A guest post, today, from someone who's almost as angry as the Mathematical Ninja. Bletchley Park's Education Officer Tom Briggs is mad as hell, and he's not going to take it any more.
I'm not a maths teacher any more1 but I still find myself having those same old conversations...
"Maths teachers have it easy: every answer's either right or wrong!"
Once the green tinge has faded from my skin and I've replaced the torn shirt (whilst my trousers, miraculously but thankfully, have remained intact but for some fraying around the edges), I'm happy to offer some calm and collected insight into something that yet another person has Got All Wrong.
From two angles2.
Mathematics, despite certain education ministers would lead you to believe, is not a big list of facts that you have to remember. It's not about recalling that $5 \times 6 = 33$3; it's not about being able to state Pythagoras' Theorem with your eyes tied behind your back; it's not about being able to tell people how many cantaloupes uncle Jim purchased, or even how much change he got.
It's not about the answer. The thrill, as with so many things, is in the chase. It's about the route you take, the thought processes, the trying and failing and retrying. Mathematics is not a list of numbers to memorise; it's about training your brain to look at a problem and finding a way of tackling it that isn't just waiting for someone else to do it.
Back in World War Two, some of those clever boffins at Bletchley Park weren't interested in the answer at all; yes, they cracked those codes, solved those ciphers for King and Country, but they also did it for the same reasons some of us do the sudoku in the Sunday papers: it's good, solid, fun. A problem to solve. The thrill was in the chase, and the actual message content - the answer - was left for someone else to finish off once the process to unfurl it had been discovered.
If it was all about the answers then there wouldn't be a problem with copying. Heck, I'd encourage it.
Even when the answer matters4, there's usually a tangled web of possible routes that allow you to get there. I've seen two students tackle the same problem in markedly different ways on many occasions. Who's right? They both are, as long as the answer fits the context. I've seen one student spot and take the shortcut, answering a question in seconds, while another takes a more mathematically scenic route, getting to the answer half a lesson later. In many cases, whilst the former student can turn the handle, the latter ends up with a much better handle to turn: how do you develop your local knowledge better? Head straight to work and back every time, or get a little lost now and then?
If you think that there's one and only one answer to any given mathematical question, then I'd love to hear the story of how your imagination got removed5. "What's the square-root of 25?" has two answers even if you're restricting it to just the boring ones. The answer to "What's six multiplied by seven?" could be forty-two, $2 \times 3 \times 7$, $101010$, or any of a number of representations depending on how you're going to use the answer.
For any mathematician worth their salt, answering the question "Jim has eight cantaloupe melons in one hand, and thirteen in the other. What does Jim have?" with "bloody big hands" isn't funny, sarcastic or flippant. It's big, and it's clever, and more importantly it opens the door to figuring out just how big Jim's hands must be...
... which is where the real maths lies. Discovery and exploration that can be done with a pencil and a bit of paper.