What does this analogy tell us about using notations of fractions?
Of course, this is merely an analogy, but I think it’s an analogy that helps us understand why students need to go through the natural process in order to fly.
Otherwise sure, they can compute, the butterfly will get out of the cocoon. So if the goal is for them to compute, then telling them the procedure is fine. If the goal is to get the butterfly out, then cutting the cocoon is fine. But if the goal is for children to go beyond calculation, to fly, then the cutting clearly would not make them fly.
It’s an analogy, but it kind of helps us understand why we are doing all these troublesome things. It was so difficult to do that lesson, and you will see Year 7 students really struggle to come to terms with all of those things that we talk about.
But there’s an easy way. Just take a pair of scissors, cut the cocoon.
The calculation will be done. The butterfly will get out, but the problem is that’s all that’s going to happen. All they can do is compute. All that’s happened is the butterfly got out, but maybe the butterfly will die soon.
So yes, they can compute, but the computation may decay, or be forgotten or misinterpreted.
“Oh, I remembered, the teacher said something about dividing becoming multiplying and inverting something…”
And they might now invert the first fraction instead of the second fraction.
So I suppose that is the danger of cutting cocoons, or in our case, teaching them procedures not backed by conceptual understanding.