and naturally I turned to Cuisenaire rods:S claim: U can add the same number to both #s in a sub prob and get the same difference.His representa. @MikeFlynn55 pic.twitter.com/GRmDUqG4f5— Kristin Gray (@MathMinds) May 13, 2015
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@kassiaowedekind @Simon_Gregg @MathMinds @Zakchamp @MikeFlynn55 From Mason & Burton: pic.twitter.com/iIACvLMFok— Tracy Johnston Zager (@TracyZager) May 21, 2015
@Simon_Gregg I like the practical, intuitive approach this enables. Caroline Ainsworth has done some great work on this - on NCETM website— WellingtonPrepHead (@WellingtonPHead) May 17, 2015
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?