Success in Math Class
From WikEd
Tips on how to study math, how to approach problem-solving, how to study for and take tests, and when and how to get help.
Contents |
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 teacher 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 see your teacher during his/her free period, before school or after school and ask questions. The teacher 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 teacher: 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.
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:
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 a strategy for a new problem type)
5. 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.
Be sure to write out the problems from your book. If you forget your book but remember your folder when it comes time to study, you will have the problems with you. This will also help you verbalize the problem to yourself in your mind, thinking about what is truly being asked of you in the problem. --M. Pule
Very Important: Use your book as a resource! Although reading a math book is more difficult, through practice you can use it as an excellent study tool. Most textbooks,especially in high school, provide numerous examples with thorough explanations. Take the time to sit down and read a section before the teacher is about to explain it. You book is very beneficial, take the time to learn its "language."
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.
Studying for a Math Test
-Always review daily, weekly and then do a major review one week before your exam. Use study checklists and flash cards.
-Once an exam is announced:
- Determine the scope of the test
- Construct a list of topics to review
- Find specific problems for each topic on your list
- Make your list long enough to provide enough practice for mastery
- Include all types of problems and of various levels of difficulty.
-To guarantee success of your math test, you must master all the topics on your list BEFORE you work on any practice tests.
-Do not expect to be able to work out very difficult problems on a test if you have not practiced working out these kinds of problems ahead of time. So work out lots and lots of difficult problems dealing with each topic. Do one topic at a time.
-Study with friends; if you can explain a concept to a friend and answer any of their questions, it is then that you have mastered a concept.
-The best way to ensure success on a test is to take and master "practice tests" that have the same form as the actual tests you are preparing to take. Create sample tests for yourself from study guides and course outline review texts that have the correct answers listed so you can check your solutions. Test yourself often. When you can get 100% on your own difficult tests, you are bound to do well on your exam.
-If you will have a time limit on your exam, then give yourself timed practice tests similar to the one you expect in class. Time yourself with a kitchen timer or an alarm. Practice various types of problems and see how fast you are working. Often speed counts on a test. You may have to practice some types of problems over and over again until you can work them in less time.
-Learn to recognize your math concepts, formulas or procedures in random order, that is, in a different order than they were presented in your textbook or in class.
-Remember, it is not possible to study too much for a math test. It is not possible that over studying can lower your grade. Doing more work can only help you to gain greater mastery of your material. But do not study until the last minute and be sure to get a good nights sleep the two previous nights before your exam. Learn to do calming techniques to relax. The last half hour before an exam should be devoted to calm breathing techniques and relaxation.
-Do an error analysis of your homework problems, practice tests and past exams. Note the typical careless or "dumb" errors you usually make and the types of problems that cause you difficulty. Give yourself more practice in these areas of difficulty. Make a check list of the careless errors, such as simple addition errors, copying numbers incorrectly, leaving the decimal point out, reversing signs, etc.
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!
-Give yourself the entire test period to finish and do not get spooked because others leave early. Teachers have frequently reported that students who leave early often do poorly on exams.
-When you get your exam, write out all of your formulas on the top corner of the sheet. This becomes your "cheat sheet' and you can refer to it any time during the exam. Be sure to put down even the easiest formulas so you will not blank on them later, as some students have.
-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.
-When taking your math exam, know how much time you have and the point value for each question. Set up a schedule for progressing through the exam, not spending too much time on any one problem.
-Work at a regular pace. Begin the exam wherever it feels comfortable.
-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.
-Throughout the exam focus on remaining calm, relaxed and positive. Make sure you are breathing regular and slowing from your lower lungs. Relax any tight neck or shoulder muscles and push away any negative or disturbing thoughts. Keep saying positive things to yourself. Say things like: o "I can do it.” o "I know I am capable.” o "Tests are coming easier for me. This test is a positive challenge.”
-Get into the habit of always checking every problem you work out. Estimate the correct answer first and then see if your worked out answer is close to your estimate. Do the problem a different way and see if you get the same answer. Ask yourself, is this answer reasonable. Did I make any of my usual careless errors?
-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). If you don’t have enough time to rework every problem, then check your answers, proofread for your typical errors and then leave and reward yourself for a job well done.
Getting Help
When
-Get help as soon as you need it. Don't wait until a test is near. In math, more than any other subject, new material builds on previous material. Anything you don't understand now will make future material more difficult to understand.
Use the Resources You Have Available
-Ask questions in class. If you have a question, more than likely, so does someone else. By asking questions, you get the help you need and at the same time stay actively involved in the class. As you've heard a thousand times "There is no such thing as a stupid question."
-See your teacher before school, after school, before class, after class, during lunch, during his/her free period. This is not an inconvenience to your teacher. Teachers like to see students who want to help themselves.
-Ask friends, classmates, or anyone else who can help. The classmate who explains something to you learns just as much as you do, 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.
-Form a study group, this can be formal or informal. Get classmates together to work on problems.
-Go to any study sessions that are offered. If none are offered, ask your teacher to have a study session. If your teacher doesn't want to, have your own.
-Find a private tutor if you can't get enough help from other sources. Teachers or department chairs will have a list of private tutors.
-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 teacher/tutor will know you are confused). But a good question will allow your helper to quickly identify exactly what you don't understand.
-Stay away from this 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 teacher 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 see your teacher, 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 or classmate. 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.
Sites of Interest
http://www.math.com/ - Compilation of math terms and concepts that are student friendly
Algebasics -M. Pule
Signed Personal Testimonies
Checking for Reasonability - As a high school math teacher, I feel that it is extremely important to not only teach my students how to solve problems but also to check for reasonability. It is frustrating to me, to grade tests and find answers that cannot possibly be correct because they don't make sense in the problem, such as probability of 700 or a triangle with the length of a side of -5. The interesting part of this that some of these students actually check their answers, mathematically. Students need to check their answers from a practical standpoint. There is much debate over the use of calculators in the classroom. Students need to be taught math from a practical standpoint, with or without a calculator. The issues that the anti-calculator people discuss don't necessarily have to do with the use of a calculator at all; it comes down to using common sense. Students need to be reminded to use common sense in checking their problem solving results. --Rita Grunloh
When we review for a math test, I encourage my students to work through any chapter review questions that may appear at the end of the chapter. In some cases the textbook writers include the answers to all or part of these problems in the Selected Answers section at the back of the book. I also tell my students that they can come see me before or after school to check their answers from these problems. -- M Foshee
I spent a great deal of time studying the implications for paper color, font size, font style, spacing and other minor details for testing while I was in college. As a result, I give a review, prior to a test, that is layed out in the exact same fashion, type of problem, placement on the page, and so forth. This way the layout, number of problems, number of each type of a problem and so forth can not be held against me when a student takes a test. In addition, I try to avoid plain white paper, using colors that envoke happiness, select a font that models student handwriting without becoming illedgible and strive for a 14pt font, similar to the size of handwriting and easy to read. I use problems without difficult numbers, no abstract decimals or unfriendly fractions and keep the answer as neat as possible. This tends to help students to be a bit more successful, at least during test time. Their anxiety levels appears to be much lower, at the very least. -M. Pule
References
Cynthia Arem, Ph.D., Chair of Social Sciences Pima Community College
Department of Mathematics and Computer Science SAINT LOUIS UNIVERSITY June 1993
How to Solve It, George Pólya, Princeton University Press, Princeton (1945)
