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Techniques for Studying Math 2 Math is different from other subjects! Math requires different study processes. In other courses, you learn and understand the material, but you seldom have to actually APPLY IT to learn. You have to DO math problems to learn math. Math is a linear learning process. What is used one day is used the next, and so forth. (In history, you can learn Chapter 2, skip Chapter 3, and still do OK on Chapter 4. In math, you must understand the material in Chapter 1 before you go on to Chapter 2.) You must keep up with the instructor: attend class, read the text, and do homework everyday. Falling a day behind puts you at a disadvantage. Falling a week behind puts you in deep trouble. Math is much like a foreign language. It must be practiced EVERY DAY, and often the vocabulary is unfamiliar. Math in college is different from math in high school. Instead of going to class everyday, in college you go only two or three times a week. What took a year to learn in high school is now covered in only fifteen weeks. In college, exams are spaced further apart and so cover more material than high school tests. College instructors may not even check your homework. You are responsible for making sure that you understand the material. I. Math Anxiety A. Math is the course that causes the greatest anxiety for students. That means you are not alone if you have math anxiety. Yes, believe it! Fortunately, there are several strategies for reducing math anxiety. 1. Overcome negative self-talk. a. Self-talk involves the things you say to yourself. It has a great effect on your ability and motivation. Listen and become consciously aware of the types of things your say to yourself. Make the decision to replace your negative self-talk with positive self-talk. Learn to say: 1) I know I can do this. 2) I have studied and I know what I’m doing. 3) I will not give up until I’ve answered every question. 4) I believe that I will be able to remember what I’ve studied. 5) Seeing that makes me remember what else I studied about this idea. b. Ask questions! If anything is unclear, or even if you want to confirm that what you think is correct, ASK! Besides, there are usually other students wanting to know the answers to the same questions you have. c. Consider math a foreign language – it must be practiced. d. Don’t rely on memorization to study mathematics. Math is learned by DOING problems, not by trying to memorize them. Each problem is different. The ability to solve problems must be learned by doing. e. READ your math text book. Yes, actually read it! The text will give you another perspective on the material and may even offer tips which were not covered in class. Additionally, it is a great way to reinforce what you are learning in class. Penn State University Center for Academic Achievement 200 University Drive; Schuylkill Haven, PA 17972 (570) 385-6140 Location: Ciletti Memorial Library (Lower Level) Website: www.sl.psu.edu/caa Hours: M-F, 8:00 a.m.-5:00 p.m./Evenings: M-F, as scheduled C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 03/27/08 ray 3 f. Get help the same day that you find you don’t understand something. g. Be relaxed and comfortable while studying math. h. Develop responsibility for your own successes and failures. You found the correct answer because YOU knew how to do the problem, not because it was just an easy one. You had an answer wrong because YOU need more practice with that type of problem. Do You Have Math Anxiety? A Self Test Rate your answers from 1 to 5; add them and check your score below. (1) = Disagree, (5) = Agree. 1. I cringe when I have to go to math class. 1 2 3 4 5 2. I am uneasy about going to the board in a math class. 1 2 3 4 5 3. I am afraid to ask questions in math class. 1 2 3 4 5 4. I am always worried about being called on in math class. 1 2 3 4 5 5. I understand math now, but I worry that it's going to get really difficult soon.1 2 3 4 5 6. I tend to zone out in math class. 1 2 3 4 5 7. I fear math tests more than any other kind. 1 2 3 4 5 8. I don't know how to study for math tests. 1 2 3 4 5 9. It's clear to me in math class, but when I go home it's like I was never there. 1 2 3 4 5 10. I'm afraid I won't be able to keep up with the rest of the class. 1 2 3 4 5 CHECK YOUR SCORE 40-50 Sure thing, you have math anxiety. 30-39 No doubt! You're still fearful about math. 20-29 On the fence! 10-19 Wow! Loose as a goose! Math anxiety is an emotional reaction to mathematics based on a past unpleasant experience which harms future learning. A good experience learning mathematics can overcome these past feelings and success and future achievement in math can be attained. ~Ellen Freedman II. Before, During, and After Math Class A. Before Math Class 1. If your math anxiety is high and you want more assurance about a math course, try auditing a course. When auditing a course, you are not a full participant in the course since you do not take tests. Your grade will be recorded as “pass” or “fail,” and you do not receive credit for the course. Auditing provides an excellent opportunity for you to practice and familiarize yourself with the material before you enroll in the course for credit. Check with your advisor or the registrar for further information on course auditing. 2. Read your textbook before class to become familiar with new terms, formulas, and concepts. For each chapter, prepare your own list of math vocabulary words. For a better understanding of the material, recite back the materials you have read. Mark any trouble spots that you might want the instructor’s help in clearing up. C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 04/05/07 ray 4 B. During Math Class 1. Come to class on time and sit as close to the front as possible. 2. Exchange phone numbers with someone in class. When you are working on math and encounter a problem you just do not get, you will have someone to call for help. 3. Attend every class. If you miss a class, ask your instructor for permission to attend the same course that is taught at a different time or day. Remember: you are responsible for material covered in a class that you missed. 4. Note-taking: a. Use a good math note-taking system (see note-taking strategies handout). b. Don’t photocopy someone else’s notes. It is important to attend class and hear everything for yourself. You may not understand what someone else wrote, or you may deem things important which someone else may leave out. c. Take notes on how to solve problems. Don’t just record the problem itself. When you look at the problem later, you may not be able to recall how it was solved. d. Copy all information that is written on the board. It is important; otherwise, the instructor would not have taken the time to write it. e. Verbalize (silently) problems the instructor writes on the board. Solve the problem or silently verbalize each solution step. f. Ask questions, ask questions, ask questions! You are paying for the course; make sure you get your money’s worth. Chances are that many other students in the class have the same question and will be relieved you asked. g. Get help early in the semester before you get too lost in the course. You know the old saying, “A stitch in time saves nine.” Get help figuring out the small stuff before it grows into a huge dilemma! C. After Class 1. Find a study buddy and set up group study times. 2. Schedule a review period after every class, even if it’s for only ten minutes. Work through examples again. 3. Make note cards or a “math dictionary” to remind you how to solve various types of math problems. a. Put each concept on a 3” X 5” note card and write an example on the back. b. List all new math terms with their definitions in a separate notebook. Review these at least once a week. 4. Take full advantage of all the math resources in the Center for Academic Achievement! It’s FREE!! a. Math tutoring Penn State University Center for Academic Achievement 200 University Drive; Schuylkill Haven, PA 17972 (570) 385-6140 Location: Ciletti Memorial Library (Lower Level) Website: www.sl.psu.edu/caa Hours: M-F, 8:00 a.m.-5:00 p.m./Evenings: M-F, as scheduled C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 03/27/08 ray 5 b. Math resource texts c. Math sample tests d. On-line math aids in the CAA computer lab 5. Read your text and do all the examples. 6. Do a bit of math review, homework, studying, and practice every day. Increase or decrease the time you spend daily on math with respect to your mastery of the material. Remember to practice working the types of problems you may be having difficulty with until you master them. III. Getting Assistance, Problem Solving, and Homework Tips A. Get assistance. 1. 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 B. Ask the right questions. 1. 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. a. Examples: 1) 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 which you don’t understand. 2) 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. 3) Good question: “Can you tell the difference between the equation of a circle and the equation of a line?” 4) Okay question: “How do you do #17?” 5) 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.) TIP - Right after you get help with a problem, work another similar problem by yourself. 2. Control the help you get. a. 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 should they 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. 1) 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. 2) 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. 3) 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. C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 04/05/07 ray 6 C. Problem Solving Skills (Homework and Tests) 1. General Problem Solving a. 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. Break these problems into smaller pieces and solve each piece – divide and conquer! b. 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. c. The practice you get doing homework and reviewing will make test problems easier to tackle. d. 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.) Understand all words, phrases, terms, and symbols. Draw a figure, if needed. 2) Next, devise a plan; that is, identify which skills and techniques you have learned that can be applied to solve the problems at hand. a) Have you seen a similar problem before? b) Break the problem into parts. c) Draw on prior knowledge. d) Spell out relationships carefully. e) Look for related unknowns. 3) Carry out the plan. If you get stuck, reorganize the elements of the problem in a new way. Check the problem to be sure a simple arithmetic mistake wasn’t made. If you are still having difficulty, try taking a small break. 4) Look back and check all your work. 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. TIPS - 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. 2. Solving Word Problems a. The term “word problem” often has negative connotations. It’s better to think of such assignments as “applied problems.” Actually, 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 Penn State University Center for Academic Achievement 200 University Drive; Schuylkill Haven, PA 17972 (570) 385-6140 Location: Ciletti Memorial Library (Lower Level) Website: www.sl.psu.edu/caa Hours: M-F, 8:00 a.m.-5:00 p.m./Evenings: M-F, as scheduled C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 03/27/08 ray 7 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. 3. Steps to Solving an Applied Problem a. First, convert the problem into mathematics. This step is (usually) the most challenging part of an applied problem. If possible, start by drawing a sketch. Label it with all the quantities stated 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 conversions of the problem into math, i.e., find equations which describe relationships among the variables and describe the goal of the problem mathematically. b. Solve the math problem you have generated, using whatever skills and techniques you need (refer to the four-step process above). c. As a final step, you should convert the answer of your math problem back into words, so that you will have solved the original applied problem. d. For further help: George Pólya, How to Solve It. Princeton: UP, 1945. 4. Some errors in reasoning to avoid: a. Failing to observe and use all relevant facts of a problem. b. Failing to approach the problem in a systematic, step-by-step manner. c. Jumping to conclusions or using leaps in logic without thoroughly checking them. d. Failing to spell out relationships fully. e. Being sloppy and inaccurate in collecting information and carrying out mental activities. D. Homework Tips 1. Do math homework every day! 2. Start your assignment as soon after your math class as you can. If you cannot begin working on your assignment right after class, take a moment to look it over. Try to make some connections between what you just learned in class and the assignment. Jot down reminders of the connections you drew, so you can more easily bring them to mind when you have an opportunity to work on your assignment. 3. Complete your most difficult homework assignments first. Usually this mean your MATH homework! 4. Copy each problem carefully. 5. Think of the steps you will use and the rules you will follow. 6. Check your notes and examples. 7. Try every problem. Take a chance! You might be right. 8. Do not skip steps at first, even if the problem has a short cut. Do all the steps until you have perfected the process. 9. Write each step below the previous one. Be neat! 10. Check your work before looking at the answer key. 11. Arithmetic error? Brush up on the basics. 12. “Plug in” your answer, or do the problem a different way. C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 04/05/07 ray 8 13. Incorrect? Check your notes and book. 14. Still wrong? Ask the instructor, a tutor, classmate, or friend. 15. Do the problem from scratch until you get it right. 16. Do an extra set of problems - practice pays off! 17. Spend as much time on homework as needed. Don’t limit yourself. When you are taking your math test, you will be more confident. IV. General and Exam Study Skills A. Good study habits throughout the semester make it easier to study for tests. How are your study habits? Take the following inventory to find out how your study skills rate. Penn State University Center for Academic Achievement 200 University Drive; Schuylkill Haven, PA 17972 (570) 385-6140 Location: Ciletti Memorial Library (Lower Level) Website: www.sl.psu.edu/caa Hours: M-F, 8:00 a.m.-5:00 p.m./Evenings: M-F, as scheduled C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 03/27/08 ray 9 Math Study Skills Inventory Rate your achievement of the following statements by placing a 3 for “almost always,” 2 for “sometimes,” and 1 for “almost never.” If you have never even thought about doing what the statement says, put a 0. Selecting a math class 1. I schedule my math class at a time when I am mentally sharp. 2. When I register for a math class, I choose the best instructor for me. 3. If I have a choice, I select a math class that meets three or four days a week, instead of one or two times. 4. I schedule the next math class as soon as possible after I have completed the current course. 5. I am sure that I have signed up for the correct level math course. Time and place for studying math. 6. I study math every day. 7. I try to do my math homework immediately after math class. 8. I have a specific time to study math. 9. I have a specific place, with few distractions, to study math. 10. I do my math homework in the lab where I can get help. 11. I am careful to keep up-to-date with math homework. 12. I study math at least 8 to 10 hours a week. Study strategies for math class 13. I read my textbook before I come to class. 14. If I have trouble understanding the text, I find an alternate text. 15. I take notes in math class. 16. I am careful to copy all the steps of math problems in my notes. 17. I ask questions when I am confused. 18. I go to the instructor or lab when I am confused. 19. I try to determine exactly when I got confused and exactly what confused me. 20. I review my notes and text before beginning homework. 21. I work problems until I understand them, not just until I get the right answer for homework. 22. I use flashcards for formulas and vocabulary. 23. I develop memory techniques to remember math concepts. Math tests 24. I preview the test before I begin. 25. Before I begin the test, I make notes on things such as formulas that I might need. 26. I begin with the easy questions first. 27. I take the full amount of time allotted for the test. 28. I carefully check or rework as many problems as I can before I turn in my test. 29. When tests are returned, I keep a log of the types of mistakes I make on tests: concept errors, application errors, or careless errors. 30. I keep up-to-date, so that I don't have to cram the night before a test. C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 04/05/07 ray 10 Anxiety 31. I believe that I can succeed in math class. 32. I have study partners in my math class. 33. I take practice tests. 34. I know several good relaxation techniques. TOTAL SCORE CHECK YOUR SCORE Total the scores from all 34 statements. 90-103: Give yourself an A. You are using the study skills you need in order to be successful in math. 80-89: Give yourself a B. You are using good math study skills. Choose a few strategies to work on each day, and you are well on your way to an A. 70-79: Give yourself a C. Your study skills are average. If you want an A, choose one or two strategies in each category to work on until you are using most of the strategies described above. Below 70: You are probably having a difficult time in math class, but math might not be your trouble! More than likely, your main problem is the study strategies you are using (or not using). Make yourself do the things on the list above. This information can be found on page 117 of The Study Skills Workbook, Third Edition, by Dr. Carolyn H. Hopper. B. General Studying 1. If your study skills need improvement (and even if they don’t), here are a few tips: a. Keep up! You need to review your notes after every class and 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 right before the exam. b. 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. c. To memorize formulas, use flashcards. d. Get a study friend and set study times. When someone else is there, you are more likely to follow through with studying math. Also, you may be able to answer questions for each other. e. Rework problems that you missed on your homework. f. 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. Penn State University Center for Academic Achievement 200 University Drive; Schuylkill Haven, PA 17972 (570) 385-6140 Location: Ciletti Memorial Library (Lower Level) Website: www.sl.psu.edu/caa Hours: M-F, 8:00 a.m.-5:00 p.m./Evenings: M-F, as scheduled C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 03/27/08 ray 11 C. Studying for a Test 1. Start by going over each section, reviewing your notes, and checking that you can still do the homework problems (i.e. 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. 2. You’re not ready yet! In the book, each problem appears at the end of the section in which you learned to do that problem; on a test, the problems from different sections are mixed together. a. Step back and ask yourself what kinds of problems you have learned to solve, what solution techniques you have learned, and which techniques go with which problems. b. 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 “instructions.” Check your verbal explanations with a friend during a study session. (It’s more fun than talking to yourself!) c. Put yourself in a test-like situation: work problems from review sections at the end of chapters and from old tests (located in the Center for Academic Achievement). It’s important to keep working problems the whole time you’re studying. d. Get lots of sleep the night before the test. Math tests are easier to tackle when you are mentally sharp. V. The Math Test! A. 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). A good test-taking strategy can make a big difference in your grade! 1. Learn how to relax before tests are taken. Take a few deep breaths, do some stretches, and, if you need it, take a nice, slow walk to the pencil sharpener to clear your head. If you think the level of distress you feel over a math test is affecting your grades, make an appointment with the campus counselor for some stress/anxiety management sessions. 2. Be aware of the time allotted for taking the math test. 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. 3. Don’t panic over not being able to remember tricky formulas while taking the test. Use flash cards to memorize the formulas. Then as soon as you are handed your math test, write the formulas in the margin or on scrap paper if that is allowed. Then your mind will be at ease about remembering them, and you can focus on the problems. 4. Do the problems in the order that suits you! Start with the problems you feel most comfortable doing. 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. Finally, try the ones you are least sure about. 5. 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. If you have not finished it in the allotted time, 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 to try. C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 04/05/07 ray 12 6. 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 may assign partial credit based on the work you show. 7. Never waste time erasing! Just draw a line through that work 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). If you are not required to fit your answer in the space provided, you can put your answer on another sheet to avoid needing to erase. 8. In a multiple-step problem, outline the steps before actually working the problem. 9. 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, as least explain how you would do it. 10. Read the questions carefully and do all parts of each problem. 11. Verify your answers. Does each answer make sense given the context of the problem? 12. If you finish early, check every problem (that means rework everything from scratch). B. After the Test 1. Immediately look up questions that caused you problems. 2. Go over the test and correct missed problems. This will strengthen your skills for the next test. 3. Check with the instructor if you cannot figure out why the problem was wrong. Do Your Best! Remember to be positive. You can do it! This publication is available in alternative media on request. Penn State encourages qualified persons with disabilities to participate in its programs and activities. If you anticipate needing any type of accommodations or have questions about physical access provided, please contact Melinda Anthony-Spolski at (570) 385-6127 or mma149@psu.edu. Penn State is committed to affirmative action, equal opportunity and the diversity of its workforce. U.Ed. SL 07-15. Penn State University Center for Academic Achievement 200 University Drive; Schuylkill Haven, PA 17972 (570) 385-6140 Location: Ciletti Memorial Library (Lower Level) Website: www.sl.psu.edu/caa Hours: M-F, 8:00 a.m.-5:00 p.m./Evenings: M-F, as scheduled C:\Docstoc\Working\pdf\3bcaa1ca-8dd8-45a6-910a-0d16a11207c0.doc rev. 03/27/08 ray

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