Table Of ContentAdvances in Mathematics Education
Michael N. Fried
Editors
Tommy Dreyfus
Mathematics
& Mathematics
Education:
Searching
for Common
Ground
Advances in Mathematics Education
SeriesEditors:
GabrieleKaiser,UniversityofHamburg,Hamburg,Germany
BharathSriraman,TheUniversityofMontana,Missoula,MT,USA
InternationalEditorialBoard:
UbiratanD’Ambrosio(SãoPaulo,Brazil)
JinfaCai(Newark,NJ,USA)
HelenForgasz(Melbourne,Victoria,Australia)
JeremyKilpatrick(Athens,GA,USA)
ChristineKnipping(Bremen,Germany)
OhNamKwon(Seoul,Korea)
Forfurthervolumes:
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Michael N. Fried (cid:2) Tommy Dreyfus
Editors
Mathematics & Mathematics
Education: Searching
for Common Ground
Editors
MichaelN.Fried TommyDreyfus
BenGurionUniversityoftheNegev TelAvivUniversity
BeerSheva,Israel RamatAviv,TelAviv,Israel
ISSN1869-4918 ISSN1869-4926(electronic)
AdvancesinMathematicsEducation
ISBN978-94-007-7472-8 ISBN978-94-007-7473-5(eBook)
DOI10.1007/978-94-007-7473-5
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2013956009
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DedicatedtoPollyEisenberg
Adevotedteacherandwonderfulhumanbeing
She wouldhavebeenproudtoseethisbook
inhonorofTed
Preface and Acknowledgements
ThisbookisdedicatedtoTedEisenberg.Itgrewoutofasymposiumwiththesame
title held in May, 2012 on the occasion of Ted’s retirement. The venue was Ben
Gurion University of the Negev where Ted spent over thirty years as a professor
of mathematics education in the mathematics department. Ted received plenty of
praise during the event, praise, needless to say, that was well-deserved. However,
forascholar,teacher,andhumanbeinglikeTed,itisafargreatertributetodiscuss
ideasthatmattertohim.Indeed,nothingcouldbemoredishearteningforascholar
like Ted thanto feelhis concernsare ignoredor belittled.Ted has confessedto us
hehasoftenfeltjustthatinrecentyears.
Ted’ssenseofneglecthasnotbeenentirelyunfoundedregardingoneconcernthat
haspreoccupiedhimwithparticularacutenessovertheyears,namely,thegrowing
distancebetweenmathematiciansandmathematicseducators.ForTed,tobeamath-
ematicseducatoronemustknowandcareaboutmathematicsitself.Itisaposition
that,forhim,isaxiomaticanduncompromising.Ontheotherhand,itisalsoafact
Tedpainfullyadmitsthatthedevelopmentofmathematicseducationasanacademic
fieldhasallowedsomeofitspractitionersattimestoputmathematicalknowledge
asideorevenpronounceitasirrelevant.
But“attimes”doesnotmean“always”and“some”doesnotmean“all.”Thetruth
of the matter, as we see it, is that mathematics education has developed to a point
wheretheplaceofmathematicswithinthefieldsimplycannotbetakenforgranted:
the question of the relationship between mathematics and mathematics education
needstobeexploreddeeplyandbetterunderstood.AndascounterweighttoTed’s
ownsensethatthereisalackofurgencyaboutthisinthefield,theverystatureof
themathematicseducatorsandmathematicianswhoparticipatedinthesymposium
andwhocontributedtothisbookunderlinesthetremendousinterestthereactually
isonallsidesaboutthisquestionwhichTedholdsdear.
Wearethusgratefultotheparticipantsofthesymposiumwhosubsequentlycon-
tributedtothisvolume,showinghowfarTed’sconcernistheirconcern.Anditmust
beaddedthattheseparticipantswerenotonlythosewhopresentedpapersandtook
part in planned panel discussions, but also those in the audiencewho asked astute
questions and brought up issues enriching the general conversation then and the
vii
viii PrefaceandAcknowledgements
ideas in this book now. Many of those audience members were especially invited
becausewethoughtthattheywouldindeedenrichtheevent.So,ourfirstthanksare
to all those who came and made that symposium such a great success. Of course,
though,therewereotherswhosecontributionitisimportanttoustoacknowledge:
• DanielBerend,MiriamCohenandMichaelLinfromtheDepartmentofMathe-
matics at Ben Gurion University, for their excellent organization of the sympo-
siumandtheirsuccessinfindingsourcesforfundingit;
• Abraham Arcavi (Weizmann Institute of Science), Hannah Perl (Israel Ministry
ofEducation),andNormaPresmeg(IllinoisStateUniversity)whohave,together
withtheeditorsofthisvolumeservedasprogramcommitteeforthesymposium;
• InaAvivwhohasbeenindefatigableandeffectiveinmakingsurethatallpartic-
ipants were cared for as best as one could imagine before, during and after the
symposium;
• TheIsraelScienceFoundation,theTrumpFoundation,theChiefScientist’sOf-
ficeattheIsraeliMinistryofEducation,theCenterforAdvancedStudiesinMath-
ematics at Ben Gurion University, the Faculty of the Natural Sciences, and the
FacultyofHumanitiesandSocialSciencesatBenGurionUniversity,allofwhom
haveprovidedgenerousfinancialsupport.
This volume would never have become possible without the symposium. It is an
amplification and expansion of the proceedings of the symposium, on which the
authorshaveworkedduringtheyearfollowingthesymposium.Thisworkwasdone
happilynotonlybecauseoftheimportanceofitscentralquestionbutalsobecause
it was an opportunity finally to present ideas to the public that we have discussed
sooftenwithTed.Morethanthesymposiumitself,inthisway,weowethiswork
to Ted’s passionate concerns. So, as Francis Lowenthal, one of Ted’s old friends,
wroteatthetopofhisowncontributionandthroughoutit,wealsosaywithpleasure,
Thankyou,Ted!
BeerSheva,Israel MichaelN.Fried
TelAviv,Israel TommyDreyfus
Contents
Introduction
1 Mathematics&MathematicsEducation:SearchingforCommon
Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
MichaelN.Fried
DialogueonaDialogue
2 MathematicsandMathematicsEducation:BeginningaDialogue
inanAtmosphereofIncreasingEstrangement . . . . . . . . . . . . 25
MichaelN.Fried
3 SomeofMyPet-PeeveswithMathematicsEducation . . . . . . . . 35
TheodoreEisenberg
4 Mathematics at the Center of Distinct Fields: A Response
toMichaelandTed . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
NormaPresmeg
MutualExpectationsBetweenMathematicsandMathematics
Education
5 MutualExpectationsBetweenMathematiciansandMathematics
Educators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
TommyDreyfus
HistoryofMathematics,MathematicsEducation,
andMathematics
6 HistoryinMathematicsEducation.AHermeneuticApproach . . . 75
HansNielsJahnke
7 ReflectionsonHistoryofMathematics . . . . . . . . . . . . . . . . 89
LuisRadford
ix
x Contents
Problem-Solving:AProblemforBothMathematicsandMathematics
Education
8 ReflectionsonProblem-Solving . . . . . . . . . . . . . . . . . . . . 113
BorisKoichu
MathematicalLiteracy:WhatIsItandHowIsItDetermined?
9 “MathematicalLiteracy”:AnInadequateMetaphor . . . . . . . . 139
E.PaulGoldenberg
10 ReflectionsonMathematicalLiteracy . . . . . . . . . . . . . . . . 157
AnnaSfard
VisualizationinMathematicsandMathematicsEducation
11 Fifty Years of Thinking About Visualization and Visualizing
inMathematicsEducation:AHistoricalOverview . . . . . . . . . 177
McKenzie(Ken)A.Clements
12 ReflectionsonVisualizationinMathematicsandinMathematics
Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
ElenaNardi
JustificationandProof
13 MakingSenseofMathematicalReasoningandProof . . . . . . . . 223
DavidTall
14 ReflectionsonJustificationandProof . . . . . . . . . . . . . . . . . 237
KeithWeber
Policy:WhatShouldWeDo,andWhoDecides?
15 MathematicsandMathematicsEducationPolicy . . . . . . . . . . 261
MogensNiss
16 ReflectionsonPolicy . . . . . . . . . . . . . . . . . . . . . . . . . . 277
NitsaMovshovitz-Hadar
CollaborationBetweenMathematicsandMathematicsEducation
17 MathematicsandEducation:CollaborationinPractice . . . . . . . 299
HymanBassandDeborahLoewenbergBall
18 Reflections on Collaboration Between Mathematics and
MathematicsEducation . . . . . . . . . . . . . . . . . . . . . . . . 313
PatrickW.Thompson
Postscript
19 WeMustCultivateOurCommonGround . . . . . . . . . . . . . . 337
JeremyKilpatrick
Contents xi
AppendixA:HomagetoTedEisenberg
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
TedasAdvisorandColleague . . . . . . . . . . . . . . . . . . . . . . . . 349
TommyDreyfus
ThankYou,Ted! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
FrancisLowenthal
AnnotatedBibliographyofTedEisenberg’sMajorPublications. . . . . 357
TommyDreyfus
AppendixB:ReprintsoftheDialoguesBetweenPresmeg,Eisenberg,
andFriedfromZDM41(1–2)
MathematicsEducationResearchEmbracingArtsandSciences . . . . 361
NormaPresmeg
DialogueonMathematicsEducation:TwoPointsofViewontheState
oftheArt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
TheodoreEisenbergandMichaelN.Fried
TheHarmonyofOpposites:AResponsetoaResponse. . . . . . . . . . 391
NormaPresmeg
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
