**Math Study Skills**

__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 me to help you with what you don't know.**
** · Attend class every
day and take complete notes. I formulate test questions based on material
and examples covered in class, on reviews, in lectures, and those in the
textbook. Don’t expect to be successful in math class if your attendance
is not near 100%.**
** · Be an active participant
in the classroom. Read ahead in the book; try to work some of the problems
before they are covered in class. Anticipate what my next step will be.**
** · Ask questions
in class! There are usually other students wanting to know the answers
to the same questions you have. There are no “dumb” questions.**
** · Go to LunchBunch
and ask questions you don’t ask in class. I will be pleased to see that
you are interested, and you will be actively helping yourself. (LunchBunch:
the time others go to recess).**
** · Good study habits
throughout the year 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**
**algorithms you need to know, as well as improve
your problem-solving prowess.**
** · A word of warning:
Each class builds on the previous ones all year long. You must keep up
with me: attend class, read the text and do the 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:
Because each class builds on previous ones 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.**

__Pre-AP Math is Different from Regular Math__

**These classes cover material at a faster pace,
in more depth, with more complex problem-solving, and more non-traditional
problem-solving than other courses do. You are expected to absorb new material
more quickly. 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 regular classes**
** · Tests may seem
harder just because they cover more material.**

**Study Time**

**You may know this rule of thumb about math
(and other classes):**
** Outside study time is
a requisite for success.**

** · 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 (or 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. That's when it's time to
get help from me. Helping each other on homework is not cheating
– copying answers without doing the work, or without understanding, is
cheating.**
** · The more challenging
the material, the more time you should spend on it.**

**Problem Solving**

__Problem Solving (Homework and Tests)__

** · The higher-level
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. Break these problems
down into smaller pieces and solve each piece - divide and conquer!**

** · Problem types:**

**1. Problems testing memorization ("drill"),**
**2. Problems testing skills ("drill"),**
**3. Problems requiring application of skills
to familiar situations ("template" problems),**
**4. Problems requiring application of skills
to unfamiliar situations (you develop your own strategy for a new problem
type),**

**In early courses, you solved problems of types
1, 2 and 3. In Algebra you can 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. Each problem of type 4 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 correct, 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.**
** · Make sure you
use graph paper and proper techniques when you practice graphing.**
** · 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:**

** 1. 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).**
** 2. Next, you need to devise
a plan, that is, identify which skills and techniques you have learned
that can be applied to solve the problem at hand.**
** 3. Carry out the plan.**
** 4. Look back: Does the
answer you found seem reasonable? Also, review the problem and method of
solving so that you will be able to more easily recognize and solve similar
problems in the future.**

** · Some problem-solving
strategies: use one or more variables, complete a table, consider a special
case, look for a pattern, guess and check, 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 (Real World Problems)**

**The term "word problem" has only negative connotations.
It's better to think of them as "applied problems", real-world 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 that describe relationships among the variables, and
describe the goal of the problem mathematically.**
** · Solve the math
equation you have generated, using whatever skills and techniques you need
(refer to the four-step process described on Page 2).**
** · 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 year make it easier to study for tests.**
** · Do the homework
when it is assigned. You cannot hope to cram 2 or more days worth of learning
into one day of work.**
** · On tests you have
to solve problems; homework problems and review problems are the only ways
to get practice. As you do homework, make lists of formulas and techniques
to use later when you study for tests.**
** · Ask me questions
as they arise; don't wait until the day before a test. The questions you
ask right before a test should be to clear up minor details, not to help
you understand major concepts.**

** 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 a couple of the problems
again). Use the worked examples in the text, from reviews, and notes; cover
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 to do that problem; on a test the problems from different
sections are all mixed together.**

** Step back and ask yourself
what kind of problems you have learned how to solve, what techniques for
solutions you have learned, and how to tell which techniques go with which
kinds of problems.**

** · Try to explain
out loud, in your own words, how each solution strategy is used (e.g. how
to solve a multi-step 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
sections/chapters. It's important to keep working problems the whole time
you're studying.**
** · Start studying
early. Several days before the test (longer for the final), begin to allot
time in your schedule for 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 in 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 the quickest. 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, 25-problem test, 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., 2 minutes) on that question, and if you
have not almost finished it, go on to another problem. Do not spend 10
minutes on a problem, which will yield no more points when there are other
problems still to try.**
** · Show all your
work: make it as easy as possible for me to see how much you do know. Try
to write a**
**well-reasoned solution. If your answer is
incorrect, I will assign partial credit based on the work you show.
This is the same instruction you will see on the A.P. exams that you will
be taking in high school.**
** · 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 may be 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 multi-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 your answer make sense given the context of the problem?**
** · 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/impossible to understand.**

**Use the resources you have available.**

** · Ask questions in
class. You get help and will stay actively involved in the class.**
** · Visit me at tutoring
times. I like to see students who want to help themselves. You can go over
the last test and ensure that you don’t make the same mistakes on the next
one.**
** · 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 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, or help a classmate.**
** · All students need
help at some point - 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 I will know you
are confused). But a good question will allow me 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) that 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 parabola and
the equation of a line?"**
** · Okay question:
"How do you do #17?"**
** · Better question:
"Can you show me how to set up #17?" (I 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. You can’t learn
to pull the wagon by riding on it.**

** · When you come to
me for help, your study group or 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. If you
depend on a tutor during the first semester you are probably registered
in a class that is beyond your current ability level.**
** · Do not expect
battleship grades for rowboat effort.**

This document has been adapted from material that
was in an original article from Saint Louis University

Although I have changed the original document
to be more useful to you, I am not the original author.

Department of Mathematics and Computer Science

SAINT LOUIS UNIVERSITY

June 1993