These suggestions are based on observations made while teaching Calculus I, II and III for many
years. They are very general suggestions and apply to most math and
science courses.
- You'll probably
find
that
the material in most university courses, including this one, is covered
much faster than it would be in high school. You'll
probably
also be asked to have a greater command of the material,
especially
in understanding the ideas (not just the computational techniques) and applying them
in new situations. This may take some adjustment in your style of
learning.
- The
instructor
and TAs
are
here to help you understand and learn the material, but actually
learning
it is your responsibility. The lectures are designed to try to
highlight
the important ideas and to give you some perspective on the material.
From there, you can digest the material as you read
the
text,
think, discuss the material with other students, and practice on problems. You
should expect that more learning will happen outside the classroom than
inside. On the practical side, a typical WU student should expect to
spend at least a couple of hours on calculus, reading and working problems, for every hour in the
classroom.
If you do this for all your courses, then being a student is easily the
equivalent of a full-time job.
- Read the text. An instructor can't possibly say everything that's there. A 50-minute
lecture would hardly be enough time just to read the day's assignment aloud, let alone
try
to highlight or clarify. Your instructor's job is to highlight and complement the
text so
that you can learn from it.
Simply relying on your lecture
notes
will (and should) only give you part of the picture. Ignoring the
text is the path to minimal understanding of the material.
- Reading a math
text takes time, focus, and active involvement.
You can't do it effectively if you're distracted every few
minutes with text messages, etc. Consider that you might spend a
few
hours reading a 100-page assignment in a literature or history
course. A
10-page reading
assignment in mathematics might take just as much
time.
Read it once quickly just for an overview. Then read
carefully
with scratch paper at hand and fill in missing steps. If the
author says "...this simplifies to...", then take time to
simplify it
for yourself!
When he writes "you should check that...", then check it
yourself!
Try to work
some
of the examples given in the text before you read the author's
presentation of the solution. Play with
the
material:
ask yourself "what if I changed the problem to...?" or "was there
any information given in the problem that I really didn't need?".
Write
down questions that come to mind. If you can't answer them, ask
your
friends. Try to understand why statements in the text are true.
One of the
marvelous
features of mathematics is that you need not (perhaps should not!)
trust
the author. If a physics book refers to an experimental result,
it
might be difficult or prohibitively expensive for you to do the
experiment
yourself. If a history book describes some events, it might be
highly
impractical to find the original sources (which may be in a language
you
do not understand). But with mathematics, all is before you to
verify.
Have a reasonable attitude of doubt as you read; demand of yourself to
verify the material presented. Mathematics is not so much about
the
truths it espouses but about how those truths are established. Be
an active participant in the process.
E.R. Scheinerman, Mathematics, A Discrete Introduction, Brooks/Cole, Pacific Grove, CA, 2000, p. xviii-xix.
- Try
the following: if you get stuck during a reading
assignment, go back to the beginning of the assignment and read again carefully. Find
the very first sentence in that reading that you don't understand. Ask someone about it. Then continue in the same way.
- Ideally, you
should read
ahead. The
lectures will be much more valuable if you try
to digest some of the material in advance. Also, you'll
know that some thing written on the board needn't be copied
down.
For
example, you'll know that those formulas the instructor is writing on
the
board are just the ones listed in the text. That frees you to
think
about what's being said and not be frantically transcribing
material
into a notebook.
- If you choose to
ignore the
advice about reading ahead, then at least read the material in the text
before you try the assigned problems. Some students start
out
with the problems and when they get stuck, they page back through the
text
looking for a formula or an example to help. This puts the focus in
the wrong place. Simply being able to "do the problems" is at best a minimal strategy for the course. You do have to do problems,
and you do want to have certain technical skills that the problems sets
can help develop, but you also want to understand the material.
Carefully
reading the text is essential for that.
- You can't learn
to
speak
French by merely sitting and listening, and you can't learn mathematics
by watching someone else do problems. You need to be involved. Do as many
problems
as you can realistically find time to do:
the more, the
better. It's useful to spend part of your study time with other people as long
as eveyone is an active player. Compare solutions to problems; ask each other
questions; make
up quizzes for each other; agree on some extra problems to try
together--such
as the review problems at the end of each chapter. Do whatever helps! But all of this
is
to improve your understanding: your study group won't be there at quizzes,
tests, or when you need your math in another course.
- There are some facts
and
formulas
that you need to have at your fingertips: that is, there are some things
you simply have to memorize. But
that's only the
beginning. Formulas are just tools and, admittedly, some assigned
problems are nothing but "drill," designed to be sure you really can
handle the tools. But when attacking a
non-routine problem, don't just hunt around for a formula. To
quote
an old but accurate text,
"(the) time
wasters
are
the formula worshipers ... who spend more time hunting a magic formula
than they would need to analyze the problem piece by piece using simple
familiar methods and calculations ... it cannot be said too often that
the ability to understand and solve problems does not come by
memorizing
formulas ... formulas are not substitutes for thought." (F.C. Dana and L.R. Hillyard, Engineering
Problems Manual, 4th edition, New York: McGraw-Hill, 1947.)
- Many
problems
on
tests
will be like problems you've seen before. Others will be
connected to problems you've seen but will be a bit
different. This isn't grounds to complain that "we never saw a
problem like
that
before." If you can only do problems exactly like ones you've
seen
before, what's the point? The purpose of problems is to
help
you understand the ideas and techniques, not merely to repeat solutions of
more problems of exactly the same type.
- Stay on top of things. I admit that it's hard to keep up with all the stuff you need to do, but it's important not
to fall behind in the course. A lot of the material we cover is
interdependent,
and if you're not comfortable with the material from two days ago,
today's
lecture might be totally incomprehensible. It's better to be working a bit
ahead -- then the lectures will be more helpful and you'll have a buffer when an
emergency
strikes. Also, there's a
limit
to
how fast you can digest material and learn to use it in your
thinking.
Avoid cramming just before tests by staying on top of things. It
makes life much more pleasant.
- Make use of
your
instructor
and the TA in your discussion session. Office hours are set aside
specifically to see students; those times are for you! Most
instructors
are also willing to set up special meetings with students as time
permits.
Your responsibility is not to waste these times. Try to solve
your
own problems first, together with your friends. What you learn that way
will
be more valuable than the same answer "given" by your instructor or
TA.
- Bring up questions quickly,
as soon as you realize there's some difficulty you can't
resolve. It's hard for any instructor to cope with "I
haven't understood
anything for three weeks"-- we aren't magicians.
- Use the Calculus
Help Room, the residential peer mentors (RPM's) and other resources available at Cornerstone: The Learning Center. Consider the opportunity (at the start of the semester) to join a Peer Led Team Learning (PLTL) group.
- Use any supplementary material or examples that are linked with the daily syllabus.
These words to
the wise should be sufficient. Best wishes as you enter the world
of calculus! |