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109 Measuring Beliefs About Mathematical Problem Solving Peter Kloosterman Frances K. Stage Department of Curriculum and Instruction Department of Educational Leadership and Policy Studies Indiana University Bloomington, Indiana 47405 Student beliefs about the discipline of mathematics and about themselves as learners of mathematics have received considerable attention in recent years. In 1985, Silver stated that beliefs about mathematics should be studied to better understand how students learn problem solving. National assessment data indicate that 83 % of seventh-grade and 81 % of eleventh-grade students agree or strongly agree with the erroneous belief, "There is always a rule to follow in mathematics" (Dossey, Mullis.Lindquist, & Chambers, 1988, p. 102). The National Research Council (1989) alluded to beliefs about the rule orientation of mathematics with their supposition that, "The public perception of mathematics is shifting from that of a fixed body of arbitrary rules to a vigorous active science of patterns" (p. 13). Beliefs about the usefulness of mathematics were also noted by the National Research Council (1989) with the claim that, "Public attitudes about mathematics are shifting from indifference and hostility to recognition of the important role that mathematics plays in today’s society" (p. 12). In brief, students and the public in general have varying beliefs about mathematics as a subject and about the individual as a learner of mathematics. Some of these beliefs make students want to learn mathematics. Unfortunately, many students also have beliefs that actually hinder interest in, and understanding of, the subject. At the present time, no instruments are available for measuring student beliefs about the discipline of mathematics or about how mathematics is learned. Mathematics instructors and researchers wishing to find out about students’ beliefs are forced to write their own instruments or restrict themselves to time-consuming interviews. If an instrument was available to mathematics instructors, many would be more willing to measure the beliefs of their students. This would allow them to determine the beliefs of their students and then modify instruction to improve beliefs if needed. Thus, the purpose of this study was to develop and validate a set of belief scales for measuring secondary school and college students’ beliefs about mathematics as a subject and about how mathematics is learned. After further explanation of the importance of understanding beliefs, four new belief scales and one previously unavailable The research reported in this article was supported in part by a grant from the National Science Foundation (Grant No. NSF- TEI-875148). The opinions expressed in this paper do not necessarily reflect the position, policy, or endorsement of the National Science Foundation. scale will be presented. Documentation of the development of these scales is also included. Beliefs and Motivation While the study of beliefs about mathematics is a topic which is currently generating interest, it is only by defining the aims of belief research that it is possible to determine the categories of beliefs that need to be studied. Some research has focused on what motivates students to leam to solve mathematical problems. An assumption behind this research is that certain beliefs result in high motivation on the part of students whereas other beliefs diminish motivation. It is also assumed that increasing a student’s motivation to learn to sol ve mathematical problems will increase the likelihood that he or she will become a good mathematical problem solver. The following example is provided as support for the validity of these assumptions. In 1981, Fcnnema, Wolleat, Pedro, and Becker studied the effects of an intervention program on high school females’ intent to enroll in optional mathematics classes in high school and college. Among other things, the intervention stressed the usefulness of mathematics outride school. This belief turned out to be an important factor related to intent to enroll in optional mathematics courses. While intent to enroll in mathematics courses is not necessarily the same as motivation to leam mathematics, they are related. Increasing a student’s belief that mathematics is useful will often increase motivation and thus achievement. The scales described in this article do not measure the extent to which students believe mathematics is useful because the Fennema and Sherman (1976) Usefulness of Mathematics scale already exists. Instead, scales were developed to measure beliefs which are related to motivation and thus achievement on mathematical problem solving. Before presenting the scales, the types of beliefs which are related to motivation will be described. Within the discussion, reasons for expecting each belief to influence motivation to leam to solve mathematical problems should become apparent. Beliefs About Mathematical Problem Solving I Can Solve Time-Consuming Mathematics Problems The first belief selected for study involves perceived ability to solve time-consuming mathematics problems. Schoenfeld Volume 92(3), March 1992

Measuring Beliefs About Mathematical Problem Solving

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Measuring Beliefs About Mathematical Problem SolvingPeter Kloosterman

Frances K. StageDepartment of Curriculum and Instruction

Department of Educational Leadership and Policy StudiesIndiana UniversityBloomington, Indiana 47405

Student beliefs about the discipline of mathematics andabout themselves as learners of mathematics have receivedconsiderable attention in recent years. In 1985, Silver statedthat beliefs about mathematics should be studied to betterunderstand how students learn problem solving. Nationalassessment data indicate that 83% ofseventh-grade and 81% ofeleventh-grade students agree or strongly agree with theerroneous belief, "There is always a rule to follow inmathematics" (Dossey, Mullis.Lindquist, & Chambers, 1988,p. 102). The National Research Council (1989) alluded tobeliefs about the rule orientation of mathematics with theirsupposition that, "The public perception of mathematics isshifting from that ofa fixed body of arbitrary rules to a vigorousactive science ofpatterns" (p. 13). Beliefs about the usefulnessof mathematics were also noted by the National ResearchCouncil (1989) with the claim that, "Public attitudes aboutmathematics are shifting from indifference and hostility torecognition of the important role that mathematics plays intoday’s society" (p. 12). In brief, students and the public ingeneral have varying beliefs about mathematics as a subject andabout the individual as a learner ofmathematics. Some ofthesebeliefs makestudents wantto learn mathematics. Unfortunately,many students also have beliefs that actually hinder interest in,and understanding of, the subject.

At the present time, no instruments are available formeasuring student beliefs about the discipline of mathematicsor about how mathematics is learned. Mathematics instructorsand researchers wishing to find out about students’ beliefs areforced to write their own instruments or restrict themselves totime-consuming interviews. If an instrument was available tomathematics instructors,manywouldbemorewilling to measurethe beliefs oftheir students. This wouldallow them to determinethe beliefs of their students and then modify instruction toimprove beliefs if needed. Thus, the purpose of this study wasto develop and validate a set of belief scales for measuringsecondary school and college students’ beliefs aboutmathematics as a subject and about how mathematics is learned.After further explanation of the importance of understandingbeliefs, four new belief scales and one previously unavailable

The research reported in this article was supported in partbya grant from the National ScienceFoundation (Grant No. NSF-TEI-875148). The opinions expressed in this paper do notnecessarily reflect the position, policy, or endorsement of theNational Science Foundation.

scale will be presented. Documentation of the development ofthese scales is also included.

Beliefs and Motivation

While the study of beliefs about mathematics is a topicwhich is currently generating interest, it is only by defining theaims of belief research that it is possible to determine thecategories ofbeliefs that need to be studied. Some research hasfocusedon whatmotivates students to leam to solvemathematicalproblems. An assumption behind this research is that certainbeliefs result in high motivation on the part of students whereasother beliefs diminish motivation. It is also assumed thatincreasing a student’s motivation to learn to solve mathematicalproblems will increase the likelihood that he or she will becomea good mathematical problem solver. The following exampleis provided as support for the validity of these assumptions.

In 1981, Fcnnema, Wolleat, Pedro, and Becker studied theeffects of an intervention program on high school females’intent to enroll in optional mathematics classes in high schooland college. Among other things, the intervention stressed theusefulness of mathematics outride school. This belief turnedout to be an important factor related to intent to enroll inoptional mathematics courses. While intent to enroll inmathematics courses is not necessarily the same as motivationto leam mathematics, they are related. Increasing a student’sbelief that mathematics is useful will often increase motivationand thus achievement. The scales described in this article donot measure the extent to which students believe mathematicsis useful because the Fennema and Sherman (1976) Usefulnessof Mathematics scale already exists. Instead, scales weredeveloped to measure beliefs which are related to motivationand thus achievement on mathematicalproblem solving. Beforepresenting the scales, the types of beliefs which are related tomotivation will be described. Within the discussion, reasonsfor expecting each belief to influence motivation to leam tosolve mathematical problems should become apparent.

Beliefs About Mathematical Problem Solving

I Can Solve Time-ConsumingMathematics Problems

The first belief selected for study involves perceived abilityto solve time-consuming mathematics problems. Schoenfeld

Volume 92(3), March 1992

110Beliefs About Problem Solving

(1985, 1988) argued that many college students think allmathematics problems can be completed in five minutes or less.It is reasonable that students would have this belief given thattypical mathematics lessons in elementary and junior highschool require completion of 10 or more exercises so thatproblems which require sustained thought are rare, and mostprecollege textbook word problems are of the 1- or 2- stepvariety that can easily be solved in one or two minutes(Nibbelink. Stockdale, Hoover, & Mangru. 1987). Schoenfeld(1985, 1988) suggested that students who believe problemsmust be solvable in five minutes or less conclude that one shouldgive up on any problem which cannot be completed in fiveminutes. Students with no motivation to solve problems thatthey cannot solve quickly will have difficulty in college-levelmathematics courses. For this reason, it is important to considerstudents’ beliefs about their ability to solveproblems which takemore than a minute or two to complete.

There are Word ProblemsThat Cannot be Solved with Simple,Step-by-Step Procedures

National assessment data clearly indicate students feel thereare rules to follow in mathematics. As noted above, mosteleventh-grade students agree with the statement, "There isalways a rule to follow in mathematics" (Dossey et al., 1988. p.102). TheNational Research Council (1989) stated."Aschildrenbecome socialized by school and society, they begin to viewmathematics as a rigid system of externally dictated rulesgoverned by standards of accuracy, speed, and memory" (p. 7).For computational procedures, there are always rules to follow,although it is important that students understand why the rulesthey follow actually work. With 1-step word problems of thetype commonly found in elementary and middle schoolmathematics textbooks, students are often taught to look for"key words" to help reduce a problem to applying a rule. Fornon-routine word problems, however, following rules is oftennot possible. Charles and Lester (1982) stated that one criteriafor a true mathematical problem is that, "The person has noreadily available procedure for finding the solution" (p. 5). Inother words, good problem solvers must be motivated to solveproblems for which there are no memorized rules to follow.Because students whobelieve all mathematical problems can besolved by applying rules will give up or apply an inappropriaterule when no appropriate rule can be found, it was important todevelop a scale to measure belief in the existence ofrules. Thus,the second belief studied was that there are word problems thatcannot be solved with simple, step-by-step procedures.

Understanding Concepts is Importantin Mathematics

The third beliefselected for study involved the importance of

conceptual understanding in mathematics. Eighty-ninepercentof the eleventh-grade students in the national assessmentsampleagreed or strongly agreedwith the statement, "Knowingwhy an answer is correct is as important as getting the correctanswer" (Dossey et al., 1988, p. 102). Schoenfeld (1985,1988), however, noted that many college students believe theyare not capable of creating mathematics, and thus they believethey should accept procedures without trying to understandhow they work. Forty-eight percent of the eleventh-gradestudents in the national assessment sample agreed or stronglyagreed with the statement, "Learning mathematics is mostlymemorizing" (Dossey etal., p. 102). Students who do not careabout why an answer is correct will have little motivation to

attempt real mathematical problems. In addition, there comesa point in the study ofmathematics when memorizing discretebits of mathematical information is no longer sufficient forkeeping track of everything on which one is going to be tested(Tobias, 1978). In brief, students who do not feel it is importantto understand why a particular algorithm works and who relyon memorized procedures to answer problems are settingthemselves up for eventual failure. In contrast, students whotake the time to understand why a procedure works will knowthey can leam mathematics and thus will be motivated to try tolearn. Thus a scale was created to measure students1 beliefsabout the importance of understanding in mathematics.

Word Problems are Importantin Mathematics

The fourth belief selected for study involved perceptions ofthe importance ofword problems as opposed to computationalskills. This belief is somewhat different than the other three asit focuses on the relative importance of two types ofmathematical content. Many recent documents speak of theimportanceofproblem solving. T^^CurriculumandEvaluationStandards for School Mathematics by the National Council ofTeachers of Mathematics (1989) stressed the importance ofteaching problem solving while advocating diminishedemphasis on paper-and-pencil computation. Nationalassessment data indicated, however, that many more studentsleam to compute with whole numbers than leam to solveproblems (Carpenter, Lindquist, Brown, Kouba, Silver, &Swafford, 1988). The National Research Council (1989)report warned that increased emphasis on computation andback to the basics is not the answer to the nation *s problems inmathematical instruction. Students who believe computationis the key to mathematics learning will have less motivation tobe good problem solvers than students who feel that solvingword problems is important. Because problem-solving skillsare more important than computational skills in most college-level mathematics courses, it is important to consider studentbeliefs about this issue.

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Beliefs About Problem Solving111

Effort Can Increase Mathematical Ability

The final belief selected for study involved the extent towhich students feel that effort and study will makethem smarterin mathematics. Like the other selected variables, this variableis closely tied with motivation because many students havecome to believe they lack a mathematical mind, and thus theyshould not be expected to excel in mathematics (Tobias. 1978).Other students, however, believe that anyone can learnmathematics and improve mathematical ability with sufficienteffort. Obviously, students who feel they cannot improve theirmathematical ability by studying are not likely to work toincrease their problem-solving skills; therefore, students*beliefabout theextent to which effort can increasemathematicalability was considered. Additional discussion of the relationbetween beliefs about ability and motivation to solve problemscan be found in Dweck and Bempechat (1983) and inKloosterman (1988).

Instrument Development

The purpose of this study was to construct and validateLikert-type scales foreach ofthe fivebeliefs aboutmathematicalproblem solving discussed above. Anunpublished scale alreadyexisted for the belief that effort can increase ability inmathematics but that scale had only been used with seventh-grade students (Kloosterman, 1988). Thus. it was necessary todevelop scales for the first four beliefs and then to validate allfive scales with college-level students.

Ten statements were written for each of the four beliefs forwhich no scale existed. The statements were written so thatstudents could respond to each statement using a Likert-typeformat of strongly agree, agree, uncertain, disagree, orstrongly disagree. Some of the items were written with apositive wording (e.g.. "Math problems that take a long timedon’t bother me.") and some were written with a negativewording (e.g., "If I can’t solve a math problem quickly, I quittrying"). In all cases, students with high motivation to learn tosolvemathematical problems wereexpected to agree orstronglyagree with positively worded items and disagree or stronglydisagree with negatively worded items. Forexample, agreementwith the statement, "If I can’t solve a math problem quickly, Iquit trying," would be indicative of low perseverance and thuslow motivation.

Once items were written for each of the four scales, sixmathematics educators (professors, graduate students, andclassroom teachers) read through the items to ensure that theyrelated to the intended constructs. Theitems werethen randomlydistributed throughout a 40-item instrument and administeredto 61 first-year college students enrolled in a remedial (non-credit) college mathematics course at a midwestem publicresearch university. Data collection took place during the firstweek of the course. Students’ responses to the items wereanalyzed using the reliability procedure from SPSS^ After

specification of the 10 items for each scale, the reliabilityprocedure provided the inter-item correlation, the squaredmultiple correlation, and the internal consistency reliability(Cronbach’s a) for each scale after a given item was removedfrom it. For each of the pilot scales, inter-item statisticsindicated improvement with deletion of three to five items perscale. So that all scales would have an equal number of itemsand because scale statistics were similar regardless of whetherthree, four, or five items were deleted, four items were deletedfrom each of the scales. Individual items were scored byassigning the number 1 to the least positive response (stronglydisagree if the item was positively worded and strongly agree ifthe item was negatively worded) and so on up to the number 5for the most positive response (strongly agree if the item waspositively worded and strongly disagree if the item wasnegatively worded). A total score for the six items on each scalewas determined by adding the values reported for the six itemson the scale. Using this scoring procedure, scores on each scalecould range from 6 to 30. Means, standard deviations, andinternal reliabilities (Cronbach’s a) of the 6 item scales areshown in Table 1. Correlations among the four scales were allnonsignificant at the p<.05 level.

Table 1

Means,Standard Deviations,and Reliabilities (Cronbach’s a)After Pilot Testing (n = 61)

Scale

Difficult ProblemsStepsUnderstandingWord Problems

Mean

18.215.624.016.8

S.D.

4.22.83.33.4

Cronbach’s a

.81

.60

.71

.63

Difficult Problems = I can solve time-consuming mathemat-ics problems

Steps = There are word problems that cannot be solved withsimple, step-by-step procedures

Understanding = Understanding concepts is importantWord Problems = Word problems are important in

mathematics

To obtain valid scale statistics for the final version of thescales, the six items from each scale and the six items from theEffort Can Increase Mathematical Ability scale (Kloosterman,1988) were randomly distributed to form a single instrumentcalled the Indiana Mathematics Belief Scales (see AppendixA). Because perceived usefulness of mathematics is animportantcomponentofmotivation, six items from theFennema-Sherman (1976)Usefulness of Mathematics Scale (seeAppendixB) were also included.

The final version of the scales was then administered to anew sample of 517 college students for calculation of scalestatistics. Of the individuals in the new sample, 273 were

Volume 92(3), March 1992

Beliefs About Problem Solving112

enrolled in a remedial mathematics course while the remainderhad successfully completed two or three college mathematicscourses for elementary teachers and were enrolled in anelementary mathematics methods course in the School ofEducation. The second testing was again done during the firstweek of a semester.

Means, standard deviations, and reliabilities (Cronbach’s a)were computed on the final version of the scales and are shownin Table 2. One item was deleted from the UnderstandingConcepts is Important scale because the deletion, while leavingthe scale with only five items, resulted in a substantialimprovement in the inter-item statistics of the scale. Anotheritem was substituted at a supplemental administration of thescales so that the Understanding scale would contain six items.The sample for the supplemental administration of theUnderstanding scale consisted of 88 students enrolled in theelementary mathematics methods courses. Table 3 showscorrelations among each of the five scales. As can be seen inTable 3, there were statistically significant correlations amongseveral of the scales although all of the inter-scale correlationswere relatively small.

Table 2

Means, Standard Deviations,and Reliabilities (Cronbach’s (x)Based on Final Sample (n = 517)

ScaleMeanS.D.Cronbach*s a

Difficult ProblemsStepsUnderstanding"Word ProblemsEffort

20.516.525.318.822.4

3.73.42.83.03.8

.77.67.76.54.84

Difficult Problems = I can solve time-consuming mathemat-ics problems

Steps = There are word problems that cannot be solved withsimple, step-by-step procedures

Understanding = Understanding concepts is importantWord Problems= Word problems are important in

mathematicsEffort = Effort can increase mathematical ability

"All statistics for the Understanding scale are based on thesupplemental sample of 88 students. The statistics for the 5-item version of this scale for the sample of 517 are mean =20.4, S.D. =2.8. a =.73.

Table 3

Inter’scale Correlations Based on Final Sample (n = 517)

Scale

1. Difficult Problems2. Steps3. Understand^4. Word Problems5. Effort

2

-.04

3

.23*-.02

4

.14*

.06

.18*

5

.29*-.27*.19*

-.05��

Difficult Problems = I can solve time-consuming mathemat-ics problems

Steps = There are word problems that cannot be solved withsimple, step-by-step procedures

Understanding = Understanding concepts is importantWord Problems = Word problems are important in

mathematicsEffort = Effort can increase mathematical ability

*p < .05^Correlations are based on the 5-itcm version of this scale.

Discussion

As noted, beliefs about the nature ofmathematics and abouthow mathematics is learned have been gaining considerableattention in recent years. The most common method ofmeasuring beliefs has been though interview and observation.This is anexcellentmethod ofmeasuringbeliefs as considerabledetail is attainable and the chance for social desirability bias issomewhat less than with self-report scales. Unfortunately,interviews and observations are very time consuming and thusare not practical for teachers of large classes or researchersworking with large samples. Thus, in this study, easilyadministered scales for measuring beliefs were constructedand validated.

Scales 2 and 4 measure beliefs about the discipline ofmathematics (there are word problems that cannot be solvedwith simple, step-by-step procedures; word problems areimportant in mathematics) while scales 1. 3, and 5 measurebeliefs about the individual as a learner of mathematics (I cansolve time-consuming mathematics problems, understandingconcepts is important in mathematics, effort can increasemathematical ability). These beliefs were chosen for studybecause they should help to explain motivation to leam to solvemathematical problems on the part of secondary school andcollege students. While four of the scales have been used onlywith college students, the type of issues they address areimportant in middle and high school mathematics and thus thescales should also be appropriate for these populations. TheEffort Can Increase Mathematical Ability scale has been usedpreviously with seventh-grade students (Kloosterman, 1988).Reliability of this scale was almost identical for both the

School Science and Mathematics

Beliefs About Problem Solving113

seventh-grade and the college populations.Upon examination of the reliabilities of the five scales (see

Table 2), it can be seen that the reliability coefficient of theimportance of word problems scale was considerably lowerthan any of the other coefficients. While there are a number ofpossiblereasons for this, two explanations seem mostplausible.The first is that two of the items mention word problems inrelation to computational skills, three items mention wordproblems without regard to computational skill, and one itemmentions application ofcomputational skill but not specificallyto word problems. It is possible that this variation in wordingsof the items caused students to respond inconsistently. Theinter-item statistics support this hypothesis only in that the itemwhich does not mention word problems at all has the lowestcorrelations with the other items. Deleting the item, however.does not improve scale reliability.A second possible explanation for the low reliability of the

importance of word problems scale involves interpretation ofthe term word problem. Because students should be taught tosolve word problems and because students who think wordproblems are a wasteoftime will not want to learn to solve them,this scale is important. The meaning of the term "wordproblem," however, is not consistent across mathematicsteachers or textbooks. In somecourses, the only word problemsstudents see are ones where all one has to do is select the correctnumbers from a problem and perform a computation. In somealgebra courses, word problems are predominantly age, coin,or mixture problems where solutions can be found by writingtwo equations with two variables and solving. In collegecourses, word problems can require a number of steps andconsiderable perseverance. Students who have taken a varietyof mathematics courses and thus have been exposed to variousdefinitions of the term word problem, may have differingbeliefs about each type of word problem they have seen. If thisis the case, responding inconsistently to statements about wordproblems would not be surprising.

Another issue to be considered is the independence of thescales. Whiletheinter-scalecorrelations were all nonsignificantfor the pilot sample, several correlations were signi ficant for themain sample. As these correlations were all less than 0.3, theoverlap of the scales is minimal and less than scale overlap inother Likert-type attitudinal instruments (e.g. Fennema-Sherman Mathematics Attitude Scales, Fennema & Sherman,1976). The significant negative correlation between solvingword problems with step-by-step procedures and effort as ameans ofincreasing mathematical ability is interesting becauseit indicates that those student who felt word problems could bereduced to step-by-step procedures tended to be the same oneswho felt effort would improve their ability. Schoenfeld (1985)talked about students who persist at solution strategies that arenot getting them anywhere. The negative correlation betweenthese beliefs suggests that many students feel that the beststudents in mathematics are the ones who can find an algorithmfor solving any word problem they come across. In real

mathematics, algorithms are usually not available but in themathematics courses taken by these students it is possible thatstcp-by-stcp solution ofword problems was the key to success.

On a different note, one issue that has not been addressed iswhat an instructor should do once beliefs have been measured.Obviously, positive beliefs are key to the development of goodproblem-solving skills. To date however, little research hasbeen done on how resistant mathematics beliefs are to change.Experienceindicatesthatcounterexamples canbevery effectivefor overcoming negative beliefs. A student who believes he orshe is not able to solve time-consuming mathematics problemsneeds to experience success at solving such problems. Astudentwho believes that most ifnot all mathematical problemscan be solved by applying memorizing rules needs to be givenproblems where a little common sense is more beneficial thana full battery of rules.

In addition to counter-examples, class or small groupdiscussion of beliefs is a good way for highly verbal students toreflect on their own maladaptive beliefs. This technique isparticularly effective in dealing with students who believe thateffort is of little value for improving mathematical ability.While this fear is not easily overcome, discussing it andexplaining that anyone who tries can leam is a logical steptoward alleviating the problem.

Beliefs do not change easily. Many students are quitecontent with the way they view mathematics and themselves aslearners ofmathematics. Students say, "Don’tconfusemewiththe reasons; just tell me what steps to follow" or "These wordproblems are too hard. Why don’t wejust skip them?" Whileit is easiest to give in to students and tell them the steps or skipthe word problems, such actions confirm their unfavorablebeliefs. Students need to believe they can do time-consumingproblems, they need to see the limits ofstep-by-step proceduresin mathematics, they need to see that time spent understandingconcepts is time well spent, they need to see that wordproblemsare an important part ofmathematics, and they need to believethat effort will makethem better at doing mathematics. Teachersneed to help students develop these beliefs.

Using the Indiana Mathematics Belie/Scales

The Indiana Mathematics Belief Scales (see Appendix A)are intended for use by secondary school and college levelinstructors and by researchers who would like to determine themotivational beliefs of students. Not all the scales have to beused at once. In fact, given the potential for confusion of theterm word problem in the importance of word problems scale,this scale should only be used in a situation where the term wordproblem hasbeen explained to students orwhere wordproblemsare a continual part of the curriculum. As noted earlier,usefulness is an important component of motivation inmathematics and thus the 6-item version of the Fennema-Sherman (1976) Usefulness ofMathematics Scale (see AppendixB) may also be used when using the Indiana scales.

Volume 92(3), March 1992

Beliefs About Problem Solving114

When administering the scales, items from each scale to beused should be randomly distributed throughout a singlequestionnaire except that items from the same scale should notbeconsecutive. Students should be instructed to read each itemcarefully and indicate the response (strongly agree, agree,uncertain, disagree, strongly disagree) which best describestheir feeling for each item. While there is no time limit tocomplete the scales, students should be told not to spend toomuch time on any one item. When all 36 items (five Indianascales and the Usefulness scale) are used, administration shouldtake about 15 minutes.

Scoring ofthe scales shouldbedoneusing the scoring systemdescribed in the Instrument Development section of this article.Each scale is scored separately and there is no overall score.When interpreting results, note that scale statistics given inTable 2 are from remedial first-year college students and fromcollege students enrolled in elementary education. While eachsecondary school and college is different, one might expecthighly able students to score higher and secondary schoolstudents with no intention of going to college to score lower onsome of the scales. As noted earlier, interviews often providedeeper insight into beliefs than self-report scales and thusinstructors may wish to interview those students whose scalescores indicate unusually detrimental beliefs.

In summary, the Indiana Mathematics Belief Scales are auseful tool for researchers and instructors. Use of the scales isof little value, however, if nothing is done to help studentsovercome beliefs that inhibit learning. While overcoming suchbeliefs is not easy, it should be an important goal ofmathematicsinstruction.

References

Carpenter, T. P., Lindquist, M. M., Brown, C. A., Kouba, V. L.,Silver, E. A., &Swafford.J.O. (1988). Results of the fourthNAEPassessment ofmathematics: Trends and conclusions.Arithmetic Teacher, 55(8). 14-19.

Charles, R., & Lester, F. (1982). Teaching problem solving:What, why, & how. Palo Alto, CA: Dale SeymourPublications.

Dossey, J. A., Mullis, I. V. S., Lindquist, M. M., & Chambers,D.L. (1988). The mathematics repart card: Arewemeasuringup? Trends and achievement based on the 1986 nationalassessment, (NAEP Report No. 17-M-01). Princeton.NJ:

Educational Testing Service.Dweck. C. S., & Bcmpechat, J. (1983). Children’s theories of

intelligence: Consequences for learning. In S. G. Paris, G.M. Olson. & H. W. Stevenson (Eds.). Learning andmotivation in the classroom (pp. 239-256). Hillsdale, NJ:Lawrence Eribaum Associates.

Fennema, E., & Sherman, J. (1976). Fennema’Shermanmathematics attitudes scales: Instruments designed tomeasure attitudes toward the learning of mathematics byfemales and males. Madison, WI: Wisconsin Center forEducational Research.

Fennema, E.. Wolleat, P. L., Pedro. J. D., & Becker. A. D.(1981). Increasing women’s participation in mathematics:An intervention study. Journalfor Researchin MathematicsEducation, 12, 3-14.

Kloosterman, P. (1988). Self-confidence and motivation inmathematics. Journal ofEducational Psychology,80,345-351.

National Council of Teachers of Mathematics. (1989).Curriculum and evaluation standards for schoolmathematics. Reston VA: Author.

National Research Council. (1989). Everybody counts: Areport to the nation on thefuture of mathematics education(summary). Washington, DC: National Academy Press.

Nibbelink. W. H.. Stockdale. S. R.. Hoover, H. D.. & Mangru,M. (1987). Problem solving in the elementary grades:Textbook practices and achievement trends over the pastthirty years. Arithmetic Teacher, 35(1), 34-37.

Schoenfeld, A. H. (1985). Metacognitiveandepistemologicalissues in mathematical understanding. In E. A. Silver (Ed.).Teaching and learning mathematical problem solving:Multiple research perspectives (pp. 361-379). Hillsdale,NJ: Lawrence Eribaum Associates.

Schoenfeld, A. H. (1988). When good teaching leads to badresults: The disastersof"well-taught" mathematics courses.Educational Psychologist, 23,145-166.

Silver, E. A. (1985). Research in teaching mathematicalproblem solving: Some underrepresented themes anddirections. In E. A. Silver (Ed.), Teaching and learningmathematical problem solving: Multiple researchperspectives (pp. 247-266). Hillsdale, NJ: LawrenceEribaum Associates.

Tobias, S. (1978). Overcoming math anxiety. New York:Norton.

School Science and Mathematics

Beliefs About Problem Solving

115Appendix A

Indiana Mathematics Belie/Scales

Belief 1: I can solve time-consuming mathematics problems.+ Math problems that take a long time don’t bother me.+ I feel I can do math problems that take a long time to complete.+ I find I can do hard math problems if I just hang in there.- If I can’t do a math problem in a few minutes, I probably can’t do it at all.- If I can’t solve a math problem quickly, I quit trying.- I’m not very good at solving math problems that take a while to figure out.

Belief 2: There are word problems that cannot be solved with simple, step-by-step procedures.+ There are word problems that just can’t be solved by following a predetermined sequence of steps.+ Word problems can be.solved without remembering formulas.+ Memorizing steps is not that useful for learning to solve word problems.- Any word problem can be solved if you know the right steps to follow.- Most word problems can be solved by using the correct step-by-step procedure.- Learning to do word problems is mostly a matter of memorizing the right steps to follow.

Belief3: Understanding concepts is important in mathematics.+ Time used to investigate why a solution to a math problem works is time well spent.+ A person who doesn’t understand why an answer to a math problem is correct hasn’t really solved the problem.+ In addition to getting a right answer in mathematics, it is important to understand why the answer is correct.1

- It’s not important to understand why a mathematical procedure works as long as it gives a correct answer.- Getting a right answer in math is more important than understanding why the answer works.

- It doesn’t really matter if you understand a math problem if you can get the right answer.Belief 4: Word problems are important in mathematics.

+ A person who can’t solve word problems really can’t do math.+ Computational skills are of little value if you can’t use them to solve word problems.+ Computational skills are useless if you can’t apply them to real life situations.- Learning computational skills is more important than learning to solve word problems.- Math classes should not emphasize word problems.- Word problems are not a very important part of mathematics.

Belief 5: Effort can increase mathematical ability.+ By trying hard, one can become smarter in math.+ Working can improve one’s ability in mathematics.+ I can get smarter in math by trying hard.+ Ability in math increases when one studies hard.+ Hard work can increase one’s ability to do math.+ I can get smarter in math if I try hard.

"This item is to be used in administration of this scale but it is not included in any statistics reported for the 5-item version ofthe Understanding scale.

Appendix BFennema-Sherman Usefulness Scale0

These items are a slightly reworded subset of the Fennema-Sherman (1976) Usefulness of Mathematics scale. Statistics forthis scale when administered to the main sample (n = 517) were mean = 23.2. S.D. = 4.2. a = .86. Correlations of the scalewith the Indiana scales were .48 (p < .05) for Difficult Problems, .06 for Steps. .33 (p < .05) for Understanding, and .30(p < .05) for Effort.

Belief 6: Mathematics is useful in daily life.+ I study mathematics because I know how useful it is.+ Knowing mathematics will help me cam a living.+ Mathematics is a worthwhile and necessary subject.- Mathematics will not be important to me in my life’s work.- Mathematics is of no relevance to my life.- Studying mathematics is a waste of time.

reprinted with permission of the first author.

Volume 92(3), March 1992