Analyze Writing Assignments

When you receive a writing assignment topic, it is a good idea to start brainstorming by listing main ideas or key concepts. Put the words or phrases down on paper and just jot notes by them. Many people put circles around each thought and join them by arrows. This process may lead to a longer article.
Decide on the type of paper you are writing. This may be assigned by your teacher, but you may have to give an opinion and justify the opinion. Keep in mind the correct subjects and verbs to use.

Start researching your topic. Make sure you know what your teacher will and will not let you use. Start with the most general source first, so you can learn about the topic then work your way out to the most specialized sources. Use print materials. Before you use internet materials, make certain you know what is allowed.

Establish parameters for your topic. You do not want to be too generalized or too specialized. Make sure you pick a topic where you can find enough information.

Make a précis of your information. It may help to take this to your teacher, so you will know if you are on the right track. If it is necessary to write out your opinion or your thoughts on a topic, make sure that you have the facts to back it. Always remember to be a critical thinker and keep an open mind. Don’t overlook important research material because it doesn’t match your thesis statement.

Vocabulary Building

A good vocabulary is crucial to academic success. You will become a better student as you increase your word power. As you increase your vocabulary, you increase your reading ability that in turn should improve your critical thinking skills. A good vocabulary is the basis of understanding what you read and learn.

Memorization of lists is one of the most accepted ways of learning vocabulary. This is a good exercise for short term studying, but you often do not retain the information that you have learned for a particular class or test. Memorization is good for standardized testing.

You can often figure out the meaning of the word from reading in context. Read through the entire paragraph to see if you can get a meaning of the word. Figure out what the reading is exploring and try to learn the word.

The best way to learn a definition and to remember a word is to look it up the dictionary. Find the word, sound it out and spell it. Notice the syllables in the word. Read all the definitions listed. Always keep a dictionary on hand during your college career.

Another method of learning vocabulary is similar to concept mapping. Have a blank sheet of paper and put the word that you are trying to learn and remember in the center of the page. In another box, write down a description of the word. Be sure to keep the description short. Draw two columns. In one, list items that describe or would help you to remember the word. In the other column, write down examples that would be similar to the word. By using all these mapping tools, your chances of remembering a word are much higher.

Taking lecture notes

Lynne:
When taking notes make sure to pay attention to the teacher when they are speaking write down what the teacher says if he says it more than once. Make sure that you understand what the subject is about and if not then ask questions. But the practice to getting good grades is R-E-V-I-E-W thats right review. Review over text books and notes on a daily basis.

M.:
Do not miss a lecture, ever, no matter what. Read the chapter before the lecture and bring coffee. Professors don’t care if you bring a whole gallon if it helps you keep alert.

Prince:
Reading the chapter once before the lecture helps you understand the material and stay interested in the lecture. Also, try to stick around in the lecture room after class and fill in what you missed or left out.

Kesha:
I try to read before the lecture, but if not, I just bring my book to class and highlight what the professor talks about.

Erika:
Always sit in front and just keep taking notes. Try to write throughout the entire lecture.

Melissa:
I outline my notes as I am taking them and then later I highlight important headers.

Sanobeia:
It’s better to pay attention and write good notes in class, rather than taping lectures. Also, highlight headings in the notes because when you are studying for an exam, then you can immediately pick out important information you need to know.

Belia:
I use the Cornell Method for taking notes. In my notebook, before class, I draw a line, making a 1 1/2-inch margin on the left. Then I take notes on the right side of the paper and add questions pertaining to the notes on the left side.

Evelyn:
I underline main topics, definitions, and anything the professor repeats or emphasizes.

Kamilah:
If you miss class, get notes from someone you know is a good note taker.

Success in Mathematics

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.

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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.

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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.

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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).

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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.

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].