Problem Solving: Tips For TeachersAuthor(s): Phares G. O'Daffer, Edward A. Silver and Verna M. AdamsSource: The Arithmetic Teacher, Vol. 34, No. 8 (April 1987), pp. 38-39Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193152 .

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38 Arithmetic Teacher

ra/j Strategy Spotlight ^ Comparing Problems

Many students have difficulty "seeing" the structure of a problem, often focusing instead on less impor- tant aspects, such as the setting or context, and key words. To help students focus on structure, several different tasks can be created. You might give stu- dents a set of three problems, such as these:

a. Forty-five people will be going in cars to the baseball team's picnic. Each car holds five peo- ple. How many cars are needed?

b. Brian is bringing five cookies for each person at the baseball team's picnic. If forty-five people are at the picnic, how many cookies are need- ed?

c. Beverly is organizing a scavenger hunt at the baseball team's picnic. Each team must have three people on it. If eighteen people want to participate, how many teams will there be?

Students can be asked, 'In what ways are problems (a) and (b) alike or different?" or "In what ways are problems (a) and (c) alike or different?" The ensuing discussion can focus on key facets of the problems. For example, problems (a) and (b) both involve the same context and the same numbers, but problem (a) requires division and problem (b) requires multipli-

Edited by Phares G. O'Daffer Illinois State university Normal, IL 6I761 Prepared by Edward A. Silver and

Verna M. Adams San Diego State University San Diego, CA 92182

cation. Problems (a) and (c) involve a different con- text and number but require the same operation.

It is important to focus on both: how the problems are the same and how they are different. Doing so helps students see that the same pair of problems can be simultaneously related and unrelated, de- pending on the criteria for judgment. An awareness of the criteria for judgment then leads to a focus on those similarities and differences that are important in the solution process. Another appropriate and valu- , able question for the set of problems would be, *

"Which two of the three problems are most mathe- /{ matically related?" y

If the problems are placed on index cards, students can be asked to sort the problems into sets of mathe- matically related problems. In small groups or individ- ually, the students can explain the basis for their groupings.

Such activities can offer students an opportunity to focus on the structure of mathematics problems and the relationships among problems. This approach should provide an enjoyable change in the classroom routine and give the teacher a rich supply of informa- tion to use in planning instruction on problem solving.

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□ Tip Board D urn 1 _ I I Classroom Climate lJ Create a reflective, thoughtful atmo-

sphere as the background for build- ing students' confidence in their prob- lem-solving skills. Whenever possible, encourage students to

j~| share their individual perspectives in small groups and in class discus- sions. Observe which criteria for judging problem relatedness are be- ing used by your students.

LJ Developing Problem-solving Skills

Having students compare prob- lems focuses their attention on the important problem-solving skill of understanding the problem.

Also, in teaching problem solv- ing, using carefully developed se- quences of related problems can be helpful, so that students don't become too mechanical in their problem-solving behavior. In class discussions help students become aware of their mental operations in solving problems, especially how often they use information from re- lated problems. As Polya suggest- ed, encourage students to think of a related problem if they get stuck.

I I Give It a Try Help students build problem- solving skills by (1) observing the types of problems that are difficult for them, (2) thinking about the structural characteris- tics of the problems, and (3) creating sets of related prob- lems for students to compare. The problem sets will be more effective if you carefully design them to focus on the areas of students' difficulty. The prob- lems can be varied in many ways, such as changing the problem's context, changing the numbers, or adding extraneous information. Useful techniques for varying problems were pre- sented in the May 1986 "Prob- lem Solving: Tips for Teachers."

D Take a Look For a more detailed discussion of problem relat- edness and the difficulties students have in us- ing related problems, see 'Think of a Related Problem" by Edward A. Silver and J. Philip Smith in Problem Solving in School Mathemat- ics, the 1980 NCTM Yearbook (pp. 146-56).

/, ^ I I Problems for Discussion *ò Here is another pair of problems that

T/"J^* might be useful in discussing the struc- ture of division and multiplication prob- lems. How are these problems the same? How are they different?

a. Maria earned $10.50 in 2 hours. How much did she earn each hour?

b. Maria earned $10.50 an hour for 2 hours. How much did she earn?

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. 9

April 1987 39

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