Impressed by Caroline Ainsworth's researches into using Cuisenaire rods, I got hold of Madeleine Goutard's 1963 booklet, Talks for Primary School Teachers. Gattegno writes of Goutard in the foreword thus:
Not the least of her talents is the ability to express simply, accurately and concisely ideas that in my words remain obscure to many.And they are sometimes, his words. I'm convinced that Gattegno was proposing an approach to teaching that put the learner at the centre and encouraged their initiative, understanding, communication. But it doesn't necessarily come across in that very directive video, and his wealth of ideas to mine doesn't necessarily come across in his books:
which - despite the treasures in them - somehow put me off with their layout and wording.
And there are really all sorts of good things. Like this, taking thirteen and seeing which rods "fit" in it:
I usually go straight for representing factors:
but this made me see, there was a step in understanding I was missing out, looking at which numbers line up in the 'wall" of a number and which ones don't. This realisation came just in time, as we've just begun to look at factors and primes.
There are seventeen children in the class, so we started off talking about getting into groups, and children being left over. Then I began to show how we could look at this with the rods.
And off we went with the rods themselves:
Anyway, some snippets from Madeleine Goutard that show, much better than Gattegno does, how exploratory all this might be:
We must avoid an over-emphasis on teaching: that is, we must avoid showing the child things he can find out for himself in his own way. (p3)
... he must be given the greatest possible initiative in building the mathematical edifice. Instead of always beginning the lesson with: "Get out your rods; do this, do that"... the start may be in the form of a question:- "Who remembers what we were looking at yesterday? And what did we find...?" (even if it is thought to be insufficiently grasped, and that there will be incorrect answers). "Let's get our rods and make sure that what we are saying is true." Leave the initiative for conversion to the children themselves. "But did we have to use that particular rod? Which other one could we have used...?" Make them work out what would happen if: "We never worked out if that number had factors? Does it? Why? Which ones? Let's see if we guessed right." In this way the mind flies ahead of the facts, instead of lagging in their wake and being led to a docile submission to what is obvious. (p5)
What I want is to see children who express themselves and use their knowledge, such as it may be, creatively. (p44)
It is not the passive possession of knowledge which is important, but rather the ability to acquire it: that is, to make use of what one has already to lead to further knowledge in abundance. (p48)