Effectively For Tests
I am often approached by students
about their test performance—often, right after they get their first midterm
exam back. They are concerned about their poor performance and confused how they
could feel confident about the material, and yet do so poorly on the exam. They
say, "I came to class every day and I did the homework assignments without any
trouble. So why did I nearly fail this exam?" (I say "nearly fail" because it's
quite rare for a student to fail the exam outright if they've put in a
reasonable amount of work.)
Upon further conversation, it becomes
clear that students in this situation inevitably have failed to follow the
cardinal rule of studying for math tests:
Starting in elementary school, we get
used to doing math a certain way. This idea of what constitutes mathematics is
further inculcated throughout junior high and high school. But then we arrive at
college and, as it turns out, math in college is different than math in the K-12
system. Nothing in the lower grades adequately prepares students for the way
college math is done.
Allow me to illustrate with an
example. Remember when you learned to multiply two-digit numbers together?
Probably not, but you'll see what I'm getting at. If you're lucky—and most of us
were not—the teacher explained the significance of "carrying digits" and
shifting the second row of numbers one place to the left before adding to get
the final result. Of course, even most calculus students now would be
hard-pressed to explain what's really going on when you perform these
operations, but never mind that for now. The fact is, we probably didn't learn
this ever, and even if we did, it certainly didn't count as "learning" to
multiply two-digit numbers.
So how did you actually "learn" to
multiply two-digit numbers? You did a ridiculous number of multiplication
problems. You did pages and pages of them. You had timed tests where you had to
finish a whole column's worth of problems in 30 seconds. You multiplied until
you had worn your pencil down to a nub, your eraser down to the metal band
holding it, your fingers down to bloody stumps.
But wait. . . isn't this mindless
repetition of multiplication exercises within the scope of my dictum:
Well, yes and no. It's true that we
did lots of problems in grades K-12, but it turns out that college math isn't
quite so simple. We move a lot faster in college. You probably took
a whole year in elementary school to go from multiplying one-digit numbers to
multiplying two- and three-digit numbers. You learn about limits,
derivatives, and integrals in a matter of months.
Furthermore, college math tends to
emphasize concept over computation. It's important that you actually
understand what you're doing when you find a limit, take a derivative, or
integrate a function. So that's what professors spend a lot of class time doing.
The problem is that you have to take
tests. These are not typically essay tests, so in the end you don't really get a
chance to explain how well you understand the "idea" of calculus. Tests involve
problems with actual numbers and variables, concrete functions, and a
computational goal. Some mathematicians try to argue that if you understand the
concepts, you will automatically know how to do the computations. This is pure
bunk! It's like explaining multiplication to an elementary school student using
blocks and containers until they understand the idea, and then telling them to
sit down and do a page of multiplication problems. One must always be shown the
computational procedure, which is justified and reinforced by a solid
understanding of the underlying concept.
So what's the remedy?
But remember when you said, "I came to
class every day and I did the homework assignments without any trouble"? Isn't
doing the homework enough to practice the material that will appear on the
It's true that exams, at least in my
class, will contain problems that are very similar to homework problems you have
seen. It's also true that homework is assigned to give you the opportunity to
practice doing computations. But does this really factor in when you are
studying for a test?
To clarify my point here, let me tell
you what I think happens when you sit down the week/night/hour before the test
to study. First, you read through the relevant sections. If you're especially
studious, you might even highlight. If you're anal retentive, you will perform
said highlighting with four or more different colors.
Next, you look at the homework
assignments you dutifully completed. As you peruse each problem, you are
reminded of what you did to solve each problem (or what the professor or TA did,
which you dutifully copied into your notes) and then with a self-assured
smugness, you declare to yourself, "Yes, I remember how to do this problem."
Then you go on to the next problem, rinse and repeat. By the time you get done
reviewing the homework assignment you completed a few weeks ago, you are now
fully convinced that you are Einstein and that no problem could stop you.
Fast forward to the next hour/day/week
when you're staring at problem one on the exam. You've managed to write down the
first couple of steps, but then you're stumped. How did you get it while you
were studying but not now?
The simple answer is this: looking
at problems is not the same as
This all seems very self-evident, but
I am past even being surprised at the huge number of students who study
ineffectively. I think what happens is that when math starts getting hard in
college and tests start to cover material spanning tens or even hundreds of
pages from the textbook, we revert to study skills that we think apply. What
other subjects do you know where you have to study a wide range of material
spanning tens or even hundreds of pages from the textbook. If you said anything,
you'd be correct. In fact almost all subjects in school fit this description.
And how do you study for almost all subjects? You read and read and read some
more. And you highlight using many colors if you're anal retentive.
Now this begs the question: why are
you studying mathematics as if it were history? Why are you reading your
textbook as if it were a Joyce novel? Because that's all we've been taught to do
since elementary school when faced with an upcoming exam covering a lot of
material. Somehow we've forgotten that to learn to do mathematical computations
And when I say
I have something more in mind than
looking at problems and then writing down the answer from the back of the book.
You must force yourself to write down the problem on a blank piece of paper and
put everything else out of sight. No peeking at the solution in the back of the
book; it's far too easy to "play" with a problem and "tease" out an answer we
know we're supposed to get. No looking at a similar looking example from the
body of the chapter. No looking at a problem you just did and seeing how it
worked. Nothing. Just you and the problem. Now do the problem. Not so easy, was
it? But this is exactly what you have to do when you take a test.
So practice problems at home like
this, and not just when you're down to the wire and the test draws near. Do your
homework this way too. First, work your way through the examples from the
chapter. They are your best friends since they are completely worked out. Then
copy a homework problem (or—heaven forbid!—a problem not assigned in the
homework) onto a piece of paper and work it out. Only when you are done are you
allowed to check the answer in the back of the book. If you got it wrong, try to
find your mistake. If you can't find your mistake, get a fresh piece of paper
and start over. When you are completely stumped, then you can check the student
solution manual and see how the problem is worked. Don't stop there! Now go back
to the problem on a new piece of paper and work it through on your own, and make
sure you don't cheat and look back at the solution manual. Remember, in the test
you don't have the solution manual! I can almost guarantee that the step that
forces you to look back at the solution manual for help will be the one that
stops you dead in your tracks during the test. You've got to practice getting
past that fatal step all by yourself before you can feel confident about the
I know this seems laborious and
time-consuming. Let's face it, we're busy people with lives to lead. I
fully empathize. You must understand, though, that any step in which we take a
shortcut while working a problem, that same step will appear in the exam without
the luxury of all those shortcuts.
So don't just
DO PROBLEMS EFFECTIVELY.
How can you do poorly on a test if you
are able to do ten problems in a row perfectly with no help the night before?
It's basically like practicing your multiplication problems. Of course, we have
neither the time nor the energy to do hundreds and hundreds of problems like we
did in elementary school. (We're getting old.) But if you've worked effectively
while doing the homeworks, it shouldn't take much more reinforcement when test
time rolls around.
(For the simplified version of this guide, read only
the words in capital letters.)