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What study practices have led you to the best success in learning mathematics (or applied math, or theoretical CS) in the classroom (undergraduate/lower graduate level), especially in courses that would otherwise be "too hard" for you?

Alternatively, if you teach math, what practices have your best students devised that set them apart.

Obviously, "study hard and constantly push your knowledge on all presented material" will be at the core of a lot of this, but I'm looking for something slightly more specific.

Examples of things I've found work well:

-Finding the simplest possible version of a proof that's still reasonably short, even if imprecise, before moving on to a more full and formal version

-Studying a few core concepts of the class before entering it; it gives a complementary perspective to the instructor and gives more time for concepts to germinate in the mind

-Do textbook problems, but only hard ones that push your knowledge. Easy textbook problems, that are basically just doing operations by rote, are a waste of time.

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The Moore Method helped me learn various areas of math. It abides by the dictum that teaching is the best way of learning. One could modify it by pretending that there is an imaginary audience in the room and lecturing out loud (i.e. if you are by yourself this is an alternative to the traditional Moore Method).

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I just looked up the Moore Method, and realized that one of the hardest yet most satisfying classes I've taken actually seems guided by its principals – Elliot Jans Jan 23 '11 at 18:36
@Elliot: I guess a course could actually be guided by principals, but you probably meant principles. ;-) – Hans Lundmark Jan 23 '11 at 18:51
Thanks. I wasn't aware I was on spelling.stackexchange, but it's good to know. But really, that was a great way for a difficult class to make itself accessible to the people with the lowest level of knowledge (CS sophomore) without becoming dull for those with much more knowledge (Mathematics MSes from France) – Elliot Jans Jan 23 '11 at 19:38
I'm not a big fan of the Moore methods for several reasons. First of all,even if your students are good enough to handle it,it produces a very alienated and narrow viewpoint on the subject. Basically,one develops a completely isolated thinking process and doesn't benefit from outside ideas. Secondly,this method really works best when the subject is one whose central precepts follow directly from the definitions-such as point-set topology, most of which doesn't really require any original thinking to derive. A real analysis course would be FAR more difficult to teach this way. (continued) – Mathemagician1234 Dec 18 '13 at 18:43
(continued from above) But one of my strongest objections to the method is that in a classroom setting, it encourages competition in the lowest Darwinist sense of the word,an attitude more approprate for Wall Street stocktraders then mathematicians. I remember a friend being in a Moore method class in algebra where all the algebra texts disappeared from the library because one of the top students in the class-who worked in the library-took them all and locked them in a safe where only he had access for the length of the class and whenever he got stuck, he'd access them.(continued) – Mathemagician1234 Dec 18 '13 at 18:48

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