What is the purpose of mental arithmetic in the modern world? In what way does it benefit someone to be able to do mental calculations considering that a calculator can be bought for about the same price as a bar of chocolate – if not less.

In the past when schoolchildren have questioned the use of mental arithmetic, teachers were at least able to say “Well you won’t always have a calculator handy”. Well I think that’s just not true at all now – everyone has a calculator practically all of the time in their pocket, although it’s usually known now as a phone rather than a calculator. So what do you say to schoolchildren now?

Unfortunately, at least for mental arithmetic, there isn’t really anything that’s going to convince schoolchildren now of the use of mental arithmetic. Some will happily learn mental arithmetic but the vast majority won’t. That’s not to say that being able to do mental arithmetic isn’t useful, it’s just not as obvious how it’s useful. Considering how little time is spent on mental arithmetic in schools and the little mental arithmetic abilities that many students have now it certainly seems that teachers have just about given up on this one.

Maybe they’re right. Maybe there is no need for mental arithmetic. Pocket calculators are faster than humans, more efficient, less prone to error – why bother learning the times tables? Isn’t it just a complete waste of time? Well…yes and no.

When you consider how abundant electronic calculators are in the modern day then mental arithmetic really is mostly a waste of time for doing calculations. With a calculator it’s a cinch to multiply huge numbers together; I can tap into a calculator a list of numbers and it will do all kinds of statistical analysis for me on those numbers – whereas once upon a time it would have taken a good five minutes to manually calculate the variance of a list of numbers, now my calculator does it in less than a second. Why should anyone even bother trying to compete with that? Maybe at some time in past decades computers (of the human kind) were important – people who could do calculations quickly were essential before pocket calculators came along. I don’t know whether being a ‘calculator’ was a particularly lucrative job to have but it was a job. Which company or organisation would be crazy enough to employ someone to sit there doing manual calculations nowadays?

However, letting an electronic calculator do the work for you all the time has some downsides. Just like using a car to always get you from A to B is far easier and quicker than walking or running, you end up getting lazy and out of shape (at least physically). Similarly, using an electronic calculator all the time means that you get lazy and out of shape (at least mentally). I come across many maths students right up to A Level who struggle with mental arithmetic – and it shows. It’s not because they’re incapable it’s just that they’ve never been discouraged from using a calculator – it saved classroom time in the past but now it comes and bites them on the backside.

As it turns out – not being able to do simple mental arithmetic such as addition and subtraction of two-digit numbers and not knowing your times tables means that you lack, to a certain degree, understanding of how numbers work so you don’t have any real idea how to start extending those ideas to algebraic problems where an electronic calculator might not be able to help (though, you can buy calculators that deal with algebra now but they’re a bit more pricey).You also end up using all of your energies trying to figure out really simple things like 12×15 in the middle of a complex algebraic problem and then you don’t have any energy left to do the more complicated parts of a problem; the result is a very bumpy stop-start, stop-start solution to a problem which can be compared to driving a car and slamming on the brakes every fifty yards – you might get where you want to be but with much more wasted effort.

I don’t think it’s necessary to be able to do huge mental calculations; there are people around today who can carry out huge mental calculations. They do it either because it’s something that they can just do or because mental arithmetic is something that they enjoy doing and have spent many hours learning how numbers work in great detail. However, I can’t see that mental arithmetic will be making a comeback in the classroom any time soon – I think that maths students are impoverished because of it’s absence and lack of mental arithmetic skills makes their lives more difficult further down the line if they decide to go on to do higher level maths; but even if they don’t want to do higher level maths they will still lack the confidence of dealing with numbers in a world where numbers and statistics are everywhere you look. Sadly, I think that mental arithmetic is a dead skill.

A couple of weeks ago I wrote a post about the quickest way to learn maths – something that I could have added to that post is this; you have to enjoy what you’re doing and what you’re learning about.

So what’s Steve Vai got to do with anything? Well Steve Vai is my favourite musician, most famous for his guitar playing abilities. I don’t consider myself a musician – I can play a few tunes on a guitar (not Steve’s stuff) but I don’t understand music in any appreciable depth. However, Steve Vai is someone who has inspired me as a person for many years since I first started listening to his music around 17 or 18 years ago. I was looking through his Youtube videos last week and I came across this one; it’s an interview with Steve where he’s talking about his famous 10 hour workout that he used to undertake when learning the guitar. What really made my ears prick up is what he says at around 5.30 onwards – that his 10 hour workout was really much more than that because he was thinking about music and the guitar all the time; because he loved what he was doing it didn’t feel like work.

This took me back to when I was doing my A Level maths because this is exactly what was happening in my mind as I was doing my A Level maths. Maths wasn’t something that I just did in lessons or when I decided to get my books out to do a bit of homework or revise for an exam; maths was something that I was almost always thinking about. Because I loved the subject so much I was happy to¬†wake up thinking about maths and fall asleep at night thinking about maths and to be thinking about maths as much as possible in between. It didn’t feel like work to me – I was thinking about problems all day long; it excited me to solve problems and to learn more about the subject. I wasn’t happy to just stick to the basic A Level maths content – that wasn’t enough for me; because I was thinking about the subject all the time I could get through the basic A Level content very quickly. It never felt like I was doing work. And when I wasn’t doing maths I as wanting to be doing maths. If I’m honest – I found A Level maths fairly straightforward. Please don’t take that as me trying to be big-headed or that I see myself as some intellectually superior being; I found it straightforward because of how I felt about it. It would probably have been a very different outcome had I not enjoyed maths to the extent that I did.

The point I’m trying to make is this; if you enjoy what you do to such a level then it’s so much easier to learn and so much faster to learn. I’m not saying that everyone has to be obsessive about maths because everyone has theor own interests – what I am saying is that whatever you choose to do you have to have a drive to do it and you have to love what you’re doing if you want rapid progress. There will still be times when you get stuck, when you get frustrated and pissed off, you still have to put the time and effort in but if it is something that you genuinely love doing then you can easily get past all that and carry on.

I think Steve puts this into words very well in his video and even though he’s talking about learning music or guitar, I think what he says applies to whatever you do. If you don’t love what you do then you’re always going to be at a disadvantage to someone who does love doing it.

Here’s a link to SteveVaiHimself Youtube Channel and the SteveVaiVEVO Youtube Channel. You can also visit his website at www.vai.com

 

It’s getting ever closer to that time of year again when the revision and cramming commences for GCSE and A Level Maths exams. It’s usually around April time that people start running around like headless chickens because they’ve suddenly realised just how much they need to do to prepare for their GCSE or A Level maths exams….in a month.

But why exactly is revision so stressful? Well, I think first of all it’s because what many people are doing is not revision; I’ve just looked up the word revise in the dictionary, just to make sure that I wasn’t mistaken on it’s meaning, and as a transitive verb, revise means to examine and correct; to make anew, improved version of; to study anew; to look at again. So, revision is, I guess, looking over something that has already been learned at some point to make sure that it is still well understood or to see if there are any corrections or additions to be made. However – when it comes to ‘revision’ for exams, I’ve found that many people are, essentially, learning for the first time. So instead of taking a whole year to learn something gradually, little by little, and then revise that knowledge in the last few weeks before the exam a lot of people fall in to the trap of not bothering to do the learning little by little over a period of time but leave everything until the last few weeks and cramming as much as they can into their heads in a short period of time. This is not revision – this is learning something for the first time. Revision and cramming are not the same and it is the latter that causes the stress.

Revision, if done properly, doesn’t need to be excessively stressful. There will always be a certain amount of stress around exam periods but this stress can be minimised by using your time earlier in the year more wisely. Once you’ve learned something then your revision of what you have learned needs to begin straight away, even if that something that you learn is in September and your exam isn’t until June the next year. During my A Level studies I didn’t cram for a single one of my exams whether maths or otherwise; on the other hand I revised constantly throughout the year. This meant that when it was getting close to exam time my revision was relatively straightforward and stress-free compared to the rough time that some of my less organised peers were having; by starting my revision early in the academic year all of the topics had had time to be absorbed and understood – something that cramming can never achieve. This also meant that I was able to go into exams knowing that I would do well; a very nice feeling to have. The end result was that I did do very well in my A Level exams not because I had any extraordinary talents or was intellectually gifted but because I was disciplined about my learning and revision. (In a way it could be said that I was cramming all year, particularly for my maths exams, because I was keen to learn as much as I could about the subject that I was reading off-syllabus about all sorts of things. I encourage you to do the same…it really helps.)

So revision and cramming are not synonymous. Know the difference between the two. Cramming piles on the pressure and doesn’t give you enough chance to take in all of the knowledge or to look at things from several different angles and see the many different ways in which things might be applied. Revision is much more easy going once you get into a routine – but you have to be disciplined and start your revision early in the year. I’m writing this in early January 2017 which means that most GCSE and A Level maths exams will be in four or five months – if you haven’t started your revision yet then I strongly suggest that you start NOW!

 

Here is my monthly review video for December 2016. It was a fairly quiet month in terms of tuition because of Christmas and New Year; not much maths-y stuff going on but there was still plenty happening in other areas.

So in this video I talk about my new toy – a Moyu Hualong speedcube. I love this thing; I wouldn’t call myself a speedcuber but this is definitely giving me the nudge to learn to be one. I’ve always been a bit reluctant to go down the route of learning to be a speedcuber; the reason being that I didn’t think I was up to it because whenever I tried to speed things up with my Rubik’s Cube solving I could get to a certain point and then I would hit a brick wall at just under 1 minute. It turns out that there is a huge difference between a bog-standard Rubik’s Cube and a speedcube and I’ll show you in this video what those differences are and how they genuinely make a difference.

I’ll also talk a little bit about the books that I got round to reading which were

  • Born on a Blue Day – Daniel Tammet
  • The Real Rain Man Kim Peek – Fran Peek
  • 1984 – George Orwell

I’ll be doing another video next month and I think that I’ll have quite a bit to talk about in a month’s time because I’ve got a few projects that I’m working on so hopefully I’ll be able to give a bit of insight into what I’m up to. For the time being, though – Happy New Year and I’ll see you in next month’s video.

 

I’m going to let you in on a big secret about how to get good at mathematics. You might not learn this in school but there IS a quick way of learning mathematics; it’s been known for hundreds of years but seldom talked about and it’s this – hard work.

If that’s not really what you wanted to hear and feel disgusted that I would say such a thing then please do not continue reading this post as I don’t want to waste any more of your time; if you would like to know what my reasons are for believing that hard work is THE QUICKEST way of learning mathematics then please read on…

Unfortunately many of my students get conned during their school maths classes; they’re told “quick and easy” or “cheaty” methods of doing everything from simple multiplications and percentages to trigonometry and integration. It has to be remembered that these cheaty methods were devised by people that understood the theory in the first place – a good example that springs to mind is the CAST diagrams method for solving trigonometric equations (which I don’t encourage using). If these methods are then shown to people that don’t understand the theory from which the cheaty method comes then, sooner or later, you end up with big problems – yet this is done year in and year out in many schools by many teachers and tutors alike in an effort to try and circumvent the hard work aspect of learning mathematics. Teaching these cheaty methods from the outset eventually leads to spectacular failure and lack of understanding as it shows students how to do something in a very limited and narrow range of cases and doesn’t usually provide any kind of flexibility or ability to adapt to unfamiliar situations. Sadly, students are duped into believing that they don’t need to know the theory and develop an unhealthy expectation that maths can always be reduced to cheaty methods.

Here’s the thing – if you understand the theory you can adapt very easily to new sitiuations, solve a wider range of problems and generally enjoy the learning process more because you get more out of it. If you rely on cheaty methods you have to learn a new method of solution for each and every “type” of question that you encounter. To start with this might not be too much of a problem – at GCSE for example you will only really encounter a fairly limited range of possible questions – but further down the line at A Level the doors are flung wide open and if you don’t have some understanding you’re up the proverbial creek without a paddle.

By teaching along these short-sighted lines you encourage an expectation within a student that everything can be reduced to a cheaty method. Which it can’t. From my personal experiences as a maths tutor the largest category of people that fall victim to this way of thinking are those wanting to do the QTS Numeracy test. I’ve lost count of the number of times that I have been asked to provide some tuition for the QTS Numeracy test but insisting that I just tell them all of the “cheaty short-cut methods” to do the questions which they’ll get on the test (by the way; I don’t know beforehand exactly which questions you will get asked on the test; and even if I did I wouldn’t tell you). I’m happy to show people how to take short-cuts provided that they have a sufficiently high level of understanding in the first place. If they don’t understand the basics then we are both wasting our time and I may as well go and talk to a brick wall for a while because they will not understand when or how to apply such short-cuts.

I understand what I’m doing when I do maths but it isn’t because I learned all of the short-cut ways of doing everything. Quite the opposite – I learned to understand what I was doing by working hard and then the cheaty methods become trivial facts; in fact they almost become redundant. By understanding what I’m doing I see where these cheaty methods come from and how they work – better still I can make them even more “cheaty” if I want to in some cases. There is NO WAY to cut out the understanding when learning mathematics and the understanding can only come about through hard work. You have to be prepared to use your own brain to solve problems and not leave yourself at the mercy of some miscellaneous method that you don’t understand but which you keep your fingers crossed that you’re using it right and that it will give you the right answer. Is that really a good way to learn?

IMPORTANT!!! By trying to avoid the hard work of learning to do mathematics properly you will end up spending (wasting) more time, energy and effort trying to get to grips with loads of shot-cut, cheaty techniques that you don’t have any understanding of and will most probably forget every couple of weeks and have to keep re-learning. So hard work really is the quickest way to learn mathematics – not really what you want to hear is it? But that’s how it is.

There is a place for short-cut methods when it comes to mathematics; they can sometimes take the pain out of an otherwise lengthy and tedious calculation but they should NOT under any circumstances be a complete substitute for learning through hard work to acquire the necessary level of competence. You wouldn’t expect to become a world-champion 100m sprinter without hard work would you? And nor would any sprint coach who knew what they were talking about tell you that you could become world champion without a lot of hard work. Leading people to believe that all mathematics can be simplified to such a point where you just need to follow a nice cheaty method is cruel and if you do it and encourage it then shame on you!