In my opinion skill-automation is the key to accessing higher-level problem-solving skills in mathematics (and possibly other fields of work as well). But what do I mean by skill-automation?

I don’t know if skill-automation is an officially recognised term but I will explain what I mean when I use this term; a skill has been automated when it can be used and applied with very little, if any, conscious thought and effort. Some skills can be automated quite easily, others require much more effort to automate and there may be yet other skills that can’t be automated.

Automating skills is essential in mathematics because it means that you are able to focus your mental energies towards the more intricate aspects of a problem without getting bogged down with trivial aspects. For example, addition of fractions is a skill that can be automated quite easily (and I don’t mean that it can be just done on an electronic calculator or similar device) and it might seem like a very small and insignificant skill and yet, a failure to automate this skill means that whenever you need to add fractions together as part of a problem you have to strain to either remember the way that you’ve been shown or to try and figure it out from scratch. Both of these are a huge drain on mental energy – it’s like having to think about every single step that you take while walking or having to figure out how to walk every time you want to take a step.

The lack of automation of certain skills is something that often frustrates my students’ attempts to solve problems. Some skills that are often not automated to a sufficiently high degree are

  • Addition, multiplication, subtraction and division of whole numbers and fractions
  • Simple algebraic manipulation and re-arranging equations and formulae and solving linear equations
  • Expanding brackets, collecting like-terms and simplification of expressions
  • Pythagoras’ Theorem and trigonometry in right-angled triangles
  • Recognising and solving quadratic equations

Of course there are other skills as well. The point is that once these are automated (which doesn’t take that long) you don’t have to worry about them and you can concentrate on the more challenging aspects of a problem.

So how do you automate a skill? Well it can happen in all kinds of ways – all of which take time to a certain degree – but the main way is through practice and initially, conscious and directed effort. At first the skill will be far from automated and you will have to think very carefully about what you’re doing (think about when a child learns to walk) but through repeated exposure to situations where the skill is required then, if you are paying attention and concentrating, you will find that you start to spot certain patterns – sometimes these patterns may be very difficult to express in words but you acknowledge them all the same. You will start to take advantage of these patterns subconsciously and through necessity as problems become more challenging – the skill is starting to be automated. Continual exposure to challenging problems will cause you to see more patterns which will often be subconsciously incorporated into your problem-solving. Of course there may be a limit to this – a skill may only need to be automated to a certain degree; the law of diminishing returns will kick in so it will make further automation very time-consuming but possibly unnecessary anyway.

The key here is practice – you cannot automate a skill without a great deal of practice and this is often why many of my students have failed to automate these skills – they simply haven’t practiced enough. They complain that they struggle to add fractions together and yet they refuse to practice this skill. Would you expect to learn to play the piano without bothering with the practice? No, it would be absurd to even think that it could be done. Admittedly, adding fractions is not very exciting (and I’m not even going to patronise you by claiming that it is!) but it IS an essential skill to automate if you want to get on to solving more challenging problems; the same applies for other skills. Learning scales is maybe not a very exciting aspect of learning to play the piano – but I would guess that any piano teacher would insist on scales being learned (and automated) otherwise higher-level piano playing becomes a practical impossibility.

So if you want to get on to solving more interesting and challenging problems in mathematics then you’re going to have to get these basic skills sorted first – there’s no way around it I’m afraid!

Over the last few months I have been working hard to produce some videos that I have uploaded to Youtube covering several topics on the A-Level Maths and Further Maths courses. The video at the start of this post will take you to a Youtube playlist of my A Level Further Maths tutorial videos, and through my Youtube channel you can view all of my A-Level Maths tutorial videos (and many more videos besides).

In my videos I usually try to address specific problems which, through my years of tuition, I have noticed that people often have. In other words, the videos that I make are not just ‘going through’ a particular topic in the same way that they are covered in the classroom or in the typical textbook; I make a real effort to emphasize the specific issues that people get stuck with. I don’t really make videos (at least not any more) that just ‘go through’ a topic – if I don’t feel that I have anything positive and fairly original to add to the noise that already exists then I just don’t make the video.

I am always keen to know what problems people have with their maths studies – there’s always new issues that crop up that, even after all these years of tuition. I want to help people come to an understanding of these problems so I can help people overcome these problems. If there is anything that you would like me to make a video on then please let me know – if (and I emphasize the ‘if’) it is something that I can make an original contribution to then I will be more than happy to make a video on it for you – just let me know. On the other hand, if you just want to ask a question about anything that is presenting a problem for you with your maths studies then please get in touch and I will be very happy to help.

I would be very grateful if, after watching my videos on Youtube, you would like and share my videos and subscribe to my channel – this really helps my videos get viewed by more people and gives me a bit of motivation to keep on keeping on making more videos.

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!

I’m going to make a somewhat counter-intuitive claim about how to go about preparing for exams (or tests) which is – in order to be really well prepared for exams you need to almost completely forget about preparing for exams (or tests).

This sounds ridiculous doesn’t it? But I’ll try to explain why I think this.

Exams and tests are a completely unnatural thing when it comes to learning. The anticipation of an upcoming test or exam can cause unnecessary stress and, in some cases, outright panic neither of which are any good for learning. The anticipation of exams and tests causes people to learn at a rate that is a fraction of the rate at which they are capable of learning because they will only ‘learn’ what they expect to be examined on – repetitively going over the same few, scanty bits of information trying to cram some poorly-understood and seemingly non-sensical facts into their heads – rather than challenging themselves and delving deeper into a subject to get a much greater understanding.

And of course, why bother to learn things if you won’t be examined on them? Who in their right mind would ‘waste time’ learning things that they won’t be examined on? Well that’s exactly the kind of viewpoint you will probably take if you see education and learning as nothing more than a series of exams and tests. And no wonder people with this attitude hate learning! So would I if that’s all learning meant to me.

But isn’t learning about more than just exams? Yes, I know that exams are an important part of your life if you want to go to university or become a chartered accountant etc. so I’m not saying that exams can be completely ignored – what I AM saying is that preparation for exams should not be the main focus of your learning. If you really want to learn more, learn faster and learn better then you need to be learning because you love learning about what you’re learning about and have a genuine desire to want to go further and acquire more knowledge simply because you can.

Does it sound weird to take such an approach to learning? Of course it does because no-one really encourages people to learn in this way in school. There’s constant testing week-in and week-out and so people only learn the bits of information that they need to know for the test, and as long as they can vomit up these bits of information in the test, regardless of whether they understand (which in many cases people know but don’t understand), then they get a good mark; a couple of days later and they’ve forgotten everything that they learned, rinse and repeat! And this approach is considered to be a good learning approach!!!!

If you are genuinely interested in what you’re learning and you’re learning because you love learning, then learning is easy and effortless. A depth of understanding will come to you that will never come if you see your learning as just learning to pass a constant stream of tests and exams. If you learn because you love to learn then you will pass the tests and exams with flying colours because you will be streets ahead of the rest of the crowd – you will learn at a rate that is unimaginable if you’re learning to just pass an exam and, moreover, what you learn will be much more permanent.

When I was learning GCSE maths I was learning to pass an exam – and it was hard, hard, hard! I hated it. When I was learning A-Level maths I was learning because I was excited about what I was learning – I learned way more than I needed to pass the A-Level exams and I didn’t even think about the exams until a couple of weeks before I sat them, at which point I realised, I didn’t need to do any revision specifically for my exams because I’d been doing it all along without even realising it! The outcome was that the exams were a piece of cake!

A few months ago, I wrote a couple of posts on my website mentioning that I was learning the Polish language. I still am learning Polish and enjoying it so much more than I first imagined that I would.

Even though I don’t subscribe to a particular fixed learning style or methodology I do like to find out how other people go about doing things that I want to be able to do – some people will do things that I really like and that I can incorporate into my learning technique and other people will go about things in a way that is the complete opposite to how I would like to learn and so, even though the technique may work for them, I may choose not to follow their example.

A name that came up during my reading and research was Stephen Krashen – Krashen is (according to Google) a linguist and educational researcher. I decided to find out more about his work and I’m so glad that I did!! In fact, my only regret is that I never came across his work before.

Much of Krashen’s work is focused on language learning and language acquisition (these two concepts are distinct) which, obviously, makes it highly relevant to my quest to learn Polish – but I believe that a lot of what he says regarding how language is acquired is applicable, at least in part, to learning mathematics.

Krashen says that for language acquisition (as opposed to language learning) to take place then the individual must receive large quantities of input that is both comprehensible and interesting (he even goes so far as to say that the input should be compelling and not merely interesting). Language acquisition is viewed as being the more permanent knowledge and makes the greatest contribution to fluency in a language; it may in many cases be a completely unconscious and effortless process. Language learning is stuff like grammar drills and vocab lists which Krashen says is almost completely worthless and unnecessary in the face of language acquisition.

Is it true that large quantities of comprehensible and interesting (if not compelling) input is required for mathematics to be ‘acquired’? looking back over my personal experiences of learning mathematics then I would have to say a huge yes in response to that question. In fact, I think it confirms some things that I often recommend to my students (and which usually get ignored) such as

  • Read around the subject
  • Use a range of different resources
  • Learn because you want to and not because you have to and not just to pass an exam

If you want to read some of Krashen’s work then you can download some of his older books and some of his journal articles from his own website for free at www.sdkrashen.com. Some of his more recent books you will have to buy but I highly recommend even if language learning and acquisition is not your thing then it’s still a real eye-opener.

Some of Krashen’s books that I’ve read over the last couple of months are

  • Principles and Practice in Second Language Acquisition
  • Language Two (written with two other authors)
  • Explorations in Language Acquisition and Use

Obviously, learning a language and learning mathematics are not the same thing and I’m not saying that everything that Krashen says regarding language learning and acquisition is applicable to mathematics but I think there is a lot of stuff in there that is very relevant and can be very useful to mathematics learners.

As I mentioned in my previous post, last year I started to (seriously) learn Polish. So how are things coming along?

Of course the first thing to say is that learning Polish is not easy – but then, I’ve never heard anyone say that any language is easy to learn so it’s really no surprise to learn that Polish is no different. Since the beginning of August 2017 I have spent hundreds of hours on the language – some of these hours are very concentrated and intense, others are a bit more laid back – sometimes I will read quite intensively and other times I will read more extensively but the main thing is that I am enjoying the learning process. I’m not bothered about if I have the most efficient langauge learning method – in fact I want to have the freedom to experiment and to find my own way of learning. That’s not to say that I don’t take an interest in how other people have, and do, learn languages – it’s just that, part of the fun for me comes from the experimentation.

Of course at this point I have to draw a parallel with how I went about learning maths (that thing about drawing a parallel wasn’t meant to be a pun by the way) when I was doing my A Levels etc. I didn’t have a specific study technique (that I knew of), I wasn’t doing it because I felt forced to do it against my will – I learned about it because I loved every second that I spent learning about it; I enjoyed working through the frustrations and every little discovery that I made was thrilling – it was brilliant. And now this is how I feel about learning Polish – it can be frustrating, but I enjoy ploughing through those frustrations.

At present I’m not really in a position to judge the level of my Polish. It could be beginner, it could be intermediate – I’m not really able to make that judgement. Possibly I am low intermediate at a push – I’m planning to take the B1 Level Polish State Certificate Examination in May of this year – B1 Level is considered to be low intermediate and so, I suppose if I pass the exam then that will be objective evidence of my level but I’ll wait until that result comes through.

I spend most of my time reading the language – this is for a few reasons. The first reason is that I can read more often than anything else – if I stop off for a coffee I can just pull a book out of my bag and get on with it. The second reason that I read more is because I like reading – I read a lot in English so it makes sense for me to do this in Polish if that’s what I enjoy doing. The third reason is that I like the theories of Stephen Krashen which says that extensive reading is a very effective way of acquiring a langauge. I don’t understand everything that I read in Polish – sometimes it is very difficult – but of course, as with anything, with practice it gets easier. I try to read authentic material that is intended to be read by native speakers of Polish if I can – obviously right at the very beginning when I started learning this wasn’t really possible but it’s surprising how quickly you can make progress in that direction. It helps, also, to try to read things that you have some familiarity with – for example, the first full book that I read in Polish was the Polish translation of A Short History of Time by Stephen Hawking (Krotka Historia Czasu). Of course this wasn’t an easy book to read but I was able to follow it quite well since I was familiar with a lot of the material. On the tother hand, fiction is much more difficult for me to read because there are so many more literary words whose meanings are not clear for me just yet.

Overall things are going well with my quest to learn Polish – I like that I am making progress but I also like that I have lots to learn. It’s a challenge and it’s a challenge that I’m happy to undertake.

Over the last hal year or so I haven’t really been doing much with my website – no new posts, only a couple of new videos. What’s been going on?

Well the good news is that everything is (and has been) perfectly fine and I’m still doing maths tuition – the reason for my absence has been more to do with getting myself interested in a new (old) hobby. As much as you love doing something, sometimes it’s always nice just to have a bit of a break and not be quite as full-on as you may have been.

So what is this new hobby that I’ve been getting into? Well I’ve been learning a new language – Polish! I started around the end of July/beginning of August last year (2017) and I’ve been really enjoying it. Arguably, this is not the first time that I have dabbled with foreign languages, including Polish; like most people I studied a language at school – French – and also like most people, I hated it. I hated it so much! I couldn’t understand why anyone would put themselves through the torture of learning French in a classroom. It was so dull, so drab and so painful. I hated having to do role-playing where I had to pretend to be a tourist asking ‘ou est la gare’ is or how to say ‘ j’aime aller a la plage’.

After leaving school in 2002 and somehow managing to get a C in my GCSE French (they must have awarded me a few pity-marks, I’m certain of it) I never even thought about learning a language – I had no reason to think about it. In around 2006, though, something changed. I don’t know what it was but I felt compelled to learn French – this time to do it on my own terms. I learned how to read farily well in French, I had regular written conversations (at the time when MSN messanger was popular) with French people, I had Skype conversations in French, I learned the grammar and I loved it – because I was now doing it because I wanted to do it. I never got to a fluent level but I could get by.

A few years later I decided to start learning Polish – why Polish? Why not? This lasted for a few months but I eventually had to set it aside for a while as there were lots of other things going on at the time which meant my time was getting spread a bit to thin. Once again I went back to not even thinking about languages – until last year. I decided that now was the right time for me to do this – to learn a language and to see it done properly. Even though it would probably have been easier for me to start back learning French I felt more drawn to Polish – so I chose Polish.

So now, I want to make my goals for 2018 official – they are

  • Sit (and pass) the Polish State Certificate Exams, B1 Level, in May 2018 (already registered for this!).
  • Sit (and pass) the Polish State Certificate Exams, B2 Level, in November 2018.

For me, I hated learning languages in a classroom. I much prefer to learn the language on my own terms; I enjoy learning grammar when I can do it at my own pace; I enjoy reading authentic material in the language and not textbook-style role-play conversations; I enjoy having authentic conversations in the language (you don’t learn how to talk about which kind of radiator you want in one of the bedrooms in Polish textbooks but these kinds of things come up in real life). As it stands I don’t have any plans to learn other languages just yet – but who knows, once I’ve got to a decent level in Polish I may turn my attention to other languages as well. For the time-being let’s go one step at a time.