how to approach problem-solving,

how to study for and take tests,

and when and how to get help.

- 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.

- 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.

- 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.

- 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.

- The higher the math class, the more types of problems: in earlier classes, problems often required just one step to find a solution. Increasingly, you will tackle problems which require several steps to solve them. Break these problems down into smaller pieces and solve each piece - divide and conquer!
- Problem types:
- Problems testing memorization ("drill"),
- Problems testing skills ("drill"),
- Problems requiring application of skills to familiar situations ("template" problems),
- Problems requiring application of skills to unfamiliar situations (you develop a strategy for a new problem type),
- Problems requiring that you extend the skills or theory you know before applying them to an unfamiliar situation.

- When you work problems on homework, write out complete solutions, as if you were taking a test. Don't just scratch out a few lines and check the answer in the back of the book. If your answer is not right, rework the problem; don't just do some mental gymnastics to convince yourself that you could get the correct answer. If you can't get the answer, get help.
- The practice you get doing homework and reviewing will make test problems easier to tackle.

- Apply Pólya's four-step process:
- The first and most important step in solving a problem is to
**understand the problem**, that is, identify exactly which quantity the problem is asking you to find or solve for (make sure you read the whole problem). - Next you need to
**devise a plan**, that is, identify which skills and techniques you have learned can be applied to solve the problem at hand. **Carry out the plan.****Look back:**Does the answer you found seem reasonable? Also review the problem and method of solution so that you will be able to more easily recognize and solve a similar problem.

- The first and most important step in solving a problem is to
- 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.

- First convert the problem into mathematics. This step is (usually) the most challenging part of an applied problem. If possible, start by
**drawing a picture. Label**it with all the quantities mentioned in the problem. If a quantity in the problem is not a fixed number,**name**it by a**variable. Identify**the goal of the problem. Then complete the conversion of the problem into math, i.e., find equations which describe relationships among the variables, and describe the goal of the problem mathematically. - Solve the math problem you have generated, using whatever skills and techniques you need (refer to the four-step process above).
- As a final step, you should convert the answer of your math problem back into words, so that you have now solved the original applied problem.

*For Further Reading:*

George Pólya, *How to Solve It,*Princeton University Press, Princeton (1945)

**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.

- 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.

- 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.

- 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).

**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.

- 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.

- 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.

SAINT LOUIS UNIVERSITY

June 1993

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