Effective Mathematics Instruction for Low-Income ...

Effective Mathematics Instruction for Low-Income Students: Results of Longitudinal Field Research in 12 School Districts Author(s): Douglas Carnine and Russell Gersten Source: Journal for Research in Mathematics Education, Vol. 13, No. 2 (Mar., 1982), pp. 145152 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/748361 . Accessed: 11/02/2015 13:22 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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BRIEF REPORTS. for LowEffectiveMathematicsInstruction Field of Results Students: Income Longitudinal Researchin 12 School Districts DOUGLASCARNINE University of Oregon

RUSSELLGERSTEN University of Oregon

The present brief report attempts to first integrate and then briefly

discuss some of the empiricalfindingsfrom field researchon educational programs for low-income students. Though quasi-experimentscannot possibly offer the same precisionin either the delineationof the independent variablesor in the controlfor potentialthreatsto internalvalidity as tightly controlled classical experimentscan, they do have certainadvantages-information is gatheredin natural,representativesettings (Snow, 1974), and effects over long periods of time can be assessed (Gallagher, Ramey, Haskins, & Finkelstein, 1975; Bronfenbrenner,1975). Though importantfor any type of research, replicationswith differentgroups of children and in a range of settings is essential for validatingresults of quasi-experiments(Cook & Campbell,1979).The longitudinalevaluation of the Follow Throughprojectoffersan opportunityto examinethe effects of 10 clearly articulatededucationalmodels (cf. Stallings, 1975)on lowincome primary grade students in different settings and across several years. In discussing research prioritiesin educationfor the next 5 years, the directors'ofthe National Instituteof Education(Note 1) concluded:"The most persistent problemfacing Americaneducationtoday is the substantial gaps in mean academic performancebetween white and nonwhite students, and between middle/uppermiddleclass and poor children" (p. 1). They cited datafrom the 1977-78NationalAssessment of Educational Progress in mathematics,indicatingthe following: At all age levels and across all dimensionsof mathematicalknowledgeand skill: whites

studentswhoseparentshadnotcompleted blacksandHispanics; highschool outperformed were outperformedby studentsfrommorehighlyeducatedbackgrounds;and studentsfrom big cities and from disadvantagedurbanareas were outperformedby studentsin all other types and sizes of communities..... These differenceshave not come about recently and have shown little sign of dissipatingover time. (ibid., p. 1)

These discrepancies in achievement pose an obvious challenge to researchersin the field of mathematicseducation.Instructionalinterventions that eradicatedifferencesin mathematicsachievementbetween lowincome minority students and their middleclass peers will not only help March 1982


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ameliorate a social inequity but can also offer some concrete, useful informationfor understandingeffective practicesin mathematicsinstruction (cf. Cooley & Leinhardt,1980;Good, Grouws, & Beckerman,1978). In the late 1960s,the U.S. Officeof Educationcontractedwith Stanford Research Instituteand Abt Associates to evaluatethe effectivenessof the 10 majorFollow Throughmodels using a longitudinaldesign. For each of these models, 6 to 12 representative sites were selected following a samplingdesign created by the Huron Institute (Haney, 1977). Students were pretested on the Wide Range AchievementTest (Jastak& Jastak, 1965),and demographicinformationwas collected. At the end of the third grade, students were assessed on all subtests of the ElementaryForm of the MetropolitanAchievement Test (Durost, Bixler, Wrightstone,Prescott, & Balow, 1971).The subtests of relevance for mathematicseducation were Math Computation,Math Problem Solving, Math Concepts, and the Total Math composite. (See Bereiter& Kurland,in press; Wolf, 1978, for reviews.) Students were also tested in other academic areas. The intricaciesof the Follow Throughevaluationgo beyond the scope of a brief report. Follow Through was one of the largest and most expensive social experimentsever conducted(McDaniels, 1975),including 180 school districts and an equally diverse group of educational models. The entire Abt report(Stebbins,St. Pierre,Proper,Anderson,& Cerva, 1977)comprises over 5000pages. A relativelycomplete synthesis of the evaluation of the mathematicscomponent, including secondary analyses of the data and a discussion of methodologicalissues, has been done by Gersten and Carnine(1981) and should be accessed for a full explanationof the following summary. The evaluation attempted to look at two major questions. The first raised extraordinarilyintricate methodologicalissues. Do low-income students taught with Distar?Arithmeticprogramsand Direct Instruction teaching techniques (Silbert, Carnine, & Stein, 1981)achieve in mathematics at significantlyhigherlevels than low-incomestudentstaughtwith traditionalmethods? A storm of controversy has been aroused over the fairness of the comparisongroups, attrition,and the statistics used (cf. Becker, 1978;Bereiter & Kurland,in press; Haney, 1977;House, Glass, McLean, & Walker, 1978). These issues are beyond the scope of this paper. As Cook and Campbell(1979) indicate in their recent work on quasi-experimentaldesigns, perhapsthe key questionfor determiningthe validity of results should be, Are the results generalizableacross tests, time, settings, and populations? These latter issues can be clearly addressed. The second major question, by contrast, is simple: Is the mean mathematics achievement score of low-income students taught with Direct Instruction techniques equivalent to the national norm sample? Another area of interest is the assessment of the effects of the mathematics programon the affective domain.A finalquestionis, Are there traces 146

Journal for Research in Mathematics Education


of the effects 2 or 3 years afterthe interventionis completed?Addressing these questions in a serious empiricalfashion could be a potentiallyrich source of informationfor understandingeffective teaching/learningprocesses in mathematics. Results To answer the first majorresearch question, Abt Associates (Stebbins et al., 1977) conducted tests for both statistical significance (using analysis of covariance with various demographiccovariates)and educational significance(i.e., mean magnitudeof effect). (See Stebbins et al., 1977, or House et al., 1978, for discussions of educationalsignificance.) Since each educationalmodel was evaluatedat a differentnumberof sites with varying numbers of replications, the percentage of significant outcomes was reported.The percentageswere determinedby combining the positive and negative significantdifferencesand dividingby the total numberof comparisons.(See Gersten& Carnine,1981,for a discussionof decision rules for determiningpercentages.) A positive percentageindicates the Follow Throughsamplesoutperformedthe comparisongroupin a majority of instances. A negative percentage represents the reverse. Table 1 reportsthe resultsfor all MathSubtestsandTotalMath.Note that for the Direct InstructionModel, 66%of the comparisonsare favorablein Math Computation, 55% in Problem Solving, and 37% in Concepts. Results could not be attributedto a Hawthorneeffect or the increased fundingper child, since the same effect would have occurredfor many, if not all, of the Follow Throughmodels. Yet several of these models have net negative effects. Generalizabilitywas indicatedby the replicationsin the Abt study, as well as the U.S. Department of Education's Joint Dissemination and Review Panel's (JDRP) designation of all 11 Direct InstructionFollow Through projects as exemplary in mathematics instruction (Gersten, Note 2). Settings included primarilyblack, inner-cityareas in two large and three intermediate-sizedurbancenters, as well as an AmericanIndian reservation, two bilingual communities, and three rural communities. Samples were evaluatedfor mathematicsachievementfor a periodof 7 to 10 years. Generalizabilityacross populationscan also be assessed by looking at those students from non-low-income backgroundswho entered Direct InstructionFollow Throughin kindergarten.On the Metropolitancomputation subtest, the mean performance for these 321 non-low-income students was 4.76, correspondingto the 83rd percentile performance. Means for the Concepts and Problem Solving subtests correspondedto the 68th percentile and 75th percentile respectively. The second majorresearchquestionhad to do with the absolutelevel of performanceof low-income children. At the end of third grade, Direct Instruction students who entered Follow Throughin kindergartenhad a March 1982


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median score correspondingto the 48th percentile, less than one-tenthof a standarddeviation from the nationalmedian(Stebbins, 1976;Stebbins et al., 1977). In the analysis of five groups of Follow Throughstudents entering Follow Through in different years in 16 sites, Becker and Engelmann (1978) reported mean scores corresponding to the 53rd percentile in Total Math on the Metropolitanand the 54th percentile on the Wide Range Achievement Test. From either source (Stebbinset al., 1977;Becker & Englemann,1978),there is convincingevidence that lowincome students in the programperform at a level comparableto the national median-the goal set by NIE (Note 1). This is well above mean performance in mathematicsof third-grademinority students in Title I eligible (low income) schools, which correspondsto approximatelythe 23rd percentile. The Follow Throughevaluationalso attemptedto examine the effects of a variety of instructionalprogramson the affective domain-using a battery consisting of the Coopersmith(1967) Self-ConceptInventory, a measure of self-esteem, and the "positive" and "negative" scales of the Intellectual Achievement ResponsibilityScale (Crandall,Katkowsky, & Crandall,1965).The lattermeasureindicatesthe extent to which children attribute control over positive and negative academic experiences to themselves, as opposed to the externalenvironment.The percentagesof significantdifferencesfor the affective area appearin the last column of Table 1. Findings in the affective domain roughly parallel those in mathematics achievement. The implicationseems to be that structured approaches that focus on academic goals appear to enhance the selfconcept of children. Ethnographicresearch by Tikunoff, Berliner, and Rist (1975) found similar phenomena. This makes sense; children who have become competent in mathematics, reading, and language might well be expected to feel good about themselves. There also appear to be traces of the effects 2 or 3 years after the intervention is completed. Fifth and sixth grades in seven sites were tested on all subtests of the IntermediateLevel of the MAT. Analyses were performedon Math ProblemSolving, Concepts, Computation,and Total Math. (Details of the study were reportedin Becker & Engelmann, 1978, and Becker & Gersten, in press.) The results indicatedconsistent positive findingsin the areas of Math ProblemSolving (mean magnitude of effect of 0.27 pooled SD units for fifthgrade)and in MathConceptsbut not in Math Computation.It appears that students retain some of the generalproblem-solvingstrategiestaughtin Direct InstructionArithmetic programs and use them in the fifth- and sixth-grade material they encounter. Discussion The Follow Through evaluation indicated that the learning gap in mathematics achievement between low-income and middle-incomepriMarch 1982


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mary grade students can be bridged in a diversity of settings. There is fairly convincing evidence that an interventionpackage called "Direct Instruction" can have a consistent significantpositive impact on lowincome students in a wide rangeof settingsfor a period of 7 to 10 years. Moreover, academic success seems associated with positive self-regard. Here the limits of evaluative researchare reached. A crucial question for researchers in mathematicseducation is the delineationof which components-or combinationof components-in the Direct Instructionmodel (or other related models) are most responsible for these results. All we can do in this brief report is list possible components and share other related researchfindings: 1. Organizationof instruction.One crucialcomponentof the program was high amountsof allocated academictime (cf. Harnischfeger& Wiley, 1976) and high amounts of academic engaged time (cf. Rosenshine & Berliner, 1978). Stallings (1975) documented that in Direct Instructionand Behavior Analysis classrooms, more time was allocated to mathematicsthan with other Follow Through models and that student engagement rate was higher than with most other approaches.However, as Rosenshine(Note 3) pointed out, the Behavior Analysis model had virtually identical amounts of engaged time, and yet significantlyweaker student achievement in Math Concepts and Math Problem Solving. He suggested that factors beyond engagedtime-for example, the curriculum-are likely to be the cause. 2. Teacher performance variables. Use of an active teaching model (Good, 1980)implieshighlevels of small-groupinstructionwith consistent adult feedback, high levels of student accuracy duringlessons (mastery learning), systematic correction procedures of student errors, use of reinforcement,and rapidpacing of lessons. 3. The instructionaldesign principlesin the Distar Arithmeticcurriculum. These principles are articulated by Silbert, Carnine, and Stein (1981). Two of the key elements that differentiateDirect Instructionfrom other curriculumprogramsare consistent trainingin explicit problemsolving strategies and cumulativereview of majorskills. 4. Continualmonitoringof studentprogress with instructionallyreferenced tests, as well as the monitoring of teacher performance by structuredweekly observationsand in-serviceand preserviceworkshops (see Morimitsu,Note 4, for detailsandEdmonds, 1979,for a moregeneral discussion). NOTES REFERENCE

1. National Instituteof Education.Instructionallyeffectiveschools: Research area plan. Washington,D.C.: Departmentof Education,1980.

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2. Gersten, R. Final reports to Joint DisseminationReview Panel, National Institute of Education,for Direct InstructionFollow ThroughProjects: WilliamsburgCounty, SC; DeKalb County,TN;Flippin,AR; P.S. 137, New York,NY; Cherokee,NC; Uvalde, TX; San Diego, CA; East Las Vegas, NM; Nichols Avenue School, Washington,DC. Washington,D.C.: NationalDiffusionNetwork, Departmentof Education,1981. 3. Rosenshine, B. Direct instructionfor skill mastery.Paperpresentedat the Universityof Wisconsinat Milwaukee,Wisconsin,April 1979. 4. Morimitsu,C. Supervisionof direct instruction.Unpublishedmanuscript,Universityof Oregon, 1979. REFERENCES

Becker, W. C. The nationalevaluationof Follow Through-Behavior-theory-basedprograms come out on top. Educationand UrbanSociety, 1978,10, 431-458. Becker, W. C., & Engelmann,S. Analysisof achievementdataon six cohortsof low-income childrenfrom 20 school districtsin the Universityof OregonDirect InstructionFollow ThroughModel (Tech. Rep. 78-1).Eugene,Oreg.:Universityof Oregon,Follow Through Project, 1978. (ERICDocumentReproductionService No. ED 145922) Becker, W. C., & Gersten, R. A followup of Follow Through:Metaanalysisof the later effects of the Direct InstructionalModel. AmericanEducationalResearch Journal, in press. Bereiter,C., & Kurland,M. A constructivelook at Follow Throughresults.Interchange,in press. Bronfenbrenner,U. Is earlyinterventioneffective?In B. Friedlander,C. Sterrit,& G. Kirk (Eds.). Exceptional infant assessment and intervention(Vol. 3). New York: Bruner/ Mazel, 1975. Cooley, W. W., & Leinhardt,G. The instructionaldimensionsstudy.EducationEvaluation and Policy Analysis, 1980,2, 7-25. Cook, T., & Campbell,D. T. Quasi-experimentation: Design and analysis issues for field settings. Chicago:Rand McNally, 1979. Coopersmith,S. The antecedentsof self-esteem. San Francisco:Freeman, 1967. Crandall,V. C., Katkowsky,W., & Crandall,V. J. Children'sbeliefsin theirown controlof reinforcements in intellectual-academicachievement situations. Child Development, 1965,36, 91-109. Durost, W. N., Bixler, H., Wrightstone,J., Prescott, G., & Balow, I. Metropolitan AchievementTest. New York: HarcourtBrace Jovanovich, 1971. Edmonds, R. Effective schools for the urbanpoor. EducationalLeadership,October 1979, pp. 15-24. Gallagher,J. J., Ramey, C. T., Haskins, R., & Finkelstein,W. W. Use of longitudinal researchin the study of child development.In T. D. Tjossem(Ed.), Interviewstrategies for high risk infants and young children.Baltimore:UniversityParkPress, 1975. Gersten,R., & Carnine,D. Direct instructionmathematics:Longitudinalevaluationof over two thousand low income students (Tech. Rep. 81-1). Eugene, Oreg.: University of Oregon, 1981. Good, T. L. Classroomexpectations:Teacher/pupilinteractions.In J. McMillan(Ed.), The social psychology of school learning.New York: Academic, 1980. Good, T. L., Grouws,D. A., & Beckerman,T. M. Curriculumpacing:Some empiricaldata in mathematics.Journalof CurriculumStudies, 1978,10, 75-82. Haney, W. A technical historyof the nationalFollow Throughevaluation.Sponsoredby OE/DHEW, Cambridge,Mass.: HuronInstitute,August 1977.(ERICDocumentReproduction Service No. ED 152426) process in elementaryschools: A Harnischfeger.A., & Wiley, D. E. The teaching-learning synoptic view. CurriculumInquiry, 1976,6, 5-43. House, E. R., Glass, G. V., McLean,L. D., & Walker,D. F. No simpleanswer:Critiqueof the "Follow Through"evaluation.HarvardEducationalReview, 1978,28, 128-160.

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Jastak, J., & Jastak, S. WideRange AchievementTest. Wilmington,Del.: JastakAssociates, 1965. McDaniels,G. L. The evaluationof Follow Through.EducationalResearcher, 1975,4, 711. Rosenshine, B. V., & Berliner,D. C. Academicengagedtime. BritishJournalof Teacher Education, 1978,4, 3-16. Silbert, J., Carnine, D., & Stein, M. Direct instructionmathematics.Columbus,Ohio: CharlesMerrill,1981. Snow, R. W. Representativeand quasi-representativedesigns for researchon teaching. Review of EducationalResearch, 1974,44, 265-290. Stallings, J. A. Implementationand child effects of teachingpracticesin Follow Through classrooms. Monographsof the Societyfor Research in ChildDevelopment,1975,40. A plannedvariationmodel(Vol. 3A). Stebbins, L. B. (Ed.). Educationas experimentation: Cambridge,Mass.: Abt Associates, 1976. Stebbins,L. B., St. Pierre,R. G., Proper,E. C., Anderson,R. B., & Cerva,T. R. Education as experimentation:A planned variationmodel (Vols. 4A-D). An evaluationof Follow Through.Cambridge,Mass.: Abt Associates, 1977. Tikunoff,W., Berliner,D., & Rist, R. An ethnographicstudyof theforty classroomsof the beginningteacher evaluationstudy knownsample (Tech. Rep. 75-10-5).San Francisco: Far West Laboratoryfor EducationalResearch& Development,1975.(ERICDocument ReproductionService No. ED 150 110) Wolf, R. Review of MetropolitanAchievementTest. In O. K. Buros(Ed.), Thesixthmental measurementsyearbook.HighlandPark,N.J.: Gryphon,1978.

[Received May 1981]

Proceedings of the FifthConferenceof PME The Proceedingsof the FifthConferenceof the InternationalGroupfor the Psychologyof MathematicsEducation(held at Grenoble,France,in July 1981)are now availablefromthe North Americangroupof PMEfor $11. Checksshouldbe madeout to PME-NAand sent to JamesMoser, WisconsinCenterfor EducationResearch,1025W. JohnsonStreet,Madison, WI 53706.

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