ERIC ED557355: Interleaved Practice Improves Mathematics Learning

2015·0.23 MB·English

by  ERIC

JournalofEducationalPsychology ©2014AmericanPsychologicalAssociation
2014,Vol.106,No.4,000 0022-0663/14/$12.00 http://dx.doi.org/10.1037/edu0000001
Interleaved Practice Improves Mathematics Learning
Doug Rohrer, Robert F. Dedrick, and Sandra Stershic
UniversityofSouthFlorida
Atypicalmathematicsassignmentconsistsprimarilyofpracticeproblemsrequiringthestrategyintro-
duced in the immediately preceding lesson (e.g., a dozen problems that are solved by using the
Pythagoreantheorem).Thismeansthatstudentsknowwhichstrategyisneededtosolveeachproblem
beforetheyreadtheproblem.Inanalternativeapproachknownasinterleavedpractice,problemsfrom
thecoursearerearrangedsothataportionofeachassignmentincludesdifferentkindsofproblemsinan
interleavedorder.Interleavedpracticerequiresstudentstochooseastrategyonthebasisoftheproblem
y.
s.adl itself,astheymustdowhentheyencounteraproblemduringacomprehensiveexaminationorsubsequent
herbro course.Intheexperimentreportedhere,126seventh-gradestudentsreceivedthesamepracticeproblems
publisnated oovrebryath3e-muosunathlbpleorciokde,dbauptptrhoeapchro.bTlehmesprwacetriecearprahnagseedcosnoctlhuadtesdkiwllisthwaerreevleieawrnesdesbsyioinn,teforlleloavweeddp1raocrti3c0e
dmi dayslaterbyanunannouncedtest.Comparedwithblockedpractice,interleavedpracticeproducedhigher
salliedisse scoresonboththeimmediateanddelayedtests(Cohen’sds(cid:2)0.42and0.79,respectively).
ofitobe Keywords:learning,mathematics,interleaved,spaced,practice
et
onot
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ors
i
ationand The ongoing effort to improve students’ mathematics profi- the Pythagorean theorem to solve this problem, they must first
cier ciency has included a wide variety of interventions. Some are infer that they should use it. In other words, the solution to a
os
ssu rooted in theoretical models of cognition, some are designed to mathematics problem requires students to choose a strategy, not
Aal
aldu eliminatemisconceptions,andstillothersaimtoreducestudents’ only execute the strategy, and students often find the choice of
ologicindivi minaspthireemdabtiycswhanatxiiestyp.erHhaeprsetwheesidmespclreisbteprainnciipnlteerovfenletiaornninthga:tthies sKtriarstecghynetro, &bevmanoreMcehrrailëlennbgoienrg, 2th0a0n4;itSsieexgelecru,ti2o0n03(e;.gS.i,eKgleesrte&r,
he
ycth practiceofaskillimprovestheperformanceofthatskill.Itmight Shrager, 1984). In formal terms, the choice of an appropriate
Psof seem that this robust maxim is already widely used in the math- strategy requires students to both discriminate between different
ne
caus ematicsclassroombecausenearlyallteachersassignpracticeprob- kindsofproblems(i.e.,problemsrequiringdifferentstrategies)and
merinal lems like the ones appearing on tests. However, most practice associateeachkindofproblemwithanappropriatestrategy(Fig-
eAerso assignments are arranged in a way that simplifies the solution to ure1).
byththep ewahcehnptrhoebyleamre, atensdtetdh.is crutch is usually not available to students supTehreficcihaolliycesiomfialanrapprporbolpermiastesosmtraetteimgyesisreoqftueinreddififfifceureltnbtesctraautsee-
pyrightedsolelyfor ssotelTpuosti,osanesetiolwlnuhesytarralttyheidasnbiysymtthhaeethfceoamlsleao,twicwisnegprmporbuolsebtmlefismrsi.ntcplouidnetsotwutotdhiasttitnhcet gekxiienasdm(poelf.eg,p.,rwoCobrhldeim,pFroietbltlioesv.miFcshoo,rft&eexnaGmlalcpaklsee,er,xthp1el9ic8pi1tre;cvuSieoiseugsthlewart,oir2nd0d0ipc3rao)t.belFethomer
scoded Agirlhikes8kmeastandthen15kmnorth.Howfarisshefrom requiredthePythagoreantheorem,buttheproblemdidnotinclude
hisdocumentiarticleisinten hitsreia1rTn7hsgtikalsemrp,tirabnonegbdclape8umo2sien(cid:3)itst?hs1eo5luv2ne(cid:2)kdnbo1yw72nth).deHisPtoyawtnhceaevgeiosrr,tehbaeenfhothyreepoosrtteeumndues(ntehtsoefcaaannrsiwugsheert hiasntnyrspyatotremutgeceniteniuostsinoeis.n“SuIosnofelvftauhellegfe(oPebry.rgatxh.,,”afsgdatooucrtedeosearninnntosgtth,meiqnouudrsaeitcdmarsatooetilrcvwtehfhoeiecrqmwhuouoarltfadio,sthnatesnr,iddabinsfuogfetleortenhon)ert.
This Similarly,muchofcalculusisdevotedtointegration,andstudents
T must learn to discriminate between problems that look alike yet
requiredifferentintegrationtechniques(e.g.,(cid:4)exedxand(cid:4)xexdx).In
DougRohrer,DepartmentofPsychology,UniversityofSouthFlorida; short,studentsatnearlyeverylevelofmathematicsmustlearnto
RobertF.Dedrick,DepartmentofEducationalandPsychologicalStudies, discriminate between different kinds of problems with similar
UniversityofSouthFlorida;SandraStershic,DepartmentofPsychology, surfacefeatures.
UniversityofSouthFlorida. Yet students need not learn to choose a strategy when every
This work was supported by the Institute of Education Sciences, U.S. problem within a practice assignment requires the same strate-
DepartmentofEducation,throughGrantR305A110517(principalinves-
gy—anapproachknownasblockedpractice.Withblockedprac-
tigator:DougRohrer).Theopinionsexpressedarethoseoftheauthorsand
tice,studentsknowthestrategybeforetheyreadtheproblem.For
donotnecessarilyrepresenttheviewsoftheU.S.DepartmentofEduca-
example, if a lesson on proportions is followed by a dozen word
tion.WethankHillsboroughCountyPublicSchoolsfortheirparticipation.
problems requiring students to create a proportion, students need
Correspondence concerning this article should be addressed to Doug
Rohrer,PsychologyPCD4118G,UniversityofSouthFlorida,Tampa,FL notlearnwhichfeaturesofaproblemindicatethatitcanbesolved
33620.E-mail:[email protected] by a proportion. In short, blocked practice does not provide stu-
1
2 ROHRER,DEDRICK,ANDSTERSHIC
shers.broadly. Fsptrirogabutelregemy1.s.TahnTedhcaehsossooiccleuiatiotoefneaoancfhaapkpmirnoadtphroeiafmtepartsioctbrsaletpemrgoybwlreietmhqu.aiSnretusadptephnraottspsrmtiuaudtesetnslttesraadrtenisgctyor.imcShtiunodaoteseentbsaedtswtoreaneteongtydnieaffeneddretenoxteckchiuontodessethoaef
blied strategyifeverypracticeproblemwithinanassignmentrequiresthesamestrategy(blockedpractice).
punat
dmi
salliedisse dentswithanopportunitytochooseanappropriatestrategyonthe clude problems in a variety of formats (e.g., procedural prob-
ite
ofob basis of the problem itself, and yet they must perform this skill lems,wordproblems,open-endedquestions,andstandardizedtest
oneott when they sit for an exam consisting of multiple kinds of prob- practiceproblems),butmostofthepracticeproblemsarededicated
orsn lems. to the immediately preceding lesson. In some of the textbooks,
i
ciationerand maBthloemckaetdicsprparcotbicleemc,apnardtrlyambaetcicaaulsleysrteudduecnetstnheeeddinffoictudlitsycroimfia- e(saocmheptirmacetsicmeoarsestihgannm1e0n0t,cpornessiusmtsabolfymsoanthyatdtoezaecnhserosfcapnrocbhloeomses
sous natebetweenproblemsrequiringdifferentstrategies.Considerthis a subset of problems), and the latter portion of each assignment
s
chologicalAheindividual emxaaCTnmhhyiplsocloeep.orbokabieklseemdre2ims4ascoionlvo?ekdiebs.yHsiemrpfalethseurbtartaect1i0ono(f2t4he–c1o0o(cid:2)kie1s4.)H,bouwt ipRpnrreceivslviueieddoweulsseosrasleSgtshpsraioornunapsl15Roa%enfvdbioeefgwtwrt.oheHueepnoteowdftaeivlvweeniruta,hmnitnhdbee1sare0osrfpeercvpotiribeaowlcnetmipclsareobdpberrllaeoewmdblnseMmcfirosxomemidn-
yt
Psof students who are capable of subtraction may not be able to infer our informal survey. It is also true that many textbooks include
ne that they should subtract, possibly because the problem does not
as periodic review assignments, usually at the end of a chapter,
cu
erial includecuessuchasthewordssubtractordifference.However,if although these assignments typically include a small block of
Amson this problem follows other problems requiring subtraction, stu- problemsforeachlesson(e.g.,afewproblemsbasedonthefirst
eer dentsknowthestrategyinadvance.Infact,theycouldsolvethis
hp lesson of the chapter, followed by a few problems based on the
bytthe problemwithoutreadinganythingotherthanthetwointegers(24 secondlesson,andsoforth).Moreover,thetraditionalmathemat-
pyrightedsolelyfor astnudAdes1nid0tse).tforPousmotlvaenexocwtuhoseirnrdgwpsratouybd,leebnmltosscfwkreoidtmhophuratavcrtieinacgdeitnosgodmainsecytirmiwmeoisrndaastl.elobwes- isscouslmuttaeiobxnltesboowonkotreikasrb-ioanwockar,ey”apsiainnggewsly,haisncudhpopsulteurmdsueernnvtteseydairnoedriacreastkpeelsadcthetadottbwhyersiaetew“ctohorenki-r-
coed tween different kinds of problems, blocked practice can also im- booksrelymoreheavilyonblockedpractice than do the textbooks.
isnd pede the learning of the association between a problem and an
entnte appropriatestrategy.Forexample,theinstruction“Findtheperim- Unfortunately, the predominance of blocked practice cannot be
mi preciselygaugedbecauseteachersmayomitpartsofthematerials
docucleis eintegr”thiendliecnagtethssuonfamthbeigsuidoeussloyfththaettghievepnropbolelymgoisns(oel.vge.,dabsyqaudadre- adopted by their school or school district, or they might choose
hisarti with sides of length 3), and this salient feature of the problem theirownmaterials(e.g.,assignmentsfreelydownloadedfromthe
Ts Internet). Nevertheless, it appears that the vast majority of math-
hi presumablymakesiteasierforstudentstorecognizewhatkindof
T ematics students devote most of their practice effort to blocked
problem it is (perimeter problem). Yet, if a practice assignment
practice.Forthatreason,blockedpracticeservedasthecontrolin
includes a block of perimeter problems, students can solve each
problemwithoutreadingorattendingtotheinstruction“Findthe thepresentstudy.
perimeter.” This repeated failure to perceive the word perimeter Inanalternativeapproachthatservedastheinterventioninthe
weakenstheassociationbetweenthekindofproblem(perimeter) present study, practice problems are merely rearranged so that a
and the strategy (add the lengths of the sides). In effect, blocked portion of each assignment includes a set of different kinds of
practiceallowsstudentstocompleteanassignmentwithoutbeing problemspresentedinanintermixedorder—atechniqueknownas
awareofthekindofproblemtheyweresolving. interleaved mathematics practice (e.g., Higgins & Ross, 2011;
Blocked mathematics practice is prevalent. We inspected sev- Richland,Bjork,Finley,&Linn,2005;Richland,Linn,&Bjork,
eral commonly used middle school mathematics textbook series 2007; Rohrer & Taylor, 2007; Schmidt & Bjork, 1992). With
and found that most of the practice problems in each textbook interleavedpractice,studentsmustlearntochoosethestrategyon
appear in a block of problems devoted to the same concept or the basis of the problem itself. One hypothetical illustration of
procedure. In particular, the prototypical assignment may in- interleavedmathematicspracticeisshowninFigure2.
INTERLEAVEDPRACTICE 3
Assignment

25 26 27 28 29 30 35 40 50
1 (cid:2)(cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
4 problems
on the skill 2 (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
learned 3 (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
that day 4 (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
5 (cid:4) (cid:3) (cid:3) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
6 (cid:5) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
1 problem 7 (cid:6) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
on each of
8 different 8 (cid:7) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
skills 9 (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
hers.broadly. prleeavrionuesdl y 1101 (cid:8)▮ (cid:3)(cid:3) (cid:2)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:2)(cid:3)
s
publinated 12 (cid:9) (cid:3) (cid:3) (cid:3) (cid:3) (cid:2) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
dmi
salliedisse Fbliogcukreo2f.fouInrteprrloebalveemdsm(asqthueamreast)icasnpdraecitgihcet.pIrnobthleismhsyopfotihnetetirclaelavileldustprraatciotinc,eA(snsoing-nsmqueanrte2s)5.iFnocrluedxeasmapslme,ailfl
eofittobe Aonsseiagcnhmoefneti2g5htfodlilfofewresnatlseksislolsnleoanrnperodpeoarrtliioenrsi,nitthweocuoludrisnecolurddeufroinugraprporbiolermcosuornsep.rEoipgohrttiaodndsitainodnaolnperopproobrtlieomn
onot problemsaredistributedacrosssubsequentassignments.Theemptysquaresareplaceholdersforunidentified
n
oris kindsofproblems.
ationand
cier
sous In addition to the juxtaposition of different kinds of problems afinaltestrequiringbatterstohitpitchesofallthreetypeswithout
s
Aal within the same assignment, interleaved practice guarantees that knowingthetypeofpitchinadvance,justastheywouldneedto
ologicalindividu penrotbalsesmigsnomfetnhtess(aFmigeurkein2d).aDreodziesntrsibouftestdudoiressphaacveeddacermoossnsdtirfafteerd- dmoatiincsategsatmine.wThhicishksitnuddeonftsbdatotinnogttkenstowistahnealkoignoduosftoproabmleamthien-
he thatspacingimprovesclassroomlearning,andtheeffectofspacing advance.
ch
yt
Psof is one of the largest and most robust effects in the learning Withregardtomathematicslearning,weknowoffivestudiesof
ne literature (e.g., for recent reviews, see Cepeda, Pashler, Vul, interleavedpractice,andfourofthesestudieswereconductedina
as
cu
erial Wixted, & Rohrer, 2006; Dunlosky, Rawson, Marsh, Nathan, & nonclassroom setting. In the first of these studies, Mayfield and
Amson Willingham,2013;Küpper-Tetzel,2014;Roediger&Pyc,2012). Chase (2002) had remedial college students learn several simple
eer Furthermore,severalstudieshaveshownthatspacingcanimprove algebraic rules (e.g., x5 x2 (cid:2) x7) with a practice schedule that
hp
ythe the learning of mathematics (Bahrick & Hall, 1991; Gay, 1973; provided either interleaved or blocked practice, and interleaved
bt
yrightedolelyfor G2sp0ra0oc6tie)n,.g1In9in9tth5ee;rvRcaroleshartbeieortn&woeTfenainyctleoorrnl,es2ae0vcu0edt6i,vm2ea0pt0hr7oe;bmYleaamtizcsdsaoanfsist&higenZsmeabemrnoetwsk,sitknhide, ptehxroaacuctgtilchyetthhpeerostdwaumoceegdprosrouubpplesermioofsrsotterusdetvesnecntosrteihnse(tsehafimfseecsttnusudimzyebdueirndkonnfooptwrroneb)c.leeAimvles-
ps
coed might expand, meaning that a particular kind of problem might becausethestudywasnotdesignedtoassessinterleavedpractice,
isnd appear initially in each of several consecutive assignments and webelieveitisthefirstdemonstrationofamathematicsinterleav-
entnte thenturnupwithdecreasingfrequencythereafter,asinthehypo- ingeffect.Inalaterstudythatexplicitlycomparedinterleavedand
mi
docucleis tbheettwiceaelnilcluosntsreactiuotnivienpFriogbulreem2s.oAflttehrenastaimveelyk,inthdemspaaycibnegfiinxteedrvsaol btolofcikneddtphreacvtoiclue,mReohorfesreavnedraTlaoyblosrcu(2re00s7o)litdasug(he.tgc.o,lslpegheersotiuddeanntds
hisarti that problems of the same kind appear once every, say, week or spherical cone) and found a positive interleaving effect on a test
This two,asinthepresentstudy.(Thereissomedebateaboutwhether given 1 week later (Cohen’s d (cid:2) 1.34). This finding was later
T
equal or expanding spacing intervals are more effective, and the replicated (with the same materials but a different procedure) by
optimalchoiceseeminglydependsonavarietyofcircumstances, LeBlancandSimon(2008),whofoundalargeinterleavingeffect
e.g.,Balota,Duchek,&Logan,2007;Küpper-Tetzel,2014;Storm, ((cid:5)2 (cid:2) .32) and further observed that interleaving improved stu-
p
Bjork, & Storm, 2010). To summarize, interleaved mathematics dents’ ability to predict their test scores. Finally, Taylor and
practice has two beneficial features: problems of different kinds Rohrer(2010)taughtfourth-gradestudentshowtosolveproblems
are juxtaposed, which requires students to choose a strategy, and relatingtoprisms(e.g.,“Findthenumberoffacesonaprismwith
problemsofthesamekindarespaced,whichimprovesretention. a 5-sided base”), and they found that a session of interleaved
Theearlieststudiesofinterleavedpracticeexamineditseffects practice (rather than blocked practice) led to greater scores on a
onskilllearning(e.g.,Hall,Domingues,&Cavazos,1994;Shea& testgiven1daylater(d(cid:2)1.21).
Morgan,1979).Forinstance,inthestudyreportedbyHalletal., Most recently, Rohrer, Dedrick, and Burgess (2014) assessed
college baseball players practiced hitting three types of pitches the effects of interleaved mathematics practice in a classroom-
(fastball, curveball, and change-up) that were either blocked by basedexperiment.Seventh-gradestudentsreceivedadozenprac-
typeorinterleaved,andinterleavedpracticeledtobetterhittingon tice problems of each of several kinds that were interleaved or
4 ROHRER,DEDRICK,ANDSTERSHIC
blocked, and interleaved practice produced greater scores on a the test. This in turn guarantees that these students benefit from
finaltest(d(cid:2)1.05).Unlikethekindofproblemsusedinprevious spacedpractice.Inbrief,theinclusionofareviewensuresthatthe
studies of interleaving, the different kinds of problems were su- counterfactualusedinthepresentstudywillbemoreeffectiveand
perficially dissimilar (e.g., proportion word problems, solving a morecharacteristicofprevailingteachingpracticesthanthatused
linearequation,graphingequations),showingthatthebenefitsof inpreviousstudiesofinterleaving.
interleaving are not necessarily limited to scenarios in which The second difference between the present study and previous
studentsseeonlysimilarkindsofproblemslikethosechoseninthe interleavingstudiesisthatthepresentoneincludedanindependent
laboratorystudiessummarizedpreviously. manipulationoftestdelay(1or30days).Thiswasdonesothatwe
Thepresentstudydifferedfrompreviousstudiesofinterleaved couldassesswhetherthebenefitsofinterleavedpracticedecrease
mathematicspracticeintwofundamentalways.First,inthepres- overtime.Suchafindingwoulddramaticallycurtailtheutilityof
entstudy,thelastpracticeassignmentwasfollowedbyareview. interleaved practice, and this possibility is not a straw man hy-
Thoughseeminglyinnocuous,theinclusionofareviewaddresses pothesis. In fact, even some recommended learning strategies
aweaknessthatisinherentinadesigncomparinginterleavedand produceaboostintestscoresthatdecreasesmarkedlywithinafew
y. blocked practice. Without a review, the use of interleaved rather weeks(e.g.,Driskell,Willis,&Cooper,1992).
shers.broadl tlhasatnpbralocctikceedpprorabcletimceoifnetraicnhsikcainlldyasnhdorthteentsestht,eadseilllauystbraettewdeebnytthhee theIinr ttehaecheexrpse’riumsuenalt lreespsoorntesdanhderer,ecseeivveendtha-sgsriagdnemsetnutdsenthtsatsawwe
publinated hypothetical illustration in Figure 2. This confounding works in created.Everystudentreceivedthesameproblems,butthesched-
dmi favor of interleaved practice because shorter test delays improve ulingoftheproblemswasalteredsothatstudentsreceivedblocked
salliedisse teeqsutastceodreths.eTdheelaiyncbleutswioenenofthaerefvinieawlpirnacthtieceprpersoebnltesmtuadnydththereetfeosrte, o1roirnt3e0rledaavyesdlpatrearctbicye.anLautnera,nsntouudnencetsdrteecseti.vedareview,followed
ite
ofob as shown in Figure 3. However, even with a review, most of the
oneott practice problems appeared later in the experiment if practice Method
orsn problems were interleaved rather than blocked, although this dif-
i
ationand ferFenucrteheisrmarogrue,abtlhyeainncinlutrsiinosnicobfenaefrietvoiefwintienrlethaevepdrepsreanctticsetu.dy Participants
cier
sous means that the blocked practice condition is a more suitable ThestudytookplaceatalargepublicmiddleschoolinTampa,
s
ologicalAindividual cbboleoufocnkrteeerdfaapccrtuaumcatliu.clTeathiaivssesiisgenxbmaemceanutossreohefivtgeehnn-stgteiaavkceehsethrtseeiswrt.hstoFuodrreelnyintsshteaaanvrceielvy,ieaowln- Ftoeflaoctrhhideearts,eadancuhdriennrgsinehthaoedf2ttha0ue1ig3rh–st2evm0e1in4dtdhsl-ecghrsoacodhleocoylleaasmrs.eastThphearmerteaitcimicpsaattfheoedrm.mEataoiccrhes
chhe thoughstudentsmighthaveaddedfractionsinonlythefirstweek than5years.Theparticipatingclassesincludedonlystudentswho
yt
Psof oftheschoolyear,areviewprovidesonemoredoseshortlybefore hadreceivedapassingscore(3orhigheronascalefrom1to5)
anse ontheGrade6mathematicssectionofthe2013FloridaCompre-
cu
erial hensiveAssessmentTest(FCAT,Version2.0;FloridaDepartment
mn
Aso ofEducation,BureauofK–12Assessment,2014),whichtheyhad
heper taken in April 2013, near the end of the previous school year. A
ythe passingscoreonthistestwasachievedby58%ofthesixth-grade
bt
yrightedolelyfor ssttuuPddaeernntttissciaipntatttihhoeenpsaitnartteicthi(peFalotsirtnuigddayscDhreeopqoaulriatrmneddenbdtyooc5fu2Em%deunoctfaattthiiooenns,io2xf0th1p3ga)rr.aednet
ps
coed permissionandstudentassent,whichwereceivedfrom150ofthe
sd
in 164 students in the participating classes. Of these 150 students,
mentinte 126 (84%) attended mathematics class on both the day of the
hisdocuarticleis utshtnueadinrenndotasut.naNcewedaerrrleeyveiaevnwearlyaynzsdetudtd.heeTndhtauwysa,osft1ht2heeyfueinanraaslnonsloadumanptclteehdeitnbecseltgu,iadnnenddinog1n2oly6f
Ts
hi thestudy,and61weregirls(48%).
T
We did not ask students for information about their socioeco-
nomic background or ethnicity because of privacy concerns, but
the school district provided demographic data for the sample in
aggregate. By these data, 38% of the students received free or
reduced-pricelunch,andthesamplewasethnicallydiverse(10%
Asian, 14% Black, 22% Hispanic, 6% multiracial, and 47%
White).
Figure 3. Procedure: The 10 assignments included 12 graph problems
and12slopeproblems.The12problemsofeachkindwereeithergrouped Materials
intoasingleassignment(blockedpractice)ordistributedacrossmultiple
assignments(interleavedpractice).Thedarksquaresindicatethelocation Students received graph problems and slope problems, and no
of the graph problems. The location of the slope problems is given in studentsawthesameproblemmorethanonceduringtheexperi-
AppendixA. ment.Graphproblemsrequiredstudentstographalinearequation
INTERLEAVEDPRACTICE 5
oftheform,y(cid:2)mx(cid:3)b,wheremandbwerenonzerosingle-digit remainingeightproblemsimmediately,whichistosaythatall12
integers. Examples include y (cid:2) 2x (cid:6) 1, y (cid:2) (cid:6)x (cid:3) 3, and problemsofaparticularkind(graphorslope)appearedinthesame
y(cid:2)(cid:6)3x(cid:3)2.Eachproblemincludedtheinstruction“Graphthe assignment. With interleaved practice, the remaining eight prob-
equation”andwasaccompaniedbyaCartesiangrid.Studentswere lemsweredistributedacrosssubsequentassignments(Figure3and
permittedtouseanyappropriatemethod,buttheyweretaughtto AppendixA).
find points by substituting at least two x values into the equation Studentsreceivedthe10assignmentsonDays1,6,14,32–33,
andfindthecorrespondingyvalues.Forexample,fortheequation 33 or 35, 35 or 38, 45–46, 72–75, 81–82, and 86–88. Students
y (cid:2) 2x (cid:6) 1, the substitution of x (cid:2) 2 yields y (cid:2) 3. For slope were asked to complete each assignment before the following
problems,studentsfoundtheslopeofthelinepassingthroughtwo school day, and the final practice assignment was collected by
givenpointsontheCartesianplane.Eachproblembeganwiththe teachers on Days 87––89. On the due date for each assignment,
instruction, “Find the slope of the line that passes through the teacherspresentedthesolutiontoeveryproblemwiththeaidofa
points”followedbyapairofpointssuchas“(1,5)and(8,9)”or slide show created by the authors. As teachers presented the
“(3, 1) and (9, –4).” Students were taught to find the slope by solutions,studentswereaskedtocorrecttheirerrors.Teachersthen
y. calculating(cid:7)y/(cid:7)x,whichisknowncolloquiallyas“riseoverrun.” collectedtheassignments.Withinthreeschooldays,oneormore
shers.broadl Fsloorpeex4a/m7.pIlne,evtheerylisnleoppeaspsrionbgletmhr,otuhgehtwpooingtisve(n1,p5o)inatnsdha(8d,in9t)ehgaesr awuitthhoorustvmisaikteindgthaenyscmhaoroklsaonndtshceoraesdsigenacmhensttus.deTnhte’saasssisgignnmmeenntst
publinated coordinates,andtheslopeequaledanonzerofractionbetween–1 werethenreturnedtotheteachers.
dmi and1. Students’ scores on the practice assignments do not provide a
salliedisse vtiaolnids bmeefaosrueregiovfinlgeatrhneiinrgabsseicganumseensttsudteonttsheciorrtreeacctehdertsh.eEirvesonlui-f
ite Design
ofob teachers had collected the assignments at the beginning of class,
oneott Wemanipulatedpracticeschedule(interleavedorblocked)and thestudentsmighthavereceivedhelpfromtheirparentsorother
orsn testdelay(1or30days).Testdelaywasmanipulatedbyrandomly students. This ambiguity is typical of students’ mathematics as-
i
ationand a3s0s-idganyindgeleaayc(hns(cid:2)tud6e3ntfowrietahcinhgeraocuhp)c.laTshsistomeeiatnhterthtahteea1c-hdacylaossr spirgancmticeentass,saingdnmmeanntys.teachersencouragestudentstoseekhelpwith
cier
sous includedstudentsatbothtestdelays. Yetthescoringofthepracticeassignmentsprovidedanobjec-
s
Aal Practice schedule was a counterbalanced within-subject vari- tive measure of the fidelity of the intervention (which consisted
hologicaleindividu aprebrcoleebi.lveSemtdusdtheaenntdrsevbinelorsGcekr.eoGduprporu1apcrte1icc(eenivo(cid:2)efds5li9on)pteeirnlceplaruovdbeedldempfrosau,craticncldeasoGsfersogu(rtapwp2oh stahosesleiglsynchmoofeontlhtseretavosesatilhgeendimrtehsntauttsd)ee.anMcthso,staetnaimdchpseotrurtddaeinsntttr,sitb’huestseeeldf-scecoaocrrrhiencgotefvditshisetosl1uto0-
ch
Psyoft taughtbyTeacherA,andtwotaughtbyTeacherB).Group2(n(cid:2) tionsfurtherdemonstratedthattheteacherspresentedthesolutions
ne 67) included five classes (two by A, two by B, and one by C). tothepracticeassignments.(Ofcourse,thisperfectrateofteacher
as
cu
erial Classes were assigned to groups as follows. Two of the classes compliance might have been achieved because we collected the
Amson were designated as “honors/gifted” by the school, and these two assignments.) The scoring of the assignments also provided a
eer classeswererandomlyassignedtodifferentgroups.Theremaining rough measure of student compliance. When the graph or slope
hp
ythe seven classes were deemed by the school as being at the same problems were blocked, students averaged 81% correct. In the
bt
yrightedolelyfor ltpehaverettilwc,ioapnagdtrioneugapccshlawosisfththhtaehdseeacncolneasqstsureaasilnwntutahmsabtraetenradocofhmecrllysaswasseitsshigimnneoedraectohthaognnroeounopef. it(nhwteheyrilcaehvavewreaedgreperdtahc8et2ic%oenclcyoonrpdrreioctibtolnefom,rssttuhtdheeanltatsswtaeevrieegrhaptgaerotdfo8tfh4e%an1c2ionrptrereroclbet,laevamnedds
ps
coed The two groups scored similarly well on a test consisting of six assignment, as shown in Figure 3). Thus, by this measure, the
isnd multiple-choice problems from the Grade-8 mathematics portion interventionandthecounterfactualproducednearlyequalratesof
entnte of the National Assessment of Educational Progress, or NAEP studentcompliance.
mi
docucleis (vNs.a7ti9o%nal(SCDen(cid:2)ter2f2o%rE),dtu(1c0at8i)on(cid:2)S0ta.6ti2st,ipcs(cid:2),20.5143,),C7o6h%en(’SsDd(cid:2)(cid:2)202.1%2). proObnlemDasycsre6a6te–d6b9y,tehveesrychsotoulddeinsttrriecct,eaivneddthaislarergveiewsetasosfigrnemvieenwt
hisarti wasdesignedtopreparestudentsforastandardizedsemesterexam
Ts
Thi Procedure given to all students in the district. (The exam is taken on a
computerwithoutthepresenceofateacher,andtheteachersdonot
The study consisted of 10 practice assignments, a review ses- seethetestitemsinadvance.)Thereviewassignmentincludedtwo
sion,andatest.Eachpracticeassignmentconsistedof12problems problems on the graphing of a line and four problems on slope,
presentedontwosidesofasinglesheetofpaper.The10assign- though the problems were stated differently than the ones in the
mentsincluded12graphproblemsand12slopeproblems,andthe study.
remainingproblemsweredrawnfromunrelatedtopics(fractions, Beforethecompletionofthepracticephase,teacherscompleted
proportions,percentages,statistics,andprobability).Teacherspre- an anonymous survey about their views on interleaved practice.
sentedatutorialonthegraphproblemsimmediatelybeforegiving Eachofthethreeteachersreceivedapapercopyofthesurveyon
Assignment 1, which included the first four graph problems, and Day 66. We visited the school on Day 75 and collected an
theypresentedatutorialontheslopeproblemsimmediatelybefore envelope with their completed surveys. The survey is shown in
givingAssignment2,whichincludedthefirstfourslopeproblems. AppendixB.
However, the scheduling of the remaining eight graph and eight EverystudentreceivedthesamereviewonDay93,about5days
slope problems varied. With blocked practice, students saw the afterteacherscollectedthelastpracticeassignment(Days87–89,
6 ROHRER,DEDRICK,ANDSTERSHIC
dependingontheclass).Thereviewincludedonegraphproblem, Interleaved
one slope problem, and eight unrelated problems. The graph and
Blocked
slopeproblemswerethesixthandseventhproblem,respectively.
Students received the review assignment at the beginning of the
classperiod,andteacherspresentedthesolutionsattheendofthe e
r 80%
same class period. Teachers then collected the assignment from co 74%
S
students,andoneoftheauthorscollectedtheassignmentsfromthe
teachers at the end of that school day. The mean scores on the st 64%
e
review problems were slightly higher, but not statistically so, for T

problems learned by interleaved practice rather than by blocked an 42%
practice(94%vs.89%).Again,wepresentthesedataasameasure Me
ofcomplianceratherthanameasureoflearningbecause,aswith
the practice assignments, students were given an opportunity to
y. correcttheirsolutionsbeforeturningintheirassignments.
publishers.natedbroadl woneSeretautudeteshntoetrdspdwruoercrientogtreetshdteetidhrer1eagdoumrlai3rn0cisltadrsaasytismoneaefotteifnregta,hcaehndrteetvshti.eewFteo.arScehtuaecdrheannotdsf 1 day Test Delay 30 days
itsalliededissemi twshiseetrietnwgnooottfesssctihxdeadmtuealstehd(1ptoaronrbdelce3em0ivsdeauytnhsreealaftetteesrtdtrhteoecetriheveveideewxap)e,firsliltmeurdeentent,ststalcwlohnoo-f Fdcoeigllaouyrrsev.e4Er.srirooRnrebosafurltsthsir:seIpfnirgteeusrerlene.atv1insgtapnrdoadrudceerdrogrr.eSateeerttheestosncloinreesaarttibcloethfotresat
ofob which were drawn from the Grade-8 mathematics section of a
oneott recentNAEP(NationalCenterforEducationStatistics,2013).We
orsn askedteachersnottoinformstudentsofthetestsinadvance,and sons.Eachdifferencewasassessedbyanindependentttest,andin
i
ciationerand tceoamchpelersteddi.dTnhoettseeset abnoyoktleesttipnrcolbuldeemdsaunctoivletrhesheexepte,raimsehnetetwoasf trhejreeceteodf.Wtheetfhoeurrefcoarseeds,idthneotaussseumthpetipoonoloefdeesqtuimalatvearfioarntcheesewrroasr
sous paperwiththreegraphproblems,andasheetofpaperwiththree term for the t-statistic, and we adjusted the degrees of freedom
s
ologicalAindividual stphlroeopbeplerpmorboslbeplmeremscsew.dWeitdheitnhcereeapactaehgdepswiaxgitevh,etrahsnieodngsrtahopefhtphpaergoetbelsewtmibtsyhirntehotehrdrseeleroinpogef ua1fs-fdienacgytttthehseetodWuetlecaloycmh,–einSotaeftrtaleenratyhvweodfatirhtaeethmfeoreuttrhhnoaudnl.lbThloyhcpiksoetcdhoeprsrriesaccttteiiosctnes.rdFeisodurlnttheodet
chhe the six versions. None of the problems had appeared in either a in higher mean test scores for the graph problems, 89% (SD (cid:2)
yt
Psof practiceassignmentorthereview.Studentswereallotted18minto 18%)vs.73%(SD(cid:2)36%),t(50.6)(cid:2)2.25,p(cid:2).029,d(cid:2)0.54,
anse complete the test and allowed to use their school-supplied basic but the effect of interleaved practice on the slope problems was
cu
erial calculator.Eachtestwasscoredattheschoolonthedayofthetest positivebutnotstatisticallysignificant,73%(SD(cid:2)41%)vs.54%
Amson by two raters who were blind to condition. The two raters inde- (SD(cid:2)47%),t(56.0)(cid:2)1.67,p(cid:2).10,d(cid:2)0.43.Forthetestgiven
heper pendentlyscoredeachanswerascorrectornotandlaterresolved aftera30-daydelay,therewasastatisticallysignificantbenefitof
bytthe the few discrepancies (six in 756). Internal consistency reliabili- interleaving for both graph problems 84% (SD (cid:2) 30%) vs. 54%
yrightedolelyfor tsileosp,eaasnmdegarsauprheidngbyprCobrolenmbasc,hr’esspaelpcthiav,elwy.ere .89 and .77 for the p(pSr(cid:2)oDbl.(cid:2)e0m014s,,46%d5)(cid:2)%, t0((S5.8D67.7.(cid:2)) (cid:2)43%3.)2v8s,.p29(cid:2)%.(0S0D2,(cid:2)d3(cid:2)9%0).,8t1(6,1a)n(cid:2)d s3l.o4p4e,
ps
coed Results The results of the teacher survey are shown in Appendix B.
sd
mentiinten Interleaved practice produced higher test scores than did Atoltghiovueghthteeiarchhoenrsesrtesapsosnesdsemdeanntso,nythmeosuasmlypalendsiwzeeroefeonncolyurtahgreede
Thisdocusarticleis bs6hl4oo%cwkee(ddSDaprma(cid:2)cotdic4eer2a(%tFe)i,gbuetr(ne6e2f4i))t.o(cid:2)Sftuin2dt.e3enr9tl,seatpvesint(cid:2)egd,.18002d%,ayd(SafD(cid:2)ter(cid:2)0th.43e23r,%ev9)i5ev%ws. ttfeeuaatucchhreeerrisfinimtdweicaaanstseadnthtoahtpatttihohenes”eoardnsadhtae“wawroeouulsldpder“ecucuoslaemtitmvheeeniandttebthreveseti.nntStieotrinvlle,innetaitohcnhe
Thi confidenceinterval(CI)[0.07,0.77].Studentstested30daysafter to other math teachers.” All three also reported that interleaved
the review showed a large benefit of interleaving, 74% (SD (cid:2) practicedidnot“interferewithhow[they]usuallyteach”andthat
39%)vs.42%(SD(cid:2)43%),t(62)(cid:2)4.54,p(cid:8).001,Cohen’sd(cid:2) “going over assignments” was no harder when assignments were
0.79, 95% CI [0.43, 1.15]. A two-way analysis of variance (with interleavedratherthanblocked.However,theiropinionsweresplit
practice schedule as a within-subject variable and test delay as a when they were asked whether their students found interleaved
between-subjects variable) showed that interleaved practice was practice to be less likable, more challenging, or more time-
superior to blocked practice, F(1, 124) (cid:2) 24.43, p (cid:8) .001, (cid:5)p2 (cid:2) consumingthanblockedpractice.
.165,andthattestscoresweregreaterattheshortertestdelay,F(1,
124)(cid:2)7.69,p(cid:8).01,(cid:5)2(cid:2).058.However,theinteractionbetween
p Discussion
practice schedule and test delay was not statistically significant,
F(1,124)(cid:2)2.84,p(cid:2).09. Thepresentstudyexplicitlycomparedinterleavedandblocked
Secondarily, we analyzed the effect of interleaved practice on mathematics practice in a classroom setting and found that inter-
testscoresforthegraphandslopeproblemsseparatelyforboththe leavedpracticeproducedsuperiorscoresonafinaltestgiven1or
1-dayand30-daytestdelays,resultinginfourdifferentcompari- 30dayslater.Putanotherway,themererearrangementofpractice
INTERLEAVEDPRACTICE 7
problems improved mathematics learning in the classroom. The suggest that Saxon texts produce higher test scores than do non-
study is also the first to demonstrate that the test benefit of Saxon texts. This is because Saxon texts differ from non-Saxon
interleavingdoesnotdiminishovertimeandperhapsgrowslarger. texts in a number of ways other than the use of interleaved
Finally,apartfromitssuperioritytoblockedpractice,interleaved practice,andoneormoreoftheseotheruniquefeaturesmaywork
practiceprovidednearimmunityagainstforgetting,asthe30-fold againsttheefficacyofaSaxontext.InSaxontexts,forexample,
increaseintestdelayreducedtestscoresbylessthanatenth(from even the lessons are intermixed so that lessons on related topics
80%to74%). (e.g.,prism,cylinder,pyramid,andcone)donotappearconsecu-
Although the size of the effects might seem surprisingly large tively—a feature that is not supported by the present study. Inci-
forastudyconductedinaclassroom,theeffectsizesobservedhere dentally, Saxon texts have been criticized by some educators
areneverthelessmuchsmallerthaninterleavingeffectsobservedin becauseoftheirpurportedemphasisonthemasteryofprocedures
the laboratory. Whereas the effects in the present study were attheexpenseofconceptualunderstanding(e.g.,Kamii&Domin-
medium to large (ds (cid:2) 0.42 and 0.79), laboratory studies of ick, 1998). However, we emphasize that this criticism of Saxon
interleaving have uniformly found larger effects (d (cid:2) 1.34, d (cid:2) textsisorthogonaltotheefficacyofinterleavedpractice.Noneof
y. 1.21,and(cid:5)2(cid:2).32;seeintroduction).Inbrief,thepresentfindings theauthorsofthepresentstudyhaveeverhadanyaffiliationwith
shers.broadl avreentaionninlsotsapenstiastoiomne—onfotitsa vefiofilcaaticoyn—whoefnthiet iasdamgeovtehdatfarnominttehre- SaxInotnertleexavtbeodomksa.thematicspracticeappearstobeafeasibleinter-
publinated laboratorytotheclassroom(e.g.,Hulleman&Cordray,2009). vention. Creators of mathematics textbooks or instructional soft-
dmi Another reason for the large effects of interleaving observed ware need only rearrange practice problems before releasing the
salliedisse henertleyangduaerlasnetweehserethiastthstautdienntetsrlesapvaecdemthaethirempraaticctsicper.acTthicaet iinsh,eirn- nimexptleemdietniotant,iownitwhoouutldalatelsriongsetehmeptororbelqeumirseorrellaetsisvoenlys.lCitltalesseroffoomrt
ite
ofob additiontothejuxtapositionofdifferentkindsofproblemswithin fromteachersbecausetheyneednotaltertheirclassroomlessons
oneott an assignment, problems of the same kind are spaced across or the manner in which they solve a particular problem. The
orsn assignments. However, the review session in the present study intervention would also presumably require little or no teacher
i
ationand maltehaonutgthhattoeavelnestsheerbdleogcrkeeedthparnactthicaet cporonvdiidtieodnbpyrotvhiedeindtesrplaecaivnegd, tdrearisntianngd, tahlteholouggihcutenadcehrelyribnugyi-nitnermleiagvhetdrepqraucirtieceth(antatmeaeclyh,ertshautni-t
cier
sous practice condition (Figure 3). In brief, the large effect observed requires students to choose a strategy and not merely repeat the
s
Aal here probably reflects the spacing effect, which is an inherent strategyusedinthepreviouspracticeproblem).Thefeasibilityof
ologicalindividu bspeanceifnitgomfiingthetrlheaavveedbemeanthreemduacteicdsbpyratchtieceu,sbeuotfthaerceovnietrwibsuetisosnioonf. aitn, yinetterlvitetlnetiiosnkanlosowdneapbeonudtstohnewlihkeatbhielirtystuodfeinnttsearlnedavteeadchperarsctliickee.
he Thelargeeffectsnotwithstanding,thepresentstudyhaslimita- The few teachers in the present study indicated that they liked
ch
yt
Psof tions.Forinstance,althoughthetestproblemswerenovel,thetest interleaved practice, even before the study was complete (i.e.,
ne problems and practice problems had the same format, and the beforetheylearnedoftheintervention’sefficacy),buttheirviews
as
cu
erial observedeffectsmighthavebeensmallerifthetestproblemshad might have been biased because of their close involvement with
Amson required a greater degree of transfer. Also, the test benefit of theproject.Also,thepresentstudydidnotincludeasurveyofthe
eer interleaving might have been reduced if the review had included students.Futureresearchisneededtobettergaugebothstudents’
hp
ythe more than one problem of each kind (graph problem and slope andteachers’perceptionsofinterleavedpractice.
bt
yrightedolelyfor phinratoveberlleebmaevn)ee,dfsitiptmeradpclttyhicebeebcclooancukdseeitdiaopnmr.aoMcrteiocreienctbeornnosdaiivdteiloyrn,eimvtiroeewrmeasthienasssniuoitnndkmindoigwthhnet pproaAtcetpniatcireatlfmbrorigemahdtittshbeecfofuinscetarfciubyluataentsdaaflselamlseivubecillhistytoo,finthmteearltiehmaepvmaeacdttimcosaf,thaanenmdinatttheicirss-
ps
coed whethertheinterleavingeffectsobservedherewouldbefoundin ventionasdoesitsefficacy.Indeed,benefitsofinterleavedpractice
isnd a study with a wider variety of material and a greater number of have been consistently observed with a variety of mathematics
entnte teachers and students. Still, the ecological validity of the present skillsandwithstudentsinelementaryschool,middleschool,and
mi
docucleis sthtuedlyeawrnaisngrepahsaosneablalystegdoo3dm.oSntuthdse,natsndletahrene1d-mfroonmthttheestirdteelaacyhwerass, cporallcetgicee. Aprsovaridgeusedsthuedreen,tsthewsiethbeannefoitpspoarritsuenibtyectaousleeairnntehrloewavetod
hisarti educationallymeaningful. chooseanappropriatestrategy(orlearnthattheycannotdoit).In
Ts
hi Wealsoshouldemphasizethatthefindingsreportedheredonot short, interleaved practice simply provides students with an op-
T
suggestthatblockedpracticebeavoidedentirely.Infact,asmall portunitytopracticetheveryskilltheyareexpectedtolearn.
blockofproblemsmightbeoptimal,especiallyattheoutsetofan
assignmentgivenimmediatelyafterstudentsareintroducedtothat References
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(Appendicesfollow)
INTERLEAVEDPRACTICE 9
Appendix A
Serial Position of Each Graph and Slope Problem in the Assignments
Practiceassignment
Students/Problem
type 1 2 3 4 5 6 7 8 9 10 Review
Group1
Graph 1–4 — 1 7 4 6 10 1 10 10 6
Slope — 1–12 — — — — — — — — 7
Group2
Graph 1–12 — — — — — — — — — 6
Slope — 1–4 1 7 4 6 10 1 10 10 7
y. Note. Forexample,forstudentsinGroup1,Assignment1includedfourgraphproblemsatthebeginningoftheassignment(Nos.1–4),andAssignment
s.adl 4includedonegraphproblemnearthemiddleoftheassignment(No.7).Thetwogroupsreceivedidenticalreviewassignments,andthegraphandslope
herbro problemswerethesixthandseventhproblems,respectively.Nostudentsawthesameproblemmorethanonceduringthestudy(includingpractice,review,
blised andtest).
punat
dmi
salliedisse Appendix B
ite
eoftob Frequency of Responses of Three Teachers to Statements About Interleaved Practice
onot
n
ors
i
ciationerand Statement Sdtirsoanggreley Disagree Nneoirthdeirsaaggrreeee Agree Stargornegely
os
su
Asal 1.Goingoverassignmentswasharderformewhentheassignmentswere
aldu interleavedratherthanblocked.(Neg) 2 1
ologicindivi 2.Ginotienrgleoavveerdarsastihgenrmtheanntsbtlooockkemdo.r(eNteigm)ewhentheassignmentswere 1 1 1
he 3.UsinginterleavedassignmentsinterferedwithhowIusuallyteach.(Neg) 2 1
ch
nPsyeoft 4.Stthuadnenbtlsocthkoeudgahststihgantmienntetsrl.e(aNveegd)assignmentsweremorechallenging 1 1 1
caus 5.Studentsfoundthatinterleavedpracticetooklongerthanblocked
erial practice.(Neg) 1 1 1
mn
Aso 6.Studentslikedinterleavedpracticelessthanblockedpractice.(Neg) 2 1
eer 7.Interleavedassignmentsimprovedmystudents’learningmorethandid
hp
ythe blockedpractice. 3
bt 8.Iwouldrecommendinterleavedpracticetoothermathteachers. 3
yrightedolelyfor 190..IIawwchooiuuellvddinuugsseestiiunndtteeerrnllteesaa.vveeddpprraaccttiicceeirnaththeertfhuatunrebliofciktewdapsraacntiocpetiwoint.hlower- 1 1 21 1
ps
coed Note. Statementsregardinginterleavedpracticewereframednegatively(Neg)inStatements1–6.
sd
in
mentinte ReceivedMay13,2014
hisdocuarticleis RevisionArcecceepivteeddJJuullyy1111,,22001144 (cid:2)
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