Sunday, 7 June 2015

Proof

I've been thinking lots about children articulating claims in maths lessons. Luckily, I'm not thinking alone.A claim came up,

and naturally I turned to Cuisenaire rods:
A lot more Twitter conversations happened. Tracy Zager was tweeting out lots of pointers. For instance:
Blogs were helping me think about it too. Kristin Gray's post The Meaning of Subtraction Kassia Omohundro Wedekind's blog post Counting, Conjectures, and Claims

On Twitter I got a great tip off that I've been following up:
So I went to Caroline Ainsworth's work on using Cuisenaire rods.

Of course I tried this approach with the Year 4s (I had the whole year group working in the two adjoining classrooms because my colleague was away) and you can see the results here on the Year 4 blog.
The next day when I reminded them about this claim (which was by then up on the new Claims Board) almost everyone agreed with it. So I asked them to show why it's true on paper, using pictures of the rods and words that would explain it to someone that didn't know about it. (I mentioned proof, but didn't stress this word.)

There were a few that didn't get very far. But there were a lot of good explanations, all different in their presentation:

Some of these explanations seem to amount to a visual proof for me. OK, they are not showing a generalised case for every difference, but taking Angiolina's example just above, she's adding the same thing to the two same left hand ends of the rods. Euclid had it as an axiom, and we know without articulating it that "if equals be added to equals the sums will be equal."
To me, the quality and individuality of the children's proofs justify this activity. They're returning to something very basic, but they're looking at it through more algebraic and logical eyes. And they're taking possession of their knowledge by experiencing it physically and articulating it in words and pictures.

Tracy pointed me towards Avery Pickford's great posts about proof. (I've been playing a 2-digit version of Mastermind with the class since reading them.)

I've got lots of questions. Like:

Are these 'proofs'? Where does explanation end and proof begin? Which is the most valuable?
The understanding is the key thing; what does proof add?
How does the social dimension enhance this learning?
What is the place of un-articulated experience and learning in all this?
What other activities will lend themselves to this kind of thinking?