In Defense of Mathematics and its Place in Anarchist Education

13 Pages·2012·0.13 MB·English

EDUCATIONALSTUDIES,48:39–51,2012
Copyright�C AmericanEducationalStudiesAssociation
ISSN:0013-1946print /1532-6993online
DOI:10.1080/00131946.2011.637256
In Defense of Mathematics and its Place
in Anarchist Education
2 MarkWolfmeyer
1
0
2
y CityUniversityofNewYork
a
M
8
0
4
3
6: Thisarticlereclaimsmathematicsfromthemeasuresofprofitandcontrolbyfirst
0
at presentingananarchistanalysisofmathematics’statusquosocietalusesandped-
] agogic activities. From this analysis, a vision for an anarchist math education is
cht developed,aswellassuggestionsforhowgovernmentschoolpractitionerssympa-
e
r thetic to anarchism can insert this vision into their current work. Aspects to this
Ut
y vision include teacher autonomy, freedom from hierarchical curriculum structure
ar andmathclassasanon-coercive,happyplace.Finally,mathematicsisarguedtobe
r
b essentialknowledgeforanarchisticsocietyforthreepotentialities:insolvingsocial
Li
y andtechnologicalproblemsthroughapplication,asananalytictechnologyandfor
sit increasingindividualhappinessviatheaestheticdimension.
r
e
v
ni
U Iamsympathetictothebadreputationmathematicsoftenendures.Someofsoci-
[
y ety’swell-knownusesofmathematicscloudourunderstandingoftheknowledge
b
d anditsplaceinavisionary,anarchistsociety;similarly,thestatusquopedagogyof
e
d mathematicseducationmightsuggestthatmathematicalknowledgeshouldbeleft
a
o
nl outofananarchisteducation.Idescribethissituationwithaheavyheart,however,
w becauseIalsohappentohavepassionformathematicsasaknowledgeformyself
o
D touseandenjoy,andassomethingIcansharewithothers.Inthisarticle,Iargue
thatmathematicsfindsahomeinanarchisteducation,andagainthatmathematical
knowledge is not in conflict with anarchist society. To begin, I offer a handful
of examples from such societal uses and status quo pedagogy that work against
threecommonlyagreedonanarchistvalues:collectivism,fraternity,andfreedom
fromsocialhierarchy.Theserepresentationswillguideanunderstandingforwhat
anarchistsocietyandeducationareandarenot.Next,thearticlediscussestherole
CorrespondenceshouldbeaddressedtoMarkWolfmeyer,ColumbiaHighSchool,17ParkerAve,
Maplewood,NJ07040.E-mail:[email protected]
40
WOLFMEYER
mathematicscanplayinanarchisteducationandfinallysociety.Putanotherway,
thisarticlefirstpresentsananarchistanalysisofcurrentmathematicalbehaviors,
bothpedagogicandotherwise,andthendevelopsananarchistmathematics.
BeforeIproceedwiththeconnectionsbetweenmathematics/mathematicalbe-
haviors and anarchism, I describe briefly the anarchist theory that informs this
article. One definition describes anarchism as “a political theory which aims to
create a society within which individuals freely cooperate together as equals”
(McKay2008,19).Inparticular,Iamhighlightingthreetenetsrelatedtothisdefi-
nition:collectivism,fraternity,andfreedom.Collectivismdenotesthecurtailment
2 of property rights, especially as they relate to ownership of capital. Fraternity
1
20 describes an inclination for individuals to recognize the needs and desires of all
ay otherpeople,andaccordinglytoactinthespiritofmutualaid.Freedomindicates
M
a lack of coercive actions by any person, group, or social institution on any one
8
0 person, as well as individual autonomy within the boundaries of imposing on
4
3 another’s freedom. I review these anarchist tenets when I describe an anarchist
6:
0 matheducation,butfirstIusethemtoexposeproblematicmathematicalactivities
at insociety.
]
ht
c
e
r
Ut ANTIANARCHISTMATHEMATICALACTIVITIESINSOCIETY
y
r
a
r
b Asmuchasanarchisttheorypresentspossiblegoalsforsocietytoworktoward,
Li
y it also offers a framework with which to critique institutional arrangements and
sit activitiesinsociety.Inthissection,Iofferahandfulofmathematicalactivitiesthat,
r
e
v whenviewedthroughananarchistlens,canbeconsideredfortheircontributions
Uni tosocietalills.Specifically,Ireviewtherolethatmathematicshasplayedinthe
y [ exploitation of labor, or economic inequality, and warfare. These mathematical
b
d activitieshighlighttwotenetsfromanarchisttheory:collectivismandfraternity.
e
d The first of these representations concerns the societal use of mathematics
a
o for unequal distribution of resources. Apple (1992, 1995) suggests that mathe-
nl
w matical knowledge is often utilized for its “technical/administrative” relevance
o
D thatis“convertibleultimatelyintoprofits”(Apple1992,420).Therecentuseof
mathematics by numerous Wall Street hedge funds for grandiose profits (Patter-
son 2010) describes this relevance quite accurately. In this case, mathematical
knowledge was highly regarded for its ability to analyze, dissect, and predict
outcomesforcapitalistsseekingtoturntheirmoneyintomoremoney.Howthis
activity leads to economic inequality rests initially, of course, on Marx’s (1976)
critiqueofcapitalisminwhichlaborisnotpaidthevalueitaddstothecapitalist.
Harvey(2005) updatedthisexploitation inexplaining today’sfinancial markets:
“The strong wave of financialization that set in after 1980 has been marked by
itsspeculativeandpredatorystyle. ... Deregulationallowedthefinancialsystem
tobecomeoneofthemaincentersofredistributiveactivitythroughspeculation,
41
EDUCATIONALSTUDIES
predation, fraud, and thievery” (161). Sadly, mathematics is an integral part of
suchredistributiveactivity.
Alsoindicatingthisrelevanceofmathematicstoprofitistheoften-citedcon-
nectionbetweenmathematicsandeconomicgrowth/security/superiority.Forex-
ample,asGutstein(2006)notes,theNationalCouncilofTeachersofMathematics’
(NCTM)Standards1989framesmathematicseducationasonemeanstocontinue
USeconomicgrowth.Generally,economicgrowthisunderstoodtomeananin-
creaseinthegrossdomesticproduct(GDP),whichmeasuresgoodsandservices
output(whateverthesemaybe)anddoesnotnecessarilyindicatebalancedincome
2 or distribution of legitimate needs among the population. To this point, the last
1
20 quarter of the twentieth century saw both a steadily increasing GDP and a four
ay percentincrease(fromtwotosix)intheshareofnationalincomeofthetopone
M
percentofincomeearnersintheUnitedStates.However,the“ratioofthemedian
8
0 compensationofworkerstothesalariesofCEOsincreasedfromjustover30to1
4
3 in1970tonearly500to1by2000”(Harvey2005,16-17).Therefore,mathematics
6:
0 educationandwithitmathematicalknowledgearerhetoricallylinkedtoeconomic
at inequalityindocumentssuchastheNCTMStandards.
]
ht Before continuing with the next representation of status quo mathematical
c
re contradictions to anarchism, I highlight the first anarchist value that has been
Ut
presentedbythecapitalistuseofmathematics.Amongavarietyofanarchisttheo-
y
ar rists,mostagreeonaneconomicsystemwithcollectivistproperties,oreconomic
r
b equality.WhereasearlyanarchisttheoristslikeProudhondidnotfullyassertthe
Li
y needtoabolishprivateownershipandcapitalisteconomicorganization,Bakunin
sit laterestablishedtheanarchisttenetfora“socialrevolutionwhichtransformspri-
r
ve vatepropertyintocollectiveproperty. ... Only‘thosethingswhicharetrulyfor
Uni personaluse’wouldremainprivateproperty”(Geurin1970,56).Currentsociety
y [ witnesses mathematical knowledge as a powerful tool for some people to take
b
d fromothers,aprogramregardedtobeanticollectivist.
e
d Thenextrepresentationofmathematics’antianarchisttendenciescomesfrom
a
o another of its infamous applications: for modern warfare. During World War II,
nl
w USMilitaryofficialsclaimedthattheyoungmenandwomenwhowereenlisting
o
D lackedthemostbasicofskillsinmathandthiswouldgreatlydeterminethewar’s
outcome.Twodocumentsfromhistoryprovideaclearpictureofthis,thefirstof
whichbeingaletterfromLouisBredvold,anacademic,toCaptainF.U.Lake,in
which he asks for more information regarding the “difficulty in finding students
in American colleges other than engineering who were sufficiently prepared in
mathematics to make them available for training for commissions in the Navy”
(Garrett 1991, 191). Admiral C. W. Nimitz’s response elaborately answers this
request,makinganumberofclaimssoastodemonstratethemilitary’sneedfor
back-to-basicsmathinstructioninpublicschools:“Acarefullypreparedselective
examinationwasgivento4,200enteringfreshmanattheleadinguniversities,68%
ofthemen ... wereunabletopassthearithmeticalreasoningtest;”“Almost40%
42
WOLFMEYER
ofthecollegegraduatesapplyingforcommissioninghadnotinthecourseoftheir
educationtaken ... trigonometry;”“Requirements[forcommissioning]hadtobe
loweredinthefieldofarithmeticalattainment,”and“Mathematicsis ... necessary
infirecontrolandinmanyothervitalbranchesofthenavalofficer’sprofession”
(Garrett1991,192–194).AsauthorsbegantocitetheNimitzletter,moremilitary
officialsopenlycriticizedmatheducationandcalledforchange.Letters,writtenby
militaryuniversityofficialsanddirectedtowardteachers,parents,andsupervisors,
were published in journals such as National Association of Secondary School
PrincipalsBulletinandMathematicsTeacher(Garrett2003,288).
2 Thistrendofmilitaryinterestinmatheducationcontinuestoday.Forexample,
1
20 thedraftingprocessforthenewdefactoUSnationalmathstandards,theCommon
ay CoreStateStandards,includedfinancialsupportfromtwolarge-scaleengineering
M
firms who happen to provide weapons to a number of nations: Boeing and Bat-
8
0 telle. These firms provided monetary support to Achieve, Inc., the not-for-profit
4
3 whichwasorganizedbytheNationalGovernorsAssociationandChiefsofSchool
6:
0 StateOfficersfordraftingnationalstandards(Achieve,Inc.2010).Theirfunding
at suggeststhatmathematicalknowledgeisneededtoengineermilitaryproductsthat
]
ht willbepurchasedbynationsforwarfare.
c
re Relatedtosociety’suseofmathematicsforwarfare,theknowledge alsofalls
Ut
prey to societal attempts to subjugate populations. Gould’s Mismeasure of Man
y
ar (1996) recounts the activities of many innovators of statistical methods whose
r
b primary goal was to scientifically prove White supremacy. These efforts con-
Li
y tinue in modern times and extend to include both classist and racist arguments,
sit notably with the much discussed work of Herrnstein and Murray (1994). Ad-
r
e
v vanced mathematical thinking dominates the perspectives in this and similar
Uni works,leavinginthemouthsofthosewithradicalsympathiesasourtastevis-a`-vis
y [ mathematics.
b
d The trend that mathematics aids in weapons engineering and subjugation of
e
d populationsbringstothefrontthesecondagreementamonganarchiststhatIhigh-
a
o lightinthisarticle:fraternityandmutualaid.Suissa(2010),quotingPatriciaWhite,
nl
w describes fraternity as an attitude comprising respect for all peoples’ needs and
o
D individuality.Inotherwords,theself-satisfactionofothers,orothers’happiness,
is of paramount concern to individuals in anarchist society. Kropotkin’s (2006)
mutualaid,derivedfromevolutionaryevidenceinhumansandotherspecies,puts
forthbenevolenceasaprimarydeterminantofindividualandcommunitysuccess.
Anarchistsviewwarfareasantitheticaltothefraternalspirit.Analysesofwarfrom
anarcho-pacificists,suchasdeLigt,suggestthatarmiesandwarsbetweennations
maintain the rulers’ power by facilitating hostility among the working people
(e.g., de Ligt 1938). Similarly, efforts to prove one person’s worth over another
(orindeed,thevalueofoneentiregroupasgreaterthanthatofanothergroup),as
thecasewiththeuseofmathematicsforprovingracialinferiority,clearlyconflict
withfraternityamongpersons.
43
EDUCATIONALSTUDIES
Thus far, societal uses for mathematics have exposed some ways that mathe-
maticalactivitiesworkagainsttheanarchistvision,specificallycollectivismand
fraternity. These examples do not comprise an entire list of mathematics’ dark
side, but have been selected because they are particularly contrary to anarchism
andarepopularchoicesforexpressinganegativeviewofmathematics.Fornow,I
abandonthesocietalusesforthewaysthatpedagogicbehaviorssimilarlydismiss
anarchistprinciplesandperhapsfosterthenegativitymanyholdformathematics.
Aswiththesocietaluses,Icontinuetoarticulateanarchisttenetsbywayofthese
negativepedagogicactivities.
2
1
0
2
y ANTIANARCHISTACTIVITIESINMAINSTREAMAND
a
M MARXISTMATHEDUCATION
8
0
4
3 Intheprevioussection,Ioutlinedtwomathematicalactivitiesinsocietythatcon-
6:
0 tributetosocietalills,whenviewedinthelightofanarchisttheory.Here,Iattend
at totypicalactivitiesandbehaviorsinmathclassroomsthatalsocanbeconsidered
]
ht counteranarchist. As before, I continue to use anarchist theory as a framework
c
re tostudysociety,thistimemathteachingandlearning.Specifically,Istudyideas
Ut
from both mainstream and Marxist math education within an anarchist context.
y
ar Mainstreammatheducationcountersanarchism’snotionoffraternity,especially
r
b as demonstrated by the prevalence of anxiety in students learning mathematics.
Li
y Both mainstream and Marxist math education present an issue regarding anar-
sit chism’stenetoffreedom,becauseeachlacksstudentand/orteacherautonomyto
r
e
v fullyexploremathematicalknowledge.
Uni Beginning withmainstream math education activities, thefirstpedagogic sit-
y [ uation is perhaps better described as a consequence of pedagogic activities, but
b
d is included here for extending the argument that mathematics, in this case math
e
d education,counterstheanarchistprincipleofconcernforindividualwellbeing,or
a
o fraternity.Iamtalkingaboutmathanxiety.Thisphenomenoniswelldocumented
nl
w across cultural contexts, for example Ho et al. (2000), and generally is taken to
o
D meanthedocumentedemotionalresponsesinindividualswhensubjecttolearning
orbeingtestedonmathematics.Mathclassisoftenanunhappyplaceformanyof
its students; fraternity does not seem to exist here. The attention by scholars on
itscognitiveoraffectiveaspects,asinHoetal.(2000),placestheblameforthis
experience on the students, rather than the situation in which the unrest occurs.
On the contrary, it is not unreasonable to suggest that this phenomenon exists
forthecircumstances ofmatheducation, suchastherushtolearnone aspect of
mathematicsinordertomasterthenext,orriskbeingleftbehind.
Indeed, the concern to master one idea before moving to the next presents
another aspect of math education in contrast to anarchist principles. Curriculum
structureinmatheducationishierarchical,whereashierarchyandanarchismare
44
WOLFMEYER
antithetical.TheintroductiontotheCommonCoreStateStandardsforMathemat-
ics (2010) includes the following quote from Schmidt and Houang: “standards
andcurriculaarecoherentiftheyare‘articulatedovertimeasasequenceoftopics
and performances that are logical and reflect, where appropriate, the sequential
or hierarchical nature of the disciplinary content from which the subject matter
derives’“ (3). The argument that mathematics learning sequentially builds from
one topic to the next should be familiar to anyone who has completed a stan-
dard math education program. Although it may be true that some mathematical
knowledgedoesbuildfromsimplertomorecomplicatedideasinalinearfashion,
2 it is an extraordinary, although commonplace, idea that all mathematics and an
1
20 individual’s mathematical development will progress in one direct fashion. For
ay instance,studentsmustmasterthedivisionoffractionsbeforebeginningtosolve
M
algebraic equations. This particular example is chosen because it simply has no
8
0 mathematicallogicbehindit:Divisionoffractionsisnotnecessaryforastudent
4
3 to understand how to solve an algebraic equation.1 However, writers of the cur-
6:
0 riculumconsistentlyconstructthisandotherfalsehierarchiesamongelementsof
at mathematical knowledge that facilitate a hierarchy of students. Certain students
]
ht continuetomastereachofthesteps,whereasotherswhomissaparticularoneare
c
re doomedtomissalltheresultantknowledgehigheralongthishierarchy.Lockhart
Ut
(2009), apublished mathematician who also teaches high school, also disagrees
y
ar with the hierarchy in curriculum. He calls it the “‘ladder myth’—the idea that
r
b mathematicscanbearrangedasasequenceof‘subjects’eachbeinginsomeway
Li
y more advanced” (56). Instead, he favors a variety of topics/inquiries that arise
sit fromteacherandstudentinterest.
r
e
v Beyond the curriculum, hierarchy also exists among the adults invested in
Uni pedagogicactivities.Mostclearlythisisseenwiththeactofteachercompliance
y [ withstandardsforcurriculum,whichnotablydoesnotoccurinhighereducation
b
d andhappenslesssoinotherK–12contentareas.Teachersareexpectedtodevelop
e
d lessonsthatwillsatisfycurriculargoalsnotdecidedbythem,andmathematicshas
a
o more rigidstandards than other subject areas. Math teachers are considered less
nl
w able tomakesuch decisionsthanmathematicians andmath educators. Indeed, a
o
D commonresearchagendaformatheducationistheendeavortoprovewhatmath
teachersdonotknow.Researchonthistopiccomesfromsuchinfluentialscholars
inmatheducationasMa(2000),whoservedonthefederalgovernment’sNational
MathAdvisoryPanelin2008.Citingwhateverdeficitsteachersofmathmayhave
assertsauthorityoverthemandreinforcestheneedforrigorouscontrol.Fromthe
anarchistperspective,thislackofautonomyforteachersmaypointtothereason
thatMaandothersfindteacherknowledgedeficits.
Oftenreferredtoasthedefiningfeatureofanarchismisitsprincipleoffreedom
fromhierarchy.Bookchin(2005)writesofhierarchyas“thedominationofyoung
by the old, of women by men, of one ethnic group by another, of ‘masses’ by
bureaucratswhoprofesstospeakintheir‘highersocialinterests,’ofcountryside
45
EDUCATIONALSTUDIES
bytown,andinamoresubtlepsychologicalsense,ofbodybymind,ofspiritby
a shallow instrumental rationality, and of nature by society and technology. ...
Hierarchy is not merely a social condition; it is also a state of consciousness, a
sensibility toward phenomena at every level of personal and social experience”
(68–69).Anarchismexposesthevarioussocialpracticesthatsubjectpeople(and
other living things) to the control of other people. Status quo math education
practicepromotesBookchin’s“sensibilitytowardphenomena”initspresentation
ofcurriculum,asdescribedpreviously.Studentsmoveuptheladderinarace-like
fashionwith“somestudents‘ahead’ofothers,andparentsworrythattheirchild
2 is‘fallingbehind’“(Lockhart2009,56).Ultimately,studentsarerankedbyhow
1
20 highupthehierarchyofknowledgetheyclimb,therebyfunctioningtosortpeople
ay intoaboveandbelow.2
M
Standing against hierarchical practices in society highlights one major dif-
8
0 ferencebetweenanarchismandMarxism.AlthoughMarxismexposeseconomic
4
3 hierarchiesandseekstoreplacethesewitheconomicequality,theprojecttoerad-
6:
0 icate other hierarchies is not considered, and what is more, Marxism asserts the
at need for a hierarchy in the educational process that will move society toward
]
ht equality. Marxist education relies on an enlightened elite who hold what they
c
re consider an objective truth for how society currently functions and how society
Ut
will be transformed. It “is seen as primarily the means by which the proletarian
y
ar vanguard is to be educated to true (class) consciousness. Once the revolution is
r
b over,itseems,therewillbenoroleforeducation.”Ontheotherhand,anarchist
Li
y education“isaimednotatbringingaboutafixedend-point,butatmaintainingan
sit ongoingprocessofcreativeexperimentation”(Suissa2010,39).
r
e
v Gutstein(2006)represents theMarxist educational perspective inthecontext
Uni of mathematics education, when he draws upon Freire’s critical pedagogy for
y [ example.Hisgoalof“liberationfromoppression”(22)utilizesapedagogycom-
b
d prising “teaching mathematics for social justice” (29). Aspects to the pedagogy
e
d include (a) “reading the world with mathematics” (26), or looking at racial and
a
o economic inequality with mathematical analyses, (b) “writing the world with
nl
w mathematics” (26–27), or seeing the power in mathematics for social change,
o
D (c) “developing positive cultural and social identities” (28–29), or learning both
thelanguage/cultureofpowerandpersonallanguage/culture(asinDelpit1995),
(d) “reading the mathematical word” and “succeeding in the traditional sense,”
(29–30)orlearningthestandardizedmathematicscurriculumtoperformwellon
testsand(e)“changingone’sorientationtomathematics,”(30–31),orappreciating
mathematicalpowerasbothitsdominantroleinsocietyanditscapacitytochange
theworld.Gutsteinusedtheseobjectivestodevelopseveralclassroompractices,
andhediscussestheirsuccessinhisownclassroom.
AnarchismhasalottosayaboutGutstein’s(2006)approach.Inhisproject,he
envisionsanenlightenedleaderwhodesignsaneducationforspecificgoals.Al-
thoughauthorityisnotnecessarilyinconflictwithaspectsofanarchisteducation
46
WOLFMEYER
orchild-rearing,Gutstein’sprescribedexperiencesforhisstudentsremovethean-
archistprocessofcreativeexperimentationfromtheeducationalprocess.Gutstein
controls his students’ use of mathematics; they are expected to learn and know
mathematicsprimarilyforitscapacitytocritiqueracialandeconomicinequality
insteadofotherpossibilitiesrelevanttobothitsnatureandapplication.Fromthe
anarchistperspective,Gutstein’sactivitycanprovokeresistancefromatleastsome
studentsandcanperhapsdevelopnegativerelationshipswithmathematicsand/or
social justice in some individuals, an outcome contrary to Gutstein’s goals. The
limitedviewofmathematicsuseresonateswithSuissa’ssecondnoteonMarxist
2 education,that“oncetherevolutionisover,itseems,therewillbenorolefored-
1
20 ucation,”orinthiscase,nouseformathematics(Suissa2010,39).Ifstudentsare
ay indoctrinated to view mathematics as primarily useful for analyzing oppression
M
andforplayingthepowergame,thenoncetheyachievethegoalofliberation,they
8
0 may not understand the continued use for mathematics. Furthermore, by adopt-
4
3 ing the traditional hierarchical math curriculum, Gutstein’s project continues to
6:
0 promote hierarchies and fails to critique such authority established outside the
at classroomwalls.Astheteacher,heacceptstheauthoritytowhichheissubjected,
]
ht and this acceptance, along with the hierarchical structuring of the knowledge to
c
re belearned,indoctrinatesstudentsinhierarchicalphenomenadescribedearlier.
Ut
To be sure, Gutstein is to be applauded by anarchists for his excellent work
y
ar developingsocialjusticelessonsforthemathematicsclassroom.Hehascertainly
r
b taken society to task for its problematic relationship with mathematics, which I
Li
y havesuggestedbytheexamplesIincludedatthebeginningofthisarticle.However,
sit thelackofstudentautonomyinhispedagogyis,indeed,toosignificantforthoseof
r
e
v uswithanarchistsympathies.Suissa(2010)discussestheseissuesmoregenerally
Uni inoutliningdifferencesbetweenMarxismandanarchismandinherarticulationof
y [ aphilosophyofanarchisteducation.Sheremindsusthatanarchismisthepolitical
b
d philosophythatdiscussesbothindividualfreedomandsocialequality.Individual
e
d freedom must be of equally paramount concern, yet one individual’s freedom
a
o cannottakeawayanother’s,hencethestaunchoppositiontocapitalism.However,
nl
w individualsaretobeotherwisefreetogovernthemselves.
o
D Intheeducationalcontext,thisdancebetweenindividualityandequalityexists,
as well. Tolstoy, a religious anarchist, put the words “Come and Go Freely”
abovethedoorsofhisexperimentalschoolatYasnayaPolyana(Tolstoy2000,1).
However,Gutstein’sstudentsdonotgetthechancetochoosewhethertheywant
to learn both the mathematics he is teaching and the social context in which he
is teaching it. To be sure, Gutstein’s efforts do embrace one aspect of anarchist
education.Suissa(2010)arguesthatanarchisteducationdoesnotrefrainfrom“the
very attempt by educators to pass on any substantial beliefs or moral principles
tochildren”(98).SoGutstein’sworkproperlyaddressesthisaspecttoanarchist
education,butIarguedoessowithtoomuchauthorityandtoolittlefraternity.He
suggests the math classroom’s primary function as liberatory pedagogy, yet this
47
EDUCATIONALSTUDIES
limits student exposure to mathematical knowledge. In turn, students have less
potentialtogainavarietyofmathknowledgeand,asIarguelater,precludessome
studentsfromdevelopingahappy,self-fulfillingrelationshipwithmathematics.
DEFININGANARCHISTMATHEDUCATION
Intheprevioussection,IconsideredtheelementsofMarxistmatheducationthat
embraceanarchisttenetsandthosethatdonot.Marxistmatheducationincludes
2 exposuretoanarchistmoralsofequalityandfraternity,butdoessoattheexpense
1
20 of student autonomy. What then, would an anarchist math education look like?
ay First, in taking a cue from Goldman that education “must insist upon the free
M
growthanddevelopmentoftheinnateforcesandtendenciesofthechild”(quoted
8
0 inSuissa2010,77),nostudentshouldbeforcedtolearnmathematicsashappens
4
3 inbothmainstreamandMarxistpedagogy.Anearnestefforttodevelopsuchinnate
6:
0 forces,however,requiresanarchisteducatorstopresentmathematicsinavariety
at ofwaysandcompriseitsvariousbehavioralformssothatstudentscandetermine
]
ht if they would like to acquire the knowledge. The term mathematics captures a
c
re widevarietyofcognitiveandphysicalbehaviors,threeofwhicharemathematics
Ut
astheartofabstractreasoning,mathematicsasabstractandautomaticprocedures,
y
ar andmathematicsasanappliedscience.Beforedetailingtheirdifferences,Iwant
r
b to present two caveats: (a) None of these are suggested to be more authentic
Li
y mathematics than the other; each is mathematics, and (b) these conceptions do
sit containcommonelements,thusinteractingandintersectingwitheach.
r
e
v Each of the three behaviors agree that mathematics can consider a variety of
Uni topics(e.g.,numbers,geometricfigures)buteachrequiresadifferenttypeofeffort
y [ whenundertakenbyanindividualorgroup.Forexample,inthemathematicalarena
b
d commonlycallednumbertheory,mathematicsasabstractprocedurestakesplace
e
d when some friends who are out to dinner add up their tab and divide it by the
a
o number of people to determine how much each person must pay; mathematics
nl
w as the art of abstract reasoning takes place when an enthusiastic student taking
o
D anelementarynumbertheorycourseattemptstoprovethateveryintegergreater
than 1 can be written uniquely as a product of primes (called the Fundamental
TheoremofArithmetic);andmathematicsasanappliedsciencetakesplacewhen
a team of computer scientists might use modular arithmetic and large prime
numberstodevelopapublickeycryptosystemtousewhenneededtokeepdigital
information private even when intercepted by a third party. In the first example,
thepartyisindeedapplyingarithmetictoasituation,butIhesitatetosaythatthis
ismathematicsasappliedscience.Theapplicationisautomaticwithoutconscious
reference to mathematical properties or theorems, whereas computer scientists
areactivelyworkingwithmathematicalpropertiesandtheoremstodevelopnew
applications.
48
WOLFMEYER
These three behaviors are not intended to capture all of mathematics but do
exhibititsvariety.Anarchistmatheducationwouldallowstudentstobeexposed
tothevarietyofmathematics,toseewhethercertainaspectsaremoreinteresting
foranindividualthanothers.Studentsandteachersarefreetochooseamongthe
mathematical behaviors that aremostinterestingtothem, possiblyresolvingfor
themselvesthe“MathWar”(Schoenfeld2004,253–254)debateoverskillsversus
concepts.Thisdebatehasfocusedlittleonwhethersomestudentspreferlearning
mathematicalskillsandalgorithmsbyrote,whereasotherspreferprovingmath-
ematicalideas.Iwouldbesurprisedifotherexperiencedteacherswoulddisagree
2 with my observation that students, indeed, often favor one of the mathematical
1
20 behaviorsoveranother.Differentfromthemathwars,anarchisteducationwould
ay placenocomparativevaluationononemathematicalbehaviorovertheother.
M
Lockhart(2009)commentsonwhatheperceivesasasadomissionoftheab-
8
0 stract reasoning behavior in today’s schools. Most students do not get a chance
4
3 toknowthatmathematicscanbe“dreamyandpoetic”;“radical,subversive,and
6:
0 psychedelic”;andadisciplinethatallows“freedomofexpression”(23).Lockhart
at presentsmathematicsasanart,andinthissensemathematicseducationwill,for
]
ht somestudents,beappreciatedforitsaestheticqualitiesbecausetheartist(math-
c
re ematician) plays in completely imagined worlds. This resonates with Marcuse’s
Ut
(1978) assertion that “art breaks open a dimension in which human beings, na-
y
ar ture,andthingsnolongerstandunderthelawoftheestablishedrealityprinciple.
br ... The autonomy of art reflects the unfreedom of individuals in the unfree so-
Li
y ciety”(72).BothtraditionalandMarxistapproachestomatheducationlackthis
sit autonomyofartbyinsteadcontrollingstudentmathematicalbehaviors;authority
r
e
v chooseswhichbehaviorsarefavored(usuallyabstractproceduresandapplications)
Uni andlimitsthesebehaviorstoonlyspecificavenuesofinquiry.Inananarchistmath
y [ educationprogram,theartofabstractreasoningwouldbeoneavenueforstudents
b
d toexploreinmathematics.
e
d Inananarchistmatheducationpractice,freedomfromhierarchywouldinclude
a
o a teacher’s capacity to choose her own path for the class experience. Aspects
nl
w of a moral education, such as those in Gutstein (2006), as well as the aesthetic
o
D dimension would probably be a part of her thinking. However, end goals would
notnecessarilybedeterminedinadvance,althoughtheycould,dependingonher
particulardispositionandpedagogicapproach.Forthosewhoareworriedabout
accountability to cover material, an anarchist education might include advanced
methods of accountability via group decision making, subjecting one teacher’s
performance to review by other math teachers and the students and parents that
areinvolved.
Suissa(2010)makestheimportantpointthatperspectivesonanarchisteduca-
tionoftencloudwhateducationwilllooklikewithinastatesocietythathopesto
becomestatelessversusaneducationinanalreadystatelesssociety.SofarIhave
perhaps described the anarchist math education in a stateless society, so I’d like