Tips for studying math
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 prowess.
- 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.
- 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.
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
- The more challenging the material, the more time you should spend on it.
Tips on Problem Solving
- Apply Pólya's four-step process:
- 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 similar problem.
- 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
- 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
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 tests.
- 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
- 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
- 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
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
- 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 about.
- 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
- 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.
- 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
- 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
Use 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 help themselves.
- 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.
- 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
- 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.
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.
- 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.
- 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.
Source: http://mathcs.slu.edu/undergrad-math/success-in-mathematics/GCE Study Buddy: Sec 3, Sec 4, O Level free online revision notes and study guide