Here's another dialogue, continuing my work on exploring questioning. Student C is trying to solve this problem:
(x.√x)^(-1) = (x.x^(1/2))^(-1) = x.x^(-1/2)
Me: I’m not sure that’s right… the idea is good, but something funny’s happened here [pointing]… What’s your thinking here?
[This is a genuine question, in that I can't quite make out on first inspection how he has arrived at the result, but also I think it will be useful for him to articulate exactly what index laws he is using at each step in order to realise the mistake he has made]
C: It’s all to the negative 1.
Me: I agree with that. That’s important. So what happens with this then?
C: That’s negative a half.
Me: What’s the rule you’re using to do that? What’s the index law you’re using?
C: Square root is a half, and to the power -1 is -1/2
Me: Do you mean this one? [I write (a^m)^n]
Me: I agree with that, but what happens to this? [pointing to the first x] That would be to the power negative one aswell.
[This is an example of a much more direct exchange than the other examples I have given. I have chosen to provide the correction, rather than asking a question. Is this more or less useful to the learner?]
C: Yes… yes.
Me: So what would it be in total?
Me: Yes, that’s what I think it is, which makes sense because that’s x^3/2 [pointing to the denominator in the original expression] because it's x^1.x^(1/2), but it’s on the bottom...