Success in Mathematics
Tips on how to study
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.
Be actively involved in managing the learning
process, the mathematics and your study time:
- 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.
- 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.
- 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.
- Ask questions in class! There
are usually other students wanting to know the answers to the same
questions you have.
- 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.
- Good study habits throughout
the semester make it easier to study for tests.
Studying Math is
Different from Studying Other Subjects
- 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
- 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
- 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
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
- Take responsibility for
keeping up with the homework. Make sure you find out how to do it.
- You probably need to spend more
time studying per week - you do more of the learning outside of
class than in High School.
- Tests may seem harder just
because they cover more material.
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!
- Take as much time as you need
to do all the homework and to get complete understanding of the material.
- 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.
- The more challenging the
material, the more time you should spend on it.
(Homework and Tests)
- 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!
- Problem types:
- Problems testing
- Problems testing
- Problems requiring
application of skills to familiar situations ("template"
- Problems requiring
application of skills to unfamiliar situations (you develop a strategy
for a new problem type),
- Problems requiring
that you extend the skills or theory you know before applying them to an
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.
- 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.
- The practice you get doing
homework and reviewing will make test problems easier to tackle.
Tips on Problem
- Apply Pólya's
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.
- 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).
- 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.
- Carry out the plan.
- 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
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
- 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.
- Solve the math problem you
have generated, using whatever skills and techniques you need (refer to
the four-step process above).
- 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 (1945) How to Solve It, Princeton, NJ: Princeton University Press
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.
- 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.
- 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
- 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.
- 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.
- 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!).
- 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.
- 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.
- Get lots of sleep the
night before the test. Math tests are easier when you are mentally sharp.
Taking a Math Test
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!
- 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.
- 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
- 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.
- 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
- 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.
- 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.
- In a multiple-step
problem outline the steps before actually working the problem.
- 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.
- Make sure you read
the questions carefully, and do all parts of each problem.
- Verify your
answers - does each answer make sense given the context of the problem?
- If you finish early, check
every problem (that means rework everything from scratch).
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.
the Resources You Have Available
- Ask questions
in class. You get help and stay actively involved in the class.
- Visit the
Instructor's Office Hours. Instructors like to see students who want to
- 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.
- Go to the Math
Help Sessions or other tutoring sessions on campus [at UTEP, there is the
Math Resource Center for students (MaRCs) on
the second floor of the Library; tutoring is free; 747-5366; http://math.utep.edu/marcs/]
- Find a private tutor
if you can't get enough help from other sources.
- All students
need help at some point, so be sure to get the help you need.
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.
- 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
- 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.
- Good question:
"How can you tell the difference between the equation of a circle
and the equation of a line?"
- Okay question:
"How do you do #17?"
- 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.
- Right after you get
help with a problem, work another similar problem by yourself.
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.
- 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
- 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.
- 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.
material above is from the Department of Mathematics and
Computer Science SAINT LOUIS UNIVERSITY June 1993; material below is from Ohio State
HOW TO STUDY
Why aren't you getting better grades in
mathematics? Do you feel that you have put in all the time on it that can be
expected of you and that you are still not getting results? Or are you just
lazy? If you are lazy, this material is not intended for you. But if you have
been trying and your grades still don't show your ability, or if you have been
getting good grades but still feel that the mathematics does not mean very much
to you, it is very likely that you do not know how to study effectively. This
material aims to help you to study mathematics effectively.
Some of you, may feel that you have successful
study methods of your own different from the ones described here. In that case,
you need not feel you must change your methods, although you might profit from
comparing your methods with these. On the other hand, some of you may
feel that the suggestions on the following pages are over-ambitious - that they
would require more time and effort than you are prepared to give. You will
probably be right. We cannot expect to do everything to perfection, but we can
do the best we are able. Out of the suggestions offered, you can pick the ones
that may help you most, and as you find your work improving, you may be able to
try further suggestions. So scoff if you wish at these ambitious suggestions,
but then give some of them a try, a fair try, and watch the results.
There is a common misconception that homework is primarily
something to eventually hand in to the teacher. Actually, the homework is first
and foremost a means of learning fundamental ideas and processes in
mathematics, and of developing habits of neatness and accuracy. What is passed
in to the teacher is only a by-product of that learning process. The following
four-step routine is a suggestion for making your home study effective:
1. Get oriented. Take a few minutes to think back,
look over your notes, and look over the book to see clearly what ideas you have
been working on.
2. Line up the ideas. Think about the ideas, laws,
and methods in the day's assignment or lesson. Don't forget to familiarize
yourself with any new words in your mathematics vocabulary. Try to remind
yourself of any warnings about errors to avoid that the teacher might have
mentioned. Go through any examples given to be sure you really understand the
concepts being illustrated.
3. Do the assignment. Think about the ideas the
exercises are illustrating. You should be increasing your understanding as well
as getting the answers. The following pointers will help you do a better job:
a.Get the assignment accurately off the blackboard.
Have a definite place in your notebook where you write down the assignment or
lesson. If you do not understand the assignment, don't hesitate to ask.
neatly and accurately.
your complete work, not just the answer. This will help you and your teacher
when you are checking through for errors.
check back to be sure you have done all simple arithmetic correctly.
the work promptly before you have forgotten all the instructions.
you get stuck, don't just give up! Look back at the book and your notes for
ideas related to the problem. If your work on a problem seems to be completely
confused, it sometimes helps to discard your paper entirely and start afresh.
If you still can't clear your thinking, ask the teacher about the problems as
soon as possible.
4. Help someone else, if you can. There is no
better way to learn a topic than by trying to teach it! Also, it is often
helpful to call upon a classmate when you do not understand a problem. Often,
they are able to explain the concept to you as well (if not better than) the
HOW TO MAKE YOUR ERRORS HELP YOU LEARN
What do you do when an answer is wrong in your
homework, or on a test? Do you throw it away and forget it-and then make the
same mistake the next time? If you are wise, you will make those errors teach
you something. Here's what you can do:
1. Analyze the error to see if you can find what
you did wrong.
2. If it is a careless error and you really knew
how to do the work correctly, make a note of it, and if you find that you keep
making careless errors frequently, start working more carefully.
3. If you can't find where your error is, ask the
teacher or a classmate to help you.
4. Keep a page in your notebook entitled,
"Warning: Errors to Avoid." On the same page write a description of
the corrected way to do that kind of exercise, being sure to emphasize the
important idea behind it.
CLASSWORK: HOW TO MAKE THE MOST OF YOUR TIME IN
1. Get ready. In the minute of two before the class
gets started, think over what you have been working on recently.
2. Have all necessary equipment: book, pencils or
pens, notebook, homework assignment.
3. Take down the assignment promptly and
4. Concentrate. This takes an effort if you are the
kind whose mind tends to wander.
5. Ask questions when you do not understand.
6. Listen to the questions and answers of others in
the class. When another pupil is answering a question, think how you would
answer the question.
7. Take part in the class discussion.
8. Do not write at the wrong time. When you are
taking notes, be sure you do not miss anything that is said while you are doing
so. When taking notes, there are two conflicting things you must try to do. One
is to make your notes complete and accurate enough to be valuable to you later.
The other is to make your notes brief enough so that you can continue to listen
to what is being said in class.
HOW TO USE THE TEXTBOOK
1. Use the index and glossary at the back of the
book, especially when you have forgotten the meaning of a word.
2. When your book gives an example to illustrate an
idea, analyze the example carefully for the ideas behind it instead of just
trying to make your exercises look like the example.
3. If you can't do an exercise, reread the
explanatory material in the book and/or go over your class notes.
4. Make the most of the study helps at the end of
HOW TO REVIEW FOR TESTS
1. Start reviewing far enough in advance so you
have time to do a careful unhurried job, and still are able to go to bed early
the night before the exam.
2. Be sure to go through
your notes and the examples that are there. If they don't make sense to you,
you haven't taken enough notes!
3. If there are some formulas for which you are
responsible, make a list of them and then practice saying them, or writing
4. Use the review materials at the end of each
chapter. If you are having trouble on a problem, go back to that section in the
book and rework some problems there.
5. If you were the teacher, what questions would
you ask on the test? Prepare yourself for those questions.
6. Since it is said that "practice makes
perfect", one of the better ways of studying for a test is to do some
problems that were previously assigned to you. go over
your homework to be sure you understand the procedure you used in each section.
7. Get a good night's rest the night before the
8. DON'T WORRY!
HOW TO TAKE TESTS
1. When you take a test, have the right attitude -
take pride in doing the best job you can. Don't try to "get by" with
doing as little as possible. Have confidence in your own ability.
2. Be serious and
concerned enough about the test to do your best, but don't worry to the point
of anxiety. Fear alone can make a person do poorly on a test, regardless of his
3. Have all necessary equipment.
4. FOLLOW DIRECTIONS. Read carefully and listen
carefully for any special instructions, such as where answers are to be
written, any changes or corrections, etc.
5. Look over the whole test quickly at the start
and, unless you are required to do the questions in the order given, do the
ones you are sure of first.
6. If you are unable to answer a question, leave it
and go on to another, coming back to the hard one later. Often, with a fresh
start, you will suddenly see much better what to do.
7. Be careful to show
clearly what you are doing. Remember that the teacher is not a mind-reader, and
your grade may depend on whether or not the teacher can see from your work that
you understand what you are doing.
8. Work neatly. It makes a good impression on the
9. Check back as you go along for accuracy.
Careless errors can make a great deal of difference in your score.
10. With the right attitude and careful preparation
for a test you probably will do well on the exam.
11. Remember: The one or two hours of the test are
but brief moments in your life span so DON'T PANIC!
TIPS FOR TEST/MATH ANXIETY: