active learning:

encourage students to think about -how- they learn.
that is, in high school you can get by w/ easy memorization and
mechanics; often you can get by without really needing to work.
(imagine a boxer: if you're a big guy, you can beat the snot out
 of most people w/o really trying -- but when you go up against
 a pro, you gotta train, you gotta pay attention.)
(in HS I'd just read the text on the sofa, go to class, do the
 homework the night before, and ace stuff)

-- here you're gonna do much deeper stuff, so start paying
attention to how you learn (in case you don't already)

The two main arenas for learning are:
- the book
- the lecture

NB: it's often helpful to make a -space- of your own;
EG: a study space at home or in the library, for reading/working;
  carving out your desk in the lecture hall, or making any desk your
  own, etc.

Homework scheduling:
- better early than late
  You may think you're so cool b/c you do the homework the night
  before it's due.
  Trust me, it's -so- much cooler to do it the afternoon you get it.
  It's particularly cool when the sexy boy who sits at the front of
  class and always knows the answers comes to you and says "hey, I
  couldn't get this one, wanna work on it?" and you've already done
  it.
  
  NB: I'll try to make homework be on stuff we've -already- covered,
  so you can get started right away.
  Many people assign homework on stuff that -will be covered- in the
  coming week, which totally encourages putting homework off.
  I'll avoid this.

Particular things that can help:
- book
  + reading
    Some things work better written than said;
    some people learn better reading than listening.
  + marginal notes
    writing your own -reactions- and -thoughts- and -clarifications-
    can be really helpful. Some people swear by highlighting; I find
    this usually too passive and brain-dead.

    In fact, I find it even better to -rewrite- sections of the book
    (this is like re-writing your notes), but this can take -ages-.
- lecture
  + listening
    Some things work better said than written;
    some people learn better listening than reading.
  + thinking
    I see lecture as largely a time set aside for you to think about a
    topic.
  + talking
    If you don't understand something in the reading or in lecture,
    a great way to understand it is to ASK A QUESTION.
    (NB: feel free to interrupt -- if the lecturer doesn't pause
     enough, that's their fault, not yours.)
    It's even better if you -think- first: this clears up many
    questions. If you still can't figure it out, then ask.
- taking notes
  ...but note that many people -shut off- when taking notes.
  It's really -hard- to -think- and take notes at the same time, and I
  tend to prefer -thinking-, unless you're going over the notes later,
  when you'll have more leisure to think.
  (and as noted, I don't do this much)
  + thinking
    I don't write down much of what's on the board,
    and I write down stuff that's not on the board.
    This is partly b/c I don't go over my notes much:
    I try to -think- during lecture, and use my notes as a sketch pad
    to work out my thoughts.
    Even if you haven't time to think it through (b/c you're following
    along, say), I find it VERY helpful to write a QUESTION to myself,
    or at least a ???, to indicate that there's something I want to
    think about.
- rewriting notes
  this takes -ages- to do properly, and you'll end up
  rewriting them many times if you keep studying;
  hence I like using a computer.
  It's worth it -if- you continue in the subject: you'll be happy in
  Grad School that you did.
  But it's -hard- to find the -time-, and if you don't study that, or
  even if you don't study -exactly- that, it's likely not very
  helpful. (EG, for me, maybe 4 or 5 undergraduate classes it would
  have been really worth writing/rewriting my notes. My POV is that
  you'll have time in grad school, but this is a useful way of
  understanding something that you -don't- understand: for now you
  should use it as a special technique, for when you're especially
  confused: it's a way of putting in more hours, and helpfully.)
  That is, I tend to only -really- understand a topic when I've
  rewritten it for myself.

Note that ideally all these will be possible; in particular, it's best
if you have -lectures- that mirror the -text-.
Unfortunately, writting notes and a coherent text takes -much- longer
than lecture, and texts are -far- more rigid; hence the only time
lecture precisely mirrors the text is when:
- the lecturer is lazy or uncreative
  (NB: the lecturer can be very creative but lazy, or very industrious
   but uncreative)
- the text precisely fits the lecturer,
  esp. if they're the author.
This is not to say that they should be -identical-, but rather should
compliment each other.

One advantage of a good text is that you don't need to take many
notes (b/c it's all written down), and you can use class time to
just transcribe your -thoughts-.
Also, texts are generally better organized and more legible than your
notes.

OTOH, big calc books are often so -fat- that you have to skim them,
which teaches you to zone out. In more serious math books, you can't
skip stuff so easily.

----------------------------------------------
NB: just b/c you learn okay reading, or better reading than listening,
does not mean that you can casually skip lecture, for a few reasons.
1: new material
   I often cover material in lecture that's not in the text;
   this is inevitable if I want to teach well, unless I've already
   written up my notes.

   Thus if you skip lecture, you won't learn this and you'll get very
   confused.

2: focus time
   Lecture is good also as a scheduled time to think math;
   if you don't come to lecture, you should schedule time to think
   math on your own, or with a study group.
   Many people who skip lecture do so b/c they have trouble with
   time management, and thus don't make up this lost time.
----------------------------------------------
Note that people have different ways of learning;
I've outlined some above, but I repeat/extend;
be aware of these, and use whichever ones you 
- listening
- speaking/dialogue
  (lecture, section, office hours, study groups;
   explaining it to someone else is also good)
- reading
- writing
  (I find this very powerful, but it takes ages.
   When absolutely, positively need to understand something,
   write it in your own words.)
- drawing pictures and diagrams
  either of concrete situatitions, or the overall structure of a
  theory, or whatever
- doing problems
- verbal:
  putting stuff in words, having special words, thinking of something
  -as- a language -- these all can help.
- mneumonics
  (stupid or otherwise,
  either ones I give you or ones you make yourself)
- relating this to other knowledge/fitting it into your scheme
  (this is very powerful, but requires lots of knowledge already)
- doing something physical/active that expresses it
  (this is denigrated in higher ed, partly b/c it's not terribly
   suited for most abstract stuff, partly b/c it's seen as
   "kid's stuff" -- but some examples can be helpful.)
  (I find this particularly good for learning vocab, say)

Look, the basic point is to, someway or another, form lots of
connections in your brain.
The above suggest certain -faculties- that you can -leverage-; note
that they are largely -language- based, which is why:
- when some budding mathematicians asked a big 19th century
  mathematician what they should study, he said "Languages".
- SAT-Verbal scores are an okay indicator of general learning skillz
