Problem Solving: Tips For TeachersAuthor(s): Phares G. O'Daffer, Edward A Silver and Verna M. AdamsSource: The Arithmetic Teacher, Vol. 34, No. 9 (May 1987), pp. 34-35Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194228 .

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Problem totoing Tip) For Taachao

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It is fairly common for students to believe that every prob- lem has one and only one answer, that there is a correct algorithm or procedure to follow in solving any mathematics problem, and that all mathematics problems should be solved quickly, if they can be solved at all. It is important to offer experiences that broaden students' conceptions of mathematics and problem solving. One useful technique is to present open-ended problems such as the following:

A. A rectangle has a perimeter of 36 units. Its area is larg- er than 50 square units. What conclusions about the rectangle can you draw?

B. Using the information given, write and solve as many problems as you can. Thu and Marina each ride their bikes to school. Thu lives 8 blocks from school. Marina lives 12 blocks from school. It takes Thu 16 minutes to ride his bike to school each morning.

C. Suppose you have a pocket full of change. It costs $0.60 to buy a soda from the soda machine. What coins should you use to get rid of as much change as possi- ble?

Problem A is a good example of a problem that does not have one unique answer. Several different rectangles with whole-number dimensions have a perimeter of 36 units and an area larger than 50 square units. In dealing with this problem, students will need to consider more than one possibility. One approach to the problem would be to consider all rectangles that have a perimeter of 36 units.

Edited by Phares G. O'Daffer Illinois State University Normal, IL 61761 Prepared by Edward A Silver and Verna M. Adams San Diego State University San Diego, CA 92182

How many rectangles are involved? Can the student draw pictures of them? What is the area of each rectangle? How many of the rectangles have an area greater than 50? For more advanced or capable students, it would be appropri- ate to suggest that the lengths of the sides do not neces- sarily have to be whole numbers.

Students are rarely given a chance to pose original mathematics problems. Problem B gives students an op- portunity to generate their own problems. The problems that they generate could then be used as a whole-class or small-group assignment. In addition to furnishing practice in the computational skills involved in the solution of the prob- lems, the assignment could get students to focus on some or all of the following questions: Can the problems be solved with the information given? If so, what is the an- swer? If not, what other information is needed? What as- sumptions did you make to solve the problem? (E.g., Does Marina ride her bike at the same rate as Thu? Does Marina or Thu have to ride up or down a hill?)

Problem C is a good example of a problem that is really many different problems. Different students might approach the problem in different ways. The different approaches and solutions might be the result of different interpretations of what it means to "get rid of as much change as possible." Students could focus on the weight, volume, surface area, or the number of the coins used to purchase the soda. It is also possible to differ about what it means to "have a pock- et full of change." For example, students might put con-

34 Arithmetic Teacher

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straints on the situation, such as the specific number of quarters, nickels, or dimes that might be in one's pocket. After the students have had a chance to explore the prob- lem, it would be valuable to have a class discussion of

these different interpretations to help students understand the importance of carefully formulating a problem before at- tempting to solve it. Only after defining the problem is it ap- propriate to focus on the required computations.

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Part of the Tip Board is reserved for techniques that you've found useful in teaching problem solving in your class. Send your ideas to the editor of the section. I

May 1987 35

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