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Thursday, January 25, 2007

Mathematics Tips: 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 PĆ³lya'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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