training the next generation (a first post in a potential thread)

so i finally have something ready to write about [1]: i'm advising a former calculus student of mine on a summer (undergraduate) research project .. and (for me, anyway) it's a scary thing.

many of my colleagues are old hands at this, i know;
so if i seem naive, it's because this is the first go and i don't know any better.

i worry, for good reason: he's studying a topic i know little to nothing about. in particular, it's not clear to me how easy or hard the problems i pose to him really are .. and as a result, i don't know how much frustration i'm throwing his way.

it depends, of course, how much the student is willing to work. if i give him a badly-posed problem, then a good work ethick can actually be bad .. in the sense that, by working with abandon for too long a time, he burns out and gets turned off by pure maths in the future.

in case it's not clear, i'm encouraging the student to make his own conjectures;
of the established theorems whose proofs he can easily understand,
i'm suggesting him to try his own variants.

in other words .. and for better or worse ..
i'm insisting that i don't give him orders;
he'll have to train himself to think like a pure mathematician,
but i'll be there if he needs advice or guidance.

i was worried about his technical chops before .. until i realise that if his proof-writing skills require work, then this is potentially the best way that he can practice them: by working with a topic that interests him.

let's hope these aren't another example of famous last words ..!

[1] as you can see at the end of this post, i'll be tagging these thoughts with the handle "Σ:nextgeneration" .. and the usual disclaimer follows: unlike other maths blogs out there, i'm not out to train or educate maths-inclined people out there, at least not directly. instead, i'm going to show you, through my mistakes, what not to do.