There are some truths in mathematics that are true because they are true because they are true. For example, if I have a right-angled triangle in the plane, the square on the hypotenuse has to be equal to the sum of the squares on the other two sides. Other truths are true in a different way. It’s true to say that this:
is a square, but it’s not true in the same way that the statement of Pythagoras’s Theorem above was true. And of course, saying that the statement is Pythagoras’s Theorem is true, but not in the same way that the Theorem itself is true.
Confused yet? Great!
The notion of different types of truth has been around for a very long time, and has been recast using many different descriptions. I’m rather fond of Dave Hewitt’s designations “arbitrary (socially agreed names and conventions) and necessary (properties and relationships)”, so calling that red shape a square is arbitrary, because I could call it anything I liked, I just choose to follow the social convention to call it a square, but Pythagoras’s theorem is necessarily true, it is a property of right-angled triangles, it is a relationship I can derive for myself.
These necessary truths are vital to mathematics, in fact, perhaps in some sense they are mathematics. And in teaching mathematics, the NRICH philosophy draws on lots of great thinkers in maths education and comes to the conclusion that these truths are something that children should have the opportunity to explore and discover for themselves. They will never be able to discover that a square is called a square without some external influence (an adult, older child or dictionary telling them that a regular quadrilateral has the name “square”).
But despite my strong feeling that it is the necessary truths that are core to mathematics, I also think that educating children into the conventions of mathematics is important. Part of being a mathematician is being able to speak a common language with other mathematicians. This means knowing the definitions, being fluent in the notation, understanding the conventions.
I ranted a bit on Facebook earlier about the questions like “20 + 20 x 0 + 1″ that have been popping up, with a popularity contest where people vote on what the answer should be. An idea that came out of that discussion was that one of the reasons that people don’t remember BIDMAS, BODMAS, PEMDAS or whatever it’s called in their local language is that they don’t see a need for such a convention. For me, the link between arithmetic and algebra means that order of operations is firmly embedded – if I was evaluating 2 + 3n, of course I would do 2 + (3 x n), so if I’m doing 2 + 3 x 4 I think of it in the same way – in my mind, the “three times four” is grouped together. Given the widespread lack of awareness that mathematicians have a convention for order of operations though, I think if I have need to write a calculation down for others I will use extra brackets just to be on the safe side!