs; it feels like there is some breathing time between each line. I have kept the equals signs on each separate line lined up which makes things look much neater. And finally I have made my final answer obvious; $x=\dfrac{5}{2}$. I have used a fraction because I personally find fractions easier on the eye than decimals but this is just down to personal preference; there isn’t anything wrong with writing $x=2.5$ here.

Example B – A bad solution

Example B – The bad solution

What’s bad about this solution? What’s good about it would take less time to list – but here’s the low-down on this monstrosity. All I see is a page of symbols – I know that this is a maths problem but that doesn’t mean that words are forbidden. Again, because everything has just been indiscriminately linked using equals signs it isn’t clear that any differentiation has taken place here; there should be a $\frac{\mathrm{d}y}{\mathrm{d}x}$ in the second line saying that $\frac{\mathrm{d}y}{\mathrm{d}x}=3x^{2}+6x-9$ as this solution incorrectly suggests that $y=3x^{2}+6x-9$. This might seem like a minor point but there is a world of difference between $y$ and $\frac{\mathrm{d}y}{\mathrm{d}x}$.

I have already mentioned that everything is connected with equals signs which makes the left hand column of working complete nonsense because $3x^{2}+6x-9$ is not equal to $x^{2}+2x-3$. It is acceptable to cancel down by $3$ as has happened here provided that you are dealing with an equation such as $3x^{2}+6x-9=0$ and not simply the expression $3x^{2}+6x-9$. Fortunately in this case we do eventually need to set our quadratic equal to zero which makes things appear to work out, although it is still wrong.

What isn’t made clear here is WHY we set the quadratic equal to zero. This is something that I see done a lot and sometimes I have to question whether the person who has written the solution knows what they’re doing, whether they’re aping what they have been shown to do but without understanding or have they just guessed what to do and got lucky? All three can, and do, happen.

The question asks for the coordinates of the turning points: where are they? I can see $x$-values and what seem to be corresponding $y$-values but they are not together as coordinates as they should be.

The second derivative is calculated and then, again, everything is connected using equals signs (believe me, this is more common than you might care to think). Am I to think that $\frac{\mathrm{d^{2}}y}{\mathrm{d}x^{2}}=-12$? or am I to think that $\frac{\mathrm{d^{2}}y}{\mathrm{d}x^{2}}=12$? More to the point why should I think that the second derivative,¬†$\frac{\mathrm{d^{2}}y}{\mathrm{d}x^{2}}$ is equal to either of these when it also says that¬†$\frac{\mathrm{d^{2}}y}{\mathrm{d}x^{2}}=6x+6$? It isn’t clear what is happening here in the solution. When I see this I have to ask, “Do I know that the person who has written this solution knows that they are using the second derivative test?”

Finally – the question asks for the turning points (the coordinates of which weren’t made clear anyway) to be classified as minimum or maximum points. If you use the second derivative test to determine whether a turning point is a maximum or minimum you MUST make it clear which points are maximum points and which are minimum points and WHY you have made that decision.

I could go into more detail on this solution but it would go on for pages and pages – the upshot is that this solution stinks! But I see this kind of presentation very often. So here is a better presented solution (though, it could still be improved on in many ways depending on who you ask)

Example B – A better solution

In this solution is have put right many of the things that I noted were wrong with the bad solution so I won’t go into any more detail about it. It can’t be considered a ‘perfect’ solution as there isn’t any such thing but I think it is objectively better than the bad solution discussed above.

Presentation is a very important aspect of your mathematics – it might not be a chapter in you textbook and it might not be something that you spend a great deal of time on in the classroom as you have enough to get through in the short time that you have in lessons. Presentation is something that YOU have to work on yourself – but being conscious of your presentation is a very good habit to develop.

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.