Applying Number Sense to Problem SolvingAuthor(s): Barbara J. Dougherty and Terry CritesSource: The Arithmetic Teacher, Vol. 36, No. 6, FOCUS ISSUE: NUMBER SENSE (February1989), pp. 22-25Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194458 .

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Applying Number Sense to Problem Solving

By Barbara J. Dougherty and Terry Crites

NCTM's Commission on Standards for School Mathematics (1987) has identified problem solving and num- ber sense as important components of an effective mathematics program. This emphasis is generating attempts to understand the problem-solving process better and to incorporate the results into classroom practice. In keeping with the thrust, this article discusses the interrelationships be- tween problem solving and number sense in light of difficulties experi- enced by students participating in the problem-solving process.

Earlier research on problem solving is reflected in the content found in most textbook series by the use of some variation of Polya's (1973) four- step plan. Essentially, the four steps are (1) understand the problem, (2) develop a plan, (3) carry out the plan, and (4) look back. Although it builds a framework for the problem-solving process, progression through these steps does not guarantee students' success. In fact, many students ap- proach problem-solving tasks appre- hensively.

At least two steps of the solution process can cause trouble for stu- dents. On the one hand, the four prob- lem-solving steps may be memorized and recited, but students may still be

Barbara Dougherty teaches at the University of Missouri, Columbia, MO 65211, in preservice mathematics education and works with in- service staff development. Her interests lie in effective teaching methods for mathematics, particularly in problem solving. Terry Crites is chairman of the mathematics department at William Woods College, Fulton, MO 65251, where he teaches undergraduate mathematics and mathematics education courses.

unable to begin the search for an an- swer in the develop-a-plan step. This inability results in high frustration and an aversion to problem-solving tasks. On the other hand, a student may arrive at an answer but be unable to assess its reasonableness in the look- ing-back step. Hence, an illogical an- swer is left as the final solution.

These problem-solving difficulties deserve consideration with respect to their relationship with a student's number sense and associated skills involving operation sense, estimation,

and mental computation. The ability to use number sense concerning the magnitude of numbers, to make qual- itative and quantitative decisions about how to proceed in a specific problem-solving situation, and to judge the reasonableness of an answer (Sowder 1987) plays an integral role in problem solving. And number-sense concepts are complemented by a stu- dent's understanding of how specific arithmetic operations affect numbers, that is, by the student's operation sense. Also, the application of num-

Fig. 1 The dart problem

I If six darts are thrown at the target and all the darts hit the target Λ rings, how can these scores be made: 8? 12? 27? 28? 56?

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ber-sense ideas in estimation and mental computation further enhances a student's success in the problem- solving process.

Relating number-sense and opera- tion-sense concepts to the context of a specific problem can often help a stu- dent put a problem into perspective and ease the transition from under- standing the problem to developing a plan. These same concepts can also be applied in the looking-back step to judge the reasonableness of results. Integrating the number-sense and op- eration-sense concepts throughout these three steps can help develop a continuity of the problem-solving steps and can help students become better problem solvers.

How can number sense help the student begin a solution process? Before applying some computation as part of the solution process to a word problem, the student should antici- pate the magnitude of the answer with consideration of how an operation af- fects the size of the numbers in- volved. Consider the problem in figure 1. Successfully solving this problem first requires that the stu- dents understand how scores are ob- tained. The teacher can check stu- dents' comprehension by having each student draw, or in some way de- scribe, what could happen in one turn of throwing the six darts. Students could then see that to arrive at a total score, the only acceptable operation is addition (or a combination of addi- tion and multiplication). It is clear that no other computation will work, since both subtraction and division give an- swers smaller than the whole numbers involved. Displaying the scores ob- tained on these trials may prove use- ful when students are later confronted with the question concerned with get- ting an odd number as a score.

Whether working with these prelim- inary trials or specific questions in the problem itself, students using num- ber-sense concepts will notice that the magnitude of the scores has bound- aries. It is not possible to get any number less than 6 (all 6 darts in the 1 ring), nor is it possible to get a score greater than 54 (all 6 darts in the 9

February 1989

ring). Teachers may want to empha- size this idea of boundary so that students see the establishment of so- lution parameters as a part of the problem-solving process. The inclu- sion of this procedure helps to insure the detection of answers that are un- reasonable because of their magni- tude. It also helps develop self- monitoring procedures for the student to follow when little or no teacher guidance is available.

Once the students have had the opportunity to demonstrate their un- derstanding of the problem, more spe- cific questions can be undertaken. For example, how can a score of 8 be made? Using number-sense concepts, students will realize that it is not pos- sible to use target scores greater than 5 when using six darts to get a score of 8. Once this idea is established, stu- dents may use the strategies of guess and test or making a table to check the combinations. In doing so, they find that the only solution is one 3 and five l's.

When working with other scores, such as 28, no target rings can be readily eliminated. Instead, students may rely on number-sense concepts to determine the relationship between the number of darts landing in a target ring and the magnitude of that ring's score. For example, if the first dart lands in the 9 ring, at most only one other dart can land there. Otherwise, the score of 28 will be reached or exceeded before all the darts are thrown. Thus, all combinations using one 9 or two 9s can be explored. Using this thought process for each ring, students can plan a systematic attack to derive possible combina- tions for 28.

The solutions to obtaining scores of 27 and 56 are directly affected by number-sense ideas. At first, students may believe that 27 is a possible score, as it falls in the range of 6-54. This notion is refuted by the fact that the addition of an even number of odd numbers results in an even number. Hence, no odd-number scores are possible. (At this point, the students may use any list of six sample dart throws made in the understanding step to have more conclusive evi- dence that all the scores are even

numbers.) Teachers should use this opportunity to discuss results such as 27 and 56 in an effort to enhance existing number-sense concepts con- cerning the even-odd idea. However, the use of this idea may suggest to the student that 56 is a plausible score because it is even, but earlier it was established that the boundaries on all possible scores were 6 and 54. There- fore, a score of 56 is not possible.

In summary, number sense helps students establish the type of number and the magnitude of the anticipated solution. The type of number that can be obtained as a score has been deter- mined to be strictly an even number with a magnitude between 6 and 54. Hence, any other possibilities can im- mediately be disregarded. For those scores that are possible, number- sense concepts help students identify correct combinations.

How can number sense help a student reject an unreasonable answer? On completion of the carry-out-the- plan stage, the student should evalu- ate the result for reasonableness, a process that is frequently ignored. Two types of errors may be discov- ered in this looking-back step. An- swers may be of the wrong type, such as answering with a fraction when a whole number is appropriate, or may be unreasonable due to their magni- tude, resulting from such errors as a computational mistake, a misplaced decimal point, omission of zeroes, or miskeying a calculator.

Considering the problem in figure 2, even though the student has correctly performed the appropriate computa- tion, it is unreasonable to have a final answer involving a fractional part of a person. To eliminate this error, the student should ask himself or herself if the answer is the type of number expected when the problem was first encountered. In this problem the an- swer is "No, I expected a whole num- ber." Students with number sense would realize that the decision to round to 854 516 or 854 517 is not an issue, because of the magnitude of the numbers involved. Using an estimate in the data suggests reporting the an- swer as 854 500, since it contains the

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Fig. 2 In May, about 26 490 000 passengers travel by air in the United States. On the average, how many passengers travel each day?

854 516.12...

3ψ26 490 000

The answer is 854 516.12 people.

Fig. 3 In one day, 2 844 metric tons of copper is mined in America. If each ton is worth $1 687.20, what is the total value of the copper mined?

J V~ $1 687.20 χ 2 844 =

$47 983 968.00 Wow' That's a lot of money.

same number of significant digits as the problem data.

Checking for reasonableness of the magnitude of the answer often in- volves mental computation, estima- tion, and number sense. If the student estimated the answer to the problem in figure 3 as 3 000 x $1 700 = $5 100 000, she or he would see that the calculated solution contains the wrong number of digits to the left of the decimal point. This fact should signal the student that a recalculation is in order. Particularly in problems

24

involving multiplication, division, or large numbers, it is important to de- termine the order of magnitude for the final answer because of the numerous chances to make mistakes.

What is the teacher's role in developing number-sense skills with problem-solving tasks? Some students may need help in be- coming accustomed to approaching problem solving by using numerical and operational reasoning. The

teacher can give assistance by asking questions that guide students through appropriate thought processes and help them develop a more specific plan for solving problems.

Consider the following problem: A farmer has 50 animals, hens and rabbits, in a barnyard. They have a total of 140 legs. How many hens and how many rabbits does the farmer have?

The teacher may question the stu- dents in the following manner:

1 . What type of number will we get for an answer?

2. What's the largest number of digits possible in the answer?

3. What's the largest number of hens or rabbits the farmer could have?

4. Can the farmer have 50 hens and no rabbits or 50 rabbits and no hens?

5. Can you guess how many rabbits and hens the farmer has? The first three questions are partic-

ularly valuable during the understand- ing and looking-back stages. Through these questions the students can focus their attention on the magnitude and number type of their answer. For ex- ample, students would be aware of the impossibility of having half a hen or rabbit, indicating that the answers should be integers. Further, since no negative numbers are feasible, the an- swers must be whole numbers. No three-digit answers are possible ei- ther, as the total number of animals is a two-digit number. These questions also help assess students' comprehen- sion of the problem.

The last two questions help stu- dents develop a plan. Once the stu- dent has made a guess in answering the fifth question, a method must be found for checking the guess. To com- plete the develop-a-plan step, the stu- dents may choose one of these strat- egies: guess-test, make a table, or set up an equation.

Opportunities to apply number sense become prominent in variations of this problem. For example, would it be possible for the farmer to have rabbits and hens having a total of 153 legs? (Assuming, of course, that no rabbit or hen had lost any legs in a

Arithmetic Teacher

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barnyard fight!) This is an excellent opportunity for the students to dis- cuss the result of adding or multiply- ing even numbers. Or, if the 50 ani- mals had a total of 146 legs instead of 140 legs, would the farmer have more hens or more rabbits than in the orig- inal problem? Could there be more of both? These questions help students see the relationship between the num- bers given in the problem and their place in the computational frame- work.

As previously discussed, the dart problem in figure 1 presents a context for the application of number-sense concepts. Even so, extensions of this problem offer an even richer context for use in developing students' num- ber sense. Extensions could include the following questions:

1 . Are any even scores between 6 and 54 impossible using six darts?

2. What would happen if we used five darts instead of six? Could you answer the questions in the same way with five darts as you did with six darts? Why or why not?

3. What if the possible scores in the target rings were all even num- bers? How would that restriction affect the scores you could obtain?

These questions and the discus- sions that result create an atmosphere for the communication of number- sense ideas in a problem-solving con- text. Standard computational con- texts are appropriate settings for this type of discussion, but frequently a correct answer becomes more impor- tant than concepts or meanings. Thus, in this problem-solving situation, a student's grasp of the effects of oper- ations can be nurtured and expanded to lay a better foundation for further work in more abstract areas, such as algebra, as well as to promote success in solving problems.

Summary Good number sense can be very help- ful to students who are solving prob- lems involving computations. It proves useful both in establishing the magnitude and the expected type of number for the answer and in helping to select the appropriate computation for the problem, that is, in acquiring

operation sense. With the further ap- plication of estimation and mental computation, the likelihood of obtain- ing an unreasonable answer is de- creased.

Many students bring these concepts with them to a problem-solving situa- tion, but teacher guidance may be needed to help them apply their skills successfully and appropriately. As students discuss the application of number-sense concepts to a word problem, insights can be gained that help remove the mysteriousness of

the problem-solving process and form a more solid foundation for under- standing the "hows" and "whys" of numbers. References Commission on Standards for School Mathe-

matics of the National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Working Draft. Reston, Va.: The Council, 1987.

Pólya, George. How to Solve It. Princeton, N.J.: Princeton University Press, 1973.

Sowder, Judith Threadgill. "Relating Mental Computation, Number Sense, and Computa- tional Estimation." Proposal for grant from National Science Foundation, 1987. W

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