How to study?
The university is not like high school. To do well in your classes you
will have to do much work on your own. This means reading supplementary
texts, seeking out help from a TA and the professor, going to the library,
working with fellow students. The rule of thumb is that for every hour
of class time you will have to spend at least two hours
studying on your own, and it has to be time well spent, rather than
"spinning the wheels", just staring at the pages of the book. Here
are a few hints you should consider following.
Find the right place and time
You won't absorb facts and concepts if you don't concentrate on what you
are doing; and to concentrate most people need a stretch of time of at
least a half hour, and an environment free of distractions. Kick out your
roommate, turn off that TV, sit at a comfortable desk with good light.
Or go to a coffee house, a park, or wherever you think you can find some
peace. Use your favorite pen, clean notebook -- things you feel
comfortable with.
Find the right mood within you
What is your first thought when you sit down to study? Is it "heck, I
have to do all this by tomorrow and it's so useless"?
If so, then your brain will treat
the things you are trying to learn as its enemies, and will get rid of
them in the next day or so. You don't have to sing with joy before opening
the book; it's enough to remember why you went to college. The only good
way to study is to study for yourself. The grade, your parents, the
financial aid and your first job are all secondary.
Divide the work into fragments
You cannot assimilate a whole chapter of a textbook in one session; you have
to break things up into small pieces. Most concepts (especially in math) are
built up from simpler ones in a very systematic way. It is impossible to
even begin to learn them without mastering the earlier ones. After reading a
paragraph of text, reflect on it; try to relate it to what came before, and
to predict where it might be leading.
Search for the meaning
Knowledge is not a random collection of unrelated facts and ideas.
Read sentences, not words. Read paragraphs, not sentences. Always remember
the context in which they are written. Think about the meaning of
what the text says, not about the way it says it. Look at a formula as
a statement which is telling you something, and not as a bunch of symbols
pieced together for no reason. Write it out as a complete sentence: "the
slope of the tangent line to a graph of a function at a point where the
function is differentiable equals the derivative of the function at that
point", or "the definite integral from a to x of a
continuous function f is an antiderivative of f(x).
Think, think, think!
Studying isn't cramming in formulas and statements -- it's
understanding them. The
fact that you have read something doesn't mean that you have comprehended
and learned it! You have to rehash the material in your head, look at it
from different angles, experiment with it, until you feel you could have
written the particular section of the text yourself. It is also
infinitely easier to remember concepts that you understand than a bunch
of facts that appear to be meaningless and unrelated.
Don't get stuck on anything for too long. If you are at a loss
as to what a given concept means, go to the library and look at three books
on the subject; one of them is bound to have an explanation which will
appeal to you. Or call up a friend and ask if she can explain the
topic.
Experiment
When you see a mathematical statement, try to write down several concrete
examples. A physicist who tries to understand how gravity works takes a
stopwatch and drops a brick from a cliff, then drops an apple, etc. In
mathematics when you see a formula such as
(f(u(x)))' =
f'(u) u'(x),
you should immediately write:
((x2+1)2)'
= 2(x2+1)2x,
(sin (cos (t)))' = cos(cos t)(-sin t),
(e3x+1)' = e3x+1 3,
and so on. Mathematics is all about abstraction, but you won't
understand or learn the abstract very well without knowing what it
says about concrete examples.
Test yourself
When you feel you are ready, pick a few problems which are answered in the
book, copy them onto a sheet of paper, and pretend you are taking a test.
Check the answers. If any are wrong, go back and redo the problems more
carefully. If you still made mistakes, start reading about the topic from
scratch. Invent similar problems of your own, and solve them.
Pretend that you are making up a quiz for the whole class.
Be patient
The nature of science is such that given three problems which look
similar, one might take 30 seconds to solve, another 5 minutes, and the
third half a century. Hopefully your final won't have the third type on
it, but you may very well see the second kind. Don't stop working on a
problem until you have tried several approaches and you are really stuck.
Ask someone for a hint (not a solution), and try again.
Be correct and precise
Mathematics is a language. It has its rules, similar to the rules of
English grammar, punctuation and spelling. You wouldn't turn in an English
paper with "i ain't know The what i shoulda knew" in it, would you?
Then don't write things such as f = x2 + 1
= d/dx
f x2 2x on your homework, on
the test, or even when you are doodling on the margin! See the
separate handout on writing correct mathematics.
You may also want to look at a much more comprehensive
guide
by Peter Alfeld.
Please send comments, suggestions, corrections to
behr@math.niu.edu.