Research on Problem Solving: Implications for Elementary School ClassroomsAuthor(s): Marilyn N. Suydam and J. Fred WeaverSource: The Arithmetic Teacher, Vol. 25, No. 2 (November, 1977), pp. 40-42Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41190284 .

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Research on Problem Solving: Implications for Elementary School Classrooms

By Marilyn N. Suydam and J. Fred Weaver

Teachers and pupils both know that verbal problems cause problems. Be- cause of this, and because it is consid- ered to be an ultimate goal of mathe- matics instruction, researchers have devoted much attention to problem solving over the years. This research has focused on characteristics of the problems, characteristics of those who are successful or unsuccessful at solv- ing problems, and teaching strategies that may help children to be more suc-

Once an elementary school teacher, Marilyn Suy- dam is now on the mathematics education faculty at the Ohio State University and a research asso- ciate at the ERIC Center for Science, Mathemat- ics, and Environmental Education. She is inter- ested in the interpretation of research for classroom application. Fred Weaver is a member of the faculty of the School of Education at the University of Wisconsin - Madison, working in curriculum and instruction and with a special inter- est in research in mathematics education.

40 Arithmetic Teacher

cessful. Recently attention has begun to be focused on the heart of the prob- lem - the strategies that children use in solving problems, the process of prob- lem solving.

From research we have learned about a variety of points connected with problem solving. Instead of giving many details about each study, we have summarized the main findings very briefly. (For additional information on the studies, or for references to the re- search reports, see the references at the end of this article.) You might want to see how many of these points agree with the conclusions you have reached on the basis of your experiences with children. You might also want to note those things you have learned that do not appear on this list; we are aware that the list is not totally comprehen- sive.

Children are prob- ably a little more successful at solving problems with fa- miliar settings, but problems with unfa- miliar settings do not seem to cause undue difficulty.

At least one study with black chil- dren from a lower socioeconomic envi- ronment indicated that there was no significant difference in achievement between problems from a textbook and problems written by children, using fa- miliar settings and people.

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Generally, it has been concluded by many researchers that children like a variety of problem settings. And it seems important that children be inter- ested in the problems, as well as in solving them.

In problems, the op- eration that appears to be easiest is addi- tion, followed by subtraction, multi- plication, and divi- sion.

A problem that involves only one of the four operations is generally less dif- ficult than a problem that involves two operations.

When the data in a multistep prob- lem are in the order required for solu- tion, higher scores can be expected than when the data are not in the order in which they will be used.

The time needed to solve a problem is less when the question is placed at the beginning of the problem rather than at the end; however, achievement will probably not differ significantly for either positioning of the question.

When pupils work on sets of verbal problems in small groups of two or four pupils, they can solve more prob- lems than those who work alone, but the groups might take a longer time on each problem than pupils working alone.

Group solutions to problems may be no better than the independent solu- tions of the most able member of the group, if he or she is perceived by the group to be most able. '

There is some evidence that group discussion in order to reach agreement on how to proceed results in signifi- cantly better achievement than being told how to solve the problem.

Systematic teaching of a variety of prob- lem-solving pro- cedures aids chil- dren in developing problem-solving strategies.

Giving pupils many opportunities to solve problems has frequently been suggested as being of great importance.

Encouraging children to solve prob- lems in a variety of ways appears to aid children in becoming better problem solvers.

Having pupils write an equation or mathematical sentence for a problem can be helpful. Writing equations that fit the problem situation (expressing the real or imagined actions in the problem) and using equations that em- phasize the operations by which the problem may be solved directly each appear to have some advantages.

Emphasis on isolated word cues (such as left or in all) can be mis- leading, for attention is directed away from recognition of the relationships inherent in the problem that may be crucial to its solution. Some discussion and illustrations of how word cues may be misleading could help, however.

Problems with extra, irrelevant data are more difficult than problems with- out extra data.

Problems with materials, diagrams, or some other type of visual aid are generally easier than those without such aids.

Instruction on what process to use and on why that process is appropriate will generally result in higher scores than merely solving problems without discussion. Emphasize what needs to be done and why it needs to be done rather than just obtaining an answer.

Many other specific techniques have been reported by researchers to be helpful; among those suggested are the following:

1 . Provide a differentiated program, with problems at appropriate levels of difficulty.

2. Provide many and varied situa- tions that give children opportunities for structuring and analyzing situations that really constitute a problem and not just a computational exercise.

3. Have pupils dramatize problem situations and their solutions.

4. Have pupils make drawings and diagrams, using them to solve prob- lems or to verify solutions to problems.

5. Have pupils write their own prob- lems, formulating them for given con- ditions.

6. Present problems orally. 7. Use problems without numbers. 8. Have pupils designate the pro-

cesses or operations to be used. 9. Have pupils note the absence of

essential data or the presence of unnec- essary data.

10. Have pupils test the reason- ableness of their answers.

1 1 . Use a tape recorder to aid poor readers.

1 2. Present some problems in sepa- rate sentences rather than in the usual paragraph format.

Opportunities should be provided for children to determine the question to be answered, select specific facts nec- essary to solution, and choose the ap- propriate process. However, rigid ad- herence to a formal analysis procedure (that is, requiring pupils to answer a specific set of questions in a specified order) does not appear to be effective.

Researchers gener- ally conclude that

1. ¡Q is signifi- cantly related to problem-solving ability;

2. sex differences do not appear to ex- ist in the ability to solve problems; and

3. socioeconomic status alone does not appear to be a significant factor.

Computational difficulties appear to be a major deterrent to finding correct answers when solving problems, with reading a secondary cause of difficulty.

November 1977 41

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Higher levels of problem-solving ability are often associated with higher levels of computational and reading ability, but much of this apparent rela- tionship may be the result of the corre- lation of these abilities with IQ.

Among the reasons commonly found for why children make mistakes as they solve problems are the following: 1 . errors in reasoning 2. ignorance of mathematical prin-

ciples, rules, or processes 3. insufficient mastery of computa-

tional skills 4. inadequate understanding of vocab-

ulary 5. failure to read to note details

Many researchers have proceeded on the assumption that if we can ascertain what successful problem solvers have in common, we may be able to help those who do not do as well. Some char- acteristics of good problem solvers have been identi- fied.

In addition to skill in computation, reading comprehension, and higher IQ scores, among the many factors that may characterize those good at solving problems are the following: 1 . ability to estimate and analyze 2. ability to visualize and interpret

quantitative facts and relationships 3. ability to understand mathematical

terms and concepts 4. ability to note likenesses, differ-

ences, and analogies 5. ability to select correct procedures

and data 6. ability to note irrelevant detail 7. ability to generalize on the basis of

few examples 8. ability to switch methods readily 9. higher scores for self-esteem and

lower scores for test anxiety

More impulsive students are often poor problem solvers, while more re-

42 Arithmetic Teacher

flective students are likely to be good problem solvers.

Good and poor achievers in prob- lem solving differ on many aspects of reading.

Activities stressing certain reading skills, such as selecting main ideas, making inferences, constructing se- quences, and following directions, may improve problem-solving achievement.

Specific instruction on quantitative vocabulary may be helpful for some pupils.

Creative or diverg- ent thinking is a successful strategy, but is used by rela- tively few pupils. Blind guessing and trial-and-error are considered to be the most unsuccessful strategies.

Having a pupil think aloud as he or she solves a problem

may help you to diagnose the par- ticular reason why a pupil is having difficulty.

A new factor has entered many class- rooms recently, the hand-held calcu- lator. Problems in current elementary school curricula are often included merely to provide practice on particu- lar computational skills. With the use of the calculator, however, there can be more focus on problem solving "for problem solving's sake." The focus can be on strategies and process when the calculator is used, with less emphasis on computation within the problem- solving context. More real problems can be used and the range of problems extended. Research has not yet consid- ered the effect of the calculator on problem solving. There is only some preliminary evidence from schools in which the calculator is used during mathematics instruction that problem- solving achievement on standardized tests may increase. We need to refocus the curriculum and see what the advan- tages of the calculator can be for help- ing children to learn how to solve prob- lems of all types.

References

Suydam, Marilyn N., and J. Fred Weaver. Using Research: A Key to Elementary School Mathe- matics. Columbus, Ohio: ERIC Information Analysis Center for Science, Mathematics, and Environmental Education, 1975. (Also available fromNCTM.)

Suydam, Marilyn N. A Categorized Listing of Research on Mathematics Education (K-12), 1964-1973. Columbus, Ohio: ERIC Information Analysis Center for Science, Mathematics, and Environmental Education, 1974.

The annual listing of research in the Journal for Research in Mathematics Education. D

Corresponding Society The National Council of Teachers of Mathe-

matics (NCTM) is interested in the development of closer relations and communications with pro- fessional associations of mathematics teachers in other countries. With this in mind, the NCTM Board of Directors has authorized an affiliation category called "Corresponding Society."

This title is meant to describe a relationship between NCTM and organizations of mathe- matics teachers to include the following: (a) An official exchange of professional journals

between NCTM and the foreign society; (b) Each organization, through its professional

publications, providing relevant information to its members about the other's professional

meeting dates, publications, journals, mem- bership, and so on;

(c) Each organization inviting members of the other organization that are already in the country'to be honored visitors to its annual meeting.

Professional associations of mathematics teachers in foreign countries interested in this type of relationship with NCTM should write to

Representative for International Mathematics Education

National Council of Teachers of Mathematics 1906 Association Drive Reston, Virginia 22091 U.S.A.

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