Has anyone noticed the new 20mph speed limits cropping up in town centres? What a drag eh?! Well, it's still a lot faster than walking, cycling, or even a horse and cart. But never mind that, it's annoying! Actually there's some reasonably interesting maths that goes into explaining the choice of speed limits in highly pedestrianised areas of the UK. You'll notice, if you ever drive down an English country lane, that the speed limit there is often around 60mph, which realistically you'd only attempt on those roads if you're either a very very good driver, or just plain suicidal. So why is on a nice wide, straight road which you could comfortably cruise at 45mph, has some half-wit set the speed limit at 30mph or even 20mph? The cheek!
The maths behind this revolves around two equations. First of all, the equation for kinetic (i.e. moving) energy, this is given by: Ke = 1/2 m v^2 I.e. it is a half the mass times the velocity squared. And another one to do with work done (i.e. how much energy is consumed): WD = F x Which is just force times distance over which the force is applied. These two equations are pretty simple to understand. For the first one just think about which has more moving energy - a heavy object moving very fast, or a light object moving very slowly? Obviously the first one. As for work done, would you rather apply a large force (i.e. push hard) over a large distance, or a push lightly on something for a small distance? I know which I'd choose!
We need to make a reasonable assumption, and that is all the work done by the brakes takes off the same amount of kinetic energy. In truth probably some kinetic energy is lost through friction and wind resistance, but by far the biggest factor is the brakes so we'll just ignore the other things to make things simpler.
Knowing this, we can plot a graph of how much the speed will decrease after applying the brakes over a certain distance. The kinetic energy before braking is 1/2 mv^2. Let's call this K. To work out how much speed it loses we know it loses Fx amount of energy. So it's new kinetic energy is K - Fx = 1/2 mv^2. Rearranging v in terms of x gives: v = (2/m (K - Fx))^0.5 If we assume a car has constant braking force, then we can plot how the speed changes over distance.
To be honest you can skip the maths section and just jump straight to this bit. The important thing is the shape of the curve you get of speed over distance. The key feature of it is that the speed actually reduces quite slowly at first, and only in the last bit does it all go down:
The black vertical line is where the pedestrian could be, 20m from where the driver started braking. Notice how at the lower starting speed (the red line), the pedestrian only gets hit at a speed of one unit. Increasing the initial speed by just 20% more than doubles the final speed the pedestrian gets hit at. At just 60% faster, the final speed is over 4X what the slower speed is.
Apparently the statistics are something like this: Collisions involving cars that were going at 30mph have an 80% mortality rate. Collisions involving cars that were going at 20mph have a 20% mortality rate. (These may not be 100% accurate as it's only from memory but it's something like that).
(Published on 3 Aug 2017)
Our education system is in desperate need of a dramatic overhaul. There are so many ways in which its failing students that it`s hard to write them all in one article. And in my experience it doesn`t have much to do with the teaching staff. There are of course exceptions but in the main I think most teachers work very hard and have good intentions. The problem we've got is that the entire orientation of the education system is incorrect. It is, in effect, a giant, state-funded CV filtering system. Let me justify that comment if you will.
One example I cite to back that claim up is the concept of "grade inflation". This is a concern from (presumably government officials, although I`ve never actually seen anyone campaigning strongly on this issue) some people that grades constantly creep up higher and higher. This would only be a problem if you were concerned about the ability of the schools to differentiate between student`s intelligence and ability. In actual fact the grade you get is not based exactly on the score you receive on the paper, but in where you stack up in relation to the rest of the country that year. So if you`re in the top 10% of scores, you get an A* (or level 9), and so forth. The entire system is geared towards differentiating between "the smart" students, and the, well... not so "smart" ones (apparently).
Compare that to something like, the driving test. The driving test is designed to get students up to a certain level of competence, and those that make the grade pass the test and those that don`t fail. THAT is true education. We have a level of ability we want to attain, and we let the students know what that is, then we educate them until they reach that standard and then we give them a certificate when they`ve achieved it. They are then a qualified driver. Why shouldn`t it be the same for Mathematics? We should decide on what level of maths we require as a society, then set that standard, let the students know what it is, and educate them until they reach that standard. Once that`s all done we should pat them on the back and give them a certificate showing they have the required ability in mathematics. Job Done.
In other words we should be aiming to get every student up to an A* / Level 9. We should be getting 100% of students get the top grade, or at least target that. But whenever schools get anywhere close there are cries of "Grade Inflation", and "Exams are getting too easy" etc. and the goal posts get moved once again and all the work teachers invested into teaching students certain things is thrown on the garbage heap and they are forced to start again. Imagine, that. Imagine in your job, right now, every time you start producing good work, and hitting your targets, your boss decides to arbitrarily make your job more difficult, removes all the tools you developed for doing the current job, and makes you start from scratch. AND THEN claims that it`s because your job was too easy anyway and had nothing to do with what you did. I mean, a salesperson, after having made millions in sales, is forced to give up his address book, throw away all the product information he`s collected over the years, burn his sales pitch and learn a new admin system. Not because he was doing anything wrong, but because he was doing too well at doing what he had already been asked to do.
It is really quite staggering. There`s a second problem here, and that is even if we moved over to this type of education system, we`re still far from being optimum. For starters there needs to be a decision made as to what skills people really need, and why we are educating them. Of course there is the general stuff around having a wider understanding of the adult world and being a good citizen, but mainly I think the government spends billions each year to try and make more productive workers that can earn more money and pay more tax. It would be an unsustainable system if they didn`t. So what then, has inverse trigonometry got to do with anything? I love maths and very much enjoyed studying it, (and to be honest I wouldn`t stop anyone studying it if they were interested as it is such a fascinating subject), but we don`t really need people to know most of it to be productive workers in the economy. Really we don`t.
I`ve tried in the past to convince students there is some value in this stuff (when they`ve asked me the standard "what`s the point in this stuff?" question), and the more I think about it, the more I think they`re right. There is no real point, except that it`s interesting. There is no other reason. Unless you specifically need it for a job, but then that should be vocational training like any other job (accountancy for example). So you wind up with students not interested in the subject, being forced to learn something they`ll never need (and forget immediately after the exam), changing every 5-10 years how it`s taught anyway so the teachers have to constantly re-do work they`ve already done, all for what? It costs billions and there is absolutely no point whatsoever. It achieves nothing. We need a total rethink of what skills people actually need in the work place, and base the courses around that. And then just train people to get those skills so that employers see that if they`ve got what`s required to do the job (rather than trying to infer it abstractly from someone`s academic scores). And we should leave people free to study academic subjects if they wish (the ones that are interested will do it anyway).
What we`ve got now is totally bananas. So if your child is struggling, it might be in part because they are caught in a completely nutty system. Nevertheless, the system is there and you need to play the game. The grades do matter and that still is important (even if only because everyone agrees that they are!). So my approach to tutoring is very much, if they`ve got to do it anyway I try to make it as interesting as possible. I try to show the students the inherent beauty of the subject so they can get a glimpse of what I see when I look at Mathematics. I don`t try and con them into believing that there`s any point in it other than that. And I do take the grades very seriously, ultimately after the years of toil if you haven`t got the piece of paper to show for it then it really has been a waste of time.
(Published on 7 Jul 2016)
There's a few interesting things about Maths that you might not appreciate at first glance. Most people think of numbers as soon as they hear the word maths, but there's a lot more to it than first meets the eye.
The first thing to realise is that it is a completely made up subject and has no basis in reality whatsoever. Ok so that might be an over-statement, but really Maths is not as scientific as the other sciences. For example think about the idea of zero, one of the most basic and fundamental concepts in Maths. Where have you ever seen zero? "Aha" you say, "I've got nothing on my table. That's zero!" Well the thing about that is can you take that zero and put it in a bag and take it to work? Can you give someone else zero cars, can you touch zero elephants? Zero isn't an actual thing at all, it is a complete absence of things. And therefore doesn't exist in the real world, but can only exist in your mind. It is a purely imaginary concept. And to be honest while it seems quite ordinary and natural is has some unpleasant side-effects. For example you cannot divide by zero without getting into all sorts of trouble.
Take for example the following:
8/2 = 4
This says that eight divided into two piles give you four in each pile. If they are equal (which they are), then if I double both of them then they should remain equal. Now dividing eight by two then doubling it should leave me with eight, as half then double doesn't change anything, so I get
8 = 2 times 4 = 8
Now let's try and use the same logic with zero. Given that it's not obvious what dividing by zero should be, I'll just make a guess to start off with and see if I'm right. Let's suppose:
8 / 0 = 4
Then again, if they are equal, multiplying both by zero should keep them the same, and also diving 8 by zero then multiplying it by zero shouldn't change anything so we get:
8 = 4 times 0
Anything times zero is just zero, since you have zero lots of that amount, so we get:
8 = 0
Which is a bit of a snag. We could repeat the trick with any number, and get results like:
5 = 0
21 = 0
In fact it would result in all numbers being equal to zero. The entire system of maths would collapse in on itself like a black-hole. Not good. So we get a first peek into what maths is really about here, we have a certain amount of choice over how we do maths. See maths is just a collection of ideas and arguments, and I've made quite a convincing argument here that we wouldn't want to allow division by zero. It's not actually that we cannot divide by zero, it's more that if we allow that then maths wouldn't be the way we desire it to be. You might come back and say that how can it be such a matter of opinion, I thought maths was about ultimate truth?! Well I'm so confident in the argument I put forward, I think it is so convincing that basically anyone would agree that allowing division by zero is too problematic and that we should just skip it. It doesn't really matter if it is an opinion if your opinion is so strong and backed up with such good logic that no one would ever disagree with you.
And there is another quite bizarre thing about maths. It is all completely made up, deals with things that don't really exist, and yet somehow seems to get universal agreement despite age, race, language or up-bringing. Even some animals can understand that basic idea of counting, it seems to be a universal property of the universe that everyone can understand even though nobody can see it, we can only imagine and discuss it together. No other totally imaginary subject gets such broad agreement. For example take any fiction work, there will be lots of people arguing about lots of different aspects of it, interpretations of the ends of certain films etc. Yet when it comes to maths, as much a work of fiction as anything else, it is case open and shut in a few short lines of reasoning. Maths is so universal, we have even inscribed mathematical bits and bobs onto satellites we send out into space, assuming the only thing that would be easily understood by aliens is some maths.
And there are even bigger problems than just dividing by zero. Maths is actually full of strange contradictions and other unfortunate events, which I'll go over in my next post. Then we'll see that someone (Kurt Godel) decided to sit down and figure out where all the problems were coming from, and what he discovered was possibly more disturbing than any of the individual problems that prompted his research. So much so that the discoveries drove him mad. His paranoia was so great that he refused to eat anything his wife had not prepared, convinced he would be poisoned. When she was away for a period of time he actually starved to death. Read on next time if you dare!
(Published on 13 May 2016)
I'm making a big assumption here, but as a mathematician I've calculated that most people either don't like maths or weren't any good at it at school; so I've got a very good chance that the person reading this is not great at maths, and a pretty decent chance they even consider themselves to be incompetent at it. I've recently developed a theory as to how that state of affairs has come to be. And to illustrate it, I'm going to introduce an entirely new subject I've developed, called complexology. It is a very important subject that everyone should learn, you'll need it in the world of work and in your career, and there'll be some pretty difficult exams on it that you'll need to study for and pass before you can do anything else in life. Basically, first of all you take something called a zag. These zags have different symbols, like so: '#+%&*(
The rules are as follows:
##% = *&
@ = #**
^%^ === @@#
and so on.
Ok time for your first test:
If I take the following zags, and imply them over a k-plane, what is the resulting fido-metric? The answer is of course &. I won't bore you with the proof of such a trivial example so let's move on to chapter two.
Ok I think you get the idea. See the great problem with maths, and also it's greatest asset, is that it is almost entirely symbolic. Let's look at the definition of a symbol: "a thing that represents or stands for something else" However, what I have found is often not made clear enough, is the link between the symbol and the something else. Without that crucial link, the symbols are about as meaningful as the mess that is complexology, and it's about as fun to study as power-sanding your own leg.
What should come first is a study of the something else, and later on we should introduce the symbols as a convenient way to refer to things already looked at and understood. What happens often though is the introduction of the symbols and the rules between them, aaand then a test about those things and move onto the next chapter. So in actual fact nothing has been studied there whatsoever except a bunch of meaningless relationships between meaningless symbols. What a big waste of time.
Let's look at an actual example of real maths. Take a look at the following grid of dots, I have put a certain number of dots there. At this point, I refuse to even name that quantity (although I'm sure you know what it is), to make the point that what is important is NOT the name or label which we give to the quantity, but the actual quantity itself. It's like constantly objectifying women and never getting to know the real person - only thinking about what they look like. So probably for the first time in a long time, maybe even ever, I want to introduce you to this guy:
He is a pretty interesting dude, not least because he can be rearranged in the following way:
See how he (I've just noticed I've defaulted to considering it to be a male here, but whatever) can be put nicely into two neat rows, each row the same size as the previous (and yes I did just use the label two, but I realised this will get very difficult if I refuse to use any names for numbers, which I think illustrates my point that the names are useful when trying to communicate or refer to different mathematical objects). Amazingly though, he can also be arranged like this:
How about that. Fits equally neatly into three rows. But what about 5 rows:
Oh dear. That didn't work did it. Well it's still pretty good going. But why is that, why is it possible to put him into 2, 3, or 4 rows, but not 5? Strange isn't it. What's even stranger, is that if I take just one dot away, and look at the guy that lives next to him, i.e. this guy:
Will only ever fit into one row. Try as you like, it won't go neatly into any equally sized rows other than one. That's pretty crazy isn't it, like how can we take just one dot away and rather than the nice jolly cooperative chap we got before, we're introduced to a stubborn, obnoxious, uncooperative bore? It still baffles me. Yet this rather intriguing problem, which has confounded the greatest minds to ever grace the face of the earth, for centuries, can be quite boringly and inconsequentially be stated in the language of standardly educated mathematics:
"Ahem, listen up everybody. A prime number is a number that can only be divided by one and itself. Ok class, make sure you remember that definition because it's going to be on the next class test. Also you need to know that the following numbers are prime: 2,3,5,7,11." And in my opinion that is why you suck at maths: you've probably never even come close to studying it in the first place. What you think you're not good at isn't even maths at all, it's a paper thin veneer with nothing underneath; masquerading as the bright, vibrant, solid block of wood that it should be.
(Published on 12 Apr 2016)