There were two instances in today's Higher maths lesson in which the learners worked as-if something was the case. I think this is a very tricky concept. Here is the first example, in which K (spontaneously) chose y to be 5 and y_0 to be 10:
I thought this was a remarkable thing to do, as it requires a thought something along the lines of "Well, supposing that y and y_0 are these values, what would the answer be?" Underlying this approach is a sense of doubt, voiced by J: "Are we allowed to do this?"
I can recall times when I was learning maths of thinking it was very odd that I could sometimes choose values for certain variables, and then solve the problem as if it was the case. Sometimes it is valid to do this, and sometimes it is not; we talked about why it was valid to do this here.
Both students had previously been stuck on the question below, but could then solve it immediately:
He asked whether he was allowed to cancel e's in this way. I asked whether he could justify what he had done. He replied that he could do it because everything was multiplied together in the fraction, and that by cancelling he was dividing top and bottom by e.
In my experience, learners find it difficult to recognise when they are 'allowed' to cancel terms within fractions, and when they are not. But what matters is whether they can justify what they have done mathematically, developing what Gattegno calls 'inner criteria' for making choices.