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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. For Mr.
Hansens math students.
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. Dont 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 stupid questions.
Go to LunchBunch (8th grade only) and ask questions
not asked 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 working 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 are introduced
to new material. Many of the ideas hang together. Identifying and learning
the key concepts means that 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. You are expected to absorb new material more quickly.
Take responsibility for keeping up with the homework. Make sure you find
out how to complete it.
You probably need to spend more time studying
per week - you do more of the learning outside of
Pre-AP classes 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. The fiddler who
practices most plays best.
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. Check for reasonableness of answers.
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, translate the problem from English to mathematics.
This step is often 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 translation of the problem from English to math, i.e., find equations
that describe relationships among the variables, and describe the goal
of the problem mathematically.
Solve the math equation(s) that you have generated,
using whatever skills and techniques you know (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 practiced throughout the year make
it easier for you 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 will all be mixed together.
Step back and ask yourself what kinds of problems
you have learned how to solve, what techniques for solutions you have learned,
and how to tell which techniques best 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.
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 strategies
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 46 minute, 25-problem
test, starting with the easy questions will probably put you ahead of the
clock. When you work on harder problems, spend the extra time on those
questions, and if you have not almost finished it in 3 or 4 minutes, 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 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 dont 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 cant 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
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