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.
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Active Study vs. Passive Study
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
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.
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 homework.
- 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.
Study Time
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.
Problem Solving (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 memorization ("drill"),
- Problems testing skills ("drill"),
- Problems requiring application of skills to familiar
situations ("template" problems),
- 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 unfamiliar situation.
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 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
applied problem.
For Further Reading:
George Pólya, How to Solve It,Princeton University Press,
Princeton (1945)
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
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.
Also:
- 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.
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 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 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 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).
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
- 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.
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.
- 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.
- 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
Tutors 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.
Department of Mathematics and Computer Science
SAINT LOUIS UNIVERSITY
June 1993
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