## Monday, 8 December 2014

### A short conversation

I found this conversation thought-provoking.

I'm not experienced in university maths (I did science), but from what I do know I want to say that there is a continuity, and not just out of respect for the efforts of young children.

Of course, Colin is correct: there are an awful lot of sums in schools, hopefully less than there were when we were at school, but still, many people do think that sums are the thing. And it must get boring getting thought of as a difficult sum doer!

But, but... there is another side to school maths, and one that I would like to see take over as people's picture of school maths.

Take this Kindergarten Interlude that Joe Schwartz posted on the other day.
Here are children in a comfortable situation, getting to know number, its constancy, its patternability, its many possibilities. I like how Joe says, he is there to "poke, push and experiment".

Now this is well-trodden ground, but children are building up their knowledge here, step by step, tentatively at times, in leaps at other times. Sometimes the knowledge will be certain: however you arrange three bears, they will always be three. Other times it will be a hunch: you need four bears before you can have a pattern.

Or take Kristin's post on Articulating Claims in Math, The elementary kids are doing "sums" here, but they're doing a lot more, they're making generalisations about the way sums work:
Now you might argue that proof involves more than this. That it involves formal generalised description. Or that it must be a shared and socially verified knowledge.

But I would say the exploration comes first, then the hunches, then more exploration, then the knowledge, then the sharing.

Consider Polya's treatment of Nicomachus's theorem. How would you first discover this? Polya writes:
Mathematics presented with rigor is a systematic deductive science, but mathematics in the making is an experimental inductive science.
And that is what is happening in many excellent maths classrooms, and could be happening in even more. I like Polya's example, not least because the nine year olds in my class had a go at exploring this last year.
A different kind of activity comes before proof. You get a sense of it in this video:

And proof is not about formal language either. There's the famous Bhaskara proof:
As Keith Devlin says about symbols, "They no more are algebra than a page of musical notation is music."

So, I want to make the case, here as I do in my teaching, that there is a continuity, there is in the best classrooms a relationship with what kids are doing and what mathematicians do.

What do you think?

1. I think I agree - there are instances in the best maths classrooms where kids are doing something that mirrors what mathematicians do. I would like it to happen more though! Often, I've seen either inductive experimental maths going on, OR deductive rigorous stuff, but missing the synthesis of the two - discovering something, having a hunch, and then finding a way to explain, convince oneself, convince others, and give that new knowledge a status higher than just a pattern spotted without any certainty.

We can touch on the beginnings of proof and start telling that story with fairly young kids (my area of expertise is age 11 up, but I've seen good examples of foundations of proof in primary maths).

2. Thanks for your thoughtful comment, Alison. And thanks also for your "random thought" that got the conversation and my train of thought started!

I like Kristin's "Claims Board" lots. And as you say, there are grades of certainty. I feel like I want the kids to have more meta-knowledge about the degrees of knowledge they have: from noticing, onto wondering, to having an idea or a hunch, onto being fairly sure, to really knowing, and having others with enough expertise agreeing with your argument. There could almost be a different board for different levels of knowledge.

(Feel like this needs a blog post in itself!)

1. Interesting discussion! Firstly, there are many maths classrooms where pupils are not encouraged to act as mathematicians, not just in the Early Years. The EY is not just number and my research and experience indicates that 4- and 5- year olds are very capable of thinking and behaving mathematically given a rich context, time to engage and space to reflect (I am starting a blog on this in the New Year!).
A few weeks ago, I gave a group of 8- and 9- year olds Cuisenaire and the problem 'The first is half of the second, the second is 2/3 of the third'. Can you tell which 3 rods I have? (Is there a solution?). It took them a while to unravel the problem (of course I didn't explain it, but I did read it out word-for-word when asked). First of all most found rods that could and couldn't be halved. They then pondered what 2/3 might mean in this context. Then 2 pairs had 2 different solutions - white, red, Lt green, and Lt green, Dk green, blue - magic! (Is there more than one solution?). After 30 mins they were excitedly predicting that sets of 3 numbers "Had to be in a times table - even the hundreds times table", and agreed the solutions were "endless".
A little bit of maths as mathematicians, what do you think?

2. Thanks, Helen. Good to hear you're starting a blog on this in the new year!

Seems like a great use of Cuisenaire rods. I've not really tried using them like that, but I'll try and give that kind of question a try with my Year 4s and see what they make of it.
I'm often surprised by how young children can go up a level of abstraction. Paradoxically, I think having a concrete or a visual resource as a starting point can help.
And I think quite a bit of maths as mathematicians!