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As a successful student of mathematics, I have encountered
a few insights, general strategies, and bits of advice which I personally
found helpful during my first ten years of studying mathematics. As an
active researcher in mathematics, these lessons continue to guide me. As an
instructor of others in the study of mathematics, I have collected these
on this webpage to share with my students, current and future, as well as
anyone else who reads them.
Learning Mathematics
The first step in being a successful math student is knowing how to study.
If this sounds obvious, let me say that it really isn't. I didn't really learn
how to study until my first semester of graduate school, and then it came as
a complete epiphany, motivated by desparation. Both in high school and as an
undergraduate I put very little effort into school, nothing more than the
minimum to get by. It wasn't that I wasn't interested in learning, far from
it; rather, I had a perception of "studying" as a kind of cheating, like
cramming for a test. I had always gotten by simply by sitting in class,
absorbing lectures, and not really thinking too much about it.
Graduate-level math classes changed all that. Sitting and hoping to absorb
material does not work with graduate math classes. The situation forced me
to learn how to study effectively. Once I learned how to study effectively,
I realized I could have had a much higher GPA as an undergrad than I ended
up with, had I known how and had the motivation to study mathematics
effectively. To be fair to my high school and undergraduate teachers,
the blame for this rests squarely on my shoulders; though ultimately, it
just took material that was sufficiently difficult to make me see the
light.
By "studying mathematics" I mean learning mathematical ideas and techniques.
I do not mean "cramming in preparation for a test," nor am I referring to
techniques for memorization. Memorization is a poor learning technique at
best; it no more helps you learn mathematics than buying a Japanense
dictionary enables you to speak Japanese.
So, how does one study mathematics? The answer is both simple and elegant:
read the text very carefully, one sentence at a time. Do not go on to the next
sentence until you have fully understood the current one. It helps to take
notes, writing a summary of the ideas you're reading or seeing in a lecture.
Then, once you've finished reading or once the lecture is over, take a blank
sheet of paper (or a stack of them :) and write an explanation of the ideas as
though you're writing a lecture. It's not important whether you actually give
the lecture to anyone; writing and rephrasing the ideas forces you to clarify
the ideas in your own mind. It is also extremely useful at exposing gaps in
your understanding, which you can then remedy by rereading the relevant
portions of the text.
Depending on the level of the mathematics text and your own level of
understanding, this technique may always not be necessary in its full form;
however, learning mathematics requires the patience to take small steps and
make certain that you understand each step before continuing on. Many small
steps add up to a large result.
Taking Mathematics Tests
A well-written math test is designed to test your understanding of the ideas
and techniques you've learned in class. This typically involves working
problems in lower-level courses and proving theorems in higher-level courses.
Ideally, if you have a firm grasp of the material and are comfortable with
your understanding, you can start by stating the problem and continue writing
each step explicitly, proceeding until you've solved the problem or proven the
result.
Of course, it's often the case that you may not immediately see how to
complete, or possibly even how to start a given problem. This can be
indicative of not having learned the material effectively, but it isn't
necessarily. Even when you know the material well, it can be hard to see how
to start a problem, especially in a test situation where you're already
feeling under pressure from limited time.
In general, the best cure for test anxiety is a healthy dose of confidence
in your understanding of the material, which comes from effective studying
and doing practice problems. So many people have test
anxiety specifically in mathematics classes that the condition has its own
name, "math anxeity." I'm sure this is a result of the popular perception of
mathematics as prohibitively difficult, but it is really quite backwards.
Mathematics is unique among human activities in the there is no uncertainty
about the results of mathematics. If you are careful at each step in a
mathamtical solution or proof, then your final result will be every bit as
certain as what you started with. Be patient and careful, and you have every
reason to be 100% confident in your results.
That said, what if you don't see how to start? First and foremost, you
should not consider cheating or trying to fool your math teacher. Aside from
the obvious point of dishonesty, trying to fool a math teacher on a test is
unlikely to work and likely to have repercussions. Every math teacher was once
a student, and we are familiar with the techniques students use to cheat (not
necessaily having used them ourselves, of course, but having learned them
they same way today's students do).
Answers listed without work are not worth much, since
the point is not getting the actual answer but showing that you know
the method of getting the answer. When two people who sit together turn in
the same incorrect work, it is hard to avoid drawing the conclusion that
cheating is going on, and this both a disappointing and angering situation
for an instructor to be in. My own policy: if I catch a student cheating on
a test, quiz, or final exam, that student gets zero points for that test,
quiz or exam.
OK, so suppose you don't know how to start a problem. What should you do?
First, don't leave the problem blank, because that guarantees that you get
no credit for it. Write something down, but not just anything; read the
problem and write down something that's true. If you don't see how to proceed,
write down something else that's true. This is always the way to start a
problem; start by summarizing what you know.
Next, read the problem carefully and state clearly what the problem is
asking for. Very often, in the course of explaining what the problem is
looking for, you will see how to use the information you've been given
to answer the question. If not, read the problem again and see if
there's anything more you can say based on what the problem says. If
you're still stuck, then move on to the next problem and if you have time,
come back when you've completed the ones you know how to do. There's no
shame in admitting to not seeing how to proceed, and such an admission
is far less insulting to your instructor than trying to fake an answer.
Avoiding Common Errors
In every area of mathematics, especially in the first courses on algebra
and calculus, there are certain common mistakes that students tend to make,
at least until they've lost points on tests, quizzes or homework several
times and learn the hard way. Each of these mistakes can indicate simple
carelessness or can be indicative of a deeper problem, a misunderstanding of
how the rules of mathemtics work. You can avoid making these common mistakes
by first understanding why they are mistakes, and then being careful with
each step you do when working problems.
Common Algebra Mistakes
- forgetting to distribute
- distributing an exponent
- cancelling terms instead of factors
- misunderstanding fractions
- misunderstanding negative and fractional exponents
Common Calculus Mistakes
- forgetting use the product, quotient and chain rules
- forgetting to add C when solving indefinite integrals
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