assessing reasonableness: some observations and suggestions

ASSESSING REASONABLENESS: SOME OBSERVATIONS AND SUGGESTIONSAuthor(s): Joe Garofalo and Jerry BryantSource: The Arithmetic Teacher, Vol. 40, No. 4 (DECEMBER 1992), pp. 210-212Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195314 .

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ASSESSING

REASONABUNESS: SOME OBSERVATONS AND SUGGESTIONS

Joe Garofalo and Jerry Bryant

friend Pat received a telephone bill of slightly more than $60 000 and called the telephone

company to report the error. When the company's representative suggested that she may have made more calls than she remembered during the month, Pat pointed out that it would take a very large number of calls to run up a bill of $60 000. The representative agreed and told ^^^ Pat that her bill would be inves- ^tt^^H tigated. The representative then ̂ ^^Щ told her that in the meantime she ̂ ^^T^ should pay the full amount of the ̂ ^Ш bill and that when the telephone ̂̂ Ш company determined the cor- ̂ ^B rect amount due, any excess pay- ̂ ^K ment would be credited toward ̂^K her next bill! ^B i

How can we explain the tele- ̂ ^A i phone company representative' s ^^K л apparently unreasonable direc- VK m tions? Very easily. The repre- ̂ ^Ш щ sentative had been trained to re- ^^ft ■ spond to billing complaints in a ^^9 И certain way, and in this instance ̂ K ■ she followed the usual step-by- ̂^m Щ step routine as taught. Since the ̂ ^B 1 telephone company rarely ^H Я makes such extreme errors, the ^EL^X routine "works" almost all the ^ЗЯ^Я time. Hence, the representative need only follow it blindly and need not evaluate its appropriateness on a case-by- case basis. Because this representative was unaccustomed to assessing the reasonableness of the outcome of the routine, she failed to do so in this situation in which it was so clearly necessary.

In a number of ways the behavior of this representative is similar to the behavior of

some students who give apparently unreasonable solutions to word problems. Con- sider the following examples:

1 . A number of third graders gave "70" as the answer to the problem "Tom and Sue visited a farm where there were chickens and pigs. Tom pointed out that there were 1 8 animals, and Sue pointed out

минеи

that there were 52 legs. How many chickens and how many pigs were at the farm?" (Lester and Garofalo 1982).

2. Many thirteen-year-old students gave "31 remainder 12" as the answer to the problem "An army bus holds 36 soldiers. If 1 128 soldiers are being bussed to their training site, how many buses are needed?" (Carpenter et al. 1983, 656).

Like the telephone company representative, these students followed familiar step-by-step routines. Their customary procedure for solving word problems

consisted of two steps. The first step was to determine an appropriate operation to carry out using the given numbers, and the second step was to perform the actual calcu- lation. Because this two-step approach to solving word problems has usually been successful, the students, like the representative, had come to follow the procedures unthinkingly, sensing no need to evaluate ^ the final outcomes.

^^^ The students attempting to ^^^b solve the first problem were ^^^F accustomed only to one-step H^^P addition and subtraction prob- ■^Ш^ lems, and only those with a ^^^r one-number solution at that. ^^^B Not surprisingly, many of them

ИдМИМ ^V simply added the numbers and ■J gave the result of the addition as ^ the answer without assessing jB the reasonableness of its form V or its magnitude. The students H attempting to solve the second W problem were accustomed

merely to giving as an answer the result of a carried-out arith-

metical operation without any interpreta- tion or reality check, and that is precisely what they did for this problem.

Schoenfeld (1989) explains such student behaviors as being shaped by the context of their mathematics education, that is, by the "routine, day-by-day prac- tices of their mathematics classrooms" (p. 85). If word problems given in school usually have answers that can be obtained by a single application of an arithmetical operation, then some students learn that they can be fairly successful just by com- puting with the given numbers and giving

Joe Garofalo teaches at the University of Virginia, С 'har lotte sviile •, VA 22903. His primary interests are in problem solving and teacher education. Jerry Bryant is a doctoral student at the university. He is interested in language and mathematical thought.

210 ARITHMETIC TEACHER



as the answer the result of that computation without interpreting it or assessing its reasonableness. When students are "trained" on such tasks, it is not difficult to see how unreasonable behaviors and answers can make sense to students in the context of school.

Recent national reports concerning mathematics education have emphasized the assessment of the reasonableness of results as an important part of problem- solving experiences. NCTM's Curricu- lum and Evaluation Standards for School Mathematics (1989) addresses this issue in several standards for early and middle grades. The first standard for each of grades K-4 and grades 5-8 is "Mathematics as Problem Solving," and toward that standard students are called to "verify and interpret results with respect to the origi- nal problem" (pp. 23, 75). This theme recurs in another grades T ' I K-4 standard, "Whole Num- /^^'| ber Computation," which in- V^^4 eludes "select and use compu- 4^^H tation techniques appropriate W^H to specific problems and deter- f^^^ mine whether the results are 1^^^ reasonable" (p. 44). Finally, ж^НЕ

under the grades 5-8 standard f^J, "Computation and Estimation" * г ", is listed "use estimation to check % * r *f the reasonableness of results" £ ЦЯ (p. 94). IT Д

Many students, of course, ¿Зшт assess their final results, and к f^*3Ç some do so appropriately. How- |' Wm ever, some students tend to as- Umm sess their results by way of spu- rious criteria. For example, some fifth-grade students who thought that the chicken-and-pig problem could be solved by division abandoned that idea once they realized that 1 8 does not divide 52 evenly. The assessment criterion they used was based on the divisibility of the numbers and not on the meaning of the operation, nor even on the fact that the solution called for two numbers. A second example points out an equally troublesome assessment criterion. Several seventh graders, when faced with the problem "There were 347

people at a $ 1 50-a-plate luncheon to raise money for charity. Expenses were $5 000. How much money went to the charity?" abandoned their plan to subtract the expenses ($5 000) from the product of 347 and $150 (52 050) because those two numbers were not "close enough." They assessed the appropriateness of subtraction on the basis of the relative size of the numbers and not on more meaningful criteria (Garofalo 1992).

Just pay the $60 000 bill.

Although these criteria may not seem very reasonable to us, they are often very effective and meaningful to students because they reflect the patterns and culture of their mathematics classrooms. The use of these criteria can help them succeed in school even if they are not based on an understanding of the operations. If the word problems given in classes and in textbooks are always solvable by divisions that do not involve a remainder or by subtractions that involve numbers that are "close enough," then assessments such as

these can help students obtain correct answers, or at least avoid some incorrect ones.

How can we influence students to assess the reasonableness of their answers and, when they do, to use more appropriate criteria? Here are a few suggestions:

1. Give students a wide variety of problems to solve. This variety should include one-step, multistep, and nonroutine problems. The mixture should include problems with single answers and problems with multiple answers. Answers should include integers and fractions. Exposure to a rich variety of problems will encourage students to analyze the conditions of problems and discourage them from per- ceiving and generalizing from inappropri- ate patterns in problems. This practice will also help students approach problems more

¿fejfe^y flexibly. This same variety of Wím problem types should, of

И course, be included in assess- k H ments of students' progress, I 0 both formative and summative. Hk И Teachers will need to develop HfeN В a collection of problems from Wk ж A outside their textbooks for these WE ̂ lr purposes. Many good sources ^A for problems are available from 1^^^^ NCTM and from commercial ^^^^k publishers (see the monthly ^^^^B "Reviewing and Viewing" sec- ^H^^B tionof the ArithmeticTeacher).

H^^L 2. Facilitate students' dis- ЦМ ̂̂ B eussions and interpretations of >V^^B the meaning and importance of щШ^^ш problem conditions. Encour- WS^^L age students to make sense of Ij^^^V the problem situation. Conduct- ^^^^B ing whole-class discussions of ^^^^m problems is one way to do so 2J2r <see Charles and Lester [ 1 9821

for a discussion of this approach). This practice will help students understand problem conditions better and put students in a better position to make appropriate assessments of their final results.

3. Encourage students to estimate answers before carrying out calculations. These estimates should include both the form and the magnitude of the results. These actions will help students assess their final results. Additionally, present activities that foster number sense and encourage knowledge of reasonable ranges

DECEMBER 1992 211



of cost, weight, capacity, time, and other quantities.

4. Encourage students to assess the reasonableness of their results by asking them to judge whether their answers "make sense." Requiring students to write out the answers to word problems in a complete sentence is one way to begin. Beyond that, ask students to communicate why they think their answers make sense and explain their criteria for making such evalu- ations. The act of describing or defending their thoughts and judgments can some- times lead students to realize when they have made an error. This approach can also help the teacher get more information and insight about students' thoughts, be- liefs, and assessment criteria.

5. Become aware of faulty assessment criteria that are common among students. Then choose problems for discussion that

Textbook problems may mislead students about reasonableness.

fly directly in the face of these faulty criteria. Bring these faulty criteria to the attention of students as stimuli for discussion about related metacognitive issues. Some students may not be aware of the inappropriateness of the criteria they do use, and some may not be aware of the criteria they need to use.

If the job training of the telephone company's representative had incorporated some of these suggestions, perhaps our friend Pat would not have had to face the unhappy prospect of sending in a check for $60 000. As it turned out, Pat was able to resolve her dilemma with the telephone company, but only because she exhibited two problem-solving behaviors that we should also emphasize with, and model for, our students - persistence and the use of multiple approaches!

References

Carpenter, Thomas P., Mary M. Lindquist, Westina Matthews, and Edward A. Silver. "Results of the Third NAEP Mathematics Assessment: Secondary School" Mathematics Teacher 76 (December 1983):652-59.

Charles, Randall I., and Frank Lester. Teaching Prob- lem Solving: What, Why and How. Palo Alto, Calif.: Dale Seymour Publications, 1982.

Garofalo, Joe. "Number-Consideration Strategies Stu- dents Use to Solve World Problems." Focus on Learning Problems in Mathematics 14 (Spring 1992):37-50.

Lester, Frank, and Joe Garofalo. "Some Metacognitive Aspects of Elementary Students' Problem Solving Performance." Paper presented at the annual meet- ing of the American Educational Research Asso- ciation, San Francisco, April 1982.

National Council of Teachers of Mathematics. Cur- riculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.

Schoenfeld, Alan H. "Problem Solving in Contexts)." In The Teaching and Assessing of Mathematical Problem Solving, edited by Randall I. Charles and Edward A. Silver, 82-92. Reston, Va.: Lawrence Erlbaum Associates and National Council of Teach- ers of Mathematics, 1989. w

2 1 a ARITHMETIC TEACHER



assessing reasonableness: some observations and suggestions

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