I thought I would take this opportunity to pass on some advice for GCSE and A-Level Mathematics students and, to a certain extent, any mathematics students.

I suppose the big question that everyone wants to ask when they’re preparing for exams is something along the lines of, “How can I do well in my Maths exams?” So here’s some things that I think are important in order for you to do well in your Maths exams. These are written in no particular order but just as they come to me.

  1. Give yourself time to find a solution to a problem. Sometimes this might be 20 minutes, half an hour or even an hour. I know you don’t have this time to spare in an exam but you’re NOT in the exam now! You have to learn to solve problems and you have to start slowly like anything else. Sometimes problems take time to solve – the solutions don’t just leap out of the page at you. Be prepared to work hard for a solution.
  2. Don’t be overly reliant on a calculator. Yes I realise that you can use a calculator in your exam but, I’ve already said it once and I’ll say it again, you’re NOT in the exam now. If you jump on the calculator at each and every opportunity then you will not develop the understanding that you need in the same way that you can’t learn to play a musical instrument by getting someone else to play it for you.
  3. Don’t be overly reliant on mark-schemes and ‘the answers in the back of the book’. You need to learn to determine for yourself whether your solutions are correct. This will develop your understanding and build your confidence. If you can see for yourself that something is correct then you know you have the marks in the bag. Try solving the same problem in different ways to see if you arrive at the same answer – if so, then you probably have the right answer. If not, try to figure out what’s gone wrong yourself.
  4. Challenge yourself. You may be fantastic at using the cosine rule and you might be able to solve quadratic equation like nobody’s business but are you prepared to combine these, or indeed any combination of ‘topics’ in the same problem? If not then you may not be challenging yourself. It is better to spend an hour working hard to solve a single problem than to spend that hour factorising oodles of quadratics or doing some other repetitive ‘type of problem’.
  5. Be interested in the subject. Unfortunately this is something that you can’t really fake – either you are interested in maths or you’re not. If you’re not then you will have to either try to get interested or accept that things are just going to be more difficult for you. Sometimes you get interested in something by accident (like I did with mathematics) – just be open-minded. Enjoy the subject!
  6. Don’t be afraid to go beyond the specification. Sometimes people will happily scour a specification and spend a couple of hours coming up with a reason why they don’t need to learn something. And yet, it would only have taken them 15 minutes to learn it! The thing is, learning something beyond the specification might help you understand things more clearly – so even though it’s not expected knowledge, sometimes it still comes in very useful.
  7. Concentrate on solutions. Mathematics is NOT about just writing the correct number or expression somewhere on the page – an answer without a valid solution is absolutely worthless. You need to concentrate on your solutions and make sure you understand how the parts fit together. If you understand this then the correct answers will naturally follow. If you’re ONLY interested in the answers then I suggest that you simply copy out the answers from the mark scheme or textbook that you’re using.
  8. Don’t just learn to copy someone else’s solutions. Your teacher will give you worked examples. Your textbook will contain worked examples. DO NOT try to learn these solutions as simply a series of bullet-point steps. Learn to understand the solutions and why they work – you will find that your workload drops significantly!
  9. Take your mind off the exam. Yes you will have exams – but if everything you learn is motivated purely by the fact that you have an exam then your learning will end up being laborious and very inefficient. Be more concerned with just ‘problem solving’ – solving challenging problems on a regular basis. If you do this then you will automatically learn the skills that you need for your exam and you may just need a bit of fine-tuning nearer the exam period.
  10. Work hard. If you want to do well then YOU will have to work hard. How hard depends on all sorts of things but don’t try to dodge the hard work. Working hard doesn’t necessarily mean working looooong into the night (although it might) but using your time wisely. It is both quality AND quantity to a certain degree, but quality ALWAYS trumps quantity. There are no guarantees that you will ace your exams – but you can give yourself much more certainty by working hard!

So that’s it for now – I hope that you find something useful here. There are loads of other things that I could add to the list but I’ll leave thode for now; I think these are the main points that I want to make right now. I would love to hear from other people what their advice is on doing well. Good luck with your learning over the coming year!

Over the last couple of months I have been working hard to make and upload quite a number of videos to Youtube. Some of the videos that I have uploaded cover some GCSE Maths topics – the above link is to a playlist on my Youtube channel of all of my GCSE Maths tutorial videos.

These videos cover a range of topics including finding the $n^{th}$ term of quadratic sequences, solving inequalities, transformations of graphs and many other topics. These videos have often been made with the intention of clarifying often overlooked aspects of GCSE Mathematics and sometimes extend slightly beyond the GCSE Maths specification but that’s not to say that the content of the videos is not understandable to GCSE Maths students – I am careful to explain (in language that a GCSE maths student can understand) WHY things work the way they do rather than just HOW to do something. So for example – WHY is the number in front of $n^{2}$ in the $n^{th}$ term of a quadratic sequence equal to half of the second difference of the sequence? The answer to this question is provided in my video on quadratic sequences. WHY does the transformation $f(x-a)$ represent a translation of $f(x)$ in the $x$-direction? You can find out by watching my videos on transformations of graphs.

I am uploading new videos every week to my Youtube channel and I encourage you to watch some of my videos if you need help with your GCSE maths. If there are any topics that you need help with and you would like me to make a video on it then please let me know and, IF I feel that I have something original to contribute rather than just repeating something that is already out there and been done-to-death by dozens of others, then I will make a video on that topic.

Foundations of Analysis by Edmund Landau is a great little book which I’ve mentioned before in a couple of my posts; Landau’s book gives a detailed account of the construction of the Real Numbers starting from the Peano Axioms.

It is tempting to take the real numbers for granted to a certain extent but one of the major developments in mathematics towards the end of the 19th Century was that the real numbers can be effectively built up from five basic axioms – the axioms for the natural numbers. These axioms, in a sense, capture the most essential properties of the natural numbers – the essence of the natural numbers – starting from these five axioms and about ninety pages later we arrive at a system of numbers that coincides with our intuitive notion of the real numbers.

The video link in this post is a link to a series of lectures that I have made covering the contents of Landau’s book up to the point where the construction of the real numbers is fully complete. The basic number system is the natural numbers which is extended into the rational numbers. The irrational numbers are added before, finally, defining the real numbers. How this all happens is remarkable (although, it wasn’t without controversy originally) and through reading Landau’s book you will never ever see the real numbers in the same way again.

Over the last couple of months I have finally got round to something that I have been wanting to do for a few years but have, until now, just never had the time to do – that is making some videos which give an introduction (albeit a rather detailed introduction) to some of my favourite historical works in mathematics and geometry.

One of the works that I have made a series of videos on is Apollonius of Perga – Treatise on Conic Sections. Apollonius was a geometer (sometimes called the Great Geometer) in Ancient Greece possibly around the time of Euclid (indeed he appears to have had some familiarity with Euclid’s work). This is one of my all-time favourite works in classical geometry; Apollonius is not nearly as well-known as Euclid but the influence that Apollonius has had on mathematicians and scientists such as Isaac Newton over the last 2000 years can’t be underestimated. Apollonius’ work is virtually unknown in modern mathematics and geometry but suffice it to say that without Apollonius’ work much of modern geometry may not have existed.

Apollonius’ Treatise is a collection of seven books (originally eight books but one is no longer extant and indeed, books 5, 6 and 7 do not exist in their original Ancient Greek) covering the theory of the conic sections, that is, of circles, ellipses, parabolae and hyperbolae. In a modern-day setting these would usually be dealt with using coordinate geometry; however, coordinate geometry wasn’t a thing at the time of Apollonius and so everything is dealt with in the Treatise using ratio, proportion and a technique, that is all but completely forgotten in modern geometry, called application of areas.

The reason that I wanted to make this series of videos (which covers the first two books of Apollonius) was to give people an idea about how geometry has been done in the past and to show what can be achieved with what would be considered quite primitive techniques nowadays. The techniques may be fairly rudimentary in and of themselves, but the ways in which Apollonius applies those techniques is anything but rudimentary – I would g so far as to say that very few modern mathematicians and geometers would be able to use these techniques with the confidence and dexterity that Apollonius uses them. Of course it would work the other way round as well – probably Apollonius would not be able to use modern-day techniques (possibly from beyond the grave) as well as modern-day geometers.

So why bother reading Apollonius? After all, there is nothing there that from a modern perspective is going to make you into a better geometer. The only real reason to read it is if you are interested in it for whatever reason; if you are interested in the historical development of mathematics and geometry. But even though you wouldn’t really be looking to use these techniques, you can get some insight into the creative mind that produced the work. You can start to see how these things have been visualised in the past. It’s very tempting nowadays to just think of a conic section as just an equation on the page – but in Apollonius, the conic sections are actual geometrical shapes which need to be visualised in order to appreciate their properties.

There are quite a few parts to my series of videos – I have tried to cover as much ground as possible without simply repeating everything in the books. The particular translation that I have used is Thomas Little Heath’s translation which dates from 1896. There are pros and cons to using this version but alas, that would be the case whichever version you used.

Cheerio for now!