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 – 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)

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.