Invention or Discovery?

With snow flurries outside the windows and the prospect of an imminent school closure, you'd think nobody would be feeling particularly focused in year 7 maths, especially given the topic. Order of operations (or BIDMAS/BODMAS) must be one of the dullest topics in the spec, especially on a snow day. However, it was one of the best lessons this week, and I'm trying to work out why.

We always start with a question, such as 5 + 2 x 3 and ask what answers we could give. What would a calculator say? (Interesting, because it varies depending on the calculator.) With such a bright group, we raced through to 'because you do multiplication before addition' and 'because of the rule'. Then, because it's almost reflex by now, I asked 'Why?' It was a particularly bright girl, let's call her Sparky, answering, and she was flummoxed. Why do we have a rule? Why is it THAT rule?

S: 'Because that gets you the right answer'
Me: 'What do you mean, Sparky?'
'Huh?'
'What do you mean, 'the right answer'?
'...'

What makes one answer right? It stopped me in my tracks as well, as the whole class paused to get philosophical.

S; 'The rule says what's right'
Me: 'Why?'
S: 'Well, you've got to have a rule, or you won't know what's right'
Other student: 'Yeah - and people would do it differently'

We'd quickly worked out that the rule gave consistency, so we knew what people meant when they wrote that notation. But we couldn't stop there...

Me: 'So this is just a rule we picked?'
Students: 'I guess...'
Me: 'Is everything in maths just rules?'
Students: 'No, some of it's obvious, it's got to be true'
Me: 'What else is a rule that people decided?'
Students: 'Numbers?'

Then we spent 20 minutes talking about different number systems, and how the way we write numbers is an invention, like BIDMAS, but other things are discoveries. They'd done Roman numerals in history, but never made the link. The point that the rules can also be different was nicely made when a student explained that whenever you have a smaller number before a bigger number, you subtract, giving the example of IV. I wrote 15 on the board, and said 'So that means 4, right?'

They really love the ideas of having some rules, that varied between cultures (we got onto Egyptian, Chinese, Babylonian and Arabic numerals, all through them), and also having discoveries that didn't vary. It is so strange to me that this is the first time these students have thought about this - the idea of discovery and invention in maths. I think it might help a lot for those students who think maths is just 'a load of random rules'. It wasn't on the lesson plan, but I think they enjoyed themselves nearly as much as I did, and I think it was far more valuable than a worksheet.