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I once taught a successful class to second graders, on euler's theorem for polyhedra by handing out colored cardboard models and letting them count the facets, etc... This would seem a suitable topic …

this is similar to some other answers. when i took basic graduate algebra from Maurice Auslander he handed out 16 pages of very terse notes the first day that he said was our Fall semester final exam …

Here is an excerpt from my class notes, inspired by the discussion in Joe Harris' book on desingularizing curves as an application of Bezout in space. The degree of a curve is the number of intersect …

I don't seem to have the option to edit my answer, but I wanted to correct myself: it is the scheme structure of the limit that determines the possible nearby varieties in the family, not vice versa. …

After many years of teaching the MVT is various ways unsuccessfully, I came to consider it more useful and practical to try to convey the corollaries of the MVT rather than the statement. …

Joe Harris's book Algebraic Geometry derives from his experience teaching algebraic geometry first by concrete examples at Harvard and Brown, but very little theory, which he said seemed to work well. …

Pete I agree that locally free is a good geometric intuition for projective. In my algebra class I gave an exercise for the students to prove that the module of tangent vector fields on a 2-sphere is …

It is worth noting that there is a lot of historical precedent for teaching it as a limit, which occurs already in Euclid. I.e. …

Please forgive these very naïve remarks. I am enjoying the chance to learn something about flatness in trying to contribute to this question. First of all, since as pointed out here, the definition …

These different goals suggest rather different approaches to teaching. E.g. if you want large numbers of students to go on, your course must be easy and give good grades. … Come to think of it I have written an essay on this topic, "On teaching". Perhaps i will link to it or even post it here. http://www.math.uga.edu/~roy/ …

I am going to argue that Dieudonne’ is actually using limits of Riemann sums to define his “Cauchy” integral. Dieudonne’ hides his use of Riemann sums within the proof of existence of primitives of r …

i suggest the proof archimedes wanted on his tombstone and its relatives. i.e. since two solids with the same horizontal slice area at every height have the same volume, hence by pythagoras, the vol …