PROBLEM SOLVING: LOOK BEYOND THE RIGHT ANSWERAuthor(s): ROBERT L. McGINTY and LAWRENCE N. MEYERSONSource: The Mathematics Teacher, Vol. 73, No. 7 (October 1980), pp. 501-503Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962129 .

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PROBLEM SOLVING: LOOK BEYOND THE RIGHT ANSWER

The answer to a problem needs to reflect more than the stated conditions.

By ROBERT L. McGINTY and LAWRENCE N. MEYERSON

Northern Michigan University Marquette, Ml 49855

Too often when students are solving a

problem, their only concern is to get the

right answer. Typically, when students solve problems such as, "If 6 candy bars cost $1.50, then how much would 15 candy bars cost?" they arrive at the correct answer of $3.75. However, there is a large class of such problems that requires a decision

making process to obtain a complete and

meaningful solution. For instance: "Sup pose a 16 oz. can of food (Brand A) costs 29 cents, and a 28 oz. can of food (Brand B) costs 41 cents. Which is the better buy?" Although we can determine the unit prices to see which is the better buy based on

price, other decisions should be made. What about quality of the two brands? If the larger were cheaper, what about left overs? Will they be eaten or thrown out?

Also, in certain items (vegetables, fruits, etc.), what percent of the weight is water or

juice? How much sugar or oil? Therefore, price alone is only one consideration.

In order to help students focus on other

aspects of problem solving, it might help to

put a diagram (see fig. 1) on the chalkboard or bulletin board. It is not intended that the

parts of the diagram be exhaustive or mu

tually exclusive, nor that all the factors are used in solving a particular problem; but rather the figure includes several global as

pects of problem solving. Each category has broad implications in everyday Ufe sit

uations, and some of these implications will be explored in the upcoming examples.

The typical approach to problem solving is to translate the problem into mathemati

Problem Solving

Decision making

I Value judgments

I Implications

Other possibilities

I Satisfaction

Practicality

i Reasonableness

-Feasibility Mathematical? Method of?Knowledgespecific

factors solution to the problem

Fig.

cal terms and then attain a numerical solu tion. However, this is not enough. We must also involve students with such components as decision making, quality of the product, reasonableness of the results, implications for use, practicality, and values. By meld

ing the mathematical, social, and ethical

components of a situation, we can help stu dents understand the complexities of the

problem-solving process and become better

problem solvers themselves. Several areas of the mathematics cur

riculum afford an opportunity to explore these many facets of problem solving, and one such common area involves ratio and

proportion.

Quantity

a. Suppose a bag of grass seed covers 400

square feet. How many bags would be needed to uniformly cover 1850 square feet? Should the person buy 5 bags and save the leftover?figuring prices will rise

October 1980 501

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next year? Buy 5 bags and spread it thicker? Buy 4 bags and spread it thinner? Similar situations exist with fertilizer, paint, recipes?again requiring decisions.

(Thanks to Art Coxford for this example.) b. Suppose a wage earner makes $250 for

working 35 hours. How much would be made by working 45 hours at the same rate? Maybe the real question should be, How much difference is there in take-home

pay for the worker? Often, because of the different tax brackets, the amount of take home pay is not proportional to the extra hours worked. Also, the worker must de cide if giving up a Saturday is worth it; or

perhaps nothing was planned for Saturday so it does not matter?another type of deci sion.

c. Magnitudes can make proportions "unproportional." To look at an extreme

case, suppose a person who makes $20 000 a year saves $1000; how much could be saved if the person made $1 000 000 a year at the same rate? Whereas the amounts are

proportional, the magnitude of the differ ence is quite startling?consider just the amounts of interest that could be earned on the savings. Also, it is improbable that the rates of saving would remain constant.

d. Along these same lines, suppose a bas ketball player makes 4 baskets in 10 shots. How many points could be made if 30 shots were taken? Again, the rate probably would not be constant, and the salary made

by the player certainly would not be pro portional.

e. Or consider the problem about the boy who can eat 4 pancakes in 5 minutes. How

many could be eaten in 15 minutes? Just think about that for a minute?absurd?

How could the rate in a problem like this ever remain constant? (Thanks to Gene Krause for this example.)

Time

/. Time problems pose special consid erations. If a worker can do a job in 7 hours and another worker can do the same job in 5 hours, how long would it take the two workers to perform the job together? Im

plicit in problems of this type is the fact that the proportion of the job done by each

worker remains constant. But often two

people can do a job in less time than is pro portional because of the nature of the work. (However, if you are paying for a job to be done, you may feel it takes two

people longer to complete the work.) Or better yet, if you are paying for the work, why not just hire the person who can do the work in the shorter time and forget about the other worker altogether?

People just do not row a canoe upstream.

g. A typical problem found in texts is,

Suppose a person can travel 45 miles in 50

minutes; how long would it take this person to travel 650 miles at this same rate? Be sides the unlikelihood that a person could

maintain the same speed over that great a

distance?the magnitude is pronounced? just try driving 650 miles and it seems like forever.

Another example of this type is, If a ca noe can go upstream at the rate of 5 miles

per hour and downstream at the rate of 10 miles per hour and the entire trip takes 3

hours, how long does the canoe travel up stream? Have you ever tried to row a canoe

upstream? People just do not row canoes

upstream, let alone at a constant rate for

any period of time. If we are going to use

problems of this type, let us use ones that are at least realistic; for example, airplanes with headwinds. As teachers, we should be

making value judgments as to what kinds of problems we will ask our students to solve. If we expect worthwhile responses from our students, it is imperative that we

have problems that are worthy of their re

spect. In short, we have to recognize the nonmathematical factors that need to be considered in decision making.

Spuriousness

A. Many advertisements fall under this

heading. In some ads, there is a statement

502 Mathematics Teacher

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something to the effect: "It might cost more but I am worth it." The implication is that if you do not use the product, then you are not worth very much. Or perhaps you use the product; then you will be worth as

Is a product better, or proportionally better?

much as the person in the ad. But the real

question that should concern the consumer

is, Is the product better, or proportionately better? This is a decision that each person must make individually. Other ads pro claim that "while other brands cost less, our brand can really be cheaper per unit of

work because our brand can perform the

job better." Then in small print appear the

conditions under which this can happen

(often conditions not normally encountered

by an average person); for example, wash

ing many dishes at one time. These appli cations of proportion directly appeal to a

decision and value judgment on the part of the consumer. Although we can calculate which brand is a better buy dollarwise, we

must also decide on the worth of that prod uct in our own particular situation.

These are just a few instances, of a par ticular type of problem, where the mathe

matics is only part of the answer to that

problem. Discussing alternatives can help students appreciate that, in some problems, unwarranted implicit assumptions have been made; in others, explicit assumptions in the problem itself are unrealistic; and

that in many cases value judgments and further inquiry preclude a final solution.

This scrutiny has several important con

sequences: students will realize that an an swer is important but not an end in itself; mathematics can be related to social areas

of the curriculum; problem solving can be

strengthened and broadened; and meaning ful dialogue can result. Teachers must be alert to the types of questions to ask so that the problem is posed in the proper social

perspective. By doing so we can strengthen

students' perception of mathematics as a

tool to help make intelligent decisions in their daily affairs.

A Crucial 4? In 1978, after Panasonic got a 70 percent share of

the U.S. market for electric pencil sharpeners, a sur

vey convinced the company that "the writing pressure of the average American is far stronger than that of the average Japanese." So Panasonic changed the

sharpening angle from 16? to a blunter 20? that it felt would produce a stronger point and reduce lead

breakage. The new models proved satisfactory for 99.98 percent of the American customers. But a num ber of architects, draftsmen, accountants, artists, and others did complain; and even though the complaints amounted to only 0.02 percent of sales, Panasonic de cided to go back to the old 16? angle. (Washington Post, 5 May 1980)

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October 1980 503

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