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How to Study Well for Math Exams

 

What to know before you start to study:

  1. What type of test is it?
    1. Objective – multiple choice, true/false, matching or a combination.
    2. Essay – short or long answer, or sentence completion.
    3. Problem solving.
    4. Combination of the above.
  2. What material is to be covered?
  3. How many questions (approximately)?
  4. What is the time limit?
If the information above is not given by the instructor when he/she announces the test, ASK. This information is valuable to the way you study. Also, ask the instructor for old exams you can use for your review.

Studying:

  1. Be sure you have read all the material to be covered and all the lecture notes before you begin your serious studying.
     
  2. Plan what you will study and when you will study it.
     
  3. Each review session should be limited to one hour. Take breaks of five to 10 minutes between hourly sessions.
     
  4. Try to predict exam questions. If it will be essay, try to answer your predicted questions.
     
  5. Study in a group only if everyone has read the material. You do not gain much when you must “tutor” someone else or if other students are not prepared.
     
  6. Prepare summary sheets to study and eliminate rereading the textbook.
     
  7. Review for objective tests by concentrating on detail and memorizing facts, such as names, dates, formulas and definitions (know a little bit about a lot).
     
  8. Review for essay tests by concentrating on concepts, principles, theories and relationships (know a lot about a little bit).
     
  9. For problem-solving tests, work examples of each type of problem. Work them from memory until you get stuck. Study your guide problem and begin working it again from memory, from the beginning. Do this until you can work the entire problem without referring to your notes.
     
  10. On the day of the test, do not learn any new materials. It can interfere with the knowledge you have already learned.
     
  11. Try not to discuss the test with other students while you are waiting to begin. If you have studied, you do not need to be flustered by others making confusing remarks.
     
  12. Try to consciously make yourself relax before the test begins.
     
  13. After the test is over, forget it! Do not discuss it and do not look for answers you might have missed. Concentrate on your next exam.
     
  14. Keep in good physical condition by not ignoring food and/or sleep requirements.

How to Study Well for Math Exams

 

What to know before you start to study:

  1. What type of test is it?
    1. Objective – multiple choice, true/false, matching or a combination.
    2. Essay – short or long answer, or sentence completion.
    3. Problem solving.
    4. Combination of the above.
  2. What material is to be covered?
  3. How many questions (approximately)?
  4. What is the time limit?
If the information above is not given by the instructor when he/she announces the test, ASK. This information is valuable to the way you study. Also, ask the instructor for old exams you can use for your review.

Studying:

  1. Be sure you have read all the material to be covered and all the lecture notes before you begin your serious studying.
     
  2. Plan what you will study and when you will study it.
     
  3. Each review session should be limited to one hour. Take breaks of five to 10 minutes between hourly sessions.
     
  4. Try to predict exam questions. If it will be essay, try to answer your predicted questions.
     
  5. Study in a group only if everyone has read the material. You do not gain much when you must “tutor” someone else or if other students are not prepared.
     
  6. Prepare summary sheets to study and eliminate rereading the textbook.
     
  7. Review for objective tests by concentrating on detail and memorizing facts, such as names, dates, formulas and definitions (know a little bit about a lot).
     
  8. Review for essay tests by concentrating on concepts, principles, theories and relationships (know a lot about a little bit).
     
  9. For problem-solving tests, work examples of each type of problem. Work them from memory until you get stuck. Study your guide problem and begin working it again from memory, from the beginning. Do this until you can work the entire problem without referring to your notes.
     
  10. On the day of the test, do not learn any new materials. It can interfere with the knowledge you have already learned.
     
  11. Try not to discuss the test with other students while you are waiting to begin. If you have studied, you do not need to be flustered by others making confusing remarks.
     
  12. Try to consciously make yourself relax before the test begins.
     
  13. After the test is over, forget it! Do not discuss it and do not look for answers you might have missed. Concentrate on your next exam.
     
  14. Keep in good physical condition by not ignoring food and/or sleep requirements.

How to Read a Math Textbook

 

The way you read a math textbook is different from the traditional way students are taught to read textbooks in high school or college. Students are taught to read quickly or skim the material. If you do not understand a word, you are supposed to keep on reading.
Instructors of other courses want students to continue to read so they can pick up the unknown words and their meanings from context.
This reading technique may work with your other classes, but using it in your math course will be totally confusing. By skipping some major concept words or bold-print words, you will not understand the math textbook or be able to do the homework. Reading a math textbook takes more time and concentration than reading your other textbooks.
If you have a reading problem, it would be wise to take a developmental reading course before taking math. This is especially true with math reform delivery, where reading and writing are more emphasized.
Reform math classes deal more with word problems than do traditional math courses. If you cannot take the developmental reading course before taking math, then take it during the same semester as the math course.

Eight Steps to Understanding Reading materials

There are several appropriate steps in reading a math textbook:
Step 1 - Skim the assigned reading material. Skim the material to get the general idea about the major topics. Read the chapter introduction and each section summary. You do not want to learn the material at this time; you simply want to get an over- view of the assignment. Then think about similar math topics that you already know.
Example: Skimming will allow you to see if problems presented in one chapter section are further explained in the next chapter sections.
Step 2 - As you skim the chapter, circle (using pencil) the new words that you do not understand. If you do not understand these new words after reading the assignment, then ask the instructor for help. Skimming the reading assignments should take only five to 10 minutes.
Step 3 - Put all your concentration into reading. While reading the textbook, highlight the material that is important to you. However, do not highlight more than 50 percent of a page because the material is not being narrowed down enough for future study. Especially highlight the material that is also discussed in the lecture. Material discussed both in the textbook and lecture usually appears on the test. The purpose for highlighting is to emphasize the important material for future study. Do not skip reading assignments.
Remember: Reading a math textbook is very difficult. It might take you half an hour to read and understand just one page.
Step 4 - When you get to the examples, go through each step. If the example skips any steps, make sure you write down each one of those skipped steps in the textbook for better understanding. Later on, when you go back and review, the steps are already filled in. You will understand how each step was completed. Also, by filling in the extra steps, you are starting to over learn the material for better recall on future tests.
Step 5 - Mark the concepts and words that you do not know. Maybe you marked them the first time while skimming. If you understand them now, erase the marks. If you do not understand the words or concepts, then reread the page or look them up in the glossary. Try not to read any further until you understand all the words and concepts.
Step 6 - If you do not clearly understand some words or concepts, add these words to the notetaking glossary in the back of your notebook. Your glossary will contain the bold print words that you do not understand If you have difficulty understanding the bold-print words, ask the instructor for a better explanation. You should know all the words and concepts in your notebook’s glossary before taking the test.
Step 7 - If you do not understand the material, follow these eight points, one after the other, until you do understand the material
Point 1- Go back to the previous page and reread the information to maintain a train of thought.
Point 2 – Read ahead to the next page to discover if any additional information better explains the misunderstood material.
Point 3 – Locate and review any diagrams, examples or rules that explain the misunderstood material.
Point 4 – Read the misunderstood paragraph(s) several times aloud to better understand their meaning.
Point 5 – Refer to your math notes for a better explanation of the misunderstood material.
Point 6 – Refer to another math textbook, computer software program or video tape that expands the explanation of the misunderstood material.
Point 7 – Define exactly what you do not understand and call your study buddy for help. Point 8 – Contact your math tutor or math instructor for help in understanding the material.
Step 8 - Reflect on what you have read Combine what you already know with the new information that you just read. Think about how this new information enhances your math knowledge. Prepare questions for your instructor on the confusing information. Ask those questions at the next class meeting.
By using this reading technique, you have narrowed down the important material to be learned. You have skimmed the textbook to get an overview of the assignment. You have
carefully read the material and highlighted the important parts. You then added to your notetaking glossary unknown words or concepts.
Remember: The highlighted material should be reviewed before doing the homework problems, and the glossary has to be learned 100 percent before taking the test.

Knowing When to Take Notes

 

 
To become a better note-taker you must know when to take notes and when not to take notes. The instructor will give cues that indicate what material is important Some such cues include:
  • presenting usual facts or ideas
  • writing on the board • summarizing
  • pausing
  • repeating statements
  • enumerating; such as, “1, 2, 3″ or “A, B, C”
  • working several examples of the same type of problem on the black- board
  • saying, “This is a tricky problem. Most students will miss it.” For example, 510 is “undefined” instead of “zero.”
  • saying, “This is the most difficult step in the problem.”
  • indicating that certain types of problems will be on the test, such as coin- or age-word problems
  • explaining bold-print words
You must learn the cues your instructor gives indicating important material. If you are in doubt about the importance of the class material, do not hesitate to ask the instructor about its importance.
While taking notes, you may become confused about math material. At that point, take as many notes as possible, and do not give up on note-taking.
As you take notes on confusing problem steps, skip lines; then go back and fill in information that clarifies your misunderstanding of the steps in question. Ask your tutor or instructor for help with the uncompleted problem steps, and write down the reasons for each step in the space provided.
Another procedure to save time while taking notes is to stop writing complete sentences. Write your main thoughts in phrases. Phrases are easier to jot down and easier to memorize.
Abbreviations
E.G.for exampleDshows disagreement with statement or passage
CF.compare, remember in contextREFreference
N.B.note well, this is importantet aland others
\thereforebkbook
ƑbecausePpage
Ìimplies, it follows from thisetc.and so forth
>greater thanVsee
<less thanVSsee above
=equals, is the sameSCnamely
¹does not equal, is not the sameSQthe following
( )parentheses in the margin, around a sentence or group of sentences indicates an important ideaComm.Commutative
?used to indicate that you do not understand the materialDis.Distributive
0a circle around a word may indicate that you are not familiar with it; look it upA.P.A.Associative Property of Addition
Emarks important materials likely to be used in an examA.I.Additive Inverse
1, 2, 3to indicate a series of factsI.P.M.Identity Property of Multiplication

10 Steps to Improving Your Study Skills

 

Improving your study skills can be the great educational equalizer. Effective studying is the one element guaranteed to produce good grades in school. But it is ironic that students are almost never taught how to study – effectively – in school.
Example: An important part of studying is note-taking, yet few students receive any instruction in this skill. At best, you are told simply, “You had better take notes,” but not given any advice on what to record or how to use the material as a learning tool.
Fortunately, reliable data on how to study does exist. It has been scientifically demonstrated that one method of note-taking is better than another and that there are routes to more effective reviewing, memorizing and textbook reading as well. The following are 10 proven steps you can take to improve your study habits. I guarantee that if you really use them, your grades will improve.

1. Behavior modification can work for you.

Use the association learning concept. Attempt, as nearly as possible, to study the same subject at the same time in the same place each day. You will find that, after a very short while, when you get to that time and place, you are automatically in the subject “groove.”
Train your brain to think math on a time-place cue, and it will no longer take you 10 minutes a day to get in the math mood. Not only will you save the time and emotional energy you once needed to psych yourself up to do math, or whatever else, it will also help you remember more of what you are studying.
After studying, reinforce yourself by doing something want to do (watch television, go to a party). Experts know that positive reinforcement of a behavior (such as studying) will increase its frequency and duration.

2. Do not study more than an hour at a time without taking a break.

In fact, if you are doing straight memorization, do not spend more than 20 to 30 minutes at a time. Here is the rationale behind taking such small bites out of study time.
First, when you are under an imposed time restriction, you use the time more efficiently. Have you noticed how much studying you manage to cram into the day before big exams? That is why it is called “cramming.”
Second, psychologists say that you learn best in short takes. In fact, studies have shown that as much is learned in four one- hour sessions distributed over four days as in one marathon six- hour session during one day. That is because, between study times, while you are sleeping or eating or reading a novel, your mind subconsciously works on absorbing what you have learned. So it counts as study time, too.
Keep in mind when you are memorizing, whether it is math formulas or a foreign language or names and dates, that you are doing much more real learning more quickly than when you are reading a social studies text or an English essay.
The specialists say you will get your most effective studying done if you take a 10-minute break every hour. In fact, some good students study 45 minutes to an hour, and they take a five- to 10-minute break. The break is considered your reward and improves your learning over the next hour.
Dr. Walter Pauk, former Director of the Reading and Study Center at Cornell University, suggests you take that short break whenever you feel you need one. That way, you will not waste your time away by clock-watching and anticipating your break.
Another technique for keeping your mind from wandering while studying is to begin with your hardest or least favorite subject and work toward the easiest and/or the one you like best. Thus, your reward for studying the least favorite or hardest is studying the subject you like best. Try it; it works.

3. Separate the study of subjects that are alike.

Brain waves are like radio waves. If there is not enough space between input, you get interference. The more similar the kinds of learning taking place, the more interference. So, separate your study periods for courses with similar subject matter. Follow your studying of math with an hour of Spanish or history, not chemistry or statistics.

4. Do not study when you are tired.

Psychologists have found that everyone has a certain time of day when he or she gets sleepy. Do not try to study during that time (but do not go to sleep either – it hardly ever refreshes). Instead, schedule some physical activity for that period, such as recreation. If you have a stack of schoolwork, use that time to sort your notes or clear up your desk and get your books together or study with a friend.

5. Prepare for your class at the best time.

If it is a lecture course, do your studying soon after class; if it is a course in which students are called on to recite or answer questions, study before class. After the lecture, you can review and organize your notes. Before the recitation classes, you can spend your time memorizing, brushing up on your facts and preparing questions about the previous recitation. Question-posing is a good technique for helping the material sink in and for pinpointing areas in which you need more work.

6. Use the best note-taking system for you.

Quite a bit of research has been done on note-taking, and one system has emerged as the best. Use 81/2-by-11-inch loose-leaf paper and write on just one side. (This may seem wasteful, but it is one time when economizing is secondary.) Take the time to rule your page as follows:
  • If the course is one in which lecture and text are closely related, use the 2-3-3-2 technique: Make columns of two inches down the left-hand side for recall clues, three inches in the middle for lecture notes and three inches on the right side for text notes. Leave a two-inch space across the bottom of the page for your own observations and conclusions. See Figure 20 (Three-Column Note-Taking System).
  • If it is a course where the lectures and the reading are not closely related, use separate pages for class notes and reading notes, following the 2-5-1 technique: Two inches at left for recall clues, five in the middle for lecture notes and an inch at the right for observations and conclusions. (After a while, you will not need to draw actual lines.)
You have most likely taken your lecture notes in the form that evolved during your years of schooling. You have also probably evolved your own shorthand system, such as using a “g” for all “-ing” endings, an ampersand (&) for “and,” and abbreviations for many words (e.g., govt. for government and evaptn. for evaporation).
The recall clue column is the key to higher marks. As soon as possible after you have written your notes, take the time to read them over – not studying them, just reading them. Check right away, while it is all still fresh, to see whether you have left out anything important or put down anything incorrectly, and make changes.
After reviewing what you have written, set down recall clue words to the topics in your notes. These clue words should not repeat information but should designate or label the kind of information that is in your notes. They are the kind of clues you would put on “crib sheets.”
Example: To remember the information contained so far in this section on note-taking, you need just the following clues: 8 1/2-by-11, loose-leaf, one side: 2-3-3-2 or 2-5-1. As you can see, they are simply memory cues to use later on in your actual studying.
Dr. Robert A. Palmatier, Assistant Professor of Reading Education at the University of Georgia, suggests that you study for tests in the following manner:
  • Take out your loose leaf pages and shift them around so the order makes the most sense for studying.
  • Choose the first page and cover up the notes portion, leaving visible just the clues. See if you can recall the notes that go with the clues. As you get a page right, set it aside.
  • If you are going to be taking a short-answer test, shuffle your note pages so that they are out of order. (That is why it is important to use just one side of the paper.) “This approach provides for learning without the support of logical sequence,” Dr. Palmatier says, “thus, closely approximating the actual pattern in which the information must be recalled.’
  • If you are going to be taking an essay test, you can safely predict that “those areas on which the most notes are taken will most often be the areas on which essay questions will be based.”
The beauty of the “recall clue word” note-taking method is that it provides a painless way to do the one thing proved to help you remember what you have learned – actively thinking about the notes and making logical sense of them in your mind. If, instead, you just keep going over your recorded notes, not only will you get bored, but you will be trying to memorize in the worst way possible.

7. Memorize actively, not passively.

Researchers have found that the worst way to memorize — the way that takes the most time and results in the least retention — is to simply read something over and over again. If that is the way you memorize, forget it. Instead, use as many of your senses as possible.
  • Try to visualize in concrete terms, to get a picture in your head. In addition to sight use sound: Say the words out loud and listen to yourself saying them.
  • Use association: Relate the fact to be learned to something personally significant or find a logical tie-in.
Examples: When memorizing dates, relate them to important events, the dates of which you already know. Use mnemonics: For example, the phrase “Every good boy does fine,”, is used for remembering the names of the musical notes on the lines of the treble clef. Use acronyms, like OK4R, which is the key to remembering the steps in the reading method outlined in number 8, below.

8. Read and study at the same time.

It really takes less time in the long run! Read with a purpose. Instead of just starting at the beginning and reading through to the end, you will complete the assignment much faster and remember much more if you first take the time to follow the OK4R method devised by Dr. Walter Pauk:
  • Overview – Read the title, the introductory and summarizing paragraphs and all the headings included in the reading material. Then you will have a general idea of what topics will be discussed.
  • K – Key Ideas – Go back and skim the text for the key ideas (usually found in the first sentence of each paragraph). Also read the italics and bold type, bulleted sections, itemizations, pictures and tables.
  • R1- Read -your assignment from beginning to end. You will able to do it quickly, because you already know where the author is going and what he/she is trying to prove.
  • R2 – Recall – Put aside the text and say or write, in a few key words or sentences, the major points of what you have read. It has been proven that most forgetting takes place immediately after initial learning. Dr. Pauk says, “One minute spent in immediate recall nearly doubles retention of that piece of data!”
  • R3 – Reflect – The previous step helps to fix the material in your mind. To cement it there forever, relate it to other knowledge; find relationships and significance for what you have read.
  • R4 – Review – This step does not take place right away. It should be done for the next short quiz, and then again for later tests throughout the term. Several reviews will make that knowledge indelibly yours.

9. Make up a color and sign system for text and notes.

For your text, Dr. Palmatier suggests:
  • Red for main ideas
  • Blue for dates and numbers
  • Yellow for supporting facts.
  • Circles, boxes, stars and checks in the margins can also be utilized to make reviewing easy.
  • Make your own glossary of the words and concepts you do not know.
In your notebook, underline, star or otherwise mark the ideas which your teacher tells you are important: thoughts to which you are told you will be coming back later, items which you are warned to be common mistakes. Watch for the words – such as therefore and in essence – which tell you what is being summarized. Always record examples. In fact, in such subjects as math, your notes should consist
mainly of your teacher’s examples.
Pay close attention in your note-taking until the last minute of class time. Often, a teacher gets sidetracked and runs out of time. He/she may jam up to a half-hour’s content into the last five or 10 minutes of a lecture. Get down that packed-few minutes’ worth. If necessary, stay on after class to get it all down.

10. Do not buy underlined textbooks.

Of course, if the book does not belong to you, you will not be underlining at all. But if you underline, do it sparingly. The best underlining is not as productive as the worst note-taking.
Over-underlining is a common fault of students; only the key words in a paragraph should be underlined. It should be done in ink or felt-tip highlighter, and it should be done only after you have finished the “OK” part of your OK4R reading.
If you are buying your books secondhand, never buy one that has already been underlined. You may tend to rely on it, and you have no idea whether the hand that helped the pencil got an “A” or a ” F” in the course! If, due to availability or finances, you have to buy an underlined textbook, mark it in a different color.
Research has proven that it is not how much time you study that counts but how well you study during that time. In fact, in at least one survey, students who studied more than 35 hours a week came out with poorer grades than those who studied less.
Remember: Use your study time wisely, and you too will come out ahead.

21 Tips For Improving Math Study Skills

 

  1. Spend as much time on math homework as needed.
     
  2. Complete your most difficult homework assignments first.
     
  3. Read ahead in the math textbook and prepare questions for the instructor.
     
  4. For each chapter, prepare your own list of math vocabulary words.
     
  5. Find a study buddy and set up group study times.
     
  6. Develop practice tests and time yourself while taking them.
     
  7. Read ahead in your textbook and make an informal outline.
     
  8. For practice, do all the example problems in the text.
     
  9. While doing homework, write down questions for the instructor/tutor.
     
  10. Be aware of the time allotted while taking a math test.
     
  11. Make sure you attend every math class.
     
  12. Schedule a study period after your math class.
     
  13. Verbalize (silently) problems the instructor writes on the board. Solve the problem on paper or silently verbalize each solution step.
     
  14. Interview instructors before actually signing up for their course to compare your learning style to their instructional style.
     
  15. Make note cards to remind yourself how to solve various math problems.
     
  16. Get help early in the semester before you get too lost.
     
  17. For understanding, recite back the materials you have read in your math textbook.
     
  18. Take notes on how to solve difficult problems.
     
  19. Copy all the information that is written on the board.
     
  20. Do math homework every day.
     
  21. If you miss a class, ask your instructors for permission to attend the same course that is taught at a different time or day. Remember: You are held responsible for material covered in classes that you have missed.

Substitution and Memory Strategies

     
Substitution The substitution strategy is used for solving math problems, especially when the student is unclear about some component of a math equation or cannot set up the appropriate math equation to solve a word problem. With substitution, one simply replaces the unknown part of a math equation or problem with something known. Applications and examples of the substitution strategy are given below (D. Applegate, CAL). Fraction Math students are often confused when trying to solve math problems with fractions. Try substituting the decimal equivalent of the fraction whenever possible (as long as the decimal is not repeating). Simply divide the numerator by the denominator to get the decimal equivalent of the fraction. For instance,
1
2 (x + 4) = 14 0.5 (x + 4) = 14 0.5 (x) + 0.5 (4) = 14 0.5x + 2 = 14 0.5x = 12 x = 24

Variables Sometimes the meaning or function of variables in an equation is unclear. In this case, substitute an actual number for the variable(s) and work out the problem. The numbers don't necessarily have to "make sense" mathematically - they are just used to help you logically figure out the steps of the problem. Then follow those steps to solve the actual problem with the variable(s). For example,
Given I = Prt
Find t in terms of the other variables. Substitute numbers for the variables except t.
10 = 30 * 2 * t How would you get the numbers on one side?
10 = 60 * t
10 = t
60 What steps did you follow to get t by itself?
Multiply 30 and 2 to get 60, then divide both sides by 60. Use those steps to solve the real equation.
I = P * r * t
I = (Pr) * t
I = t
Pr

Word problems Students commonly experience difficulty with word problems, especially how to set up the equation using the informaton given in the question. Try substituting the unknowns or variables with actual numbers to help set up the equation. For instance,
Question: Two numbers add up to 15. If the larger number is twice the smaller number, what are the two numbers? Answer: First we need to assign variables. From the problem we know the relationship between the two numbers: the larger number is twice as big as the smaller number. If the smaller number is x, then the larger number is 2x. Now we need to write an equation using the variables plus the other information provided in the question. But how? Try substitution. Pretend one of the numbers is 2. If the two numbers add up to 15, as the problem states, the other number must be what? 13. How did you get this? This was determined by subtracting the pretend number from 15: 15 - 2 = 13. Now generalize. One number is equal to the total minus the other number. In other words, one number equals 15 minus the other number. This is your equation in English! Now you just have to put it into an algebraic expression. Our two numbers are x and 2x. We replace these into our English equation to get the math equation we need to solve the problem:
one number equals 15 minus the other number
x = 15 - 2x ... or ...
2x = 15 - x
Either equation will give the correct answer. Now just solve to find your answers!
Memory strategies Math courses often require that four types of information be remembered by students on quizzes and exams. Strategies for encoding and retrieving terms and definitions, symbols, math equations, and problem solutions are described here (D. Applegate, CAL). Terms and definitions Key words Highlight and focus on key words in the definitions. This reduces the amount of information to be remembered and helps one to identify words that may be omitted in fill-in test questions. Association Once the key words have been identified, try to associate the term with the key words. You can use phonetic associations, vivid visual associations, associations with prior knowledge, or other associations. Some examples are:
  • The numerator is the top number in a fraction, whereas the denominator is the bottom number in a fraction. Remember that "numerator" and "top" go together because they begin with letters that are close to each other in the alphabet. Similarly, "denominator" and "bottom" also begin with letters that are close together in the alphabet, plus the letters "d" and "b" look very similar in form.
  • A polynomial is a series of one or more terms that are added or subtracted, such as 3x + 2y - 4. To associate this word with its definition, try this visual association: Picture a prison inmate in a black and white striped outfit whose prison term involves adding and subtracting a bunch of parakets named Polly.
Flash cards Flash cards are useful for registering definitions of terms into memory. Write the term on one side of the card and the definition on the other. Use the flash cards to test your recall. Practice recalling the definition when given the term and visa versa. Running concept lists Make a running concept list by writing all terms and definitions on notebook paper divided into two columns. The terms go in the left-hand column and the definitions with highlighted key words are written in the right-hand column. Fold the paper or cover one column to test your recall of the terms and their definitions. Symbols Characterization Try drawing or visualizing math symbols as characters in order to remember their meaning. For example,
  • A cursive M stands the for mean of a population. Draw or picture in your head a bunch of angry-looking M's to remember this symbol.
  • In the equation I = Prt, the P stands for the principal (amount of money) invested. Draw or picture in your head a large P that will remind you of your school principal - a face in the loop of the P and arms holding a ruler or some other significant object. Have little dollar signs floating around the P to help you remember the symbol represents a sum of money.
Flash cards Symbols and their meanings may be summarized on flash cards and reviewed periodically to store them in memory. Running concept lists Make a running concept list by writing all symbols and their meanings on notebook paper divided into two columns. The symbols go in the left-hand column and the meanings are written in the right-hand column. Fold the paper or cover one column to test your recall of the symbols and their meanings. Math equations and rules Association Try phonetic, visual, and other associations to remember math equations and rules. The goal is to associate the math equation or rule with something you already know or something with which you are familiar. For instance,
  • This association based on fundamental moral principles helps one to remember the rules for multiplying signed numbers (REFERENCE). "Good" things in this association represent positive numbers and "bad" things represent negative numbers.
    • A good thing happening to a good person is good.
      [positive times positive equals a positive]
    • A good thing happening to a bad person is bad.
      [positive times negative equals a negative]
    • A bad thing happening to a good person is bad.
      [negative times positive equals a negative]
    • A bad thing happening to a bad person is good.
      [negative times negative equals a positive]
  • The rules for converting decimals to percents may be remembered using a variety of associations.
    • Use common experiences in the association: Think of common percentages we see in our everyday lives, such as sales (50% off and 20% off) or runaway inflation rates (100% or 150%). These are big numbers. Decimals are small numbers (0.5, 0.2, 1.0 and 1.5). How do you make a large number smaller? By dividing. How do you make a small number larger? By multiplying. So to change from percents to decimals (large to small), you divide by 100. And to change from decimals to percents (small to large), you multiply by 100.
    • Use alphabetic associations to remember the rules: To change from percent to decimal, you move the decimal point two places to the right. When you start with a percent you move to the right - p and r are close in the alphabet. To change from decimal to percent, you move the decimal point two places to the left. When you start with decimal you move to the left - decimal ends in l and left begins with l.
  • Use a variety of associations to keep straight the equations for the perimeter (P = 2L + 2W) and area (A = L * W) of a rectangle.
    • Associations based on real-life experiences can be used to remember the equations. When ordering fence to go around the perimeter of your yard, you would order so many feet or meters - the units are raised to the first power. How do you keep the units of something in the first power? By adding - so use the equation with the addition sign. Now, when ordering carpet to cover the area of your room, you would order so many square feet or square yards - the units are raised to the second power. How do you get units to the second power? By multiplying - so use the equation with the multiplication sign.
    • A simple association based on the length of the equations might help you to keep them straight. The word perimeter is a long word and it corresponds to the longer of the two equations. The word area is a short word and it corresponds to the shorter of the two equations.
Flash cards Math equations and rules may be summarized on flash cards and reviewed frequently to store them in memory. Running concept lists Make running concept lists of math equations and rules using notebook paper divided into two columns. The names of the equations or rules go in the left-hand column and the mathematical expressions are written in the right-hand column. Fold the paper or cover one column to test your recall of math equations and rules. Problem solutions Problem solutions refer to the correct order of steps required to successfully solve math problems. Herrman, Raybeck, and Gutman (1993, p. 192) offer the following suggestions for registering and remembering solutions to math problems. Associations (D. Applegate, CAL) may also be used. Rehearsal Repetitious review of the steps for solving a problem aids in registration in long-term memory. The effectiveness of this strategy is enhanced when rehearsals are done frequently and when rehearsals are made active by vocalizing, listening to recordings, or writing. Practice Working several practice problems for each solution set aids in registration. Try working sample problems from the book or problems for which answers are indicated in the book. Check answers to insure accuracy. Solve forwards and backwards Registration in long-term memory is enhanced when problems are solved forwards and backwards. Work the problem to find the answer, and then take your answer and work back to the original problem. Procedure cards Try using procedure flash cards to register problem solutions in long-term memory. On one side of the card write the type of problem and/or give an example. On the other side write the steps in English for solving the problem and actually show the steps for solving the example. Explain problem to someone else Remembering is enhanced when one explains or "teaches" the problem solution to another person. Try working with another student in the class, with a tutor, or with a friend or family member. Carefully and thoughtfully go through the solution process, step by step. Find an empty classroom and "teach" by writing the steps on the chalk board. Frequent review Review the solution often. Take flash cards with you to review while waiting in line or between classes. Explain the problem solution to a friend while walking to class. Frequent reviewing aids registration of information in your memory. Mnemonics Problem solutions may be registered in memory using mnemonics. Take the first letter of each step and form it into a cue word or cue phrase. The classic math mnemonics are:
  • Foil
  • This cue word stands for the steps in multiplying two binomials: multiply the First terms, then multiply the Outer terms, then multiply the Inner terms, and finally multiply the Last terms.
  • Please Excuse My Dear Aunt Sally
  • This cue phrase helps in remembering the order of operations: Parantheses, Exponents, Multiplication, Division, Addition, and Subtraction. Combine it with a mental image of your aunt doing something rude in an operating room to enhance your memory.
Past experience To remember the problem solution during a testing situation, think of specific practice problems that were similar to the test problems. Key words and associations Use visual associations or associations with real-life experiences to remember the key words in the steps for solving a particular problem. For instance,
  • Problem: Find the equation of a line that passes through the points (8, -3) and -2, 1).
  • Key Words: equation of line, through two points
  • Steps in the Solution: find the slope, use the point-slope formula, solve for y
  • Visual Association: Picture the slope equation at the top points of two mountain peaks [step 1], go down the mountain slope to the point-slope formula [step 2], and move to the Y of a clear mountain stream to find your equation [step 3].

Success in maths

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
  • 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.
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 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 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

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
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).

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 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 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 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.

General Math Tips

     
Read the book Read carefully over the assigned sections and look carefully at the sample problems. Decide if you benefit more by reading before or after the instructor covers the material. More information about reading math texts will soon be provided in a separate section of this page. Develop a sound math foundation Because most math courses are cumulative, in other words new concepts are added to and build upon previous concepts, it is very important that the early material be mastered thoroughly. Similarly, mastery of material from previous courses makes success in later courses more likely, so continually review and practice concepts from prior math classes. Time management Complete all readings and especially homework assignments as soon after they are announced as possible. And definitely complete all assignments before new material is covered since math is cumulative. This insures that the inforamtion is fresh in one's mind and linked to prior, more fundamental information. Do your assignments early enough that you can get help with the things you do not understand. Calculator Learn how to use your calculator effectively and efficiently, especially if exams are timed and you have trouble completing tests in the allotted time. Check with the instructor about suggestions for the appropriate calculator to purchase for a class. Be sure the machine comes with an instruction manual and read the manual. Learn how to use important function keys. Get in the habit of carrying the calculator with you. It is better in the long run to become proficient with your own calculator rather than borrowing other people's calculators. Show your work Avoid the temptation to skip steps when solving a problem unless you are quite clear about how to proceed. This is a good habit to get into with your math homework. And definitely don't skip steps on an exam no matter how well you know the material. Why take chances (unless you're running out of time)? Showing your work allows you to locate logical or calculation mistakes more easily, and sometimes partial credit is given for the correct portions of an answer. Organize your work and write legibly Write all numbers and variables clearly so they may be easily distinguished. Pay particular attention to 4 and 9, 1 and 7, x and y. Spaces are as important in math equations as are the numbers and variables themselves. Allow enough space between different terms in an equation so it is easy to distinguish them. Be sure to line up terms in each step of the solution, and write steps one below the other rather than to the right or left. Use lined paper or graph paper to help organize the problems on your page. Don't scrunch! Use plenty of paper to work each problem. Recycle the paper at the end of the term if you are concerned about wasting paper. Support services and materials Find out about the support services and materials available to you. Support services include workbooks, study groups, self-help videos and cassettes, peer tutors, professional tutors, and instructors' office hours. Using the resources from the start of the course may help your confidence and get you off on the right foot. Minimally, make use of these resources as soon as you feel uncomfortable with the material - do not wait until it is too late! Preparation and supplies Being prepared for each course involves several important factors:
  • complete any previously assigned homeworks
  • compile a list of questions about the previous assignments to ask the instructor
  • preview the material to be covered that day
  • take your textbook and/or workbook to class
  • carry the proper supplies to each class - calculator, pencils, erasers, lined or graph paper, etc.
Information organization Math information - including definitions, symbols, equations, and steps for solving problems - may be organized using flash cards, running concept lists, flow charts, and matrices (D. Applegate, CAL). Flash cards Flash cards are useful for organizing all forms of math information. Two examples are given below. Running concept lists Running concept lists organize all forms of math information. Flow charts Flow charts are useful for organizing sequential information such as the steps for solving a problem. Matrices Matrices may be used to organize math symbols, equations, and definitions.
 

TERM

DEFINITION

EXAMPLE

numerator

top number in a fraction

the 1 in
1
5

denominator

bottom number in a fraction

the 5 in
1
5

reciprocal

the inverse of a fraction (flip it)

2
3 the reciprocal is 3
2

integer

any member of the set of positive numbers, negative numbers, and zero

1, 2, 3,
-1, -2, -3,
0

PROBLEM

EQUATION

perimeter of rectangle

P = 2L + 2W

area of rectangle

A = L * W

volume of a rectangle

V = L * W * H

perimeter of square

P = 4s

area of square

A = s * s
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