Mathematical Problem-Solving Processes ofElementary Male and Female Students

Leah P. McCoyDepartment of EducationWake Forest University

This study examinedtheproblem solving behavior ofyoung elementary schoolstudents. Ninety second-and third-grade students were observed and interviewed as they solved three problems. Conclusionsincludedthat the childrenreadily attemptedunfamiliarproblems andthat they usedsystematic solutionprocesses. No significant gender differences were observed.

Mathematical problem solving is a topic thathas received considerable attention in recent years.A revolution is under way in mathematics educationwith a major goal being that students will under-stand and be able to apply mathematics in a widevariety of situations. In this view, "Problem solvingshould be the central focus of the mathematics cur-riculum" (National Council of & Teachers of Math-ematics, 1989,p.23).

Young children have a natural curiosity thatmotivates them to enjoy novel problems (NCTM,1989). Problem solving is not an easy topic toteach to elementary students. At the very age thatthey are most dependent on concrete representationsof all concepts, we ask them to adopt an abstractgeneral approach to problems. The problem solving"technique" most often found in elementary math-ematics curricula is something similar to Polya’s(1957) steps for problem solving:

1. Understand the Problem,2. Devise a Plan,3. Carry Out the Plan,4. Look Back.

In a study of specific problem-solving be-haviors of lower elementary students, Maurer(1987) observed that students will try imaginativelyto deal with "problems," and will usually followtheir process and construct a solution, even thoughit may be incorrect because of "bugs." One sourceof these "bugs" is that when children begin schoolthey often have considerable informal mathematicsexperience and partial understandings of manymathematical concepts which form inaccurate and/or incomplete schema for solving problems(deCorte & Verschaffel, 1985). Similarly, they maylack basic fact knowledge which may handicapthem with some problems (Bransford, Hasselbring,Barron, Kulewicz, Litdefield and Goin, 1988).

Some students simply look for any numbersand perform a quick operation to generate ananswer with little thought (Bransford, Hasselbring,Barron, Kulewicz, Littlefield, and Goin, 1988). Inan extensive study of ninth-grade students,Kantowski (1977) concluded that there was widevariation*in the amount of analysis of the problems,and that most of the students did not "look back,"Polya’s fourth step. Schoenfeld (1988) alsodescribes the lack of "sense-making" analysis inyoung students’ problem solving.

There is some evidence from a study of fifth-and seventh-grade students (Charles & Lester,1984) that children may deal with problem solvingtasks by getting teacher assistance instead ofsolving the problem on their own. This is the "Ican’t do it" approach, and is characterized by thestudent refusing to attempt an unfamiliar problem.

There has been considerable discussion ofgender differences in mathematics. Generally, thestudies of gender differences in mathematicalproblem solving report that boys scored higher thangirls on problem solving measures (Fennema, 1990;Fennema & Sherman, 1978; McCoy & Dodi, 1989),but recent research has reported complex interac-tions between gender and other variables. Theserelated and intricately intertwined variables includelearner characteristics (Fennema & Tartre, 1985;Tartre, 1990); belief systems, including anxiety,confidence, and motivation (Fennema & Sherman,1977; Garafalo & Lester, 1985; Kloosterman, 1988,1990; Leder, 1990; Meyer & Koehler, 1990); andclassroom practices (Fennema & Petersen, 1986;Koehler, 1990; Leder. 1990). Thus, there are manyaspects of the relationship of gender and mathemat-ics that have yet to be explained.

What we know is that young children initiallylike to solve problems. Even though their knowl-edge is somewhat limited, there is evidence that

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they enjoy problem solving. We do not have arecord of observation of their particular problem-solving processes. The purpose of this study was toexamine and describe the problem-solving behaviorof young elementary school students.

Methodology

Participants in the study were 90 students inGrades 2 and 3 at nine public and private elemen-tary schools in a metropolitan area. Potential par-ticipants were randomly selected from the classes ofteachers who volunteered to participate, and those90 students returning parental permission slipsbecame the sample in the study.

Problems used in the study were from Krulikand Rudnick (1987). Problems in this resource arerecommended for use with students of various agesand experience levels. There were no reliability orvalidity data for these problems; however, this maybe less important as they were used in this studyprimarily as a context for observing problem-sol-ving behavior. Five problems were initially chosenand pilot tested with a small group of second-gradestudents; two were subsequently eliminated becausethey were too difficult. The three problems used inthe study are presented in Figure 1.

Figure 1. Problems.

July 4 is a Tuesday. Your birthday in on July23rd. On what day of the week is your birth-day?

1.

How many triangles can you find in the follow-ing figure? A

2.

Three children had a race. Meg was betweenJoe and Ann. Ann beat Joe. Who came in first,second, and third?

3.

In an individual, private session with a trainedobserver, each participant talked aloud as he or shesolved the three problems read aloud by the ob-server. Observers did not provide assistance orcues, but only asked probing questions when stu-dents forgot to talk aloud. Each session was tapedand transcribed and the resulting protocols wereanalyzed by the trained observer and by the re-searcher. Additional observations by the observer("teacher") provided qualitative data describing the

students’ problem solving performance as sug-gested by Silver and Kilpatrick (1988).

The analysis was concentrated on the followingquestions:

Was the student successful in solving theproblem?""Did he or she follow each ofPolya’s Steps?""What process did he or she use?"

Results

The number and percent of problems solvedcorrectly by the total group and separated by genderare presented in Table 1. While only 2% of theparticipants correctly solved all three problems,28% solved 2 correctly, 43% solved 1 correctly, andonly 27% were not able to solve any. There wereno significant gender differences in number ofcorrect solutions (t (88) = .673,;? = .50)

Table 1.

Correct Problems

MalesFemale Total

3 correct2 correct1 correct0 correct

Total

13 (28%)21 (46%)11(24%)1 (2%)

46

11(25%)18 (41%)14 (32%)1 (2%)

44

24 (27%)39 (43%)25 (28%)2 (2%)

90

Each protocol was examined for evidence ofeach of Polya’s four steps in each of the 270problems. These results are presented in Table 2.Of the 270 total problems that the children at-tempted, 89% used Understand the Problem, over70% used Devise a Plan and Carry Out the Plan,and 34% Looked Back. There were no significantgender differences.

Table 2.

Problem Solutions mth Evidence of Polya’s Steps

Devise Carry LookPlan____Out Back

202(75%) 194(72%) 92(34%)104 (75%) 99 (72%) 50 (36%)98 (74%) 95 (72%) 42 (32%)

UnderstandTotal 240 (89%)Males 116(84%)Females 114(86%)

Total N = 270, Male n = 138, Female n = 132

Problem solving process was classified as eithersystematic or nonsystematic. All typical heuristic

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approaches such as "Draw a Picture, Make a List,Guess and Check," etc. were considered systematic.Nonsystematic was the label given to guesses wherethe student stated that his or her answer was a guessor could not give any explanation of process. Athird category under process was "no attempt." Re-sponses in this category were similar to "I don’tknow," or<<! can’t do it." Process data are summa-rized in Table 3.

Table 3.

Number and Percent of ProcessesNon- No

Systematic systematic AttemptTotal 196 (73%) 59% (22%) 17 (6%)Males 103 (75%) 31% (22%) 6 (4%)Females 93 (70%) 28% (21%) 11 (8%)

Total N = 270, Male n = 138, Female n = 132

The number of participants who used a system-atic process for problem solving was significantlygreater than the number who either used a non-systematic process or made no attempt «88) =.741, p = .46). There were no significant genderdifferences.

The protocols were further analyzed to identifyand describe the exact processes that were used. Inaddition to the expected solution methods, severalchildren displayed quite creative processes.

For Problem 1 (The Day of the Week Prob-lem), four categories of systematic processes wereidentified: diagrams, counting, arithmetic, andguessing. The most common process was drawinga diagram. Students drew various representationsof a calendar, filling in the days according to theconditions of the problem. Most diagrams weresuccessful; however, some students did not knowthe days of the week. One student labeled July 1 onSunday because "it was the first of the week andthe first of the month."

The second process observed was counting.Some students wrote the seven days of the weekand then pointed in sequence from Tuesday andcounted from 4 to 23. Several students wrote 4through 23 and then wrote the days of the week tocorrespond. Students wrote days and correspondingdates, and one student orally counted through themonth ("Tuesday is the 4th, Wednesday is the 5th,etc."). One student counted backward from 23 to 4and then labeled with days. One student counted(successfully) with his fingers, and tried to hide hismethod from the observer.

The third process was arithmetic. The mostsuccessful arithmetic-using students added incre-ments of 7 to 4 until they were close, and thencounted days forward to 23. Others added incre-ments of 7 until they were past 23, and then countedbackwards two days. Several students subtracted23 - 7, and then counted 19 days beginning withTuesday. Two students found the numbers in theproblem and "computed" them, one adding and onesubtracting, and both giving inappropriate numeri-cal answers.

The fourth process was guessing, and eventhough most guesses were incorrect, they wereinteresting. The most common (nonsystematic)guess was the student’s own birfhdate day for thecurrent year. Another student was somewhatsystematic: "The 23rd; well, Wednesday is thethird day, so it must be a Wednesday."

For Problem 2 (The Triangle Problem), theonly major process identified was counting. Manystudents counted only the nine small triangles.Some counted the nine small and the large outlinefor a total often. Only a few counted the small,medium and the large triangles.

For Problem 3 (The Race problem), five pro-cesses were identified: diagrams, lists, visualiza-tion, logic, and guess and check. Several studentsdrew a diagram to represent the race, using stickpeople or letters to represent the children. Somestudents did not grasp the meaning of "between,"and drew the children in a diamond or in a circle.One student wrote down letters to represent thechildren, and then drew arrows to move them in thecorrect order.

The second process identified was listing thenames. Several students wrote the names in theorder presented in the problem and then erased ormarked them out and moved them to the properorder.

The third process utilized in this problem wasvisualization. Students said they "pictured it in myhead," or "drew a picture of the race in my head,"or "imagined them running the race," or "saw it inmy mind." One student quickly gave the answer,and when questioned about process, he said, "I readmy mind."

An impressive number of the young children inthe sample used logic to solve this problem. Onestudent said, "Meg is in the middle, and Ann beatJoe, so Ann must be first and Joe last." None of thelogic solutions were really logical. One studentsaid that Joe must win because boys can run faster

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than girls. Another student said, "Ann beat Joe sothey are first and second, so Meg must be third.You ignore the second sentence in the problem,because it doesn’t make sense in the problem."

The fifth, and final, process observed in thisproblem was guess and check. Several studentsguessed at the order and then reread the problem tosee if it "fit." They then made adjustments andchecked again, until they were satisfied.

Conclusions

The results of this study provide importantknowledge about the problem-solving behaviors ofsecond- and third-grade students. This study clearlysupports the assertion of the National Council ofTeachers of Mathematics (1989) that many youngchildren are good problem solvers. They readilyattempted the problems; on only 6% of the 270problems were the responses classified as "no at-tempt." This is in contrast to the results of Charlesand Lester (1984), who found many more "no at-tempts," but with older students in Grades 5 and 7.It appears that the younger students have not be-come "school-wise," and thus are less inhibited andmore willing to use their creativity and devise aunique solution to an unfamiliar problem.

In addition to being innovative, the childrenwere able to solve the problems. Seventy-three per-cent of the participants got the correct answer for atleast one of the three problems. In many of theproblems which were incorrect, good processeswere observed and the specific "bugs" which causedthe errors could be identified. This agrees withearlier studies which blamed many errors on "bugs"rather than procedural difficulties (Bransford, et al.,1988; deCorte & Verschaffel, 1985; Maurer, 1987).The students used Polya’s four steps in solving theproblems, even though, like KantowskTs (1977)older sample, they also frequently neglected look-ing back.

Perhaps the most significant information fromthis study is the observation that the children usedsystematic solution processes. There was clearevidence that they used seven problem solvingheuristics: diagrams, lists, visualization, counting,arithmetic, logic, and guess and check. Theseheuristics were mathematically and logically ap-propriate, even though they were sometimes unsuc-cessful in producing the correct solution, due tolack of factual or procedural knowledge.

We would expect young children to use con-crete processes such as diagrams, lists and counting.

We also would expect them to use arithmetic be-cause it is familiar, and to use guess and check be-cause it involves a somewhat natural instinct. Butthe use of visualization and logic by 6- and 7-year-olds is an interesting result that suggests higherlevel, abstract thinking that we normally do notassociate with children this young. This apparentspontaneous use of abstract processes by youngchildren is an area that should be further explored.

No significant gender differences were ob-served in either number of correct solutions, use ofPolya’s steps, or use of systematic processes. Eventhough earlier studies (Fennema, 1990; Fennema &Shennan, 1977; McCoy and Dodi, 1989) found gen-der differences in mathematical problem solvingskills, their subjects were all older. This result sup-ports the current theories that mathematics anxietyand gender differences in all areas of mathematicsachievement are a product of cultural learning, andif children are treated equally at earlier ages, mostof the observed gender difference will disappear (cf.Fennema, 1990). In the current study, perhaps theyoung children had not been in school long enoughto "leam" that mathematics is an area where boys"should" outperform girls.

It appears that young students, both male andfemale, are good mathematical problem solvers inthe early years of school. They are able to solvenonroutine problems, and even when they do not

get the correct answer, they exhibit an innate abilityto construct appropriate solution processes. Per-haps problem-solving skill is not something thatshould be taught as much as it should be nurtured.If children are naturally curious and logical, thenwe must provide them with experiences to buildtheir background knowledge and problem-solvingskill.

Two areas are recommended for further re-search. First, we need to examine the developmen-tal aspect of problem-solving abilities as childrenprogress through school. Second, gender differ-ences and/or interactions of gender with othervariables should be further studied.

References

Bransford, J., Hasselbring, T., Barron, B.,Kulewicz, S., Littlefield, J., & Goin, L. (1988).Use of macro-context to facilitate mathematicalthinking. In R.I. Charles & E. A. Silver (Eds.),The teaching and assessing of mathematicalproblem solving (pp. 125-147). Reston,VA:National Council of Teachers of Mathematics

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Charles, R. I. & Lester, F. K. (1984). An evaluationof a process-oriented mathematical problemsolving instructional program in grades five andseven. Journal/or Research in MathematicsEducation, 15, 15-34.

deCorte, E. & Verschaffel, L. (1985). Beginningfirst graders’ initial representation of arithmeticwork problems. Journal of MathematicalBehavior, 4, 3-21.

Fennema, E. (1990). Teachers’ beliefs and genderdifferences in mathematics. In E. Fennema &G.C. Leder (Eds.), Mathematics and gender(pp. 169-187). New York: Teachers CollegePress.

Fennema, E. & Petersen, P.L. (1986). Teacher-student interactions and sex-related difference’sin learning mathematics. Teaching andTeacher Educations, 2(1), 19-42.

Fennema, E. & Sherman, J. (1977). Sex-relateddifferences in mathematics achievement, spatialvisualization and affective factors. AmericanEducation Research Journal, 14(1), 51-71.

Fennema, E. & Tartre, L.A. (1985). The use ofspatial visualization in mathematics by girls andboys. Journalfor Research in MathematicsEducation, 16(3), 163-176.

Garafalo, J. & Lester, F. K. (1985). Metacpgnition,cognitive monitoring, and mathematical perfor-mance. Journalfor Research in MathematicsEducation, 76(3), 163-176.

Kantowski, M. G. (1977). Processes involved inmathematical problem solving. JournalforResearch in Mathematics Education, 8(3), 163-180.

Kloosterman, P. (1988). Self-confidence andmotivation in mathematics. Journal of Educa-tional Psychology, 80, 345-351.

Kloosterman, P. (1990). Attributions, performancefollowing failure, and motivation in mathemat-ics. In E. Fennema & G.C. Leder (Eds.),Mathematics and gender (pp. 96-127). NewYork: Teachers college Press.

Koehler, M. S. (1990). Classrooms, teachers, andgender differences in mathematics. In E.Fennema & G.C. Leder (Eds.), Mathematicsand gender (pp. 169-187). New York: Teach-ers College Press.

Krulik, S. & Rudnick, J.A. (1987). Problemsolving: A handbookfor teachers (2nd ed.).Newton, MA: Allyn & Bacon.

Kuhn, D. (1989). Children and adults as intuitivescientists. Psychological Review, 96(4), 674-689.

Leder, G. C. (1990). Teacher/student interactionsin the mathematics classroom: A differentperspective. In E. Fennema & G.C. Leder(Eds.), Mathematics and gender (pp. 149-168).New York: Teachers College Press.

Leder, G. C. & Fennema, E. (1990). Genderdifferences in mathematics: A synthesis. In E.Fennema & G.C. Leder (Eds.), Mathematicsand gender (pp. 188-200). New York: Teach-ers College Press.

Maurer, S. B. (1987) New knowledge about errorsand new views about learners. In A. H.Schoenfeld (Ed.), Cognitive science andmathematics education (pp. 165-187).Hillsdale, NJ: Eribaum.

McCoy, L. P. &-Dodl, N. R. (1989). Computerprogramming experience and mathematicalproblem solving. Journal of Research onComputing in Education, 22(1), 14-25

Meyer, M. R. & Koehler, M. S. (1990). Internalinfluences on gender differences in mathemat-ics. In E. Fennema & G. C. Leder (Eds.),Mathematics and gender (pp. 60-95). NewYork: Teachers College Press.

National Council of Teachers of Mathematics.(1989). Curriculum and evaluation standardsfor school mathematics. Reston, VA: Author.

Polya, G. (1957). How to solve it (2nd ed.). Gar-den City, NJ: Doubleday Anchor.

Schoenfeld, A. H. (1988). Problem solving incontext. In R. I. Charles & E. A. Silver (Eds.),The teaching and assessing of mathematicalproblem solving (pp. 82-92). Reston, VA:National Council of Teachers of Mathematics.

Silver, E. A. & Kilpatick, J. (1988). Testingmathematical problem solving. In R. I. Charles& E. A. Silver (Eds.), The teaching and assess-ing of mathematical problem solving (pp. 178-186). Reston, VA: National Council of Teach-ers of Mathematics.

Tarte, L.A. (1990). Spatial skills, gender, andmathematics. In E. Fennema & G. C. Leder(Eds.) Mathematics and gender (pp. 27-59).New York: Teachers College Press.

Note: The author’s address is Leah P. McCoy,Department of Education, Wake Forest University, Box7266 Reynolda Station, Winston-Salem, NC 27109.

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