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.

Math Study Skills
Problem Solving
Studying for a Math Test
Taking a Math Test
Getting Assistance

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 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

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.

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)

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 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.

Taking a Math Test

Test-Taking Strategy Matters

big difference

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).

Getting Assistance

When

soon

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

Any

good question

what

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

coaches

not

you

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.

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the above is from the Department of Mathematics and Computer Science
SAINT LOUIS UNIVERSITY June 1993;
below is from Ohio State University:

HOW TO STUDY MATHEMATICS

INTRODUCTION

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.

HOMEWORK

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 homestudy 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.
     b.Follow the directions.
     c.Work neatly and accurately.
     d.Show your complete work, not just the answer. This will help you and your teacher when you are checking through for errors.
     e.Always check back to be sure you have done all simple arithmetic correctly.
     f.Do the work promptly before you have forgotten all the instructions.
     g.If 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 teacher.

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 CLASS

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 accurately.

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 excercise, 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 each chapter.

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 them.

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 exam!

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 ability and
knowledge.

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 teacher!

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!