Mathematics and Virtual Culture: An Evolutionary Perspective on Technology and
Mathematics Education
Author(s): David Williamson Shaffer and James J. Kaput
Source: Educational Studies in Mathematics, Vol. 37, No. 2 (1998 - 1999), pp. 97-119
Published by: Springer
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DAVIDWILLIAMSONSHAFFERandJAMESJ. KAPUT
MATHEMATICSAND VIRTUALCULTURE:AN EVOLUTIONARY
PERSPECTIVEON TECHNOLOGYAND MATHEMATICS
EDUCATION'
ABSTRACT. This papersuggests thatfrom a cognitive-evolutionaryperspective,computationalmedia are qualitativelydifferentfrom manyof the technologiesthathave promised
educationalchange in the past and failed to deliver. Recent theories of humancognitive
evolutionsuggest thathumancognition has evolved throughfour distinctstages: episodic,
mimetic, mythic,andtheoretical.This progressionwas drivenby threecognitive advances:
the ability to 'represent'events, the developmentof symbolic reference,and the creation
of external symbolic representations.In this paper,we suggest that we are developing a
new cognitive culture: a 'virtual' culture dependenton the externalizationof symbolic
processing. We suggest here that the ability to externalize the manipulationof formal
systems changes the very natureof cognitive activity.These changes will have important
consequences for mathematicseducationin coming decades. In particular,we argue that
mathematicseducationin a virtualcultureshouldstriveto give studentsgenerativefluency
to learn varieties of representationalsystems, provide opportunitiesto create and modify
representationalforms, develop skill in making and exploring virtual environments,and
emphasize mathematicsas a fundamentalway of making sense of the world, reserving
most exact computationand formalproof for those who will need those specializedskills.
1. PRINTING PRESS OR STEREOSCOPE: AN INTRODUCTION
The historyof the printingpress and its transformativeimpacton Western
cultureis a well-known story.The introductionof the printingpress, and
with it mass-producedand widely-availablebooks, led to the standardization of vernacularlanguages, the spreadof literacy,the developmentof
new literaryforms (includingthe 'novel'), and ultimatelycontributedto
the growthof a middle class and with it many of the social and economic
transformationsof the last few centuries(McLuhan,1962).
A slightly less well-known episode in the history of technology is the
stereoscope. The stereoscope was first described in 1838 as a means to
capture three-dimensionalimages by directing separatephotographsof
the same scene to each of the viewer's eyes, thus simulatingbinocular
vision. The trade in stereoscopic images boomed after Queen Victoria
viewed them at the Crystal Palace exhibition in 1851; by 1853 a New
YorkTribunewriterproclaimedwith excitementthat 'the day must soon
Ai EducationalStudies in Mathematics 37: 97-119, 1999.
?C)1999 KluwerAcademicPublishers. Printed in the Netherlands.
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DAVIDWILLIAMSONSHAFFERAND JAMESJ. KAPUT
come when nearlyall importantphotographicpictures... will be produced
double, that is, by couples, [for] stereoscopicreproduction,in all the exact truth of living nature' (New YorkDaily Tribune,1853). But history
proved this journalistmisplaced in his enthusiasm,as still photography
and stereoscopic images were replacedfirst by motion picturesand later
by television and video as the dominantvisual medium. Stereoscopicimages today are the preserveof collectors andspecialists,or relegatedto the
statusof children'stoys.
We raisethese examplesbecausediscussionsof computationalmediain
educationoften takeon the tone of the enthralledjournalist(Oppenheimer,
1997), with advocates claiming that the introductionof computersinto
educationwill bringprofoundandlastingchanges-andarguingthatputting
computersand internet-accessinto schools should be our highest priority
in educationalpolicy. At the same time, skeptics look to recent history
and suggest that technologies come and technologies go without making
any dramaticimpacton educationalpractices.These skepticspointout that
some of the same kindsof rhetoricthatwe hearaboutcomputerstodaywas
used in the past about motion pictures,radio, film strips, television, and
other 'new media' (Cuban, 1986; Tyack & Cuban, 1996; Oppenheimer,
1997).
This paper looks at the introductionof computationalmedia from an
even longer-termperspective.Recentworkby psychologistMerlinDonald
(1991) arguesthathumancognition has developedover evolutionarytime
througha series of four distinct stages: startingwith episodic (ape-like)
memoryand progressingthroughmimetic,mythic, and theoreticalrevolutions. This papersuggests that viewed in termsof the growthof symbolic
understanding,computationalmedia are the manifestationof a fifth stage
of cognitivedevelopment-andthus in theirlong-termimpacton ourculture
and society more like the profoundly-transformative
printingpress than
the relatively-insignificantstereoscope.As mathematicseducatorswe are
particularlyinterestedin the critical role that mathematicshas played in
the developmentof this new stage of cognition,and on the implicationsof
this new cognitive culturefor mathematicslearning.
2. FOUR STAGES OF MENTAL EVOLUTION: AN OVERVIEW
In his book Origins of the ModernMind (Donald, 1991), Merlin Donald
arguesthatanatomicalevidence of humanevolutionarydevelopment(both
currentand in the fossil record) suggests that human culture has gone
throughfour distinct stages of development.2Donald suggests that each
of these stages of cultural developmentwas driven by a specific cognit-
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MATHEMATICSAND VIRTUALCULTURE
99
ive advance, and that these changes in cognition led to changes in brain
developmentas well as new kindsof communicationandsocial interaction.
Thefirst stage Donald outlines is essentially that of primate(ape-like)
cognitionwith origins amongearly primatesmorethanthreemillion years
ago. This stage is based on 'episodic' thought,which Donald describesas
thinkingbasedon literalrecallof events.Apes can rememberdetailsof, for
example, a social interaction,and can even recall those details in context
- thus an ape might 'remember'that a largermale is dominantbecause
he can recall a fight where the dominantmale won. But, as Donald and
studiesof primatebehaviormakeclear,apes do not representeventsin any
way. They do not attachlabels to events, meaningsto events, or generalize
from events. They do not process events otherthanstoringtheirimages in
episodic memory.Donald arguesthatapes who have learnedrudimentary
sign languageare essentially storingand using the signs in much the same
way as they wouldprocessany kindof conditioning- they remembersigns
as responses leading in certain circumstancesto pleasure or away from
pain (p. 154).
Episodic cognition provided a basis for social interactionby giving
early hominids the ability to recall previous events and respond accordingly. This rudimentarysocialization was extended by the development
of the fundamentalability to 'represent'events datingfrom homo erectus
about 1.5 million years ago. Donald describes this as 'mimesis,' or 'the
ability to produce conscious, self-initiated,representationalacts that are
intentionalbut not linguistic' (p. 168), comprisingthe second stage. For
example, following the gaze or pointing gesture of anotherrequiresan
understandingthat their gestures are referringto something of interest.
Or, more dramatically,reenactingor replayingevents using objects (like
a child who spanksa doll aftergetting a spankinghim or her self) shows a
basic ability to process events and to communicateabout them to oneself
and to others.
Donald arguesthat this ability to representevents was not (and is not)
dependenton language. The morphologicalchanges requiredfor the development of speech are quite dramatic,and thereforeunlikely to occur
without some evolutionarypressurefavoringthe ability to communicate
using language. Donald believes that the evolution of language was dependenton this priorcognitive development:namely,the developmentof
symbolic reference.Donald distinguishesiconic representation,wherethe
representationsharessome propertywith the thing being referredto, from
symbolic representation,where the symbol can be any arbitrarygesture,
sign, or sound.3 The first symbols were probably,accordingto Donald,
standardizedor ritualizedgestures.Fromsimplevocalizationto morecom-
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DAVIDWILLIAMSONSHAFFERAND JAMESJ. KAPUT
plex articulation,languagedevelopedas the most efficient system for creating and communicatingsymbols of the world. In otherwords, the social
need to communicateideas (symbolic representationof events) drove the
evolutionarydevelopmentof language,not the otherway around(p. 255).
The development of language markedthe arrivalof a 'mythic' culture
based on narrativetransmissionof culturalunderstanding,comprisingthe
third stage beginning about 300,000 years ago (see also Bruner, 1973,
1986, 1996).
In her recent book on language and development,KatherineNelson
(1996) accepts Donald's categorizationof stages of mental development,
but argues that in individual (as opposed to evolutionary)development,
the relationshipDonald describesbetween representationand languageis
reversed.That is, Nelson arguesthat languagedrives individualdevelopment of symbolic competence. Language provides an external structure
thatscaffolds a child's abilityboth to representevents, andlaterto develop
narrativeand categorical understandingof the world. Although Nelson
does not say so explicitly, this reversal makes sense for a child who is
raised in a culturethat has already developedlanguage.4In otherwords,
it seems reasonablethat evolutionarydevelopmentof a cognitive ability
and individualdevelopmentof the same ability might differ - and that
the evolutionarydevelopmentof a new form of representationmight have
profounddevelopmentalconsequences.We will returnto this idea laterin
the discussion.
Thefourth stage Donald identifiesis that of 'theoreticculture,'or culture based on writtensymbols and paradigmaticthought.Again, Donald
arguesthatthe principalchange here was in the developmentof new cognitive ability ratherthan a new means of expression - in this case, the
need to work with complex phenomenadrovethe developmentof external
representationsbeginning 30-50 thousandyears ago. The record-keeping
needs of commerce and astronomydrove the creationof externalsymbol
systems (p. 333ff), of which mathematicalnotationswere probablythe first
(Donald, 1991; p. 287; see also Schmandt-Besserat,1978, 1992, 1994).5
The existence of externalrepresentations,accordingto Donald, made it
possible for humans to begin reflecting on the interrelationshipsamong
recordedideas in an analyticfashion. Donald refersto this use of notation
to augmentthinkingas the 'externalmemoryfield' or EXMF.The accumulation of recordedknowledge,which Donaldcalls 'externalsymbolic storage' or ESS, providesa largercontextin whichanalyticthinkingtakesplace.
Donald suggests thatmodernscientificculturedevelopedfrom and depends on the existence of externalnotationsfor thinkingand of external
recordsfor ideas. Muchof schoolingis aboutlearningto access partsof the
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MATHEMATICSAND VIRTUALCULTURE
101
culturalrecordandto manipulatethem using the tools of externalworking
memorysuch as writingand mathematicalnotation(p. 320). Ourtheoretic
culture is dependenton large-scale storage of informationas a kind of
databasefor analyticthinking,as well as on a set of externaltools thathelp
us control the flow of this informationto our biological processors- that
is, to our minds, which evaluateand transformthatinformation(p. 329).
Donald arguesthat all of these ways of thinking- episodic, mimetic,
narrative,and theoretic- exist in our minds simultaneously,and that we
move among them and use them in a fluid way. So, for example, a songand-danceinvolves both mimetic and linguistic representation,often in a
mythiccontext.The processof scientificdiscoveryis a dialog betweenparticular(episodic)eventsandgeneral(theoretic)models.Donaldrefersto this
combiningandshiftingof representationalperspectivesas the 'hybridmind.
3. INDIVIDUAL AND CULTURAL DEVELOPMENT: FOUR THEMES
There are four themes that emerge from this summaryof Donald's work.
The firstis that,at the evolutionarylevel, changesin cognitiondrivechanges in representationratherthan the other way around.At each stage in
development,a new way of thinking about (modeling) the experienced
world graduallycreatesnew means of instantiatingthatmodel. Language
evolves as a consequenceof symbolicthinking,not the otherway around;a
point of view consistentwith the deep analysesofferedby Deacon (1997).
The second idea that emerges from Donald's work (or, at least from
this overview of it), is that our current,'theoretic'culturedependson the
externalstorageof information.We use externalmediato recordideas and
to act as an externalmemory buffer while we are processing ideas. The
generation,translation,andtransformationof ideas aredone internally,but
depend on the presence of sharedexternal informationand the external
tools to augmentour workingmemories.6
A third idea from Donald's work is that this theoretic culture based
on external storage of informationarose, in large part, from the need to
deal with quantitativeinformation(Schmandt-Besserat,1978, 1992, 1994;
Donald, 1991). Whetherit was records of harvests,of business transactions, or of the movementsof celestial bodies, the storageandcomputation
of numericalinformationwas a driving force in the developmentof our
currentculture.
A fourthand final point worthnoting here comes from Nelson's study
of Donald'stheoriesin the developmental(ratherthanevolutionary)realm.
This is the idea that new cognitive processes affect the way older modes
of thoughtemerge in individualdevelopment.The 'hybridmind' does not
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DAVIDWILLIAMSONSHAFFERAND JAMESJ. KAPUT
102
just referto the interactionof modes of thinkingwithin an individual.The
presence of variousmodes of representationwithin a culturechanges the
way we learnto understandour worlds as individuals.
The remainderof this paper explores these ideas in relation to our
posited fifth stage of the developmentof humancognition.Computational
media make it possible to externalizenot only information,but also the
processing of information.This has profoundimplicationsfor the nature
of humancognitionin generaland mathematicslearningin particular.7
4.
INFORMATION
AND PROCESSING:
A NEW MEDIUM
Donald is quite explicit in his claim that the major cognitive development involved in the creationof theoreticcultureis the appearanceof the
'externalmemoryfield' as a externalmemoryloop-thatis, as an externalization of workingmemory(p. 329ff).8 Cognitivetheorists,and particularly
those whose informationprocessing perspectivematches Donald's analysis (see, e.g., Block, 1981; Akin, 1986; Rowe, 1987; Kosslyn & Koenig,
1992; Simon 1996), refer to short term or working memory as a kind of
scratchpad or datastoragebufferfor mentalprocessing.Workingmemory
holds pieces of informationfor processing,but is not necessarilythe part
of the mind thatdoes the actualtransformationof information.
Whetheror not one believes that mentalactivity can be as neatly segmentedas such an informationprocessing perspectivesuggests (andwe do
not), it is clear thatDonald is arguingthattheoreticculturedependsnot on
externalprocessingof information,buton externaldevices to storeinformation as a substitutefor long- and short-termmemory.That such storage
has cognitive consequences is clear. When a writermakes an outline for
a paper,it helps organize his or her thinking.But the transformationof
outline into text is still at every level a function of the workingof his or
her biological mind.
Or is it? Clearly, when a person is working with pen and paper,the
externaltools are recordingthe productsof his or her thinking.Thatthese
inertproducts(i.e., the outlineor emergingtext) feed back into thinkingis
obvious. But when a personwriteswith a wordprocessor,before finishing
a paper he or she can also run it througha spell-checkerand grammarchecker.These programsalterthe text (or more accuratelyin most cases,
make suggestionsfor alteringthe text) based on rules of standardusage for
formal spelling and writing. The computerrunninga spell-checkeris not
just recordinga person's thinkingin a loop of productionand expression
of thought. It is actually performingsome of the functions that a mind
might take on in a similar circumstance- in this case, the functions of a
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MATHEMATICSAND VIRTUALCULTURE
103
(particularlyliterally-minded)editor.It is truethatthe authormightbe able
to do this editing him- or herself, but the point is that he or she does not
have to. The computernot only records the writer'sthinking,it actually
does some of the informationprocessing; similarly,a calculator- either
numericor symbolic - processesinformationoutside the user's head.
To take a more dramaticexample, when a researcherperformsa statistical analysis,he or she may ask the computerto computea multivariate
regression,or perhapsa principalcomponentsanalysis, on a set of data.
The softwareanalyzes the raw dataand, using an iterativetechnique,produces outputthat the researchercan use to understandthe structureof the
datain question- perhapseven by producinga highly visualrepresentation
of the newly organizeddata. What makes this example so compelling is
that for all practicalpurposes,the researchercould not performthe same
manipulationof that information.This is true partlybecause of the time
it would take to run the same calculationsby hand. But a researchercan
even use software to compute statistics that he or she knows how to use
and interpretbut does not know how to computeby hand.
In all of these examples(and thereare many otherspossible), the computeris doing somethingquite differentfrom Donald's model of the 'external memoryfield.' The computeris, essentially, acting as an external
processing field, takinginformationin one form andreturningit in another
form without action by the writer(or researcher)in the interim.Whether
the computer'understands'the informationin any sense is irrelevanthere.
The point is that a person can use the computeras a tool to augmentor
replacenot only memory,but substantialmentalprocessingof information
as well.
In both of these examples, the computeris externalizingmental processing by addingalgorithmsto an underlyingstoreof information.Thatis,
the computerhas an externalstore of information,and a set of procedures
for actingon thatinformation.In the case of the spell-checker,for example,
the computerhas the equivalentof (partsof) a dictionary.But it also has a
set of rules for comparingwords in a text to entriesin the dictionary.As
is the case with a traditionalprinteddictionary,it can 'remember'- but it
can also 'act.'9
To take a concreteillustration,if a great chef dies without ever telling
anyoneaboutor recordinghis or her secretrecipes, the meals die with him
or her. No one can recreateexactly the dishes he or she cooked. If he or
she writesdown the recipes,then as long as the book exists, someone (with
some degreeof culinaryskill) can recreatethe dishes. But if thatchef were
also a master engineer, he or she could (conceivably) create a machine
or machines that could reproducethe actualdishes - that is, it recordnot
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DAVIDWILLIAMSON
SHAFFER
ANDJAMESJ. KAPUT
only the informationfor the food, butactuallyreplicatethe processesfor its
production.The book of recipeswould be an exampleof Donald'sexternal
symbolic storage.The machinefor food productionwould be more like a
moderncomputer.10
This suggests that we are on the verge of a new culture in Donald's
sense of the term,one dependenton the externalizationof pieces of mental
process as well as on externalizationof pieces of representation.Computationalmedia make it possible to externalizealgorithmas well as information.11According to Donald, the developmentof an ability to represent
events createda 'mimetic' culturebased on communicationmediatedby
the exchange of physical gestures. The additionof language made possible a 'mythic' culture based on the exchange of narrativestories. The
creationof writtensymbols led to a 'theoretical'culturebased on external
symbolic storage.Continuingthe progression,we suggest thatthe computationalmedia are in the process of creatinga virtualculturebased on the
externalizationof algorithmicprocessing.12
5. VIRTUAL CULTURE AND FORMAL SYSTEMS: THE POWER OF THE
EMPTY SIGN
Donald's thesis suggests that we should look for the roots of the developmentof a fifth stage of cognitionnot in the mediaof representationbut in a
changein the way we representor model our experienceof the world.That
is, we shouldunderstandthe culturaldevelopmentof computationalmedia
by looking at the cognitive processes that made possible their creation.
The developmentof computationalmedia depends on three factors: the
existence of discretenotations,the creationof rules of transformation,and
an externalsystem capableof autonomouslyapplyingthose rules.
Nelson Goodman (1968; see also Gardner,1982) argues that a principal feature of any notation system is the extent to which the symbols
it uses are both disjointand well-defined,as opposed to syntacticallyand
semanticallydense. Put in less jargonyterms,symbol systems differin the
extent to which a given symbol can be precisely identifiedand precisely
interpreted.The canonical example is a spiked line, which might be the
trace of a function on a graphor a drawingof a line of hills. In one case,
the meaning of the markwould be unambiguousand its translationinto a
representationof numericalvalues clear. In the other,the line would hint
at spacespresentas well as spaces missing andhave shadesof meaningfor
the viewer.
Perhapsthe first, and certainlythe most well-explored,system of notation with precise symbols andunambiguousmeaningsis the representation
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MATHEMATICSAND VIRTUALCULTURE
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Figure 1. Illustrationfrom USA Todayof a graph that uses lines in both pictorial and
notational modes.
of numbers,particularlyof whole numbers.In the case of integers,from a
cultural,if not a psychological perspective,a specific symbol (or combination of symbols) represents a specific number and no other. The symbols
of the numbersystem are also one of the firstinstancesof a system of notation with rules of transformation.Algorithmsfor additionand subtraction
of numeralshave existed for thousandsof years. The process of addition
(or subtraction,multiplication,etc.) can be representedby a set of rules
for manipulating strings of numerical symbols (digits). Terminology from
our most current algorithms like 'carry the one' or 'borrow from the next
column' reflectthe extentto which we thinkof the processes of arithmetic
as a set of rules for manipulatingnumerals.The base-tenplaceholdersystem of numeralsand the algorithmsbuildupon it have been a criticalfactor
in the developmentof our culture(Swetz, 1987).
A prodigiousadvancein the developmentof mathematicswas the creationof another,moregeneralandthereforemorepowerfulset of algorithms
for representingand manipulatingquantitativerelationships:namely, the
developmentof algebraand the rules for manipulatingalgebraicsymbols
to solve equations,transformcharacterstringsinto one or anothercanonical form, and so on (Bochner,1966).
In even a relatively simple 'toy' problem, such as the one shown in
Figure 2, we can see how an algebraicsolution representsa situationas a
set of equationswithin a notationsystem and then applies transformnation
rules to the equationsto produce a solution. As the transformationrules
are being applied,in many cases the intermediatestates of solution do not
make much sense in terms of the original problem(note the bold cells in
Figure 2). 13 The rules of transformationoperateon the equations,not on
the situationthe equations represent.In Bruner'sterms, the symbols are
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DAVIDWILLIAMSONSHAFFERAND JAMESJ. KAPUT
Equation
Transformation
Interpretation
Rule
There were as many chairs as boys
C=B+G
and girls at the party.
4T=C
There were 4 chairs per table.
T/2 - 1 = B/8
When the boys sit 8 to a table, they
take up one less than half of the
tables.
T - 2 = B/4
T = B/4 + 2
Multiply both
If the boys sit 4 to a table, they use
sides by 2
two less than the numberof tables.
Add two to
If the boys sit 4 to a table and use 2
both sides
more tables for holding presents, that
uses up all of the tables.
4 (B/4 + 2) = C
Substitution of
You take four times the number of
equations
groups you get when you put the
boys in groups of 4 and add two
more groups, then that total number
of groups will be equal to the total
number of chairs at the party.
B+ 8=C
B+ 8=B+G
8=G
Distributive
The number of chairs is eight more
property
than the number of boys.
Substitution of
The number of boys and girls is eight
equations
more than the number of boys.
SubtractB from There were 8 girls at the party.
both sides of
the equation
Figure2. Solving a problemusing the notationand transformationrules of algebra.Note
that the intermediatesteps of the solution (see the bold area) 'make sense' as transformations of discrete symbols ratherthan as statementsabout the original problemsituation.
Also notice thatin the last step, the variable'B' is canceledfromboth sides of the equation,
giving us additionalinformationthat the numberof girls at the partyis not dependenton
the numberof boys at the party.
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MATHEMATICSAND VIRTUALCULTURE
107
being treatedas 'opaque:'thatis, they act as autonomousobjectswith their
own identityandrules of transformation.This is differentfrom a use based
on what the symbols stand for, which Brunerrefers to as 'transparent'
(1973).
The combinationof a discrete (i.e., unambiguous)notationand set of
transformationrules can be thought of as a formal system. Or, perhaps
more precisely, as the core of a formal system. Euclid created the first
known example of a set of explicitly defined objects and the relations
on them. But Euclid (and the rest of the mathematicalcommunity for
centuries) believed that this system was an ideal representationof some
aspectof the real world.It was not untilthe developmentof non-Euclidean
geometriesthat this notion was seriously questioned.Among many other
events, this helped lead to a twentiethcenturycrisis in the foundationsof
'truth'in mathematicsthat, in turn,led to a genuinely 'formalist'view of
mathematicsas a domain where symbols and actions on them no longer
needed concretereferents.
Davis and Hersh (1982) describe in some detail the developmentof a
formalistphilosophyof mathematics.Briefly,the formalistposition is that
mathematicalinquiryis, at heart,a 'game.' In this game, we define a set of
symbols, a set of legal stringsof symbols (axioms), and a set of rules for
manipulatingthose symbols. The game is to determinewhatpossible wellformedcombinationsof symbols (theorems)can be madefromthe starting
set and the rules of combination.14 What is particularlyimportantto note
aboutthis vision of mathematicsis thatthe symbolsdon't necessarilyrefer
to any specific externalreferent.I can interpret'line' as having a meaning
in a plane, on a sphere,or as an abstractentity with no physicalor conceptual equivalent.One might define an operationon a set of symbols in such
a way thatyou mightrecognizethe set of symbols andoperationas, say, an
abstractsemigroup.But whetheror not it is recognizedas a conceptually
familiarobject is independentof its statusas a formallydefinedsystem.
A formal system, then, is an arbitrarybut well-defined set of symbols and, most importantly,rules of transformationon those symbols.15
A critically importantfeature of such systems is their operativenaturethe existence of internallycoherent rules for transformingthe allowable
('well-formed') symbols into othersymbols.
The power of computationalmedia is in using such formal systems
to model aspects of the experiencedworld.16 In Donald's analysis, mimetic culturedevelopedfromthe abilityto representthe worldusing iconic
gestures. Narrativeculturedeveloped from the ability to use abstract(arbitrary)symbols for representingaspects of the world. Theoreticculture
developed from humans' ability to use symbols to refer to other symbols
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DAVIDWILLIAMSONSHAFFERAND JAMESJ. KAPUT
using an externalmedium.In our extension of this scheme, the next stage
of algorithmicor virtualculturedevelopsfromthe ability(1) to use operative systems of symbols thatdo not have fixed externalreference,and (2) to
effect autonomousoperationson those symbols in a dynamicmedium.'7
One might object at this point that the symbols the computeruses are
not arbitrary- that in fact they do refer to something particularin the
world, like the array of pixel values in a computer graphics image or
the entries in a balance sheet createdin a spreadsheet(Newell, 1980). It
is, of course, true that in many cases the symbols on which a computer
is operatinghave come from the externalworld. And in many cases, the
symbols thatresultfrom processingwill be interpretedas havingmeaning
in the real world - as, for example, would be the case in a simulation
program(Smith, 1996). And, of course,most such systemsarisefrommore
concretelyreferencedsystems (e.g., the matrixsystem mentionedbelow).
But the key idea is thatit doesn't matterwhetherthe symbols referto any
particularthing or not. The computationis the same whether the image
startsas a randomassortmentof colors or a digitizedimage of your cousin
Bert.The cognitiveleap is in thinkingabout 'processes'on theirown, with
an underlying set of symbols as merely a representationof unspecified
inputsandoutputsof the process- thatis, withoutreferenceto any specific
domain,concreteor abstract.18
One might also complainthat we have not said anythingin particular
about the machine that uses this formal system to externalizeprocessing
of information.Just so. There are many media with which one can externalize the storage of symbols, and while one of them (alphabeticwriting)
has been particularlypowerful and influentialhistorically,the key developments in theoretic culture are the cognitive causes and cognitive consequences of the externalizationof memory.The contemporaryhardware
and softwareof computersare not the only way that one can externalize
process as describedabove. For example,the formalsystem of linearmatrix algebraover some field can be used as a non-computerizedsystem for
externalizingpieces (albeit small ones) of mental processing requiredto
solve systems of linear equationswith coefficients in the field - you can
solve a systemof equationsby following a simplerseries of transformation
rules on the 'equivalent'matrix.Anotherparticularlysimple example is
the abacus, which involves purelyphysical actions on a highly structured
physical system to effect arithmeticoperations- with a human partner.
Whatthe computerpresentsis a particularlyeffective - and thereforeparticularlytransformative- instanceof this underlyingcognitive revolution
thatenables autonomous processing.
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Obviously, the fact that a mathematicalmodel or system might have
more than one interpretationin the 'real world' does not mean that relationships within the model are arbitrary.What is significantis that, over
the past several hundredyears, modem mathematicshas created a way
of thinkingaboutrules of transformationseparatefrom any particularset
of symbolic references.19A 'formal system,' describes a set of abstract
relationshipsand proceduresthat can producea specified result based on
any set of objects or symbols that fit its initial conditions. The ability to
specify explicitly the rules of such systems makesit possible to externalize
the processing as well as the storage of information- and thus make a
virtualculturepossible.
6. MATHEMATICS AND VIRTUAL CULTURE: SOME IMPLICATIONS
One strikingfeatureof this descriptionof virtualcultureis thatlike theoretic culture,the impetusfor a new cognitivemode came from the world of
mathematics.If theoreticculturehad its roots in the need to recordquantitativeinformation,then virtualculturehas its roots in the need to conduct
more and more complex processingof informationusing formalmathematical systems.20Perhapsmore interesting,though, is an examinationof
the consequences- andparticularlythe consequencesfor mathematicsand
mathematicseducation- thatcome aboutas a resultof the developmentof
externalizablealgorithmicthinking.
6.1. New representationalforms
Papertand Kaput(Papert,1980; Kaput, 1992) among othershave written
about the way in which computationalmedia make it possible to externalize algorithmsand thus make processes of thinkingavailableas explicit
objects for reflection.Whetherthis takes place throughthe 'construction
by example' of scriptsfor automatingactions(as in the Geometer'sSketchpad), throughthe explicit programmingof procedures(as in Logo), or
throughthe recordingof editablealgorithmsfrom actions on physical manipulatives(Kaput, 1996a), the existence of an externalrepresentationof
an algorithmthat can be built, tested, discussed, and changedby students
makes it possible to talk aboutmentalactivitiesthatwere once difficultto
describe,much less investigate.
Computationalmedia also create new representationalforms, such as
dynamicgeometryenvironmentsor manipulableCartesiangraphsseparate
from algebraicnotations(see Kaput& Roschelle, 1998). Such new representations enable students to work on problems using differentcognitive
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modes, allowing students,for example, to take more concreteapproaches
to abstractproblems (see Turkle & Papert, 1990; Papert, 1993). In general, we might say that computerswiden the availablerange of different
approachesfor generating,collecting, processing,andinterpretinginformation. One can solve a problem of linear relations using algebra and a
pencil. Or find a graphic solution using graphingcalculator.Or use iterated estimation with a spreadsheet.Or use a geometric representation
in a dynamic geometryenvironment.Or use a motion simulation.Or use
symbolic manipulationsoftware(see Fey, 1989).
This representationalpluralism (see Turkle & Papert, 1990; Papert,
1993) suggests that one of the key changes in mathematicseducationwill
be a move away from the traditionalfocus on algorithmicfluencyin a few
extremelyconcise and notationallycompressiverepresentationalsystems
(notablyarithmeticandalgebra).The pedagogyof mathematicsin a virtual
cultureshouldlogically shift towardsfluencyin representingproblemsituations in a varietyof systems, and towardsstudents'ability to coordinate
among representationsas well as create and interpretnovel ones. Kaput
(1986) has suggested in earlier work that one of the importantfeatures
of computationalmedia in mathematicslearning is their ability to help
studentssee the relationshipamong differentrepresentationsof the same
mathematicalsituation. Others have similarly suggested that mathematics is (or should be) more about the ability to move among and use a
variety of representationsthan about facility in performingarithmeticor
algebraicmanipulations(Confrey & Smith, 1994; Davis & Hersh, 1982;
Yerushalmy,1991).
The associationof 'mathematics'with 'fluencyin arithmeticandalgebraic algorithms'makes perfect sense in the context of a theoreticculture.
Theoretic culture is about using external symbolic memory to support
complex mental processing.Thus doing mathematicsin theoreticculture
is clearly aboutdoing the manipulations(whethersimple additionof numbers or complex integrationby parts)that move from one characterstring
(externalsymbolic representation)to the next. Virtualculture,on the other
hand, is about off loading such symbolic processing from the biological
mindto some externaldevice. Froma virtualperspective,the mathematics
is not in performingthe formalmanipulations,but in imaginativelyframing the problemin a way that uses clear and compelling representations,
often interactively,leading to efficient and convincing solutions through
interpretationof externalaction by technologicaltools.
To take a different example, from a theoretic perspectivereading is
not about committinga text to memory.That is the role of the external
symbolic storagesystem. Readingis aboutdecoding a text, makingsense
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of it, andbeing able to use the informationit containsin a meaningfulway.
A curriculumthatfocused for yearson students'abilityto memorizeprose
would be seen as pedagogicallybackwardin a theoreticculture.Similarly,
from a virtual perspective, mathematicsis not about calculations. That
is the role of the external symbolic processing system. Mathematicsis
about understandinga problem,representingit in an externalprocessing
system, and being able to use the informationproducedby the external
calculationsin a meaningfulway - as RichardHammingput it: 'The purpose of computationis not numbers,but insight.' Curriculathat focus on
students' ability to performroutine calculations are thus regressivein a
virtualculture(see also Dorfler,1993). This is not to say thatstudentsin a
virtualcultureshouldnot learnto do some calculationsby handor in their
heads- just as studentsin a theoreticculturestill commitsome particularly
importantideas to memory,and still learnto do some calculationswithout
pencil and paper.As Donald suggests, the mind is a hybrid of its many
cognitive cultures.But the balance of mathematicseducationin a virtual
culture would reasonably shift towards an emphasis on representational
versuscomputationalfluency.21
6.2. New pedagogical approaches
Another implicationof virtual culture for mathematicseducation is that
computationalmedia make it possible to think of mathematicseducation
in a much more inductiveand naturalisticway. This is not to say that the
concept of mathematicsas a deductive enterpriseshould be abandoned
- althoughas Davis and Hersh (1982) point out, mathematics'claim to
deductive'truth'is increasinglyshaky.Rather,it suggeststhatstudentscan
profitablyexperience mathematicsas an experimentalenterpriseleading
to the need and desire for more formaljustification.That is, they can encountermathematicalideas in a way thatlets them manipulatephenomena
and expose possible underlyingmathematicalrelationships.For example,
early work by Yerushalmysuggests that studentswho used the Geometric
Supposerevolved a significantlystrongerexpressed need for proof than
did theirmoretraditionallytaughtpeers (1986).
It becomes possible, in otherwords,to put a studentin a highly interactive mathematicalsituationandlet him or her 'play' as a way of developing
mathematicalunderstandingbecause the feedbackloops are much tighter
and less dependenton students' own ability to transformsymbols. It is
also possible to put a studentin a manipulablesimulationand supportthe
student in extending his or her understandingof the simulatedsituation
and of the underlyingmathematics.Just as Nelson describes the power
of languageto drive developmentonce it exists in the externalculture,so
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algorithmicprocessing can drive understandingonce it exists as a common tool in a child's environment.The ability to think about and with
'live' externalrepresentationsof processes can scaffold the development
of mathematicalunderstanding,which was a fundamentalpointof Papert's
Mindstorms(1980).
Such mathematicalplay might be scientific and experimental,as describedabove. Or it might be artisticand expressive.The ability of computationalmedia to representartistic expression within formal systems
(as, for example, in image-manipulationtools such as Adobe PhotoShop)
meansthat studentscan exploremathematicaland aestheticconcernssimultaneously (Shaffer, 1997). Part of the power of a computeris that a
sufficientlyfast formalprocess can take on artisticdimensions- just as a
sufficientlyfast staticimage can take on dynamicqualities,as in a motion
pictureor a dynamicgeometryenvironment.
6.3. New objects of study
New virtual tools not only change the way students can approachtraditional mathematicalideas, they also make it possible to address new
and differentphenomenain the world aroundus - phenomenathat were
difficultor impossibleto model using non-computationalmethods.In particular,computationaltools providevisual, manipulablemodels of dynamical systems (Heim, 1993; Cohen & Stewart,1994; Hall, 1994; Holland,
1995; Kauffman, 1995; Casti, 1996). These models approachcomplex,
often non-linearphenomenawith iterative,dynamicrepresentations- representationsthat can only producemeaningfulresults in a realistic timeframein a computationalmediumthatsupportsmassive andrapidexternal
processing. These new dynamic models make it possible to examine social, physical, and biological situations such as trafficjams, oscillating
springs, or the behavior of biological systems such as ant colonies that
embodymany stronglyinteractingelements.These kinds of situationscan
be modeled in entirelynew ways by and for learnersusing computational
environments(see Resnick, 1994).
Mathematicalexperience in a virtualculturewill thus be more intimately connected with students' wider world of experience. Inductiveexplorationswith manipulablemodeling systems will make it possible for
learnersto make morelively, more compelling,andmoreintimateconnections between 'mathematics'and what they experienceas the 'real world'
(Kaput,1996b).
In response to these new tools, new pedagogical approaches,and new
content areas, we may need to reexaminethe idea of 'mathematicalabstraction.'This is a topic thatNemirovsky(1998), Noss andHoyles (1996),
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and Wilensky (1991) have addressedin other work and that we plan to
pursue in subsequentpapers. Briefly, as these authorsvariously suggest,
we may need to make room in our notion of mathematicalunderstanding for a kind of 'concreteabstraction'that builds mathematicalmeaning
from an active web of meaningful associations ratherthan an relatively
meaninglessset of empty formalrules.
6.4. Towardsa virtualmathematics
All of the above suggests thatone of the implicationsof virtualculturefor
mathematicseducationwill be an increasedemphasis on embeddedand
situatedmathematics- on mathematicsas a way of knowing the worldratherthanon mathematicsas computationor mathematicsas formalproof.
To understandthis mathematicswe will need ways of thinkingaboutmathematicalknowledge,especially the new forms thatthemselvesrequirethe
computationalmedium,thatinclude a broaderrangeof thinkingstyles and
that are situatedmore firmly in the real-worldexperiencesof mathematics learners.Again, this is not to say that computationand proof will or
should disappearfrom the curriculum.After all, older forms of thinking
did not disappearwith the appearanceof writing,nor shouldthey have disappeared.Rather,the broadeningof mathematicalformsin a virtualculture
suggests that computationand proof are bettersuited to a supportingrole
- ratherthan their traditionalstarringrole - as mainstreamstudentslearn
to make sense of quantitativeinformationthroughmathematicalmodeling
and exploration.
7. COMPUTERS AND COMPUTATION: IN CONCLUSION
We began this paperby suggestingthat 'computationalmedia' are a transformativetechnology, more like the printingpress than the stereoscope
in terms of their long-term impact on our culture. We chose the broad
term 'computationalmedia' ratherthan the more usual term 'computer'
deliberately,because the point of this paperis not that 'computers'in their
currentform at the end of the twentiethcenturyare necessarilytransformative. Rather,we suggest thatthe abilityto externalizesignificantpieces of
the processingof information- of which the moderncomputeris only the
best example to date - has the power to transformthe way we approach
mathematics(as well as a host of otherareasof cognitionand culture).We
expect thatthis transformationwill only be fully realizedaftercomputation
moves off the desktop and is distributedmore widely in the space around
us (Roschelle, Kaput,Stroup,& Kahn, 1998). But a cognitive-evolutionary
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perspective on computationalmedia and their relation to earlier forms
of cognition suggests that computationalmedia are qualitativelydifferent
from many of the technologies that have promisededucationalchange in
the past and failed to deliver.
When and how - and perhapswhether- change will come from computationalmedia and the virtualculturethey make possible will of course
dependon questionsof politics and policy as much as cognitive and educational theory. But all technologies are not created equal. Some, like
the stereoscope, are relatively insignificantin their epistemological implications. Others,like the printingpress and computationalmedia, have
profoundcognitive and social consequences- profoundenough, in some
cases, to lead to substantialculturaland social transformation.
In making this claim, we recognize that this account of mathematics
and virtual culture is far from complete. In particular,we have focused
primarilyon individualcognition and how virtualculturechanges the way
individualstudentsmight approachmathematicalthinking.In doing so we
have only touched briefly on how a virtualculturecreates new relations
between individualand social experience,andnew culturalforms.Nor can
we reliablypredictthe dynamicsof evolutionthatwill occur in the virtual
culture,and what new representationalforms may arise (Dyson, 1997) after all, who really predictedthe worldwideweb (www)? It seems clear,
however, that in a virtual culture students will have new ways of sharing new formsof mathematicalexperiences,mathematicalrepresentations,
and ultimately,mathematicalunderstanding.We plan to approachthis issue - andto look moregenerallyat the social implicationsof virtualculture
for mathematicslearningand cognition - in subsequentpublications.We
hope that this paper has helped illuminate some currentissues in a new
light.
NOTES
1. Workin the paper,by the second author,was supportedby Departmentof Education
OERIgrant# R305A60007.
2. As might be expected,a bold andcomprehensivethesis such as Donald's will generate
a body of reactionand criticism.See, for example, (Mithen, 1997) for a critiquefrom
anthropology,and a broaderset of responses in the special issue of Behavioraland
Brain Sciences (Donald, 1993). The structureand implicationsof Donald's thesis remainintact.See, for example,the reviewof Mithen'sbook by HowardGardner(1997).
3. In this sense, Donald makes the same kind of distinction that Jerome Brunerdoes
between an iconic and symbolic stage of representation(1973). However,in Donald's
scheme, linguistic symbols at firstreferto concreteevents.For Bruner,the significance
of the symbolic stage is thatabstractsymbols can referto other symbols. See also the
discussion below of Goodman'sanalysis of symbolic systems (1968).
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4. Paperthas made a similarpoint aboutthe developmentof mathematicalunderstanding
in the context of a mathematically-richsurroundingculture(1980).
5. Other authorshave similarly argued for the record-keepingas the impetus for the
developmentof writtenlanguages.See (Kaput& Roschelle, 1998) for a more detailed
discussion of this issue.
6. Less significantin Donald's argument- but certainly of great importancein understanding 'theoretic culture' - are the social and institutionalroles of the external
inscriptions(Latour,1979). The impactof computationalmedia on forms andpatterns
of communicationwill be the subjectof a subsequentpaperon virtualculture.
7. A subsequentpaperwill addressthe questionof changes in the natureof mathematics
(content)itself.
8. In a similar way, theorists of social or distributedcognition suggest that cognition
is best understoodnot as a phenomenon 'in the mind,' but as a complex interaction between an individualand the people and things aroundhim or her (Vygotsky,
1934/1986, 1978; Bruner,1990, 1996; Rogoff, 1990; Pea, 1993).
9. Ultimately,the algorithmscome from humanbeings who have programmedthe computer- or, more accurately,who designed anothermachinethat createsand programs
the machines we use. In this sense the algorithmsare not that differentfrom the dictionaryitself, which was recordedover time by otherpeople andcomes to us as partof
the culturalsurround- Donald's externalsymbolic storagesystem. Whatis important
is not wherethe algorithmscome from as muchas the fact thatthey allow independent,
externalprocessingof information.
10. Dennett(1996) suggeststhata cookbookis actuallyrecordingalgorithms.The pointhere
is thatwhile a cookbookrecordsalgorithms,it still requiresa humanbeing to carrythem
out - and is thusan example of externalinformationstorage ratherthan external informationprocessing. It is interestingto note that muchof the currentfood-processing
industryoperatesby automationin the productionof ready-to-eat(or heat)meals- which
serves as a gustatorialreminderof thepotentialpowerof externalprocessing.
11. We should be careful here to point out that we are not advocating or assuming a
traditionalinformation-processingview of the mind or of all mental activity.Rather,
we are arguingthatvirtualcultureresultsfrom the developmentof a new meansof carrying out a particularkindof mentalfunction:namely,the executionof well-specified
algorithms.
12. In making this argument,we recognize that the creationof this 'virtualculture' is in
its infancy.In particular,we recognize thatthe creationof a true 'virtualculture'may
also requirethe simulationof directsensory information,as describedin the fiction of
authorssuch as in Gibson's Neuromancer(1984). The ability to directly experience
sight, sound,or othersimulatedsensoryinformationwould surelycomplete a 'virtual'
culture.A point of view stronglyarguingfor the importanceof more directexperience
and less heavily mediatedexperienceis offered by Reed (1996).
13. Given the relativereferential'senselessness' of the intermediatesteps in an algebraic
solution, the process of algebraic manipulationusually requires the temporarysuspension of referentialsense-makingin favor of syntacticalsense-making.Of course
it is also the case thatthe syntacticallygeneratedstrings might help reveal previously
hiddenrelationships.Such is the magic and the mysteryof formalism.
14. Douglas Hofstadterprovidesa particularlyclear exampleof such an ideal system with
his MIU game Godel, Escher,Bach (1979).
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15. A more precise definitioncan, of course, be offered relativeto a choice of logic (see,
for example, Roman, 1990), but this level of precision is beyond the scope of this
paper.
16. It is worth pointing out in this regardthat the externalrepresentationof algorithmic
processes (with or withoutformalsystems) is distinctlyandimportantlydifferentfrom
the externalexecutionof symbolic manipulation(which in the case of computational
media depends on the existence of formal systems). The cognitive advance we are
describinghere was not only in the externalstorage of algorithmsin formal systems
(which remove memory requirementson human cognition), but also in the external
operationof those algorithmsusing formal systems, which make it possible to offload
symbolicprocessing as well as symbolic storage.
17. In the mid 1970's the notion of a genetic algorithmwas inventedby John Holland
and others, wherein the programmodifies itself across iterationsby way of random
mutationsof its operationstrings.This form of externalsymbol processingamountsto
a new level of processing autonomy,differentin kind from self-modifying programs
as had been developed by John MacCarthyand others in the context of LISP during
the 1960's (see Holland, 1995).
18. This is, of course, the formal idea of a computerprogramas describedby Turingand
von Neumann(VonNeumann, 1966, Turing,1992).
19. It is worth rememberingthat operationson quantitieswere very muchtied to dimensionality and otherphysicalreferentialconstraintstill Descartes'time (Kline, 1972).
20. In suggesting that mathematicshas played a significantrole in culturaland cognitive
evolution, we certainly do not mean to imply that mathematicswas the sole cause
of cultural or intellectual change. As is appreciatedby the readers of this journal,
mathematicsoperateswithin a culturalcontext, and is only one - perhapscurrently
- factorin culturaldevelopment.
under-appreciated
21. It is worthnoting that the forms of knowledge developed in a computationalmedium
may be subtlydifferentfrom those developedin static, inertmedia (see Sherin, 1996),
just as the forms of knowledgedevelopedin mentalcalculationsmay be differentfrom
those developedusing pencil and paper.
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DAVIDWILLIAMSONSHAFFER
ProjectPACE,HarvardUniversity,
Cambridge,MA, U.S.A.
JAMESJ. KAPUT
MathematicsDepartment,
The Universityof Massachusetts,
Dartmouth,MA, U.S.A.
E-mail:jkaput@umassd.edu
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