no matter how many proofs I do on the board, hardly anyone can do one as exercise, or even begin one. they never even seem to think of starting with the definition of convergence, when trying to prove convergence.

when i tried to help them get started by asking what is the definition of convergence, it seemed nobody had bothered to learn the definition.

the exercise i assigned was a proof i had already given on the board in class, but still no one had any idea how to do it. i do not know if they just ignored what i wrote on the board, or tried and did not understand it.

these are smart, curious students, and i love them, but we are both struggling. they all have relatively high AP scores but know little about calculus, or mathematics, or effective learning. still there are a lot of bright spots, i get wonderful questions, sprinkled in with mouth droppingly odd ones. they do want to understand the stuff, but seem to have no experience either at rigorous reading or writing of mathematics.

many freshmen seem never to have seen a geometric series, which in my day was explained (without convergence details) in the 8th grade.

I am enjoying my class, and i hope I am helping them with a difficult transition to college, but sometimes wonder if my expectations are totally out of kilter. Am I doing something strange by teaching series to freshmen honors students?

when do other people try to teach these things? to freshmen? juniors/seniors? (i have no relevant experience to guide me, since as a freshman in the 1960's, without prior calculus experience, our class was taught rigorous sequences and series in the first semester, indeed at the beginning of the semester. the first homework exercise set included proving e is irrational using the series expansion.)

In high school we were taught propositional logic with quantifiers and truth tables. we never had calculus, whereas nowadays most of my students, honors and non honors, have had calculus. Still most do not understand geometry or algebra or logic or proof, and are apparently much less prepared than we were for college maths.

Is something backwards here? Why teach calculus to people who do not know any of the more basic material, and without encountering reasoning or logic? I.e.; those of us who were taught more elementary material well, seemingly were better prepared for college calculus than today's students who are taught watered down calculus badly in high school, with time for it purchased by omitting a decent algebra and geometry background.

There is no way to turn back the clock, but there is a real challenge here for us in university to try to accomodate these extremely math deprived high school graduates in college math. What do the high school students out there suggest?