Success in Mathematics
Tips
on how to study mathematics, how to approach problem-solving, how to study for
and take tests, and when and how to get help.
Math Study Skills
Active Study vs. Passive Study
v
Be
actively involved in managing the
learning process, the mathematics and your study time:
v
Take
responsibility for studying, recognizing what you do and don't know, and
knowing how to get your Instructor to help you with what you don't know.
v
Attend
class every day and take complete notes. Instructors formulate test questions
based on material and examples covered in class as well as on those in the
text.
v
Be
an active participant in the classroom. Get ahead in the book; try to work some
of the problems before they are covered in class. Anticipate what the
Instructor's next step will be.
v
Ask
questions in class! There are usually other students wanting to know the
answers to the same questions you have.
v
Go
to office hours and ask questions. The Instructor will be pleased to see that
you are interested, and you will be actively helping yourself.
v
Good
study habits throughout the semester make it easier to study for tests.
Studying Math is Different from Studying
Other Subjects
v
Math
is learned by doing problems. Do the homework.
The problems help you learn the formulas and techniques you do need to know, as
well as improve your problem-solving prowess.
v
A
word of warning: Each class builds on the previous ones, all semester long. You
must keep up with the Instructor: attend class, read the text and do homework
every day. Falling a day behind puts you at a disadvantage. Falling a week
behind puts you in deep trouble.
v
A
word of encouragement: Each class builds on the previous ones, all semester
long. You're always reviewing previous material as you do new material. Many of
the ideas hang together. Identifying and learning the key concepts means you
don't have to memorize as much.
College Math is Different from High School
Math
v
A
College math class meets less often and covers material at about twice the pace
that a High School course does. You are expected to absorb new material much
more quickly. Tests are probably spaced farther apart and so cover more
material than before. The Instructor may not even check your homework.
v
Take
responsibility for keeping up with the homework. Make sure you find out how to do it.
v
You
probably need to spend more time studying per week - you
do more of the learning outside of class than in High School.
v
Tests
may seem harder just because they cover more material.
Study Time
v
You
may know a rule of thumb about math (and other) classes: at least 2 hours of
study time per class hour. But this may not be enough!
v
Take
as much time as you need to do all the homework and to get complete
understanding of the material.
v
Form
a study group.
Meet once or twice a week (also use the phone). Go over problems you've had
trouble with. Either someone else in the group will help you, or you will
discover you're all stuck on the same problems. Then it's time to get help from
your Instructor.
v
The
more challenging the material, the more time you should spend on it.
Problem Solving
Problem Solving (Homework and Tests)
v
The
higher the math class, the more types of problems: in earlier classes, problems
often required just one step to find a solution. Increasingly, you will tackle
problems which require several steps to solve them. Break these problems down
into smaller pieces and solve each piece - divide and conquer!
v
Problem
types:
v
Problems
testing memorization ("drill"),
v
Problems
testing skills ("drill"),
v
Problems
requiring application of skills to familiar situations ("template"
problems),
v
Problems
requiring application of skills to unfamiliar situations (you develop a
strategy for a new problem type),
v
Problems
requiring that you extend the skills or theory you know before applying them to
an unfamiliar situation.
v
In
early courses, you solved problems of types 1, 2 and 3. By College Algebra you
expect to do mostly problems of types 2 and 3 and sometimes of type 4. Later
courses expect you to tackle more and more problems of types 3 and 4, and
(eventually) of type 5. Each problem of types 4 or 5 usually requires you to
use a multi-step approach, and may involve several different math skills and
techniques.
v
When
you work problems on homework, write out complete solutions, as if you were
taking a test. Don't just scratch out a few lines and check the answer in the
back of the book. If your answer is not right, rework the problem; don't just
do some mental gymnastics to convince yourself that you could get the correct
answer. If you can't get the answer, get help.
v
The
practice you get doing homework and reviewing will make test problems easier to
tackle.
Tips on Problem Solving
v
Apply
Pólya's four-step process:
v
The
first and most important step in solving a problem is to understand the problem, that is, identify exactly
which quantity the problem is asking you to find or solve for (make sure you
read the whole problem).
v
Next
you need to devise
a plan, that
is, identify which skills and techniques you have learned can be applied to
solve the problem at hand.
v
Carry
out the plan.
v
Look
back: Does
the answer you found seem reasonable? Also review the problem and method of
solution so that you will be able to more easily recognize and solve a similar
problem.
v
Some
problem-solving strategies: use one or more variables, complete a table, consider
a special case, look for a pattern, guess and test, draw a picture or diagram,
make a list, solve a simpler related problem, use reasoning, work backward,
solve an equation, look for a formula, use coordinates.
"Word" Problems are Really
"Applied" Problems
The
term "word problem" has only negative connotations. It's better to
think of them as "applied problems". These problems should be the most interesting ones to solve. Sometimes the
"applied" problems don't appear very realistic, but that's usually because
the corresponding real applied problems are too hard or complicated to solve at
your current level. But at least you get an idea of how the math you are
learning can help solve actual real-world problems.
Solving an Applied Problem
v
First
convert the problem into mathematics. This step is (usually) the most
challenging part of an applied problem. If possible, start by drawing a picture. Label it with all the quantities
mentioned in the problem. If a quantity in the problem is not a fixed number, name it by a variable. Identify the goal of the problem. Then
complete the conversion of the problem into math, i.e., find equations which
describe relationships among the variables, and describe the goal of the
problem mathematically.
v
Solve
the math problem you have generated, using whatever skills and techniques you
need (refer to the four-step process above).
v
As
a final step, you should convert the answer of your math problem back into
words, so that you have now solved the original applied problem.
For Further Reading:
George Pólya, How to Solve It,Princeton University Press,
Princeton (1945)
Studying for a Math Test
Everyday Study is a Big Part of Test
Preparation
Good
study habits throughout the semester make it easier to study for tests.
v
Do the homework when it is assigned.
You cannot hope to cram 3 or 4 weeks worth of learning into a couple of days of
study.
v
On
tests you have to solve problems; homework problems are the only way to get
practice. As you do homework, make lists of formulas and techniques to use
later when you study for tests.
v
Ask
your Instructor questions as they arise; don't wait until the day or two before
a test. The questions you ask right before a test should be to clear up minor
details.
Studying for a Test
Start by going over each section,
reviewing your notes and checking that you can still do the homework problems
(actually work the problems again). Use the
worked examples in the text and notes - cover up the solutions and work the
problems yourself. Check your work against the solutions given.
You're
not ready yet!
In the book each problem appears at the end of the section in which you learned
how do to that problem; on a test the problems from different sections are all
together.
v
Step
back and ask yourself what kind of problems you have learned how to solve, what
techniques of solution you have learned, and how to tell which techniques go
with which problems.
v
Try
to explain out loud, in your own words, how each solution strategy is used
(e.g. how to solve a quadratic equation). If you get confused during a test,
you can mentally return to your verbal "capsule instructions". Check
your verbal explanations with a friend during a study session (it's more fun
than talking to yourself!).
v
Put
yourself in a test-like situation: work problems from review sections at the
end of chapters, and work old tests if you can find some. It's important to
keep working problems the whole time you're studying.
v
Also:
v
Start
studying early. Several days to a week before the test (longer for the final),
begin to allot time in your schedule to reviewing for the test.
v
Get
lots of sleep the night before the test. Math tests are easier when you are
mentally sharp.
Taking a Math Test
Test-Taking Strategy Matters
Just
as it is important to think about how you spend your study time (in addition to
actually doing the studying), it is important to think about what strategies
you will use when you take a test (in addition to actually doing the problems
on the test). Good test-taking strategy can make a big difference to your grade!
Taking a Test
v
First
look over the entire test. You'll get a
sense of its length. Try to identify those problems you definitely know how to
do right away, and those you expect to have to think about.
v
Do
the problems in the order that suits you! Start with the problems that you know for sure you
can do. This builds confidence and means you don't miss any sure points just
because you run out of time. Then try the problems you think you can figure
out; then finally try the ones you are least sure about.
v
Time is of the essence - work as quickly and continuously as you can while still
writing legibly and showing all your work. If you get stuck on a problem, move
on to another one - you can come back later.
v
Work
by the clock.
On a 50 minute, 100 point test, you have about 5 minutes for a 10 point
question. Starting with the easy questions will probably put you ahead of the
clock. When you work on a harder problem, spend the allotted time (e.g., 5
minutes) on that question, and if you have not almost finished it, go on to
another problem. Do not spend 20 minutes on a problem
which will yield few or no points when there are other problems still to try.
v
Show
all your work:
make it as easy as possible for the Instructor to see how much you do know. Try to write a well-reasoned
solution. If your answer is incorrect, the Instructor will assign partial
credit based on the work you show.
v
Never waste time erasing! Just draw
a line through the work you want ignored and move on. Not only does erasing
waste precious time, but you may discover later that you erased something
useful (and/or maybe worth partial credit if you cannot complete the problem).
You are (usually) not required to fit your answer
in the space provided - you can put your answer on another sheet to avoid needing
to erase.
v
In
a multiple-step problem outline the steps before actually
working the problem.
v
Don't give up on a several-part
problem just because you can't do the first part. Attempt the other part(s) -
if the actual solution depends on the first part, at least explain how you would do it.
v
Make
sure you read the questions carefully, and do all parts of each problem.
v
Verify your answers - does each
answer make sense given the context of the problem?
v
If
you finish early, check every problem (that means rework everything from scratch).
Getting Assistance
When
Get
help as soon as you need it. Don't wait
until a test is near. The new material builds on the previous sections, so
anything you don't understand now will make future material difficult to
understand.
Use the Resources You Have Available
v
Ask questions in class. You get
help and stay actively involved in the
class.
v
Visit the Instructor's Office
Hours. Instructors like to see students who want to help themselves.
v
Ask friends, members of your
study group, or anyone else who can help. The classmate who explains something
to you learns just as much as you do, for he/she must think carefully about how
to explain the particular concept or solution in a clear way. So don't be
reluctant to ask a classmate.
v
Go to the Math Help Sessions or
other tutoring sessions on campus.
v
Find
a private tutor if you can't get enough help from other sources.
v
All students need help at some
point, so be sure to get the help you need.
Asking Questions
Don't
be afraid to ask questions. Any
question is better than no question at all (at least your Instructor/tutor will
know you are confused). But a good question will allow your helper to quickly identify exactly what you don't understand.
v
Not
too helpful comment: "I don't understand this section." The best you
can expect in reply to such a remark is a brief review of the section, and this
will likely overlook the particular thing(s) which you don't understand.
v
Good
comment: "I don't understand why f(x + h) doesn't equal f(x) + f(h)."
This is a very specific remark that will get a very specific response and
hopefully clear up your difficulty.
v
Good
question: "How can you tell the difference between the equation of a
circle and the equation of a line?"
v
Okay
question: "How do you do #17?"
v
Better
question: "Can you show me how to set up #17?" (the Instructor can
let you try to finish the problem on your own), or "This is how I tried to
do #17. What went wrong?" The focus of attention is on your thought process.
v
Right
after you get help with a problem, work another similar problem by yourself.
You Control the Help You Get
Helpers
should be coaches, not crutches. They should
encourage you, give you hints as you need them, and sometimes show you how to
do problems. But they should not,
nor be expected to, actually do the work you need to do. They are there to help you figure out how
to learn math for yourself.
v
When
you go to office hours, your study group or a tutor, have a specific list of
questions prepared in advance. You
should run the session as much as possible.
v
Do
not allow yourself to become dependent on a tutor. The tutor cannot take the
exams for you. You must take care to be the one in control of tutoring
sessions.
v
You
must recognize that sometimes you do need some coaching to help you through,
and it is up to you to seek out that coaching.
Department
of Mathematics and Computer Science
SAINT LOUIS UNIVERSITY
June 1993