Alan Musgrave is an amazing (philosophy) teacher. He taught me to be intrigued by math by regaling us with 'gossip' about Pythagorus. He had a golden glint in his eye as he genuinely wondered about a straight line (even though he would have delivered that same lecture over 100 times before). He could capture a lecture theatre of well over a hundred people and get them to play "What If..." without anyone shouting others down - no matter how absurd our ideas were. And although the room was filled with a most diverse bunch, he had the ability to challenge our assumptions where we 'let it all go', and played with our thoughts. We would often leave the theatre giggling with delight as our brains popped and buzzed with new possibilities. He was a mind alterer, a mind expander, and a superb story teller.
He showed us that playing with ideas was delightful and for the first time I felt okay with the fact that I had 'wasted' my intermediate years by dreaming things up. My self directed (corridor) maths curriculum involved an obsession where I tried to find a triangle whose angles did not equal 180°. At other times, I had an overwhelming urge to find (or imagine) a new colour (not a shade of an existing colour) a new colour - (I blame the yellow highlighter for opening up this possibility). And I also had an 'unhealthy' obsession with money (not the value of it, but the pictures on it). The $10 note, for example, had pictures of the Mt Cook lily (which is very closely related to the edelweiss). I found these (pre-google) connections fascinating and told everyone I knew - complete with a rendition of the song from The Sound of Music whenever I saw someone with a $10 note - some liked it, others seemed more concerned with buying their dagwoods from the school canteen.
I think we should encourage the playing with our thoughts more - it's so much more exciting than just putting other peoples ones in.