It’s coming up to that time of year again for GCSE and A Level maths students – exam time! If it hasn’t already done so then the past-paper frenzy will start very soon. I’m not against the use of past-papers as I think they fulfil a very important role; I do ask my students, on occasions, to complete past papers for me between lessons as this allows me to identify any gaps in knowledge that may have been overlooked – some of them do past-papers whether I ask them to or not which I don’t really have a problem with at all. I have a problem with past-papers being used excessively and effectively replacing actual teaching which seems to happen in many schools now; textbooks being almost non-existent in the classroom after February or March and teaching to understand is completely subordinated to teaching to pass an exam which I feel is a very short-term outlook and much less effective.

But the reality is that past-papers are used and they can be very useful when used properly. I want to point some mistakes that people make when using past-papers to prepare for their exams. I don’t think one post will suffice for this so I think there will be a couple of posts. But let’s make a start…

The first mistake that people make is to be obsessed with the time-limit for the exam. I recognise that there is a time limit in all exams and that when you sit your exam for real then you have no choice but to stick to the time limit that you have. BUT…when you do your first few past-papers (maybe not applicable if you’re doing the new 1-9 GCSE maths exams as there aren’t really any past-papers) YOU DO NOT need to worry about time-limits and, in my opinion, it could be very detrimental to you to worry about it. If the time limit when the exam was actually sat was 90 minutes then forget about it – for your first batch of past-papers take as much time as you need.

When I ask my students to do a past-paper for me for the first time I will say to them, “Forget about time-limits and take as long as you need. Use your class notes or our lesson notes, refer to your textbook if you need reminding about how to do something; use the resources that you have avilable to you. Work through the paper with a friend if you want (not get them to do it for you, though). Just, whatever you do, avoid the temptation to look at the mark-scheme” Almost invariably I’m completely ignored and they will just give themselves as much time as the time-limit stipulates and try to do the whole thing under exam conditions.

And then I look at their paper the next lesson and it’s only half done – if that! Most of the problems will be poorly answered, if at all; the ones that have been answered will be virtually incomprehensible and solutions will be riddled with silly mistakes. I’ll ask them, “How much did you learn when you did this?” – I know the answer before they even open their mouths; “Well I couldn’t do this and I couldn’t do that and then I just ran out of time.” See what I mean about being ignored!

I would much much MUCH rather that this person had taken their time and figured out solutions to as many problems as they could, even if that meant that they had to take two or three times a long to complete the paper as the time-limit gives. I don’t want them to do it under exam conditions because they’re not ready for that yet. I want them to refer to their textbooks, online videos and other resources so that if they’ve forgotten how to do something then they will end up revising and learning as they go. There is NO POINT in doing a past paper under exam conditions unless you’re prepared for it.

That’s not to say that there doesn’t come a time when past-papers need to be done under exam conditions – but not from the outset. If you don’t give yourself the time that you need to start with then you will learn nothing – or at least significantly less than you would. The speed and confidence that you need for your final exam will naturally come about if you give it chance. If you try to go too fast too soon then, ironically, although you think that you’re going fast – you’re going much more slowly than if you slowed down a bit to start with. A pianist wouldn’t learn to play fast by going fast to start with – but by going sloooooooooooooooooowly to start and building up speed gradually and naturally. This is no different.

There’s some more things that I want to talk about here but I think I’ll leave them for the post next week – see you then.

 

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.

 

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.

 

 

For all of you out there who just can’t get enough of me, I know you’ve all been counting down the days and now the wait is finally over – here’s my monthly review video for February 2017. So other than watching seemingly endless episodes of Peppa Pig and In the Night Garden with my daughter what else has been going on?

Well, many of my GCSE students have been sitting their mock exams over the last couple of months which has given them an idea of what to expect on the new GCSE maths exams which students will be sitting for the first time in around May or June. I think many students, and teachers for that matter, have been a bit dismayed at the new exam style – the questions are tougher, more in-depth knowledge is required and students are tested more on their problem-solving skills rather than their ability to parrot out model solutions. There was one question in particular that really caught my eye on one of the papers; it was a fairly difficult question by GCSE standards but it wasn’t impossible. In this video I have a bit of a chat about the question and, though I don’t present a fully worked solution to the problem (that would spoil the fun for you) I do give a bit of an idea as to why the conclusion (which many people made but couldn’t explain) can be justified.

I also talk about what I’ve been up to learning how to speed-solve a Rubik’s Cube (or should I say Twist Game as it isn’t an official Rubik’s Cube that I use any more?) and a couple of books that I’ve read (or partially read), in particular

  • The Art of Memory – Frances Yates
  • Mathematics, science and epistemology – Imre Lakatos

I’m still practising with my Twist Game (Rubik’s Cube is much easier to say, though) and it’s really fun learning to speed-cube. I still can’t come anywhere close to kinds of times that a proper speed-cuber can achieve but I’m making progress.

Anyway – look out for my video next month and I’m sure I’ll have loads more exciting things to talk about. I bet you can’t wait. Be excellent to each other; and Party on, Dudes.