Total length: 6 minutes
B is stuck on this question:
So far, B has tried to use ‘reversing the chain rule’ to integrate this, i.e. differentiating (x^2+1/x)^3, but has realised this will not work, as there will be x terms outside the brackets. We have just been focusing on integrating by reversing the chain rule recently, which explains some of B’s fixation.
B: I thought you could use the chain rule for this, because it has powers and brackets
Me: Yes, that’s why you’re thinking chain rule… that’s the first thing that crosses your mind, but then the second thing would be - oh, I can’t do that, so now what am I going to do?
B: Good question…
[long pause, I take my attention away, allowing him space to work on it… he has created the simpler example - integrate (2x+6)^2 - presumably following yesterday’s episode with A (not him).]
B: Reversing the chain rule works for this one, does it not?
Me: Yes, because when we differentiate (2x+6)^3 we get a number outside the brackets… but the one we’re working on can’t be chain rule, can it, because you get terms involving x, so the chain rule can’t be what it is about. You need to find a way from coming away from the chain rule… you’re ruling out the chain rule, OK, good, what else?
B: There’s basic differentiating, but that won’t help either…
[pause, I’m thinking again of the strategy ‘simplify the problem’, and - unhelpfully - have in mind that he could remove the squared and try integrating that]
Me: Let’s go back to the idea of simplifying it again, what’s another way it could be simpler? What else could you vary?
B: Vary what’s outside the brackets?
Me: OK, what would make the question easier?
B: Put an x there [in front of the brackets].
Me: Mmm… well you could, but you also have to differentiate that [pointing to 1/x]… what else could you do to make it easier?
B: Something to do with zero, no that wouldn’t help...
[I realise that I am funnelling here, and snap out of it by coming back to my three questions, which may be helpful, but also may be inhibiting me from finding a good question]
Me: What other questions might I ask; you can’t compare it with anything, what is it an example of, well, it’s not the chain rule…
B: Is there anything else you can use?
Me: You can just integrate things directly
B: Well, you can’t, because of the 1/x
[I pause again. I’m trying to find the right approach. I am aware that I could tell him how to do this here, but I don’t want to, because it feels valuable not to tell right now. This is about modelling ways for other alternatives to come to mind.]
Me: Here’s a way of thinking about this. So, the chain rule is not useful, imagine the chain rule was never invented, what would you do?
[long pause.. I’m aware that this is a bit of a guess-what’s-in-my-mind, as I know the (only) way is to expand the brackets, but still think that it is worthwhile]
B: You could work backwards?
Me: But the chain rule has never been invented, so what are you going to work backwards from? [pause] What’s it for the chain rule, it deals with things with brackets and powers… Do you remember at the beginning, and we didn’t know about the chain rule, what did we do?
B: We just differentiated it simply, the thing we had.
Me: Which was what? A massive thing…
B: You had x^3 something, plus x^2, and just sorted each one individually
Me: Can you do that?
B: I thought we were not supposed to do them individually?
Me: But could you?
B: You could…. Ohhh! Would you… you could times out the brackets, would that help? Ah.
Me: Give it a go… yes, that.
Me [later, after he has expanded the brackets, etc.]: The chain rule is invented to deal with large powers of brackets… [I have found, and am showing him a problem I presented just before learning about chain rule, in which they were asked to find stationary points of (2x-1)^5, and J had multiplied out the brackets…]
Me: It’s hard when you get stuck on one thing... I could have just told you how to do it, but I think it was important to try to find ways of switching out of a pattern of thinking, that’s what I was trying to make you aware of there
B: It’s just that we don’t usually multiply out the brackets.
Me: No, that’s right, the reason that this approach is viable is because it’s squared, so it is not too much work to multiply out the brackets directly, but if it was to the power five... it’s very unlikely you would be asked to multiply out five brackets, but you could! It’s sneaky isn’t it, all the ones we’ve done recently have been chain rule, chain rule, chain rule, but this one is not.
[Then, a few minutes later, this conversation happens]
Me: Is it frustrating when I don't tell you the answer, or do you understand why I do it?
B: Well, if you gave the answers all the time, you wouldn't find anything out for yourself.
Me: I bet sometimes you're like, come on! [Turns to A] Must be like that sometimes mustn't it, A?
A: Mm-hm. Yesterday was one of those days.
Me: You should just have said: can you just tell the answer please!? And I would have said, yeh, alright! We can have that as a deal from now on. Let me make some cards which say: Please can you just tell me how to do it. How many do you think you'll need for the rest of the year?
Me: Happy with three, A?
Me: [Makes cards, while they carry on working]