Friday 13 April 2012

My First Published Maths Article - Maths V Science

The secondary school I attended was a specialist science college, and so we had an in-house science magazine that came out every term. When I was in year 13, I submitted an article for it. This was a little bit weird because I wasn't taking any science A-levels, so I wasn't part of the science "crew". What was weirder was that the article was anti-science. It is about how maths is so much better than science.

It was the first time I'd written about maths and I loved it. I've always loved writing. I've kept a journal since I was 12 (when I first read the Princess Diaries) and I always did well in creative writing in English. Aged 17, my passion for maths was really starting to show itself. I'd been reading books by Simon Singh and Marcus du Sautoy and Ian Stewart on the bus to school and I was spending roughly 30 hours a week studying maths. Writing this piece made me realise that this was what I wanted to do: try to get people excited about maths through words. Which is what teaching is about, and also what this blog is about.

Anyway, here is the article.


A not-at-all-biased look at the strengths and weaknesses of two essential subjects*

Round One: Proof

It is an indisputable fact that mathematical proof is far more powerful and more rigorous than scientific “proof” (please note the use of ironic quote marks). This is because when a mathematician proves something to be true, we know that it will always be true and that there is not even a small, slight, snowball-in-hell chance that it will ever be false. Mathematical proof’s spotty younger brother, scientific proof, relies on evidence backing up a hypothesis. So if an enormous amount of evidence supports their claim, a scientist will say they have proved that it is right. However, consider this: we know that if you flip a coin 100 times you would expect 50 heads and 50 tails, but we know that it is possible to get all heads and no tails, or vice versa. Now if a scientist flipped a coin fifty million times and got all heads, they might well conclude that this coin will always produce heads. But how do they know that on the 50 000 001st time they flip it it won’t be a tails?
Maths: 1 Science: 0

Round Two: Problem Solving

Amongst other things, mathematicians and scientists have to solve many problems. The main difference between the problems that mathematicians solve and those that scientists solve is that, most of the time, scientific problems are of great importance, whereas maths problems are usually as useless and unnecessary as sleeping pills in a geography lesson. One of the greatest mathematicians ever, GH Hardy once said “no discovery of mine has ever made or is likely to make…the least difference to the amenity of the world”. (in actual fact, Hardy’s discoveries have been of great use, but lots of the maths that number theorists do is really comparable to thermal-underpants-in-hell in terms of usefulness). Despite this, it is clear that mathematicians are better at problem solving than scientists are. Consider the following example: you have a normal 8x8 chessboard, and you cut the top left and bottom right corner squares off. What you are left with looks like this:

So there are now 62 squares remaining. You have been given 31 dominoes, the same size as a pair of squares. Can you put all 31 dominoes on the board so that every square is covered? A scientist would solve this problem by experimenting, by trying out different arrangements. (Have a go yourself- you can make a board and dominoes out of paper. It’s a fun alternative to listening to Mr Clargo in your maths lesson). After trying out a few different arrangements with no success, a scientist would conclude that it can’t be done. However, the scientist would never be sure whether they are right or not (unless they tried every one of the millions of arrangements-which would be a very pitiful existance). Now, a mathematician would tackle this problem logically : the corners that were cut off were both black. So there are now 30 black squares and 32 white. Each domino covers two adjacent (neighbouring) squares. Adjacent squares are always the opposite colour. That means that for every white square that is covered, a black square is also covered. So to cover all 32 white squares, 32 dominoes would be needed, and we only have 31. therefore, it can’t be done. Pure genius! The mathematical way of solving problems is often quicker than the scientific way, and you can always be sure that you are right.
Maths : 2 Science: 0

Round Three: Fame and Fortune

Most of you will probably find it hard to name a famous mathematician or scientist who is alive today (and no, the guys from Braniac do not count). However, if you look way back in time, the only people from that period who are remembered are the mathematicians. So whereas most of you don’t know who Andrew Wiles** is, in a thousand years time, he will be remembered and Jade Goody will not. Again I am going to refer to the great GH Hardy: “Immortality may be a silly word, but probably a mathematician has the best chance of achieving whatever it may mean”. To be fair, a few scientists will also be remembered. I suppose. Now as for fortune, you should know that there is a lot of money up for grabs for talented mathematicians (or someone with a calculator and way too much time on their hands). If you find a prime number with loads and loads of digits, the FBI will give you $1 million. However this is easier said than done (I have wasted many a Sunday afternoon number crunching to no avail).
Maths: 3 Science: 0

There we have it; maths wins hands down. So if you think maths is boring, hard, or pointless, think again!

*please note: maths is more essential than science.
 ** English mathematician famous for solving a 358 year old problem: Fermat’s Last Theorem

By the Way...

Several things in this article were edited without my permission for the magazine. The above is the unedited version. The bit about Mr Clargo was edited out. Also, the asterisked bit "*please note: maths is more essential than science" was changeed to: "*please note: maths is more essential than science, in my opinion" which completely ruins the joke. It was obviously written in a tongue-in-cheek way, I really don't know why they had to change it and make me look like a dork.

What do you think, not bad for a 17 year old?

Emma x x x 

Wednesday 11 April 2012

"I'm rubbish at maths!"

Why is it socially acceptable to admit you're bad at maths?

In fact, it's not just socially acceptable, it's encouraged. You would feel more uncomfortable announcing that you're quite good at maths than saying you're bad at it. I'm right, aren't I?

We had an NQT/PGCE meeting the other day about numeracy. As soon as it started, there was a chorus of "Oh, I'm rubbish at maths"s from the non-maths or science teachers. One English specialist said, "actually, I like maths, I'm quite good at it", then hastily added, "I know, I'm weird!". Why do we feel the need for such an addendum?

The week before, we had a meeting about literacy. Now it will not surprise you at all to learn that the meeting did not start with a load of teachers saying "Oh, I'm rubbish at reading". I have never once heard an adult admit this. Admitting you are illiterate is very tabboo. So why are people falling over themselves to declare that they are innumerate?

It really annoys me when I meet parents of my pupils and they say to me, "I'm terrible at maths, I hated it at school". Children naturally copy their parents, so they pick up on things like this, and they think it's OK to say things like this. And as soon as a child gets it into their head that they're bad at maths, they become worse at maths. It's a self-fulfilling prophecy.

I hate it when people claim to be bad at maths. In my opinion there should be just as much stigma on being innumerate as there is on being illiterate. Maths is as much a life skill as reading and writing. Everybody in the UK should be able to do basic maths, just like everybody should be able to read and write. And if you can't, FOR SHAME.

Maybe I'm being a bit harsh. My worst subject is probably Geography. I'm sure I've said on more than one occasion, "I'm rubbish at geography". I think I've already mentioned on this blog the time my geography teacher wrote, "Is this some kind of joke?" next to an essay I'd written. I tell that story proudly. So I'm a complete hypocrite really. My head of department asked me the other day if I knew where Hull was. I admitted, unabashed, that I had no idea. I retell at any opportunity the time some pupils asked me if I knew where Bosnia was, and, looking my Bosnian student straight in the face, I said, "Of course, it's in Africa". Did I mention this student is blonde? I'm not ashamed of this. It was funny. So, yeah. Huge hypocrite.

OK, ignore everything I've said. Maybe it's obvious but I'll tell you anyway: I wrote this blog post on two separate days. The paragraph starting "Maybe I'm being a bit harsh" I wrote today, four days after the beginning paragraphs. I'm obviously in a better mood now (at the time of writing it's Good Friday, and hence the first day of my Easter break). I'll probably still publish this, in a few days, but I'm a bit disappointed that my argument fell flat on its face.

I hope you're enjoying your Easter holiday!

Emma x x x

Friday 6 April 2012

Is Zero a Square Number?

In my faculty we have an unofficial lunchtime meeting every Thursday. What do you think happens when you put several cleverer-than-thou maths enthusiasts together in one room for forty minutes, along with some "normal" people who sit back and observe detatchedly, for fear that their views will be immediately shot down as they have not used phrases like "corollarlarily" or "without loss of generality" in expressing their argument?

You get some pretty heavy meta-mathematical debates which question the very core of mathematics.

You also get some pretty pointless debates, like: is zero a square number?

Now obviously the answer to this depends on how you define a square number in the first place. Here are some of the definitions my department discussed:

*A number, n, that can be written in the form n = a x a, where a is a natural number (slight variation, where a is an integer).

*A number n for which there exists a square with area n square units, and whose length is a whole/natural number.

*A number whose square root is a whole number.

None of these definitions is particularly satisfying. When I'm teaching my students about square numbers, I tell them that if you times a number by itself the result is a square number. The implication is that by "number" I mean natural number. When I list the square numbers, I never include zero. However, I feel pretty strongly that zero is a square number.

Take definition number one. Does zero satisfy 0 = a x a where a is natural? I say yes. Other members of my faculty say no. I believe that zero is a natural number. They do not. Why do I believe this? Well my professors at uni, for the most part, included zero. I know this because when they didn't want to include zero they'd put a little + next to the blackboard-bold N. If you define natural numbers as a "counting number", ie a number that could be the cardinality of a set, then zero is without a doubt a natural number. How many cows are there in this classroom? Zero. I have counted the cows in the room and the answer is zero.

On to the next definition: does there exist a square with area zero and natural-numbered sides? Leaving aside the "is zero natural?" debate, we first have to deal with: can there exist a square with area zero? My immediate response was, "Yes! There's one right here! In fact, I've got twelve of them!" (I was a bit hysterical at this point. You must bare in mind this was the last day of term). But it's an interesting question. It's almost like, "What is the sound of one hand clapping?", a Zen koan that for some reason doesn't interest me at all (hey monks, listen up: the answer is NO SOUND. Done.)

And the third definition. This one was offered by probably the most down-to-earth of the faculty. despite having a maths degree, she's somehow managed to maintain a grip on reality other maths grads signed away during Freshers' Week. If you square root zero, do you get a whole number? Yes. You do. Zero is a whole number. But using "whole number" didn't feel precise enough. Which then led to another debate: integer versus natural versus positive natural. Is i a whole number? If it is, then -1 would also be a square number. Can you have a square with an area of -1? Why not?

What do you think? Do you have another definition of square number that would settle this argument? Do you think zero is a natural number?

Why is 1 a triangle number? You can't draw a triangle with one dot. Are the triangle numbers defined by the picture or by the formula (1/2)n(n+1)? And hence would zero be a triangle number?

Enjoy the Easter break!

Emma x x x