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.

I think it’s fair to say that maths gets quite a lot of stick; for many, school maths lessons were lessons where you catch up on a bit of sleep or spend some time looking out of the window while the teacher was droning on about algebra and trigonometry. How could anyone derive any kind of pleasure from this form of legalised torture? It’s a good question and to be honest with you I don’t know the answer as to how people do enjoy it – but some people do.

The fact is – and I’m sorry to have to say this but I’m sure you’ll agree – that a lot (not all) of the maths that you may have learned while at school and preparing for your GCSE maths exam (or O Level maths exam) or even your SATS exams was boring. Fractions are boring; percentages are boring; ratios are boring – I don’t even try to hide the fact that these topics are boring. I know that as a maths tutor I’m supposed to be all enthusiastic about this stuff and trying to inspire you – but, frankly, this stuff is completely uninspiring. This is why I don’t tutor primary maths – I don’t feel able to make fractions fun and interesting and quite honestly it makes me cringe to think about trying to teach ‘funky fractions’ so I leave it to those who want to do it and let them get on with it however they see fit.

BUT….I am enthusiastic about maths and I do try to inspire my students when it comes to maths but not through fractions or percentages or ratios. You see, these boring topics are necessary to know about if you want to get to the interesting maths. Just like when you learn a musical instrument you have to learn all of the boring scales and chords and all that lot, and if you don’t know the boring stuff then you’ll never get to the interesting stuff – well the same applies with maths. And also, just like when you learn a musical instrument there most people will drop off before they get to the interesting stuff but there are some who, for whatever reason, persevere – then again, the same applies with maths. For those that drop out early on, maths will always remain dull and boring and it will always be bewildering to them why anyone would enjoy it – but for those who persevere then they get their reward eventually.

The reality is that it can, and does, take years to get to the really interesting stuff in maths. When you’re in the primary stages of learning maths at school then you might be told that all of this stuff that you’re learning about leads on to this or that; and that might sound interesting but, sadly, it’s a long way off. You’ll have to invest many years of learning before you get to it – some do but most don’t.

So what do I find interesting about maths? Well certainly not fractions, times-tables, reverse percentages (I still don’t know what the difference is between a percentage and a reverse percentage!) ratios or converting top-heavy fractions to mixed numbers – are you still awake? What I find interesting is learning about Algebraic Topology, Group Theory, Mathematical Logic, Number Theory…I could go on; I enjoy reading works by some of the greatest mathematicians (and philosophers) who have ever lived such as Bertrand Russell, Euclid, Georg Cantor, Hermann Weyl – it’s like seeing inside their minds; I like the fact that there are still a lot of unsolved problems in mathematics and the philosophy of mathematics; I like that there is always something to challenge myself with and that I might have to spend days, weeks or months learning to understand something but then being able to see the beauty of the subject first-hand.

Maths is a fascinating subject – whether you believe me or not is up to you – but I feel in my element when I’m reading about some of the mathematical theories that have been developed. Mathematicians aren’t creative? Pull the other one! If mathematician’s weren’t creative there wouldn’t be anything like Non-Euclidean geometry; Georg Cantor would never have been able to develop his theories on Transfinite Numbers if he was creatively barren. If you want to see these things for yourself then you have to have the drive at the outset to get through the boring stuff – believe me if you get through it then you will see for yourself why maths is so interesting! But I also say (and I don’t mean to sound sneering or that I’m trying to belittle people when I say this) that maths is not for everyone – it may be that if you find maths boring then maybe your interests and talents lie elsewhere and you absolutely should be investing your time elsewhere.

As I go through problems with my students and show them possible solutions, it might be that from time to time I solve a problem that someone isn’t familiar with, though it uses theory that they do know; at this point a common response is “How am I supposed to know to do that if I get a question like that in the exam?” This question really shows the misunderstanding that people have about what goes into producing a solution to a problem.

Unfortunately, maths education within the UK school system has become very sterile over the years – though steps have and are being taken to change this with the introduction of the new GCSE maths specification and the proposed new A Level maths specification to begin in the not too distant future. I’m speaking, here, from a point of view of having observed many GCSE and A Level maths students over the last half a decade or so and a big problem for them has been that they don’t understand that maths is about problem solving; many of them have never really had the opportunity in school to solve a problem.

But what about all of the textbooks that they have the and worksheets that they get? What about all the homework that they do? Well when you look at the homework sheets that are given out, or if you look at the homework ‘problems’ that students are set they often are very routine and uninspiring. What happens (in many cases) is this: the teacher introduces a topic and gives a few worked examples; the teacher then sets a bunch of questions that effectively follow the exact pattern of the wroked examples; then move on to the next topic Sadly, this is not problem solving by any stretch of the imagination. This is what I call ‘pub-quiz maths’ – either you know the answer and can parrot it out or you don’t know the answer and you hit a brick wall.

The thing is with maths is that, even though you might not know the final answer to a problem or the method of solution, you can figure it out. You can use what you DO know to find out answers to what you DON’T know. You can’t do this in a pub-quiz: either you know what the highest mountain in Europe is or you don’t. With maths you can use what you know to come up with original solutions to problems that are not necessarily anything like you’ve come across before; and many schools will stop short of this. Many teachers will try to analyse past exam papers and distill out what the most common ‘types’ of problem are and try to show their students model solutions for each ‘type’ of problem. But then when these students are faced with an unfamiliar situation (and it might only be very slightly different to what they are familiar with) they haven’t got the problem solving experience to modify and apply their knowledge and they say, “I’ve never been shown how to do this.” Well that’s the point – you have to find a way of solving it yourself.

My tutoring style concentrates more on this problem-solving aspect of mathematics. It’s hard work for the students and for me but I can’t, and don’t, shy away from putting my students in situations where they have to apply their knowledge and solve problems themselves. This is what maths is about and this is what makes maths interesting and exciting. For me to take away that opportunity from my students would be disgraceful and a complete disservice to my students.

So the answer to the question, “How am I supposed to know to do that if I get a question like that in the exam?” is this; get the problem-solving experience that you need to solve problems confidently and independently. I can teach the knowledge and encourage problem-solving skill development but I can’t teach experience – you have to get that yourself and you’ll get it through practice. You have to solve problems to learn to solve problems and then eventually you become flexible enough with your thinking to apply knowledge very naturally in quite creative ways. When you learn to do this THEN you are doing maths.

Over the last few weeks I’ve been spending some time learning some speed-cubing techniques on the Rubik’s Cube. I’ve been able to solve the cube for about ten years now using the Petrus method and I’ve never really bothered to go any further than just a basic solution – I’ve never been inclined to do so until now.

Anyway – as I learn about speed-cubing there are some similarities that I notice between how to go about learning to speed-cube and how to learn mathematics.

When learning to speed-cube you have to start very slowly – this seems completely counter-intuitive to what you’re supposed to be doing; going fast. You have to have a solid foundation to build on and there are some basics that need to be etched on to your brain so much so that you don’t have to consciously think about them; things like learning the colour-scheme of your cube and the relative positions of the colours (very important). To start with these are a pain in the arse; you mess things up all the time; you struggle to visualise things from different angles and you struggle even more to think ahead. But with lots of practice you eventually start to account for these things without even thinking about them; with practice the basics become automatic and you can direct more of your energy planning ahead with you solution of the cube. It takes time but slowly you make progress – if you don’t get sufficient practice you don’t develop the automaticity that you really need and all of your energy is wasted on doing the basic things leaving nothing in the tank for more advanced things.

And then I realised – this is exactly the problem that many GCSE and A Level maths students have; they have very weak foundations that they’re trying to build on. They might be trying to solve some integration problem for their A Level maths homework but they’re wasting all of their energy trying to remember how to multiply fractions together! Why? Because they haven’t had sufficient practice.

I admit that fractions are not the most interesting thing to learn about; HOWEVER, if you’re not able to automatically add or multiply fractions correctly then you’ve got a problem. I think that this comes about because it is very unfashionable in schools to drill children in the basics nowadays (and has been for a long time) and this is very detrimental to progress further down the line. I know that it’s boring and a very Victorian way of going about things but do you know what – it works! It simply is not enough to show children the idea behind adding and multiplying fractions (or whatever basic skill they’re learning) and expecting that that’s all that is needed – they need drilling until they can just do it automatically otherwise they will never be able to confidently and independently tackle more complicated problems.

I see this in many GCSE and A Level maths students – their knowledge of the foundational skills is so shaky that they can’t get very far into the solution of a problem before they run out of energy. I have to be very firm with my students and make it clear to them that they will struggle to solve problems unless they get completely familiar with the basics; of course I help them here as much as I can but there does come a point where I have to insist on them taking charge of their own destiny – I can teach them how to do basic skills but they themselves have to get the experience and practice because you can’t teach experience.

When I’m solving A Level maths problems (and even back when I was an A Level maths student myself) I was able to think ahead when solving problems and almost see the whole form of the solution in my head before I’d even put pen to paper. Why? Because I wasn’t having to waste energy faffing around trying to remember whether $x \times x$ is $2x$ or $x^{2}$ or whether I need a common denominator or not when multiplying fractions. I know that some of my A Level students think I’m being funny when I say that they need to work on their fractions but I’m serious – if you don’t go slow to start with and learn the basics well then you can almost forget about speeding things up or moving on to more advanced things because you will always be several steps behind those who have solid foundations.

A really useful skill to develop when learning mathematics is graph sketching; although some of the GCSE and A Level textbooks do have a short chapter covering this somewhere amongst the pages, unfortunately the value of graph sketching is, for the most part, almost invariably overlooked. Students think that this is something that they only need to do when they are asked to do it and even then it’s clear in many cases that they don’t really understand what it is that they are doing.

An equation such as $y=x^2-5x+7$ gives a clear, unambiguous, relationship between $x$ and $y$. So what? Well this means that for each $x$ value that I put into the right hand side of the equation I will get a corresponding y value. So for example if I choose $x=3$ (there is nothing special about this choice of value for x; I could just as easily have chosen $x=5.87$ or $x=-29$), I simply replace any $x$’s on the right hand side with $3$ to get $y=3^2-5×3+7$ and so $y=9-15+7$ and therefore $y=1$.

For many GCSE and A Level students this is fairly straightforward – but they miss the important part. This $x$ value and its corresponding $y$ value form a pair of coordinates, in this case $(3,1)$ – remember that the $x$ coordinate is the first number, $3$, and the $y$ coordinate is the second number, $1$.

Again – SO WHAT?

Well if I do this for several x values then I get a few more corresponding y values and so more pairs of coordinates which I can start to plot on some $xy$-axes. The more $xy$ pairs that I find the more points I can plot; what emerges is the graph of this equation. In many cases you don’t even need to plot a great deal of points to figure out what the shape of the graph is going to look like; many equations can be grouped together into families of equations – equations that have certain similarities – and the graphs of these families of equations (quadratic, exponential, logarithmic) have characteristic shapes and behaviours.

This is important – the equation and the graph represent the same thing but in different ways; the equation determines what the graph will look like and the graph will (at least in theory) determine the equation. If I choose any point on the graph and study the coordinates of that point I will find that the relationship between the $x$ and $y$ coordinates, for all points, is the relationship described by the equation and, moreover, any $x$ and $y$ values that are related via the equation WILL, without exception, be the coordinates of a point on the graph of that equation. So now, rather than seeing an equation as a string of miscellaneous symbols on a page I can start to draw (in some cases of my own volition) the graph of an equation and from this I can see (in the literal sense) the behaviour of the equation at a glance. The equation suddenly becomes more than an equation – it has more character to it. Drawing the graph of an equation might seem like more work – why would you inflict that on yourself? But that isn’t the case. By knowing what the graph of an equation looks like you get a visualisation of the situation and this usually means that overall you have don’t have to work anywhere near as hard to get your solutions.

Sadly, many GCSE and A Level maths students never get to grips with this. Not because they CAN’T draw graphs but because it is never made clear to them that this relationship between equation and graph exists. They go on thinking that in order to draw the graph of an equation they have to have been told beforehand what the graph of that particular equation looks like. This is a disaster for them! I often ask my students to sketch the graph of an equation and to start with they may look at me blankly or say, “I’ve never been told what that looks like”. My response might usually be something like, “Well think about what the equation is telling you is going to happen for different values of $x$”. This might not hit home straight away but eventually it does in most cases and, finally, they understand that the graph (even of unfamiliar equations) can be figured out themselves by thinking about what the equation is doing at various (well chosen) values of $x$.

Once you get your head around this then you’ll find that maths becomes ten times easier.

I just want to talk about something that really holds a lot of people back when it comes to learning maths; something that stops (too) many people from achieving what they can truly achieve – the fear of failure.

The fear of failure and of making mistakes is a very common issue amongst maths students (though, I’m certain that it’s also the case in other subjects as well). Failure (or at least what is perceived to be failure in someones own mind) and mistakes are always seen as a bad thing. Sometimes failures and mistakes are bad things because the consequences may be very severe – but when you’re learning GCSE or A Level maths, the good thing is that no-one dies if you fail or make a mistake; there is no catastrophic nuclear accident; the world continues to turn and the sun continues to rise and set.

Mistakes are a necessary part of learning and doing mathematics at all levels; and the possibility that you might fail to solve a problem is something that you have to learn to accept as this will ALWAYS be the case even if you happen to be called Albert Einstein (OK, he wasn’t a mathematician but you get my drift).

I know that it feels good when you get the correct answer to something and you succeed in writing out a correct solution – but if you only continue to do what you know how to do then you will only ever be able to do what you know how to do; in other words you will stop learning. The possibility of failure is something that I very often have to work very hard to get my students to accept. Sometimes, and this is particularly true of people who have done well with GCSE maths and moved on to A Level maths, students are so used to getting things right first time and without any difficulties, once they’re presented with an actual problem that they have to solve they will give up. The thing is, it’s not that they can’t solve the problem because in many cases they can, but just that they think that if they make a mistake that they have failed miserably and that only ridicule and embarrassment will follow. They are so afraid of making a mistake that they won’t even make an attempt – they just say, “I haven’t been shown how to do that”, or simply “I don’t know what to do”.

As a mathematician (or student of mathematics) you have to be willing to make an attempt at applying your knowledge and accept that you may (and in all likelihood, will) make mistakes; you may even be unsuccessful in finding a solution for some time and you may NEVER find a solution. Have you failed in these cases? No – not at all. You have failed if you make no attempt at a solution; you have failed if you’re not prepared to make the mistakes that you need to make in order to learn.

Sometimes I have to put my students in situations where they HAVE to make an attempt at solving an unfamiliar problem and where they might make many mistakes along the way; I will not give any clues about what to do until a good attempt has been made by themselves to find a solution. Cruel of me? Not really as this is what they’ll have to do in the A Level or new GCSE maths exams so it would be cruel and wrong of me to not do this and only have them do things that they can easily do. By doing this I’m not saying that I expect them to find a full solution and I even say to them that I’m not bothered if they don’t find a solution; what I do expect is that they put themselves out there and try things out; I AM bothered if they don’t make any attempt even if they claim that they don’t have any ideas. I have to get them to understand that mistakes are a natural part of learning and doing mathematics; that no-one is going to laugh at them if they make mistakes; that their otherwise impeccable reputation will remain intact despite their mistakes; and that they will learn more from making these mistakes than by not making them.

I can see their faces contorting when they first write something that they know might be wrong – but this is what doing mathematics is about; trying things out until you find something that works. As a mathematician you’re not expected to be able to blurt out the correct answer or solution immediately upon being presented with a problem. What you are expected to do is to work to find a solution and this means making umpteen false-starts in the process. Once my students accept this fact and free themselves from the fear of failure (whether real or perceived) then I know that they’ve taken a huge step in the right direction.

 

 

Here’s a nice long video for you – I realised that the A Level exams are just around the corner for 2017 and the panic-frenzy will be really kicking in soon. But ther’s still time to get well prepared for your exams if you start your revision and everything now.

This is a video that I made to point out some of the bear-traps that people very commonly fall into when studing for their A Level Maths and Further Maths and when preparing for their exams. All of the points that I make in this video are based on my personal experiences as a maths tutor over the last several years – they are mistakes that I see people make time and time again, year in and year out and they are mistakes that could cost you quite dearly if you persist with them either knowingly or unknowingly.

Obviously I can’t cover every single eventuality but I’ve tried to focus on the main things that people do wrong. Equally obviously, there is no magic wand that I can wave to make everything better and to guarantee the result that you want. Whether you get the result that you want is entirely down to your own level of work and your own attitude but I hope that the points that I make in this video will point you in the right direction at least.

If this video was useful to you then you might also want to watch my other A Level Maths videos – one is for A Level Maths and Further Maths and the other video is for those taking the STEP Papers.

 

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.