Handbook of Cognitive Mathematics

1392 Pages·2022·26.506 MB·Russian

by  Marcel Danesi

Marcel Danesi
Editor
Handbook of
Cognitive
Mathematics
Handbook of Cognitive Mathematics
Marcel Danesi
Editor
Handbook of Cognitive
Mathematics
With264Figuresand34Tables
Editor
MarcelDanesi
Anthropology,VictoriaCollege
UniversityofToronto
Toronto,ON,Canada
ISBN978-3-031-03944-7 ISBN978-3-031-03945-4(eBook)
https://doi.org/10.1007/978-3-031-03945-4
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Preface
Mathematicsisbothacreationanddiscoveryofthehumanmind.Researchhasbeen
showing that mathematical discoveries in general seem to be located in the same
neural circuitry that sustains ordinary language and other symbol systems. Mathe-
matics makes sense when it has meanings that fit our cognitive experiences of the
world – experiences of quantity, space, motion, force, change, mass, shape, proba-
bility, self-regulating processes, and so on. The inspiration for new mathematics
comes from these experiences. Studying the relation between cognition and math-
ematicscomesunderageneralrubric–whichcanbecalledthecognitivescienceof
mathematics,orcognitivemathematicsforshort.
This handbook is intended as a reference volume for cognitive mathematics,
whichcanbedefinedmoreconcretelyasthestudyoftherelationofmathematicsto
other human faculties, from the arts to language. This is of interest to diverse
disciplines, spanning education to computer modeling. Therefore, this handbook
includesthebackgroundworkinthefieldandsectionsdealingwiththeinterconnec-
tionsbetweenmathematicsandotherfaculties.Thechaptersarewrittenbyinterna-
tionallyrenownedauthorswhoareauthoritiesintheirfields.Anoveralldiscussionof
thevolume’sfocusandareascoveredisprovidedintheintroductiontothehandbook.
In contemporary academia, the question of what is mathematics and how it is
learnedhasbeenaddressedthroughavarietyofmethodsinabroadsetofdisciplines.
Building on some of the developments in the different fields, the chapters in this
volume,whenconsideredcumulatively,discusshowmathematicsinvolvesablend
of imagination, abstraction, and notation-making processes – all of which have
profound implications for the teaching and learning of mathematics. In his 2008
collectionofstudiesonmathematicalcognition(MathematicalCognition,Charlotte:
InformationAgePublishing),JamesRoyerpointedoutthatthestudyofmathemat-
ical cognition spanned a broad field of scientific, educational, and humanistic
interests.Theverityofthisobservationisbeingconstantlyevidencedinthestudies
published in diverse journals, which show the truly wide range of approaches to
mathematics cognition, bringing together researchers from diverse fields to cast
shreds of light on this truly fascinating phenomenon from different angles. The
main implication that can be drawn from looking over works in the field is that
mathematics cognition cannotbestudied withinthe confinesofa single discipline.
The main focus seems to be discerning and explaining the neural basis of
v
vi Preface
mathematics. This handbook aims to cast a wide net to the study of mathematical
cognition,bringing together scientists and humanists,so astoenlargethepurview.
Whilethescientificstudyisfundamentallyempirical,theadditionofmorehuman-
istic disciplines to the mix allows the field to enter into more theoretical and
speculative domains, thus perhaps both opening up a debate that is not based on
empiricalissuesalonebutonamorespeculativeground–thusmirroringtheorigins
ofmathematicsinphilosophy.
The study of mathematical cognition really took off after the publication of
Lakoff and Núñez’s controversial 2000 book, Where Mathematics Comes From,
which argued that mathematics is essentially no different from language or other
symbol systems, since both share a basic modality – blending information from
differentpartsofthebraintoproducenovelinformation.Onesimpleverificationof
thisisthefactthatweuselanguagetolearnmathematicsandthatmathematicshas
many structural properties that are linguistic. The most salient manifestation of
blending can be seen in metaphor, which undergirds how we think and learn. The
questionthatLakoffandNúñezaskedwashowmetaphoricalprocessesallowusto
understandcomplexconceptssuchasinfinityandlimits.Ifmetaphorisindeedatthe
coreofmathematicalreasoning,thenitbringsmathematicsdirectlyintothesphereof
culture where it is shaped symbolically and textually. Whatever the truth, it is
obvious that the study of mathematical cognition will benefit significantly by the
participation of humanists and mathematicians in collaboration with the empirical
scientists.
In his groundbreaking 1962 study on the cognitive source of scientific theories,
the American philosopher Max Black argued, before Lakoff and Núñez, that the
genesisoftheoreticalnotionsandframeworksinthesciencesandmathematicswas
not solely the result of scientists deducing them from empirical observations or
experimental results, but also, and primarily, the result of scientists making infer-
encesandconnectionsbetweenfacts,othertheories,andeveneverydayexperience.
Indirectly,Blacklaidthefoundationsforcognitivemathematicswithhisradicalidea
fortheerainwhichitwaswritten.
Intheancientworld, actually, theidea thatmathematics was apart ofabroader
systemofthoughtwasavirtualgiven.Thetransitionofmathematicsfromapractical
counting,measuring,andgenericproblem-solvingcrafttoatheoreticaldisciplineis
traced generally to the emergence of the method of proof. The chapters of this
handbook span the interdisciplinary scope of cognitive mathematics, from the
empirical to the educational and speculative, as well as examining aspects of
mathematical methods and what they tell us about the nature of mathematical
cognition. The objective here is twofold: to show how this line of inquiry can be
enlargedprofitablythroughanexpandedpoolofparticipatingdisciplinesandtoshed
somenewlightonmathcognitionitselffromwithinthispool.
Wehopethatreaderswillfindthishandbookinformative.Webelievethatmathe-
maticians,cognitivescientists,educatorsofmathematics,philosophersofmathematics,
andalltheotherkindsofscholarswhoareinterestedinthenatureofmathematicswill
findsomethingofinterestinthisvolume.Theimplicitclaiminallthestudiesisthatin
Preface vii
ordertopenetratethephenomenonofmathematics,itisnecessarytoutilizemethods
andtheoreticalframeworksderivedfromavarietyofdisciplines.
Toronto,Canada MarcelDanesi
September2022
Contents
Volume 1
SectionI MathematicsandCognition ........................ 1
DonnaE.West
1 TheCognitiveandEpistemicValueofMathematics:Making
theWorldIntelligible–TheRoleofAbduction,Diagrams,and
Affordances ........................................... 9
LorenzoMagnani
2 PeirceandGrothendieckonMathematicalCognition:AMerging
ofthePragmaticistMaximandToposTheory ................ 49
FernandoZalamea
3 LinguisticandVisuospatialChunkingasQuantitativeConstraints
onPropositionalLogic .................................. 67
DonnaE.West
4 BlendingTheoryandMathematicalCognition ............... 89
MarcelDanesi
SectionII Ethnomathematics .............................. 111
MyrdeneAndersonandTodShockey
5 EthnomathematicsandCulturalIdentitytoPromoteCulturally
ResponsivePedagogicalChoicesforTeachersinEarly
ChildhoodandElementaryEducation ...................... 115
SandraWildfeuer
6 BundlesofEthnomathematicalExpertiseResidingwith
Handicrafts,Occupations,andOtherActivitiesAcross
Cultures .............................................. 129
VeronicaAlbanese
ix
x Contents
7 Emic,Etic,Dialogic,andLinguisticPerspectiveson
Ethnomodeling ........................................ 161
MiltonRosaandDanielClarkOrey
8 EthnomathematicsinEducation:TheNeedforCultural
Symmetry ............................................ 191
TamsinMeaney,TonyTrinick,andPiataAllen
9 EthnomathematicsAffirmedThroughCognitiveMathematics
andAcademicAchievement:QualityMathematicsTeaching
andLearningBenefits ................................... 221
JenniL.Harding
SectionIII CognitiveNeuroscienceofMathematics ............ 251
RolandH.Grabner
10 DevelopmentalBrainDynamics:FromQuantityProcessingto
Arithmetic ............................................ 257
StephanE.Vogel
11 NeurocognitiveFoundationsofFractionProcessing ............ 289
SilkeM.Wortha,AndreasObersteiner,andThomasDresler
12 IndividualDifferencesinMathematicalAbilitiesand
Competencies ......................................... 317
SaraCaviola,IreneC.Mammarella,andDenesSzűcs
13 Mind,Brain,andMathAnxiety ........................... 349
RachelPizzie
14 NeurocognitiveInterventionstoFosterMathematical
Learning ............................................. 385
KarinKucianandRoiCohenKadosh
SectionIV BiologicalApproachestoMathematics ............. 413
DanVilenchik
15 TheNeurobiologicalBasisofNumericalCognition:
Decision-MakingProcessesasaNewLineofInquiry .......... 415
LitalDachesCohenandOrlyRubinsten
16 TrustingYourOwnEyes:VisualConstructions,Proofs,and
FallaciesinMathematics ................................. 431
MargoKondratieva
Contents xi
17 NumericalAbilitiesinNonhumans:ThePerspectiveof
ComparativeStudies .................................... 469
RosaRuganiandLuciaRegolin
18 ExecutiveDysfunctionAmongChildrenwithADHD:
ContributionstoDeficitsinMathematics .................... 501
LaurenM.Friedman,GabrielleFabrikant-Abzug,SarahA.Orban,
andSamuelJ.Eckrich
SectionV MathematicsandtheArts ........................ 539
StéphanieWalshMatthews
19 TheChallengeofFormalLogicsandMetaphysicalSystemsto
Semiotics ............................................. 543
EeroTarasti
20 CulturalSymmetry:FromGroupTheorytoSemiotics ......... 573
JaminPelkey
21 Fractals,Narrative,andCognition ......................... 595
RichardRosenbaum
22 VisageMathematics:SemioticIdeologiesofFacialMeasurement
andCalculus .......................................... 617
MassimoLeone
Volume 2
SectionVI LearningandTeachingMathematics .............. 643
DraganaMartinovic
23 TheRolesofIntelligenceandCreativityforLearning
Mathematics .......................................... 647
MichaelaA.MeierandRolandH.Grabner
24 GesturesinMathematicsThinkingandLearning ............. 685
OrnellaRobutti,CristinaSabena,ChristinaKrause,CarlottaSoldano,
andFerdinandoArzarello
25 TeachingandLearningAuthenticMathematics:TheCaseof
Proving .............................................. 727
AndreasJ.Stylianides,KotaroKomatsu,KeithWeber,andGabrielJ.
Stylianides
26 WhyAreLearningandTeachingMathematicsSoDifficult? ..... 763
AlanH.Schoenfeld