I've had a few difficulties in some discussions with students recently. It comes down to this: "How do I explain something that is so blatantly obvious it doesn't need explanation?"

The problem really is that a particular concept can be obvious to me, but not obvious to a student. The danger is then that, in a lecture, I just assume that a student is implicitly happy with such a concept and I plough on forward without any hint of explanation. The consequence is a bemused student who just doesn't get what I'm talking about.

It is really, really hard for a lecturer on two counts. First, you have to recognize the fact that something might not be so obvious to a student. That's pretty tough when it's second nature to you. Secondly, you have to find a way of explaining the (to you) blatantly obvious.

Here's an example. I was asked by a student why velocity was equal to rate of change of position. He didn't get it. My initial response was "But that's just the definition of velocity - it IS the rate of change of position - what's there to get?". But that didn't cut it for the student. I had to dig deeper to see what the stumbling block was. It turns out that he had a vague understanding of velocity from his everyday experience, but not one he could put down in precise, physical terms. He was also (as many students are) uncomfortable with using vectors. Therefore, he couldn't see how to match a rate-of-change of position, described in terms of vector calculus, with his everyday concept of what velocity was. When I said "They're the same thing by definition" - that didn't help one bit. __Why__ are they the same?

Often, when someone is not grasping the blatantly obvious, there's some underlying block in their thinking. In teaching and learning literature, we talk about threshold concepts - ones that are really difficult to get, but once grasped, transform the way that someone thinks. Once someone's crossed that threshold, it might well be blatantly obvious. But beforehand, it certainly isn't. Teaching a threshold concept is very, very hard indeed, especially if you yourself crossed that threshold many years ago. Often these things are best taught by those who have only just 'got it' - i.e. students teaching students, or peer-instruction.