Must be getting close to C1 and C2 now, huh?
Well, what can I help you with this week? Just drop me a comment below and I’ll do what I can to answer it.
Why maths exams are not just stupid but actively harmful (reprise)
I’m not talking here about the stress exams put students under, although I could. I’m talking here about how exams are one of the worst possible ways of testing whether someone is a good mathematician.
Here’s the problem: maths exams are tests of a) memory and b) calculation. You can do well in almost any maths exam by getting hold of all of the past papers and working through them. That’s what I spend about 80% of my time as a maths tutor doing, because it’s the most effective method I know for getting students a good grade in the exam.
Which sucks.
It doesn’t need to be that way. The most important skills – in my opinion – for a mathematician to have are:
- Persistence
- Looking things up
- Modelling real-life problems
- Discussing ideas with others
- Communicating results clearly
Virtually none of those come up in an exam. The last one, sure; possibly modelling real-life problems (although normally you’re spoon-fed the model the examiners want) and possibly persistence (although you’re limited by how much time has been arbitrarily allotted to the exam).
However looking things up and talking with your colleagues – two of the most important skills as a mathematician, and two of the most transferable mathematical skills – are considered ‘cheating’ and will lead to you failing the exam and probably having other qualifications revoked as well.
Let me say that again: the exam system actively penalises two of the most important skills that can be learned in maths.
Meanwhile, they encourage things like last-minute cramming, following instructions blindly, and teaching to the test – which I do, because my job is to ensure that little Jimmy gets the grade he needs to get to university. (I’ll do my best to foster his curiosity and get him enthusiastic about maths, but if I do that and he still does poorly, it does nobody any good).
Convenience is no way to run an education system
Exams are designed for convenience. An examiner can sit down and grade your work (you got 80% of the available marks, so you deserve to get a letter A! have a cookie) and a university can say ‘you have the magic letters A, B and B, so you’re allowed to come here and study for more exams in much the same vein’, or so an employer can differentiate between candidates without having to bother interviewing them.
This isn’t just silly, it’s actively harmful. We end up with a workforce whose main characteristics are docility, a good short-term memory and the ability to work in silence under pressure.
And I don’t think those are the traits we should be emphasising.
The Secrets of the Mathematical Ninja: Converting degrees into radians
I’ll hold up my hands here and say: most of the secrets of the mathematical ninja are guides to showing off. Few of them have much practical use beyond making everyone else in the class look at you funny. This one is different.
If you’ve read my article on why radians rock, you’ll know exactly where my loyalties in the radians-versus-degrees debate lie – and unlike with logs base ten, there’s no way back for degrees. From C2 onwards, you need to be able to use radians with comfort and panache, if only because calculus doesn’t work in degrees.
And every year, my C2 students ask me ‘how do you convert radians to degrees, again?’ and I say ‘you don’t,’ because there’s almost never a reason to. Converting degree to radians, on the other hand, is a different story.
One radian is about the same as 57.3 degrees (more precisely, it’s 180/π). The showing-off ninja might say “Aha! 180/π is close to 180 ÷ (22/7), which is 1260/22, or 630/11 (= 57.27). That’s actually a really good approximation, if you can pull it off (it’s about 0.05% low) – but even a real-life mathematical ninja would find the sums a bit tricky on that one.
Instead, I tend to go for 400/7 (= 57.14), which is a little less accurate (-0.3%), but uses simple numbers. To get a rough radians value from a number of degrees, do this:
- Divide by 4
- Multiply by 7
- Divide by 100 by moving the decimal point left by two places.
So, to find 115 degrees in radians, you divide by 4 (29 less a quarter), multiply by 7 (203 less 7/4, 201.25) and put the point in – 2.0125. The actual answer is 2.0071, so we’re right to 3 sig fig.
If you absolutely insist on going backwards, you just do the same thing in reverse: multiply by 100, divide by 7 and multiply by 4 (in any order you like). That means 1.2 radians is about the same as 480/7 = 68.6 degrees. The right answer is 68.755. Off by two in the third sig fig; again, that’s an error of about 0.3%
So, why is this useful? It’s a good check to see whether you’ve ‘gone the right way’ in your conversion, especially if you’re not prepared to use some common sense to decide whether 5,000 radians is a reasonable angle.
Free for all Friday
Here we all are again, the weekend looming large, the sun threatening to come out, and… a pile of work to do.
Well, what can I help you with this week? Just drop me a comment below and I’ll do what I can to answer it.
The Roulette Wheel and its Distributions
You don’t see many run-down, out-of-business casinos, which should serve as a tip-off: almost nobody beats the house in the long run.
Sure, you might get a few people who come out ahead by luck or guile – check out The Newtonian Casino
But this article’s not about subterfuge, it’s about random variables, which I’m sure you’ll agree is a much more exciting topic. (If you don’t agree, buy one of the books in the last paragraph and I’ll get a few pence in kickbacks, hooray!)
Why the casinos win
How do the casinos always win? Well, it’s largely down to two things:
- For each bet you win, the casino pays slightly less than ‘fair’ odds
- The Law Of Large Numbers says that, if you run an experiment often enough, the mean of the outcomes will converge onto the ‘true’ mean of the experiment
Quick note: The Law Of Large Numbers – which is actual, proper maths – isn’t the same thing as the Law Of Averages – which isn’t; the law of averages says ‘I’ve been losing a lot recently, so I’m bound to win soon to make up for it’, as if there’s some kind of God of the roulette wheel who’d notice you were losing and say ‘oops! my bad, let me balance that out.’ There isn’t.
Now, as it happens, winning and losing over a certain number of spins on the roulette wheel is a perfect example of a binomial distribution – the probability of winning is fixed, the spins are independent, and you have a fixed number of trials. One nice thing about the binomial distribution is that if you do enough trials – more than 30 or so – you can model it as a normal distribution, which is a bit easier to work with.
I’m going to talk about red and black here, but you can do the same experiment with any of the bets available on the roulette wheel.
A thousand spins
On a European wheel, there are 18 black and 18 red numbers, plus a green 0 which is what makes the casino its money. If you were to bet one chip on red 37 times, you’d expect to win 18 times and lose 19 – on average, you lose one chip every 37 spins, or about 2.7%. That doesn’t sound like much, does it?
So, let’s imagine we do 1,000 spins. That’s plenty for us to use the normal assumption, with the mean and variance you can look up in the tables: the mean number of wins you’ll get is np, which is 1000 × 18/37, about 486. The variance is np(1-p), which is about 250 – making the standard deviation 15.8. So, how likely are you to come out ahead over 1000 spins?
You’re going to need a z-score. To come out ahead, you’re going to need to win 501 times, and the z-score is just how many standard deviations you are above the mean: 
(I’ve used the accurate numbers there rather than the rounded ones).
You look up that number in the table and see you have an 82% chance of losing – so you’d win something that you might reasonably expect to be a 50-50 chance a little more than one time in 5.
A million spins
This is how the casinos make their money. They make tens of thousands of bets a day, each with a tiny built-in advantage – if the number of spins was a million, your z-score would be up to 27 (check it!), which is so far off the table as to mean ‘impossible’ – I’m guessing your chances of coming out ahead are about 10-70 – it’s not going to happen.
EDIT: Ah, Abramovitz and Stegun
(on page 972 of my edition), meaning 
. Three interesting things there: 1) I was off by a factor of 1090, which is probably the wrongest I’ve ever been about anything; 2) I was correct to about 70 decimal places, which is probably the rightest I’ve ever been about anything; 3) it’s still not going to happen.
The casino, meanwhile, happily rakes in its large number of tiny profits and turns them into a huge one.
The Secrets of the Ninja Mathematician: An Apology To Base 10 Logarithms
I had occasion to write this letter a few weeks ago:
Dear Base 10 Logarithms,
I’m very sorry that I bullied you. I recall saying things like ‘grown-ups always use natural logs’, and that your button ought to be expunged from calculators. I was wrong to say those things, which I believed to be true; I have now changed my mind and wanted to try to patch things up between us.
Since saying those terrible, ugly words – frankly, I’m ashamed of them – I’ve discovered your tremendous power* in the world of Ninja Maths. It’s to your credit that you never got defensive about the torrents of abuse coming from me – and others – to say ‘but look at standard form!’ You kept your dignity and poise throughout. I salute you.
With kindest, humblest regards,
The Ninja Mathematician
* No pun intended!
Now, there’s a hint in there about why I felt compelled to try to make my peace with log10 – and it’s all to do with standard form. You remember playing with that at GCSE, right? Where you’ve got a number between one and ten, and then ten to the power of something? That makes it easier to work with very big and very small numbers instead of writing out huge strings of zeros.
Something you never do with standard form – and you totally should – is take logs of it. For instance, the log of 5 × 104 is – without looking at a table or a calculator – about 4.70. (With a calculator, it’s 4.698: I was 0.02% off.)
How did I get that? Well, I’m starting to learn my logs base 10. log(5) is very close to 0.7, and log( 5 × 104 ) = log(5) + log(104) = 4 + log(5), or 4.7.
If I wanted to square root that – and who wouldn’t? – I’d just have to halve the log to get 2.35, and work out 102.35. Well… that’s harder. I know it’s between 100 and 1000, because it’s between 102 and 103, but after that I can ignore the 2 completely. What’s 100.35? I’m going for 2.23, but only because I know my square roots. (Doing it blindly, I’d say 100.3 is 2, another one I’ve learnt; ‘a bit more than 200′ would be a more honest guess.)
How about the tenth root? That, I have no idea about just looking at the numbers. However, I can say 4.7 ÷ 10 is 0.47 – and I know that 100.477 is 3. I should correct that downwards slightly – maybe 2.95? There we go.
As long as it’s a positive number, the number before the decimal point is the power of 10, and working out 10 to the power of the other bit gives you the number to multiply by.
Here’s a quick table to learn, if you’re into that kind of thing:
| x | log10(x) | comment |
|---|---|---|
| 1 | 0.000 | by definition! |
| 2 | 0.301 | very close to 0.3 |
| 3 | 0.477 | 0.5 (-2.3%) |
| 4 | 0.602 | double log(2) |
| 5 | 0.699 | very close to 0.7 |
| 6 | 0.778 | very close to 7/9 |
| 7 | 0.845 | 5/6 (+1.4%) |
| 8 | 0.903 | Three times log(2) |
| 9 | 0.954 | Double log(3) |
| 10 | 1.000 | By definition! |
What other creative ways of using logs base ten can you come up with?
Free for all Friday
I believe it’s traditional to do a joke about May The Fourth be with you, but it was old when I first heard it, about 15 years ago. So I’ll kind of resist, but only in a way that still makes it clear I’d love to do the bad joke.
Well, what can I help you with this week? Just drop me a comment below and I’ll do what I can to answer it.
