359

The Standards9 Beliefs Instrument (SBI): Teachers’ Beliefs Aboutths NCTM Standards

Alan Zollman

Emanuel MasonDepartment of Curriculum and Instruction

Department of Education and Counseling PsychologyUniversity of KentuckyLexington, Kentucky 40506-0017

As aresponse to a perceived crisis in mathematics education(Crosswhite et al.. 1986; Dossey et al., 1988; EducationalTesting Service, 1989; National Council of Teachers ofMathematics [NCTM], 1990),theNational Council ofTeachersofMathematics has initiated a national reform in mathematicseducation with the Curriculum and Evaluation Standards forSchool Mathematics (NCTM, 1989). This document, thecornerstone of the reform, presents NCTM’s vision of howmathematics should be learned, taught, and evaluated in gradesK-12. The 54 standards, developed over a 3-year period,propose thebasis for abalanced curriculum thatfocuses onbothmathematical ideas and processes. They give greater emphasisto conceptual development, mathematical reasoning, andproblem solving than was given in previous curricula inmathematics. The Standards calls for changes in instructionalpractices in which the roles of both students and teacher areredefmed. The role of the student changes from a passivereceiver of information to an active participant in learning; theteacher changes from an active dispenser of knowledge to apassive moderator and manager of learning experiences. TheStandards portrays mathematics as a connected, cohesivebodyof knowledge in which teachers encourage students to beactively engaged in making conjectures and discussing ideas.

The purpose of this research was to provide an instrument(the Standards’ Belief Instrument [SBI]) to assess teachers’beliefs about the NCTM Standards, using items representativeof beliefs about the Standards. For the purpose of this study, abelief was defined as, "a judgment of the credibility of aconceptualization. Credibility ofa conceptualization has to dowith whether one accepts, rejects, or suspends judgmentconcerning a set of concepts and the interrelationships amongthose concepts" (Reyes, 1987, p. 10).

Recent research in mathematics education (Bush, Lamb, &Alsina. 1990; Fullan. 1983; Kesler. 1985; McGalliard. 1983;Silver, 1985; Thompson. 1984) indicate teaching behavior isinfluenced profoundly or subtly by what teachers believemathematics should be. For example, Thompson (1984) foundthat mathematics teachers’ views, beliefs, and preferences didinfluence their instructional practice. This is further illustratedby Ferrini-Mundy (1986) who found many inappropriateteaching practices attributed to teachers’ beliefs aboutmathematics. In addition. Bauch (1984) speculated that "a

teacher’s adherence to a particular set of instructional beliefsmight limit what a student can obtain from schooling" (p. 18).

The cited research suggests an important relationship existsbetween the teacher’s beliefs and the teacher’s behavior. Withthis background in mind, a wide acceptance of the NCTMStandards would depend on a teacher’s beliefs. This presentwork was done to answer the need for an instrument to assessteachers’ beliefs about the Standards.

Development of the Instrument

Items for the SBI wererandomly chosen from several levelsof the Standards to be representative of the Standards. Itemsmet three criteria. First, the implication of each item wasintended not to be intuitively obvious to avoid a sociallydesirable (or correct according to the Standards) pattern ofresponding. Second, the item could be clearly stated in apositive or negative manner. Third, the central idea ofthe itemcould be incorporated into a single sentence.

Theitemsoftheinstrumentwereintended toberepresentativeof the Standards, not inclusive. The purpose was to assessbeliefs underlying the Standards, rather than measurecomprehensive knowledge or achievement of specific aspectsof^Standards. In this context, another issue notaddressedbythe SBI is theappropriategrade level ofthe contentofeach item.Items were randomly chosen on the basis of theme rather thangrade level so that the SBI wouldrepresent the total elementary,middle, and secondary school mathematics program.

The 16 items of the SBI were either nearly direct quotes orthe inverse ofdirect quotes from ^Standards. Theeight itemsthatagreed with the Standards were considered to havepositivevalence; conversely, the eight items that disagreed with theStandards wereconsidered to havenegative valence. Therefore,when items were summed to obtain a total score, the negativeitems were reversed to coincide with the positive items.

The instrument was pilot tested and revised with the aid of15 teachers familiar with the Standards. As a result of the pilottesting, important words in each item were capitalized to betterinsure that respondents would focus attention on the intent ofeach item. The capitalized words were chosen as indicative ofthe meaning of the specific part of the Standards the itemrepresented. Before this capitalization was added, respondents

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Standards’ Beliefs Instrument360

appeared to emphasize aspects of items that were distracting.Respondents and reviewers saw value in this capitalization. The16 items of the SBI, the basis of each item from the Standards,and the valence ofevery item (positive if it agrees; negative if itdisagrees with the Standards), and the valence of every item(positive ifit agrees; negative if it disagrees with ^Standards)are shown in the Appendix.

The final instrument was tested for construct validity byrequesting a panel of experts composed of 17 mathematicseducators, who had either helped edit, develop, and/or writeparts of the NCTM Standards, to evaluate the items in terms ofwhether or not the item agreed with the Standards. The panelconsisted of eleven mathematics education professors, fourpast/present/future state NCTM presidents, one statemathematics consultant, and one classroom teacher who helpdevelopa videotapeon the Standards. All 17 educators indicatedthey had studied, discussed with colleagues, and implementedthe ideas of the Standards as a part of their professions.

Construct validity was further investigatedby analysis oftheconvergent and discriminant nature ofcorrelations between theSBI and other variables. The data used in this analysis were fromtwo groups of teachers who were administered the instrument.Thesegroups consisted of 61 undergraduate practicum teachersin their final semesterbeforestudentteachingand 72experiencedteachers in graduate courses.

Each subject was administeredthe SBI using a 4-pointLikenscale (1 = strongly agree, 2 = agree, 3 = disagree, 4 = stronglydisagree) and a demographic questionnaire during regular classsession. Subjects were instructed to leave blank any item theyconsidered confusing. Afterthedatawerecollected, theresponsesfor eight negative valence items of the SBI were re-aligned(response subtracted from 5.0). Thus the rating of positive andnegative valenceitems wouldbein aconsistentdirectionrangingfrom 1.0to4.0. Thecloserthere-alignedresponse number to 1.0the stronger the agreement with the Standards.

Results

Table 1 shows the mean, valence, and standard deviation foreach item. The data suggested that the teachers tended to agreewith the Standards on most of the items on the SBI, whileidentifying areas for further inquiry.

The expert panel of 17 mathematics educators was asked todetermine whether each item agreed or disagreed with theStandards. Table 2 shows these data. An inspection ofTable 2suggests strong support for construct validity in the sense thatthe experts viewed the items as representing the Standards asintended to a very high degree (^ = 229.79, df == 1, p < .001).

Informationaboutbackground, including mathematics abilityand mathematics anxiety (rated on a scale of 1 [low] to 5 [high]).familiarity with the Standards, and teaching experience werereported by the respondents on a separate questionnaire. Table3 shows the correlations between individual items and the SBItotal agreeand disagree subscores, the SBI total score, questions

about the respondents* mathematics ability and mathematicsanxiety, the respondents’ familiarity with the Standards, andthe respondents’ teaching experience. Mathematics ability,mathematics anxiety, familiarity with the Standards, andteaching

Table 1

Valence, Mean and Standard Deviation (SD) of Items on theSBI after Adjustment for Valence

Item

12345678910111213141516

Valence

-+++-+------++++

Mean

1.9551.1721.1941.4811.4811.0971.7162.3543.0452.6462.7023.1601.6472.7091.0302.326

SD

1.0180.3980.3970.6110.7340.3850.8460.9300.8400.9630.9090.8300.6760.9400.1710.985

n

134134134133133134134127131130131131133134134132

experience items wereapart ofthedemographic questionnaire.Correlations between individual items and teachers’

mathematics abilityandmathematics anxiety werelow, rangingfrom -.16 to .23. Items correlated with familiarity with theStandards slightly higher (0.5 to .33) with item 9 highest.Further, each item correlated with total agree, total disagree,and total score from a low of -.16 to a high of .64. Teachingexperience had a correlation range from -.16 to .06 with eachitem.

Total SBI score correlated with totals of the agree items(.66) and disagree items (.90). Total SBI score correlatedrelatively low with mathematics anxiety (.27), mathematicsability (-.21), and teaching experience (-.14). This lastcorrelation index (SBI with teaching experience) suggests thatbeliefs about the Standards are fairly uniform regardless ofexperience. Further, respondents reported level of familiaritywith the Standards correlated moderately with total SBI score(.44).

These patterns of correlations support the integrity of theconstruct underlying the SBI (Campbell & Fisk, 1959). Inother words, construct validity is supported when variablesthat should highly correlate with the instruments do so in theexpected direction and lower correlations are obtained forthose variables that should not correlate highly with theinstrument. This pattern is found with total SBI score being

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Standards’ Beliefs Instrument361

morerelatedwithagreeanddisagreeitemsthanwith mathematicsability, or mathematics anxiety items, or with teachingexperience.

Reliability of the SBI was investigated with internalconsistencyprocedures. Table4 displays the Spearman-Brownreliability and the coefficient of alpha for two distinct

Table 2

Expert Panel’s Analysis of the SBI (N = 17)

Expert Panel’s Classification of Items as Agree or Disagree

Item Valence Agreeing Disagreeing

123456789

10111213141516

-+++-+------++++

01717171

1701000413171716

170001601716171717134001

Valence of 16 ItemsValence

NegativePositive

6AgreeingDisagreeing

1315 130

populations.Table 4 permits comparison of reliability in two separate

groups. One is a general population of teachers (N = 123); thesecond is a trained population of 13 teachers who studied theStandards as a part of a graduate course in mathematicseducation. Despite themuch smaller sample size (A^= 13) ofthetrained group, the Spearman-Brown reliability for this groupwas much higher (.803, coefficient of alpha approaching .79)than the untrained group (Spearman-Brown .493, coefficient ofalpha approaching .65). This evidence for the reliability of theSBI also shows more consistency ofratings from subjects more

knowledgeable about the Standards and thus suggest evidencefor validity of the instrument as well.

Implications for Teaching

Beliefs regarding the new NCTM Standards should beexamined as mathematics education is reformed. At thepersonal level, the SBI can serve as a catalysis for reflectivethinkingand active decision makingregarding ourown teachingactivities.

Second, many mathematics educators possess severaltraditional, if unfounded, beliefs with respect to the learningand teaching of mathematics. Table 3 correlationsshow that mathematics ability, teaching experience, or evenfamiliarity with the Standards did not indicate that teachersagree with the consensus the Standards represents.

The SBI is not intended to assess teachers’ knowledge oftheStandards. Nevertheless, it canbe used as abasis for evaluatingteachers’ perspectives toward the underlying vision of theStandards. The SBI can be utilized as a beginning forcommunications among teachers in a school or in an inserviceconcerning the real purpose of mathematics education.

Finally, the data ofTable 4 show teachers’ beliefs apparentlycanbechanged ifpresented withproperknowledgeand sufficientexperiences. Thus the SBI can serve a valid instrument forassessing beliefs, both initially and throughout a project,program, inservice, or university course.

Discussion

The objective of this study was to develop an instrumentwhich could be used to access the beliefs of teachers about theNCTM Standards. The results of this study suggest that theinstrument (SBI) is useful in this regard. First, the instrumentwas written using the phrasing of the Standards. Second, theexpertpanel ofmathematics educators viewed the instrument asrepresenting the Standards. Third, convergent and divergentcorrelations supported the construct validity ofthe SBI. Fourth,the level ofreliability ofthe instrument implied that it produceda fairly dependable score in subjects who studied the Standards.

Theitemsoftheinstrumentwereintended toberepresentativeof the Standards, not inclusive. The purpose here was tomeasure teachers’ beliefs underlying the Standards, rather thanassess comprehensive knowledge of specific aspects of theStandards. Items were chosen on the basis of them and generalunderlying perspective towards the Standards, rather thanconcern for grade level, and thus, the SBI should represent thetotal elementary, middle, and secondary school program.

Volume 92(7) November 1992

Standards’ Beliefs Instrument36:

Tal

CoiTot,

Vai

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

Note: Leading decimals omitted.

2

)le3

relationsBetweenItemal Score, and Selected.

table (17) (18)

Itemi -11 -16n 131 130

Item 2 -03 -00n 131 130

Item 3 05 00n 131 130

Item 4 -16 11n 130 129

Item 5 -06 09n 130 129

Item 6 06-01n 131 130

Item 7 -06 10n 131 131

Item 8 -05 08n 126 125

Item 9 -10 20/? 130 129

Item 10 -08 11n 129 128

Item 11 -09 23n 128 127

Item 12 -15 09n 128 127

Item 13 -00 05n 130 129

Item 14 -03 08n 131 130

Item 15-07 04n 131 130

Item 16 -13 18n 129 128

Math ability -52n 130

Math anxietyn

Familiar with standardsn

SBI agree totaln

SBI disagree totaln

SBI total scoren

Teaching experience

s, andABackgn

(19)

071312613116

13119

1300513011

13015

13112

1263313031129

1 016

12818

12812

130251310713122129-2613132130

greea,wndC

(20)

07130341303813053130-08129321303713011

126281283212805129-0512849130511302113060130-111275412741127

ndDisa,^haractt

(21)

5712408124221242712344124-0212454124531245712463124601242912404124061240012418

124-18123-171263312326123

greeSut’ristics

(22)

481232112334123451233112312

12358123461235612364123481232012326123311230912342123-2112227122441226612390123

itotals,

(23)

0413306133-01133-16132-04132-051330013303127-0413101130-11130-16130-08132-09133-02133-06131-0013130121-16131-14129-11124-14123

Table 4

Internal Consistency Reliability of SBI before (n = 123) andafter (n = 13) Training in the NCTM Standards

Item Before Training After Training

mean if item alpha if mean if item alpha ifdeleted item deleted deleted item deleted

1 29.74 .608 23.46 .7572 30.48 .632 23.92 .7643 30.46 .622 23.92 .7644 30.15 .605 * *5 30.14 .623 23.54 .7356 30.56 .637 23.92 .7647 29.95 .588 23.15 .7468 29.34 .611 23.31 .7159 28.62 .587 22.54 .79510 29.05 .573 23.31 .71711 28.92 .600 23.23 .75012 28.53 .653 21.69 .76813 30.00 .637 23.46 .78414 28.98 .639 23.31 .76115 30.63 .636 23.85 .76016 29.33 .624 23.39 .782Spearman Brown .493 .803

*Therewasno variation in theresponses ofthe 13 subjects aftertraining and therefore the effects of this item could not beanalyzed.

References

Bauch,P. A. (1984, April). The impact of’teachers’’instructionalbeliefs on their teaching: Implications for research andpractice. Paper presented at the annual meeting of theAmerican Educational Research Association,NewOrleans,LA.

Bush. W. S.. Lamb. C.E., & Alsina, A. (1990). Gainingcertification to teach secondary mathematics: a story ofthree teachers from other disciplines. Focus on LearningProblems in Mathematics, 12(1), 41-60.

CampbeU, D. T., & Fisk. D. W. (1959). Convergence anddiscriminant validation by the multitrait-multimethodmatrix. Psychology Bulletin, 56, 81-105.

Crosswhite,F.J.etal. (1986). Secondinternationalmathematicsstudy. Washington, DC: National Center for EducationalStatistics.

Dossey,J.etal.(1988). Themathematics report card. Princeton,NJ: Educational Testing Service.

Educational Testing Service. (1989). Crossroads in Americaneducation. Princeton, NJ: Author.

Ferrini-Mundy, J. (1986, April). Mathematics teachers’

School Science and Mathematics

Standards’ Beliefs Instrument363

attitudes and beliefs: Implicationsfor in-service education.Paper presented at the annual meeting of the AmericanEducational Research Association, San Francisco, CA.

Fullan, M. (1983). The meaning of education change. NewYork: Teachers College Press.

Kesler, R. (1985). Teachers’ instructional behavior related totheir conceptions of teaching and mathematics and theirlevel ofdogmatism: Fourcase studies. Dissertation AbstractsInternational, 46A, 2606A.

McGalliard,W.A.,Jr. (1983). Selected factors in theconceptualsystems of geometry teachers: Four case studies.Dissertation Abstracts International, 44A, 1364.

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

National Council of Teachers of Mathematics. (1990).

Professional standards for teaching mathematics. Reston,VA: Author.

Reyes. L. H.. (1987, April). Describing the affective domain:Saying what we mean. Paper presented at the researchpresession to the annual meeting of the National Council ofTeachers of Mathematics, Anaheim, CA.

Silver, E. A., (1985). Research on teaching mathematicalproblem solving: Someunderrepresented themesandneededdirections. In E. A. Silver, (Ed.), Teaching and learningmathematical problem solving: Multiple researchperspectives (pp. 247-266). Hillsdale, NJ: LawrenceEribaum Associates Publishers.

Thompson, A. G. (1984). The relationship of teachers’conceptions of mathematics and mathematics teaching toinstructional practice. Educational Studies in Mathematics,15, 105-127.

Item

Appendix

The Standards’Beliefs Instrument (SBI), Valence, and Item Basis

Valence Item Basis in Standards

1. Problem solving should be a SEPARATE,DISTINCT part of the mathematics curriculum.

2. Students should share their problem-solving +thinking and approaches WITH OTHERSTUDENTS.

3. Mathematics can be thought of as a language +that must be MEANINGFUL if students are tocommunicate and apply mathematics productively.

4. A major goal of mathematics instruction is to help +children develop the belief that THEY HAVE THEPOWER to control their own success in mathematics.

5. Children should be encouraged to.justify theirsolutions, thinking, and conjectures in a SINGLE way.

6. The study of mathematics should include +opportunities of using mathematics in OTHERCURRICULUM AREAS.

7. The mathematics curriculum consists of severaldiscrete strains such as computation, geometry,and measurement which can be best taught inISOLATION.

Problem solving is not a distinct topic but a process thatshould permeate the entire program and provide thecontext in which concepts and skills can be learned (p.23).Ideally, students should share their thinking andapproaches with other students and with teachers (p.23).Mathematics can be though of as a language that mustbe meaningful if students are to communicatemathematically and apply mathematics productively(P. 26).A major goal of mathematics instruction is to helpchildren develop the belief that they have the power todo mathematics and they have control over their ownsuccess or failure (p. 29).Children shouldbeencouraged tojustify their solutions,thinking processes, andconjectures in a variety ofways(P. 29).In grades K-4, the study ofmathematics should includeopportunities to make connections so that students canuse mathematics in other curriculum areas (p. 32).The mathematics curriculum is generally viewed asconsisting of several discrete strands. As a result,computation, geometry, measurement and problemsolving tend to be taught in isolation. It is importantthat children connect ideas both among and withinareas of mathematics (p. 32).

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Standards’ Beliefs Instrument364

8. In K-4 mathematics. INCREASED emphasis shouldbe given to reading and writing numbersSYMBOLICALLY.

9. In K-4 mathematics, INCREASED emphasis shouldbe given to use of CLUE WORDS (key words) todetermine which operation to use in problem solving.

10. In K-4 mathematics, skill in computation shouldPRECEDE word problems.

11. Learning mathematics is a process in whichstudents ABSORB INFORMATION, storingit in easily retrievable fragments as a result ofrepeated practice and reinforcement.

12. Mathematics SHOULD be thought of as aCOLLECTION of concepts, skills and algorithms.

13. A demonstration of good reasoning should beregarded EVEN MORE THAN students’ abilityto find correct answers.

14. Appropriate calculators should be available toALL STUDENTS at ALL TIMES.

15. Learning mathematics must be an ACTIVEPROCESS.

16. Children ENTER KINDERGARTEN withconsiderable mathematical experience, a partialunderstanding of many mathematical concepts,and some important mathematical skills.

Summary of changes in content and emphasis in K-4mathematics-decreased attention-number: earlyattention to reading, writing, and ordering numberssymbolically (p. 21).Summary of changes in content and emphasis in K-4mathematics�decreased attention-problem solving: useof clue words to determine which operation to use (p.21).Traditional teaching emphasis on practice inmanipulating expressions and practicing algorithms asa precursor to solving problems ignore the fact thatknowledge often emerges from the problems. Thissuggests that instead of the expectation that skill incomputation shouldprecedeword problems,experiencewith problems helps develop the ability to compute (p.9).In many classrooms, learning is conceived of as aprocess in which studentspassively absorb information,storing it in easily retrievable fragments as a result ofrepeatedpracticeand reinforcement. Research findingsfrom psychology indicate that learning does not occurby passive absorption alone (p. 10).This notion is based on the recognition ofmathematicsas more than a collection of concepts and skill to bemastered; it includes methods of investigating andreasoning, means of communication, and notions ofcontext (p. 5).In fact, a demonstration of good reasoning should berewardedeven morethan students’ ability to find correctanswers (p. 6).Because technology is changing mathematics and itsuses, we believe that appropriate calculators should beavailable to all students at all times (p. 8).Young children are active individuals who construct,modify, and integrate ideas by interacting with thephysical world, materials, and other children. Giventhese facts, it is clear that the learning of mathematicsmust be an active process (p. 17).Children enter kindergarten with considerablemathematical experience, a partial understanding ofmanyconcepts, and someimportantmathematics skills(P. 16).

School Science and Mathematics