The first story describes how Cambourne learned to iron (see it here).
This story describes Cambourne's attempt to understand a high school topic.
Recently I had cause to call a colleague who teaches Maths Education at the University where I work. I needed some help with the binomial theorem so that I could help with my daughter's homework. Here is a verbatim account of the telephone conversation I had.
B.C. Hello Grahame. I need to help my daughter understand the binomial theorem, (a+b)^2=a^2 +2ab +b^2. I vaguely remember it from my high school days, but I can't remember much about it. Can you explain it in language?
G.W. That's easy. It means: If you want to raise the sum of any two numbers to the power of two then....
B.C. Hang on! You've lost me. What do you mean, 'If I want to'? Do I have a choice. Why any two numbers? What does raise to the power of two mean?
G.W. Uh oh. I'll have to go back a level with you won't I? Try this: If you want to square any two numbers...No wait a minute, That type of 'If' language bugs you doesn't it. Try this: When you want to square the sum of any two numbers then you....(long pause)
B.C. What's wrong?
G.W. I don't like saying it that way. It changes the meaning slightly and will only lead to confusion later.
B.C. What do you mean?
G.W. Well let's say you want to go beyond 'squared' and 'cubed' and you want to raise the sum of any two numbers to the power of 'n' we have no words beyond 'squared' and 'cubed'.
B.C. I'm getting more confused and lost. What do you mean 'When I want to?' When would I want to raise the sum of any two numbers to any power? What's a power? Why the sum of any two numbers? Mightn't I want to do it with any three or four numbers?
Although I'm not sure why or when I'd ever want to do that. And I'm not too sure about 'squared' and 'cubed' or 'n'. And all that conditional language. Why do mathematicians talk like that?
G.W. I'll go back another layer. Let me see: Let's suppose that you needed to add two numbers together and then multiply the result by itself... Uh oh that's even more confusing isn't it? And what's more it doesn't mean what I intended... it's imprecise.. You'll have to call me back later. I need to think about the language I use. I've never reflected like this before.
It is clear in this description that learning was elusive. What made it more difficult for Cambourne to learn the binomial theorem than to iron? Consider current classroom approaches: are they more like mentoring (learning to iron) or the game of telephone (learning the binomial theorem)? Why?