导航数学战争：数学教学分歧与争论实践指南

m(x

α+β+γπ

e^iπ+1=0

22=4

C=2πr

σ²=Σ(x−μ)²/n

det(A)=ad−bc

π

√(x²+y²)

y=ax²+bx+c□2φ=0

θ=arcta

NAVIGATINGTHEMATH

WARS:APRACTICALGUIDE

∫

TOTHEDIVIDESANDcos²θ+sin²θ=1=πr²DEBATESINFLUENCINGp+q=1

MATHINSTRUCTION

APRIL2026

i2−1

½+⅓=5/6

Σfᵢ=N

Σxᵢ/n

P(X≥k)=1−F(k−1)

d/dx[xⁿ]=n·xⁿ□1

λ·v=A·v

1/n²=π²/6

tanh(x)=(eˣ-e□ˣ)/(eˣ+e□ˣ)

CENTERONREINVENTINGPUBLICEDUCATION

2

“TheMathWars:TimedTests,MathAnxiety,andtheBattleOverHowWe

TeachOurKids”

TheSaturdayEveningPost,May28,2024

“HowADebateOverTheScienceOfMathCouldReignitetheMathWars”

TheHechingerReport,May8,2023

“Are‘MathWars’ReallytheSameAs‘ReadingWars?’”

Forbes,March15,2023

“HowMathBecameanObjectoftheCultureWars”

TheNewYorker,November14,2022

“HowWeCanFinallyEndthe‘MathWars’”

EducationWeek,August22,2022

“TheFaultyLogicofthe‘MathWars’”

TheNewYorkTimes,June16,2013

Theseheadlines,andsimilaronesgoingbacktothe1990swhentheterm

“MathWars”firstenteredthenationallexicon,reflectdecadesoffierce,

unresolveddebateoverhowmathematicsshouldbetaughtandlearned.Whatstartedasacontentiouspublicdebatebetweenadvocatesforreform-orientedandtraditionalapproachestomatheducationhascontinuedtocapturethe

media’sattentiontothisday.Overtime,thetermhascometoincludeabroadrangeofdebatesaboutwhatmathematicsstudentsshouldlearn,howit

shouldbetaught,and,morerecently,whoseexpertiseandresearchshouldguidethosedecisions.

Theconflicthasitsrootsintheearlytwentiethcentury,butitsmodern,

publicformwasshapedinthedecadesthatfollowed.In2010,theCommon

CoreStateStandardssoughttobrokerpeacebyestablishingasharedK-12

frameworkdesignedtoaddresslongstandingdivides,suchasthetension

betweenpracticingproceduralfluencyandbuildingdeeperconceptual

understanding.Instead,itreignitedoldtensions.Althoughpoliticalbacklash

hassincepushedtheCommonCoreoutofthespotlight,atleastinname,new

CENTERONREINVENTINGPUBLICEDUCATION

3

frontshaveopened.Particularmathcurriculahavedrawn

intensescrutiny

,

andamovementcallingitselfthe

ScienceofMath

hasemerged,arguingthatinstructionaldecisionsshouldbegroundedincognitivescienceandempiricalresearch,agoalsharedbymanyonboththetraditionalandreformsidesof

theMathWars.

Giventhestakes,itisnotsurprisingthattheScienceofMathhasstirred

suchintensedebate.MathachievementintheUnitedStateshas

declined

sinceits2013peak,1aslidethepandemiconlyaccelerated,andmany

district

leaders

arelookingforwaystoimproveit.Policymakers,meanwhile,are

watchingclosely

afterwitnessingwhathappenedinreadinginstruction,

whereamovementcalledtheScienceofReadingsuccessfullychallenged

decadesofpracticebuiltaroundLucyCalkins’“balancedliteracy”approachthrough

legislativemandates

,

parentlawsuits

,andareturntophonics-basedinstruction.Manyareaskingwhethermathisheadedforasimilarreckoning.

Buttheparallelhaslimits.UnliketheScienceofReading,whicheventually

achievedsomethingclosetoaprofessionalconsensus,theScienceofMath

hasfaced

strongcriticism

fromthefield’smostprominentmathorganizations,whoarguethatthemovementmisappliesresearchandpromotestoo

narrowavisionofwhatmathteachingcouldbe.WhethertheScienceof

Mathrepresentsalastingreorientationorjustanotherfrontinanunresolvedwar,themovement’sstancesandsurroundingcontroversyareworth

understandingontheirownterms.

ThedebatesgroupedundertheMathWarscoverawiderangeofissues:

discoverylearningversusdirectinstruction,whethertoacceleratestudents

withadvancedcontent,howtrackingaffectsequity,andthelinkbetween

timedpracticeandmathanxiety,tonameafew.Toooften,thesedisputes

leaveschoolleaders,teachers,andparentsnavigatingaconfusingmixof

termsandcompetingrecommendations.2Someofthesefaultlines,when

examinedmoreclosely,arelesscleanbinariesthantheyfirstappear.Butthatdoesnotmakethemtrivial.Theystillinvolveconsequentialdisagreements

aboutinstructionalsequence,defaultemphasis,andthekindsofevidencethatshouldcarrythemostweightinpolicyandpractice.

Thisguideisdesignedtohelpreadersnavigatethatterrain.Ittracesthe

1.NAEPprovidesthemostreliableavailablemeasureoflong-term,nationaltrendsinmathematics

achievementandremainsthestandardreferencefortrackingperformanceovertime.However,severalcautionsapplywheninterpretingscorechanges.BecauseNAEPusesmatrixsamplinganddoesnot

produceindividualstudentscores,itisnotdesignedtoevaluatetheeffectivenessofspecificinstruc-tionalapproachesattheclassroomordistrictlevel.Additionally,scorechangescanpartlyreflect

measurementartifactsratherthanshiftsinactualstudentlearning(e.g.,whennationalassessment

frameworksarenotyetalignedtorecentlyrevisedstatecurricula).Thepost-2013declinereferencedhereshouldthereforebeunderstoodasabroadpopulation-levelsignalwarrantingattention,notasaprecisediagnosticofinstructionalfailure.

2.SeeAppendixAforglossaryofkeyterms.

CENTERONREINVENTINGPUBLICEDUCATION

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originsofthesedivides,theirevolution,andwhattheylookliketoday.Foreachmajordivide,presentedasadichotomy,weidentifythetraditionalandreformendsofthespectrum,summarizeeachside’smainargumentsandevidence,

andnotewheretheScienceofMathmovementfallsalongthatcontinuum.

TheScienceofMathreceivesparticularattentionbecauseithasbecome

asignificantflashpointinrecentyears,shapingstatepolicy,curriculum

conversations,professionallearning,andpublicdebate.Thatfocusshouldnotbemistakenforanendorsement.Ratherthanadjudicatingthefullbodiesof

researchunderlyingeachdichotomy,theguideaimstogivestateanddistrictleadersaclearermapoftheterrainsotheycanevaluatethecompetingclaimsalreadyreachingtheirschools,determinewhichmattermostfortheircontext,andmakemoreinformeddecisions.Wheretheguideassessesspecificclaims,particularlythoseabouttheScienceofMathmovement,itdoessobasedon

themovement’sownpublishedpositions.

Tothatend,wealsoexaminehowtheMathWarsareplayingoutinpractice,exploringpoliciesenactedsincethepandemicandreviewingwhatweknowaboutmathteachers’commonlyusedpractices.Theguideconcludeswith

suggestednextstepsatboththelocalandnationallevels,withappendicesprovidingaglossaryofkeytermsandadditionaldetailforreaderswhowanttogodeeper.

CENTERONREINVENTINGPUBLICEDUCATION

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THE“OLD”MATHWARS:TWOROADSDIVERGED

Tounderstandwhymatheducationhasbeensofiercelydebated,ithelpstolookbacktotheearlytwentiethcentury,whenJohnDewey,oneofAmerica’smostinfluentialeducationreformers,describedtwoopposing“sects”in

educationwithstarklydifferentviewsonhowchildrenlearnbest.While

Deweyrejectedtheextremesofbothsides,herecognizedakeyquestionthatledtotwodivergingroads:shouldeducationbeginwiththeaccumulated

knowledgeandexperienceoftheteacher,organizedintoaformalcurriculum,orwiththeneeds,interests,andabilitiesofthestudentasthefoundationforlearning?3

Deweyherebyidentifiedoneofeducationreform’searliestfaultlines:whethercurriculum-centricorstudent-centricapproachestoteachingandlearning

shouldtakeprecedence.Thispedagogicaldivide,hewarned,shouldnotbetreatedasawarbetweenantagonists,yetthatispreciselywhathappened

overthedecades,ashisnuancedvisionhardenedintopolarizedpositionsoneducationingeneralandmathematicsinparticular.Dewey’sownviewwas

moreintegrative.Thestudentandthesubjectmatter,heargued,serveastwonecessaryanchorsfortheteacher,whousestheorganizedcurriculumasa

guidetounderstandstudents’currentcapacitiesandshapetheenvironmenttodirecttheirongoinggrowth.4Thisdividecanstillsurfacetodaywheneverdistrictsdebatewhethertoadoptacurriculumthatleadswithteacher

modelingorwithstudentexploration.ThetablebelowdrawsonDewey’s

originalessaytoshowhowthesubsequentemergenceof“traditionalist”and“reformer”stancespulledthesetwoanchorsapart,producingtheopposingpositionsonmatheducationthatcontinuetodefinetheMathWarstoday.

3.Dewey,“TheChildandtheCurriculum,”7-10.

4.Ibid.,11,30-31.

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Table1.TwoSides,TwoApproaches:APedagogicalDividethatShapedEducation

In1902,JohnDeweyidentifiedakeydividebetweentwogroupsineducationthat,over

time,articulatedincreasinglypolarizedapproachestoteachingandlearning.

“One[group]fixesitsattentionuponthe

importanceofthesubject-matterofthe

curriculum…Letthe[student]proceedstep

bysteptomastereachoneoftheseseparate

parts…The[student]issimplytheimmature

beingwhoistobematured.”5

“Notso,saystheother[group].The

[student]isthestarting-point,thecenter,

andtheend…Learningisactive…Itis[the

student]andnotthesubject-matterwhich

determinesbothqualityandquantityof

learning.”6

“Traditionalists”

“Reformers”

Historical

Polarization

MathWars

Polarization

Historical

Polarization

MathWars

Polarization

•Teacher-ledinstruction

•Proceduralfluencyoverconceptual

understanding

•Student-centeredpedagogy

•Conceptual

understandingoverproceduraldrills

•Masteryof

establishedsubjectmatter

•Explicitinstructionoverdiscovery

learning

•Curriculumbuiltfromthelearner’sinterestsand

experiences

•Inquiry-based

learningoverdirectinstruction

•Viewofthestudentasneedingto

acquireorganized

knowledgefromtheoutsidein

•Accelerationandearlyintroductionofalgebra

•Emphasison

active,hands-on,exploratorylearning

•Integrationofmathintoreal-world

contexts

•Emphasison

systematic

instructionand

sequentialcoverage

•Standardizedtestingas

accountability

•Teacheras

facilitatorratherthandirect

instructor

•Flexiblepacingandde-emphasison

timedtesting

Withthisdivideinmind,itbecomeseasiertorecognizehowtheseearly20th-centuryrootsstillshapetoday’sMathWars.Traditionalistsvalueasequencedteacher-ledprogressionofskillsandproceduresthatallstudentsmustmaster,whilereformersemphasizestudent-centeredproblem-solving,exploration,

andtheapplicationofmathematicstoreal-worldproblems.Overthedecades,successivewavesofreformandbacklashhavehardenedthesepositionsintoentrenchedcamps.Theresultisacentury-longchainofevents,witheach

shiftininfluencepavingthewayforthenext,culminatingintoday’sdebatesovertheScienceofMathandbeyond.Next,wetracekeymomentsacrossthedecadestoillustratehowthatchainunfolded.

5.Ibid.,7-8.

6.Ibid.,9.

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1950s-1970s:NewMathandColdWarCompetition

The1957launchofSputnikbytheSovietUnionjoltedU.S.policymakers

andeducatorsintoaction,fuelingfearsoffallingbehindinscienceand

technology.Inresponse,thefederallyfunded“NewMath”movement,led

byuniversitymathematiciansandcurriculumdevelopers,soughttoelevate

mathematicalrigorbyintroducingconceptslikesettheory,numberbases,

andsymboliclogicintoK-12classrooms.7Whileintendedtobuilddeep

conceptualunderstanding,theapproachwasoftenoverlyabstractandpoorlyalignedwithteacherpreparation.8Manyparentsandeducatorsalsosawitasdisconnectedfromthepracticalskillsstudentsneeded.Bytheearly1970s,

widespreadfrustrationledtoa“BacktoBasics”backlash,restoringemphasisoncomputationandfoundationalskills.9

1980s-1990s:TheRiseofReformMathandtheFirstMathWars

In1983,thefederalreportANationatRiskwarnedofa“risingtideof

mediocrity”inAmericaneducation,sparkingnationwideconcernabout

academicstandardsandeconomiccompetitiveness.Thereportcreateda

policyclimateripeforreform.In1989,theNationalCouncilofTeachersof

Mathematics(NCTM)releaseditsCurriculumandEvaluationStandards,a

landmarkagendathatemphasizedproblemsolving,mathematicalreasoning,andreal-worldapplicationwhilede-emphasizingrotememorizationand

repetitivecomputation.10Thesestandardsshapedtextbooks,curricula,

andstateassessmentsthroughoutthe1990s,leadingtothewidespread

adoptionofprogramssuchasEverydayMathematics,TERCInvestigations,

andMathLand.11However,somecriticsarguedthatthesematerialssacrificedprecision,minimizedbasicskills,andleftstudentsunderpreparedfor

advancedmath.AdvocacygroupslikeMathematicallyCorrectandNYC

HOLDemergedtoopposeNCTM-stylereforms,markingtheperiodwhentheterm”MathWars”firsttookholdinpublicdiscourse.12

2000s:SearchingforCommonGroundThroughEvidence

Inanattempttoresolvethesedisputes,thefederalgovernmentconvened

theNationalMathematicsAdvisoryPanel(NMAP)in2006.Comprisedof24expertmembers,thepanelreviewedover16,000researchpublicationsand

policyreportstoissuerecommendationsbasedonthebestavailablescientificevidenceforimprovingmathematicsinstructionandstudentperformance.Its

7.Loveless,“ATaleofTwoMathReforms,”184-209.

8.Askey,“GoodIntentionsAreNotEnough,”163-183.

9.Burrill,“MathematicsEducation,”25-41.

10.Ibid.,28.

11.Askey,“GoodIntentionsAreNotEnough,”170.

12.Loveless,“ATaleofTwoMathReforms,”184,198.

CENTERONREINVENTINGPUBLICEDUCATION

8

2008

report

calledforastreamlined,coherentPreK-8mathematicscurriculumthatemphasizesawell-definedsetofcriticaltopics,includingproficiency

withwholenumbersandfractions,topreparestudentsforearly,authentic

algebrainstruction.13AddressingcoreMathWarsdebates,theNMAPdeemedtheconflictbetweenconceptualunderstandingandproceduralfluency

“misguided,”concludingthatthetwoaremutuallysupportive.14Thepanelalsoexplicitlystatedthathigh-qualityresearchdoesnotsupportexclusiverelianceoneither“teacher-directed”or“student-centered”instruction.15Intheyears

thatfollowed,stakeholdersdrewondifferentpartsofthereporttosupportcompetingnarratives,andthebroaderconflictpersisted.

2010s:CommonCoreandRenewedControversy

The2010releaseoftheCommonCoreStateStandardsforMathematics

(CCSS-M)framedmathematicalrigorasathree-partemphasisonconceptualunderstanding,proceduralskillandfluency,andapplication.16Inmanyplaces,implementationbecameaflashpointthatrevivedfamiliartensions.Some

traditional-leaningcriticsarguedthatCCSS-M-alignedmaterialsdelayed

standardalgorithmsandreliedheavilyonvisualrepresentations,whilereform-orientedvoicesemphasizedconceptualunderstandingandapplicationand

warnedagainstareturnto“drill-and-kill”instruction.17Unfamiliarstrategies

andterminologyfrustratedmanyparents,sparkingpubliccontroversiesthat,amplifiedbysocialmedia,becameentangledwithbroaderpoliticalbacklash.18Inresponse,somestatesrenamedorrevisedtheirstandards,oftenpreservingsubstantialalignmentwithCCSS-M,andmanyunderlyingdisagreements

persisted.19

2020s:TheEmergenceoftheScienceofMath

Catalyzedbystagnantpre-pandemicstudentperformanceandinspiredbytheScienceofReading,theScienceofMath

movement

officiallyformedin

December2020andgainedtractionthroughout2021.20Ledprimarilyby

researchersfromspecialeducationandschoolpsychology,themovement

emphasizesusingempiricalevidence,particularlyfromquantitativeresearch,toguideclassroompracticeandpushesbackagainstwhatitconsiders

13.NationalMathematicsAdvisoryPanel,“FinalReport,”xiii-xvi.

14.Ibid.,xix,26.

15.Ibid.,xxii,45.

16.CommonCoreStateStandardsInitiative,“MathematicsStandards.”

17.GarelickandZimba,“TheProblemofInstructionalTime.”

18.TimeMagazine,“Dad’sRantAboutCommonCore.”

19.AchieveInc.,“StrongStandards,”2-5.

20.Barshay,“ProofPoints”;Codding,Peltier,andCampbell,“IntroducingtheScienceofMath,”6-7.

CENTERONREINVENTINGPUBLICEDUCATION

9

“pseudoscientific”practicesinmathinstruction.21Itspecificallyadvocates

forsystematic,explicitinstruction,regularpracticetoachieveautomaticity,

multi-tieredsystemsofsupport(MTSS),anddata-baseddecision-making.22

Whilealignedwithtraditionalistprioritiesliketeacher-ledinstruction,the

movementmaintainsthatitdoesnotpromoterotememorizationatthe

expenseofmeaning.23DrawingontheNationalResearchCouncil’sfive

intertwinedstrandsof

mathematicalproficiency

andtherecommendationsoftheNMAP,itarguesthatconceptualunderstandingandproceduralfluency

developbidirectionallyandsimultaneously,directlychallengingtheview

thatconceptualunderstandingmustalwaysprecedeproceduralpractice.24

Criticsfromthereformandgeneralmathematicseducationcommunities

haveraised

seriousconcerns

aboutthemovement’scitationalpractices,its

narrowdefinitionofevidence,andthegeneralizabilityofitsclaims,objectionsweexaminelaterinthisguide.25Theemergenceofthismovementhasthus

openedanewfrontintheMathWars:afundamentaldisputeoverwhat

constitutesvalidscientificevidenceandhowitshouldguidemathematicsinstruction.26

21.NCSM,“StrengtheningResearch-informedDecisionMaking,”3-5;Codding,Peltier,andCampbell,“IntroducingtheScienceofMath,”7.

22.Powelletal.,“EssentialComponentsofMathInstruction,”14-24;Codding,Peltier,andCampbell,“IntroducingtheScienceofMath,”10.

23.Barshay,“ProofPoints.”

24.Codding,Peltier,andCampbell,“IntroducingtheScienceofMath,”8;Powell,Hughes,andPeltier,“MythsThatUndermineMathTeaching,”2;Rittle-Johnson,Schneider,andStar,“NotaOne-Way

Street,”587-597.

25.NCSM,“StrengtheningResearch-informedDecisionMaking,”3-7;Cohen,Jones,andGibbons,“TheMissingMiddle?,”3-4.

26.Barshay,“ProofPoints”;Cohen,Jones,andGibbons,“TheMissingMiddle?,”1-3.

CENTERONREINVENTINGPUBLICEDUCATION

THE“NEW”MATHWARS:FIVEDICHOTOMIESANDTHESCIENCEOFMATH

Atthispoint,wecanbegintounderstandwhytheMathWarsseemboth“newandold”andwhythecurrentdebatearoundtheScienceofMathmovement

canfeelsochargedandpersonal.Atfirstglance,manyofthenewdebates

surroundingexplicitinstructionorproceduralfluencyversusconceptual

understandingappeartorecycleolddivisionsbasedontraditionalists’and

reformers’preferencesforteacher-ledversusstudent-centeredapproachestoteachingandlearning.Andindeed,Dewey’spedagogicaldividerepresents

afundamentalbeliefcapableofexplainingatleastsomeoftheMathWars’morerecentintensity:apersonviewingthechildasthestartingpointofalleducationwholearnsbestbydiscoveringmathematicswithminimalteacherguidancewouldhaveverylittleincommonwithanadvocateforteacher-ledexplicitinstruction.

Yetthegapbetweenthetwosidesisnotaswideastheheadlinessuggest.

Inpractice,manytraditionalistsandreformersoccupyoverlappingterritoryratherthanoppositeendsofanunbridgeabledivide.Mosttraditionalistsdo

notrejectconceptualunderstanding;theycontestitssequencingrelativeto

proceduralfluency.Mostreformersdonotrejectexplicitinstructionoutright;theycontestitsscope,warningthatitshouldnotcrowdoutinquiry,problem-solving,andsensemaking.Bothsidesalsoshareseveralbroadcommitments:thatstudentsneedbothfluencyandunderstanding,thatlearnersvaryintheirneeds,andthatteachersrequirestrongerpreparationandsupport.

Thoseareasofoverlapmatter.Theysuggestthatmanyrecurringdisputes

intheMathWarsarenotpureeither/orchoices.Atthesametime,the

disagreementsarestillreal.Theyfocusonwhentointroducespecific

practices,howbroadlytoapplythem,andwhatkindsofevidenceshould

guideclassroomandpolicydecisions.Theresultisnotamanufactured

conflict,butadebateinwhichoverlapcoexistswithdurabledifferencesoversequence,defaultpedagogy,andevidentiarystandards.

Thosedurabledifferenceshavebeenreinforcedbyinstitutionalinertia

onbothsides.Overdecades,eachcamphasbuiltadvocacynetworks,

professionalcommunities,andcurricularecosystemsthattendtocirculate

andvalidatetheirownpreferredarguments.Thathistoryhelpsexplainwhy

recurringdisputescanfeellarger,sharper,andmoresettledwithineachcampthantheyoftenappearfromabroadervantagepoint.

Towardthetraditionalendofthespectrum,wehavehistoricalfigures(e.g.,JamesMilgram,RichardAskey)andcontemporarycontributorstorecent

10

CENTERONREINVENTINGPUBLICEDUCATION

11

debates,suchasthoserelatedto

California’smathframework

(e.g.,Brian

Conrad,JelaniNelson)ortoadvocatefortheScienceofMathmovement

(e.g.,SarahPowell).Traditional-leaningapproachesarefurthersupportedbyorganizations(e.g.,AmericanMathematicalSociety[AMS],Mathematically

Correct,andNYCHOLD)andfeaturedinprograms,suchasSaxonMath

andSingaporeMath.Towardthereformendofthespectrum,wealsohavehistoricalfigures(e.g.,TomRomberg,GlendaLappan)andcontemporary

contributorstorecentdebateontopicssuchas

equityversusexcellence

(e.g.,AlanSchoenfeld,PhilDaro),problem-basedlearning(e.g.,DanMeyer),or

critique

oftheScienceofMathmovement(e.g.,JoBoaler).Reform-

orientedapproacheshavegarneredlong-standingsupportfromNCTM

andtheAssociationofMathematicsTeacherEducators(AMTE),alongsidehistoricalfundingsupportfromtheNationalScienceFoundation(NSF)for

prominentreformprograms,suchasInvestigationsinNumber,Data,and

Space;ConnectedMathematicsProject;Core-PlusMathematics;andIllustrativeMathematics.

ThesecloselyknitnetworkshavereinforcedwhatJulieCohenandcolleaguescall“

epistemicbunkers

,”insulatedcommunitiesthatgrantgreatercredibilitytoevidencegeneratedwithintheirownrankswhilediscountingoutside

perspectives,oftenjustifiedbydifferencesinmethodology,theoretical

orientation,orprofessionalculture.Thisselectiveengagementcannarrow

thescopeofinquiry,stifleinnovation,andentrenchpreexistingassumptions.

Structuralbarriers,suchasseparateconferences,publicationvenues,andprofessionalnetworks,furtherlimitopportunitiesformeaningfuldialogueacrossdivides.Theresultisoftenafragmentationofknowledgethat

producesincoherentorincompleteguidanceforpractice,27undermining

thedevelopmentofwell-roundedsolutionsand,ultimately,constrainingtheimpactofresearchonthepeopleandproblemsitaimstoaddress.28

Withthisperspectiveinmind,wearenowreadytoexamineseveral

instructionaldichotomiesthathaveemergedinthe“new”MathWarsof

recentyears.Ratherthanfocusingonabstractacademicdebatesabout

methodologyorepistemology,weexaminetensionsthatdirectlyaffect

classroomteaching.Thesedisputesareexaminedatsomelengthbecause

theyareoftentreatedtooshallowlyinpubliccommentary,andtheresulting

oversimplificationhasrealconsequencesforpolicy,curriculum,andclassroompractice.Foreachdichotomy,weclarifywherethecurrentdisagreementlies,oftenaroundemphasis,sequence,orinstructionaldefaultratherthanasimpleeither/orchoice,andthenpresentthetraditionalandreformendsofthe

debatespectrum.Awordofcaution:thosepositionsaredeliberately“pure,”

27.Cohen,Jones,andGibbons,“TheMissingMiddle?,”1-2.

28.Ibid.,”4,9.

CENTERONREINVENTINGPUBLICEDUCATION

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representingeachside’smainargumentsandcommonlycitedevidence.

Practitionersandresearchersrarelyholdtheseviewsintheirentirety;they

aremeanttohelpreadersrecognizethetraditionalorreformorientationof

argumentstheywillencounterinthefield,nottodescribehowanyindividualororganizationactuallythinks.Bothsidesinvokeresearchinsupportof

theirclaims,andwecitethatevidenceasreported,withoutadjudicatingtheunderlyingbodiesofliterature,ataskwellbeyondthescopeofthisguide.

TheScienceofMathmovementistheexceptiontothisoverallapproach.

Becausethemovement’spositionshaveattractedsignificantpublicattentionandarefrequentlydescribedinreductiveterms,wepresentthemovement’sstatedstancesoneachdichotomysoreaderscanevaluateitsclaimsfirsthand.Theobjectionsthatreform-orientedorganizationshaveraisedaboutthe

movementareexaminedingreaterdetailattheendofthesection.

ConceptualUnderstandingvs.ProceduralFluency

TheDivide:Fordecades,matheducationhaswrestledwithwhetherto

prioritizedeepconceptualunderstandingormasteryofproceduresandfacts.Whiletwomajorpanelreviewsofmathematicsresearchhaveaddressed

thisissuebynotingthatan“integratedandbalanceddevelopment”29

wasneededandthat“forallcontentareas,conceptualunderstanding,

computationalfluency,andproblem-solvingskillsareeachessentialand

mutuallyreinforcing,”30theirinstructionalsequenceremainedunclear.In2014,NCTMreignitedthisdebatebyissuingaconceptual-firstrecommendation,

emphasizingthat“proceduralfluencyfollowsandbuildsonafoundationof

conceptualunderstanding,strategicreasoning,andproblemsolving”31aspartofeffectivemathematicsteaching.Therefore,thecurrentdivideisnotaboutchoosingbetweenteachingconceptsorprocedures,butaboutwhichtostartwithfirst.

TheSides:Onthereformend,advocatesofaconceptual-firstapproacharguethatstudentsshouldgraspunderlyingmathematicalideas(e.g.,placevalue,

numbermagnitude)beforelearningformalprocedures.Theyworrythatroteexecutionofalgorithmswithoutunderstandingleadsto“fragileknowledge,”meaningthatstudentsmayproducecorrectanswersbutcannotexplainwhyorapplytheirknowledgeflexibly.32Someinfluentialframeworks,includingtheCommonCoreandNCTM’sPrinciplestoActions,havereflectedthisconcern,citingresearchthatsupportsaconceptual-firstapproachbasedonbetter

29.NationalResearchCouncil,“AddingItUp,”11