MULTIMATHEMACY: Anthropology and Mathematics Education

195 Pages·2015·2.12 MB·English

by  Rik Pinxten

Rik Pinxten
MULTIMATHEMACY:
Anthropology
and Mathematics
Education
MULTIMATHEMACY: Anthropology
and Mathematics Education
Rik Pinxten
MULTIMATHEMACY:
Anthropology
and Mathematics Education
123
RikPinxten
University of Ghent
Ghent
Belgium
ISBN978-3-319-26253-6 ISBN978-3-319-26255-0 (eBook)
DOI 10.1007/978-3-319-26255-0
LibraryofCongressControlNumber:2015956128
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Dedicatedtothechildrenforwhomeducation
through schooling did not (yet) yield equal
opportunities in life
Foreword
Thebookofnatureiswritteninthelanguageofmathematics,sotoldusGalileoin
his famous treatise The Assayer (1623). What Multimathemacy makes absolutely
clearishowspecific,i.e.,Eurocentricthisstatementis.Amoment’sreflectionleads
tothefollowingobservations.Firstly,itdoesnotconcernhumans,itonlyconcerns
nature,itisthereforeanoutsider’sviewand,inextremis,aGod’seyepointofview.
Secondly, it assumes that it can be expressed in written form, more specifically, a
book. And asingle book atthat. Thirdly, it takes for grantedthat there will be just
the one language and hence just the one mathematics (or should I use the capital
‘M’, as Alan Bishop suggests). It has of course taken centuries of development,
mathematical, scientific, and cultural, to arrive at this bold statement and it has
taken almost four centuries to question it, to unravel its presuppositions, and to
suggest alternatives. This book is such an alternative.
The focus of Multimathemacy is on mathematical education. There is a nice,
helpful and seminal metaphor in terms of a city, that summarizes quite neatly the
whole enterprise of the book:
Thecityshowsmanybuildings:oneimpressiveskyscraper,afewsemi-tallbuildingsanda
hugeamountofhutsandsmalldwellings.TheskyscraperisthebuildingofAM(Academic
Mathematics)withitsownlogicalstructure,itsneatlydesignedseparateroomsandastaff
lookingafterthemaintenanceandtheeventualenlargementorrehabsofthebuilding.The
staff are the mathematicians. The other taller buildings are the substantial mathematics
corpuses of Chinese, Indian and other traditions. All of themdeveloped one or the other
special branch, working on their own particular intuitions and often applying their
knowledgeinarchitecture,irrigationsystemsandsoon.Andthenonefindsahugeamount
of small dwellings in the city, harboring particular local knowledge in mathematics, as
exemplified in building, sacred doings, tapestry or pottery making and the like (EM or
ethnomathematics)
Nodoubt,abovetheentranceoftheskyscraper,Galileo’swordsareetchedinto
stone.Theeducationaldramahasbeenandstillisthatthebookofnaturewrittenin
that specific language, so cherished in the tall building, has become a standard for
thewholevillage.Andthatmustleadtoproblems,todropoutsandtomathophobia.
Whatisneededareformsofmathematicaleducationthattakeintoaccountthislocal
vii
viii Foreword
knowledgeandtheselocalpractices.Itdoesleadtoaquitechallenginganddetailed
viewonmathematicaleducation.Ittakestheformofaframework,calledFORMA,
wherein all kinds of activities, involving mathematics one way or another, can be
‘inserted’, depending on the specific cultural background of the pupils.
FORMA isshorthandfor‘FrameofReference forMathematical Activities’and
itbringstogetherthefollowing activities: (a) counting, (b)locating,(c) measuring,
(d) designing (shape, size, scale, proportion and other geometric concepts),
(e) playing as a tool for exploration, (f) explaining through underlying structures
and rules), (g) moving, especially dancing and rhythmic moves and ceremonial
actions, (h) generalizing by comparing, (i) logically operating, (j) exchanging and
market activities, (k) making music, and (l) story telling. (a) up to and including
(f)havebeenproposedbefore,mostnotablybyAlanBishop,butthesixremaining
activities, (g) up to and including (l), are new. One of the strong features of
Multimathemacy is that for each of these cases, plenty of illustrative material is
presented. Thereby FORMA is not merely a theoretical proposal, an abstract
framework or something similar, but also a concrete vision that can be applied
almost immediately in specific learning settings.
Let me briefly return to the metaphor. Although it is not the intention of this
book to deal with the skyscraper itself, nevertheless—and I am now writing as a
philosopherofmathematics—ifFORMAweretobecomeaccepted,thensurelythis
musthaverepercussionsontheinhabitantsofAM(AcademicMathematics),asthe
skyscraper is called. Probably the strongest effect will be the realization that AM
requiresitselfaspecificculturalsetting,intheverysamewaythatanyotherformof
mathematicalactivityrequiressuchabackground.Thisraisesallkindsofquestions
concerning the so often claimed universality of mathematical knowledge and
thereby its necessity. How did this universality arise? What does AM look like
without this feature? Can there be different AM cultures? Could AM have devel-
opedalongdifferentroutes?Is,inotherwords,AMcontingent?Atpresentthereare
a number of philosophers and mathematicians who take these questions seriously
and one can only hope that philosophers, anthropologists, mathematicians and
educators(andwhoeverelsewhoisinterested)willfindcommon meetinggrounds
to address these issues all over the city as it were and not merely on the roof or,
more likely, in the cellars of the skyscraper (often called ‘foundations’ by the
philosophers).
Thisisnotgoingtobeaneasytaskasonemightexpecthugeresistancetothese
novel ideas. And it will be huge as even sympathizers sometimes do not seem to
escape its seduction as the case of Hans Freudenthal illustrates. Innovative as his
ideas were on mathematical education (as this book shows), nevertheless he did
hold the belief that a universal language, including mathematics, was possible to
communicate with whatever alien life forms. He effectively though partially
developedthislanguagein1960andnamedit Lincos(Lingua Cosmica).Dareone
surmisethatitwastoohardtorealizethattheskyscraperwasjustabuildingnextto
othersinthecityandthereforetobearthisidea’sweightthegazewentupwardsinto
Foreword ix
thecosmostodreamaway?ButthenoneisremindedofthewordsofErrettBishop,
themathematicalconstructivist,perhapsabitharshbutnolesstothepoint:“IfGod
hasmathematicsofhisownthatneedstobedone,lethimdoithimself.”Probably
writing it down in Galileo’s book of nature.
Jean Paul Van Bendegem
Vrije Universiteit Brussel
Center for Logic and Philosophy of Science
Preface
How do you start yet another book? On mathematics education? I can say that the
figuresaretragic:poorpeoplearoundtheworldgetlittleeducation,letaloneagood
schooling in mathematics and the sciences, which might help them to escape from
poverty. Hence, generations of poor people continue to get born, live and die
without the promise of a decent life. Since all democratic states in the world (in
theoryallmemberstatesoftheUNO,althoughthelatterisnotreallyasynonymof
‘democratic state’) subscribe to the Universal Declaration of Human Rights, it
wouldfollowthatthehumanraceofficiallyengagesitselftoofferagoodeducation
toitsyouth.Thefactsbeliethisengagement.Mathematicseducationisbelievedto
be a pillar of any conceptualization of what could be termed ‘a good education’.
Hence, offering suggestions for the improvement of mathematics education is a
good thing to do. That could have been one opening for a book like this.
Another one is to state yet another truth: I am an ignorant learned person, who
has been teaching at good Western universities throughout his life. Yet, the con-
cepts, theories and problems of science I do not know anything about (more than
thelayperson,oreventhatissayingtoomuch)areinnumerable.Moreover,Iamnot
anexceptioninthis:itissafetosaythatallmycolleaguessharethisconditionwith
me, and the amount of knowledge we are lacking is growing every minute.
Academic lore has it that the last person, who ‘knew everything’ about the natural
sciencesalone,diednearthemidstofthenineteenthcentury.Onenetresultofthis
conditionisthattheauthorityoflearnedknowledgedwindledovertheyears:today
the media are more likely to ask a sportsman or a movie star for their opinion on
issues of religion, culture, morality or politics, and even the cost of scientific
research today than colleagues of mine. With the corporatization of academia, we
witness today that CEOs are hired as chancellors of universities, because they
pretend to know how to run a business, and a university is considered more and
more to be yet another branch of business (Sahlins 2008). In our ‘learned
unknowing’,weresearchersstilldowanttohaveasayinwhatthenextgeneration
mightbestlearninschools,becauseschoolingseemstohelporattheveryleastnot
xi
xii Preface
hinderlotsofpeopleintheircareeropportunities.Andfromthere,itenhancestheir
chances to live a decent life.
I choose the second entry for this book: I am ignorant, and yet I know some-
thing.IamconvincedthatIamratherableinposingaproblem.Ihavebeenreading
andwritingalotabouthumancreativity,asitunfoldsandcanberecognizedinsuch
human endeavours as thinking, artistic activities, religious and life stance stories
and procedures and even in political or community practices. I have been doing
fieldwork(asananthropologistbyvocation)whilebeingtrainedinphilosophy,and
hence, I have been struggling with problems of meaning, contact, communication
and interaction, with identity and with existential issues. That was and is a good
schooling, I think: you learn to think, rethink, negotiate and change the problem
you consider relevant. It may sound as a truism to most, or maybe even as blas-
phemy to some, but when I think I can pose problems, then that is largely due to
thispracticeofinterculturalnegotiationandinteraction.Strangelyenough,veryfew
people are teaching that at the learned schools, but it comes along in some disci-
plines (like anthropology or psychotherapy training) when the students are pushed
outsideandintothefieldandhavetomeetwithrealpeopleinorderto‘doscience’
aboutthem.BecauseofthispeculiarextracurricularlearningIbenefitedfrominmy
contactswithotherculturalsubjects,Inowdaretowritethebookthatisinfrontof
you.
Indeed, this book is not written by a mathematician, but rather by an
anthropologist/philosopher. Hence, the lack of knowledge in pure or ‘academic’
mathematics is obvious from the start. In the words of the famous and uniquely
creative mathematician Hardy (1967), I am not concerned with pure or creative
mathematical knowledge, but rather or at the most with what he calls ‘trivial
mathematics’ (Hardy 1967, sec. 28). Trivial mathematics deals with elementary
geometry, elementary number theory and such, which can be shown to be ‘useful’
in business calculation, in orientation in the real world, or in a more general way
allows people to become educated in mathematical skills that have a beneficial
impact in jobs, in daily life and in sustainable ways of life in the present-day
predicament. The pure mathematician is not busy with that sort of issue (which is
perfectlyalrightandinagreementwithHardy’swell-madepoint),butmathematics
asasubdomainofknowledgehasimpactandpotentialhere,andthatiswhatIwant
to highlight. So, even though I am not and cannot be concerned with problems
withinmathematicsasadiscipline,Itaketheparticularfocusontheconstraints,the
estrangement, the attraction and the wonder that I can describe when pupils from
different cultural groups come into contact with mathematics in schooling and are
giventounderstandthatthisformalwayofthinkingisextremelyimportantfortheir
chances in life and for the future of the world in general. The mathematician is,
fairlyandunavoidably,asstupidorignorantasmeinamillionotherquestionsand
conceptsoutsidehisorherownspecialization. Becausethesecolleaguesareraised
inthesamesortoftunnelviewonthedisciplineasawhole,whichissupposedtobe
abuildingblockofscienceingeneral,ofwhichnobodyhashadanoverviewforthe
past five generations or so, it is permitted that somebody—even an outsider like
me—poses some questions on the primary level of mathematics education. The