Tools and Mathematics

497 Pages·2016·10.667 MB·English

by  John Monaghan, Luc Trouche, Jonathan M. Borwein (auth.)

John Monaghan
Luc Trouche
Jonathan M. Borwein
Tools and
Mathematics
Instruments for learning
Mathematics Education Library
More information about this series at http://www.springer.com/series/6276

John Monaghan (cid:129) Luc Trouche (cid:129)
Jonathan M. Borwein
Tools and Mathematics
Instruments for learning
JohnMonaghan LucTrouche
DepartmentofMathematicalSciences InstitutFranc¸aisdel’Education
UniversityofAgder EcoleNormaleSupe´rieuredeLyon
Kristiansand,Norway Lyon,France
JonathanM.Borwein
CentreforComputerAssistedResearch
MathematicsanditsApplications
SchoolofMathematical
andPhysicalSciences
UniversityofNewcastle
Newcastle,Australia
ISSN0924-4921 ISSN2214-983X (electronic)
MathematicsEducationLibrary
ISBN978-3-319-02395-3 ISBN978-3-319-02396-0 (eBook)
DOI10.1007/978-3-319-02396-0
LibraryofCongressControlNumber:2015952621
©SpringerInternationalPublishingSwitzerland2016
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To Seymour Papert—for inspiration
over the decades

Foreword
In our book Windows on Mathematical Meanings: Learning Cultures and
Computers,CeliaHoylesandIwrote:
The role of mathematics in underpinning social and economic life stretches back to the
dawnoftheindustrialrevolutionandbeyond.Everyaspectofmodernsocietyisinfused
withthecongealedmathematicallabourofmathematicians,computerscientists,engineers
andsoon.Yetatthesametime,thismathematicsisincreasinglyinvisibletothosewho
merely share in, rather than construct, the artefacts of the culture ... It is mathematics
whichliesdormantinsidethechipsofvacuumcleaners,thewarheadsofmissiles,andthe
graphicaldisplaysofnewsbroadcasts.Eventhesimpleexchangeofgoodsandcommod-
ities,oncerelativelyamenabletomathematisation,hasbeenoverwhelmedbytheworkings
ofglobalmarketswhicharedominatedbyinvisiblemathematicalforcesthatareincreas-
inglyoutofcontrol.(Noss&Hoyles,1996,p.253)
Iamsureneitheroftheauthorswouldclaimanyspecialprescienceinpredicting
thestateofaffairswefindourselvesinsome20yearslater.Itwouldbebanal,now,
to force such an obvious point. In a world of CGI, mobile communications and
automateddronestrikes,theimportanceofmathematicsinourliveshasneverbeen
greater—ormoregreatlyacknowledged.Theworldhastrulychanged,andwithit,
theroleofmathematicshasbecomeacriticalforce.
Inschool,whiletheimportanceofmathematicsiswidelyappreciated,itisoften
unclearwhatthisimpliesorhowthisappreciationshouldmanifestitself.CeliaandI
arguedthatthecomputercouldhavesomespecialroleinchangingthat.Centrally,
ourargumentwastodowiththepossibilityofemployingcomputerstoopenup—to
make visible—the knowledge that drives the world, not just how to improve the
transmission of knowledge from teacher to student. Our friend and inspiration,
SeymourPapert,atthe 17thICMIStudy ConferenceonDigitalTechnologiesand
Mathematics Teaching and Learning: Rethinking the Terrain, encouraged mathe-
maticseducatorspresenttoacknowledgethedependenceofmathematicalexpres-
sion on tools and the ways in which mathematics is shaped by their use. Papert
challengedustospend‘just10%’ofourtimefocusingonthewhatratherononly
the how of mathematics teaching and learning. A necessary—though far from
vii
viii Foreword
sufficient—condition for change is for computers to be ubiquitous, to be freely
availableasatoolthatisgenuinelyusefultoaccomplishworthwhilegoals.
It is happening. Each of us—in the developed world at least—has a powerful
computer in our pocket. Yesterday, I attended the launch of the BBC’s Micro Bit
computer,which—accordingtoitsdesigners—isa‘pocket-sizedcomputersettobe
giventoaboutonemillionUK-basedchildreninOctober’.Thedetailsdon’tmatter:
afreecomputerforeverychildisaroundthecorner.Andatthesametime,theUK
government has mandated the teaching of programming to all children—again, a
massive opportunity for developing new ways to use mathematics to accomplish
thingsthatcouldnototherwisebedoneexceptbytheveryfew.
Sothepublicationofthisbookistimelyintheextreme.Toolsmatter,andasthe
authorsofthebookmakeclear,tools—andespeciallycomputers—cannotsensibly
be thought of as just passive artefacts that can be sprinkled on the mathematical
terrain. The computer points to possibilities. It sets us thinking about ways of
reassertingmathematicsasaculturalpractice,notmerelyaschoolishendeavourto
achievetechnologicalknow-howforthefew,buttoeducateinthebroadestsense.
Ofcoursetheadventoffreecomputersisjustthestart.Computerssealedintheir
boxes, programming courses that exclude the big ideas of mathematics and com-
puter science, failure to adequately support teachers, curricula that just miss the
point...Icouldmakeaverylonglistofpossiblenegativeoutcomes.ButI,likethe
authors ofthisbook,amoptimistic.We canseize themomenttochange what we
teach,tofocusonPapert’s10%.Thisbookwillhelp.
RichardNoss
Preface
Chapter1introducesthebook,soweusethisprefacetocommentonthegenesisof
thisbookandthedevelopmentofideasduringitswriting.
This book originates in discussions at conferences between John and Luc.
We (meaning John and Luc at the moment) had similar backgrounds: a number
of years in high school mathematics teaching and then university work in mathe-
maticseducation.Inbothsiteswe’ddonealotofexperimenting(bothpracticaland
research) using digital technology. We shared prejudices for the types of tools/
systemswelikedtousewithourstudents:graphplotters,Logo,computeralgebra
systemsandsimilartools/systemswhichourstudentscouldusetoexpressmathe-
matical relationships rather than tools/systems that might go under the name of
‘computer-aided instruction’. When we first met, in the later 1990s, we’d read
papers of the other where we’d being trying to ‘get a theoretical handle’ on what
went on in high school mathematics classrooms when teachers and students used
these tools/systems. Our theoretical frameworks were not identical (John was
influenced by activity theory and Luc by instrumentation theory), but there was
plenty of room for productive dialogue; we were both also influenced by reading
papersofandtalkingtoMiche`leArtigueandJean-baptisteLagrangewhowere,in
turn, influenced by Yves Chevallard’s anthropological theory of the didactic and
Brousseau’stheoryofdidacticalsituations.
In the early years of the new millennium, an interesting thing happened—we
realised thatmuchofourthoughtaboutdigitaltechnologyandmathematics(edu-
cation)wasaspecialcaseoftooluseandmathematics(education).Thisrealisation
probablycamequiteslowly,butonceimplantedtheimportanceoftools(andlater
the distinction between artefacts, instruments and tools) in mathematical activity
producedawayofviewingmathematicsthatwasbothsimpleandprofound(atleast
for us): of course mathematical activity is not possible without tools; why are
peoplewritingaboutissuesinlearningandteachingmathematicsandnotmention-
ing the tools used in this learning and teaching? There were, of course, papers on
tool use in learning and teaching mathematics, but few of these were research
ix