Social Constructivism, IndividualConstructivism and the Role of Computers

in Mathematics Education

Erick Smith

University of Illinois at Chicago, Chicago, IL, USA

By asking the question: `̀ Can computers do mathematics?'' this paper investigates the relationship

between computers and views of mathematics from both individual and social constructivist

perspectives. Although these two perspectives ask many of the same questions, they frame these

questions in quite different ways of viewing the interaction between individual, subject matter,

culture, and cultural tools (e.g. computers). I argue that whereas social constructivists are more

likely to take the position that computers alter the way we do mathematics, individual

constructivists would more likely say that computers changes the mathematics that we do.

Individual constructivists, by placing mathematics itself within the actions carried out by an

individual, provide a theoretical framework that allows for the richness and diversity of student

constructions that can expand our understanding of mathematics beyond the bounds of any one

particular culture.

In an interview on the PBS radio program, All Things Considered (December 13, 1991),

D. Ford Bailey Jr., son of the first black member of the Grand Old Opry, was asked what

made his father's harmonica playing so different:

Bailey: He played his ideas.

ATC: What do you mean by `he played his ideas'?

Bailey: You couldn't write them down. He just played the ideas in his head.

In a paper on the role of computers in education, Gavriel Salomon states that there has

been `̀ a shift of attention from Piaget to Vygotsky'' (Salomon, 1992, p. 25). Although

Salomon made this remark in connection with the importance of understanding the social

changes in the classroom that accompany the introduction of computers, he also captured

the essence of an important debate which has emerged in education. In this debate, a

challenge is being issued to what might be called `individual' constructivist views based on

the work of Piaget. This Piagetian view is often characterized as seeing constructive activity

as an individual process isolated from both cultural artifacts and more knowledgeable

others (Scott, Cole, & Engel, 1992, p. 191). The emerging challenge comes from social

constructivists, primarily from those who place their work in a Vygotskian tradition which

Direct all correspondence to: Erick Smith, University of Illinois at Chicago, Chicago, IL, 60607, USA; E-Mail:

erick@uic.edu

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JOURNAL OF MATHEMATICAL BEHAVIOR, 17 (4), 411±425 ISSN 0364-0213.Copyright C 1999 Elsevier Science Inc. All rights of reproduction in any form reserved.JMB

emphasizes the situated nature of learning within social and cultural settings. Individual

constructivists have responded to the challenge with the claim that the importance of social/

cultural factors has always been recognized, but that a research program which has

legitimately focused on individual constructive activities has been inappropriately inter-

preted as reflecting an individually isolated view of the learning process (Confrey, 1991,

1992; Smith & Confrey, 1991; von Glasersfeld, 1995). However, it does seem to be the case

that the social constructivists have given articulation to the importance of a social

perspective and provided focus for many individual constructivists in recent work.

In this paper, I examine this issue further by posing the question: Does a computer do

mathematics? To answer in the affirmative is to view the computer as a sort of `̀ assistant

mathematician,'' helping the student carry out mathematical activities, and, in effect,

changing the way we do mathematics. To answer in the negative is to place the computer

outside of mathematical doing and into the mathematics itself. It is to say, for example, that

the mathematical content of geometry with a computer is different from the content

without a computer, that is, it changes the mathematics that we do.

To help understand the importance of this distinction, I will first describe a distinction

between `̀ individual constructivist'' and `̀ social constructivist'' approaches to education

and research. The above two models of the role of computers in the mathematics are then

examined in relation to these differing constructivist views. Although each view can offer

a different and valuable perspective on mathematics education, I finally argue that the

individual constructivist view, which embeds mathematics in human activity, ultimately

allows for the kinds of individual and cultural diversity, that are often missing in both

mathematics instruction and in views of mathematics.

1. INDIVIDUAL AND SOCIAL CONSTRUCTIVISM

A researcher who wishes to study how it is that individuals learn must, at some point,

phrase what she believes to be the relevant questions. The act of phrasing a question is, in

effect, to situate oneself as an observer. From the claim that: `̀ Every distinction is made by

an observer'' (Maturana & Varela, 1987), I would further emphasize that the distinctions

made are a result of an observer who has chosen a perspective. Thus, distinctions are not

there for the observing, but must be constructed by the observer.1 In this case, we can learn

something about constructivist researchers by initially asking ourselves where it is that we

see the researcher placing herself as observer. I will use two general observer categories to

distinguish between individual constructivists and social constructivist. Individual con-

structivists, who include many working in a Piagetian tradition, take the view that learning

can best be understood by imagining oneself as an observer `̀ in'' the mind of the

individual. They do not claim that the researcher can ever actually know what is in the

mind of another, rather, they claim that understanding learning can best be facilitated if the

models one creates are made as if they represent what occurs in the mind of the learner.

One always imagines oneself in the place of the student looking out at and attempting to

make sense of the experiential world. Although social and cultural factors are seen as

intrinsic to learning by most individual constructivists, they are looked at as part of this

experiential world. Individual constructivists emphasize the individual constructive

activity regulated through a personal sense of `̀ viability.'' Although most constructivist

working in a Piagetian tradition would tend to be individual constructivist, those who

SMITH412

identify with radical constructivism have articulated this perspective most strongly (von

Glasersfeld, 1995).

An alternative perspective is to place the observer within a social setting, observing the

individual as social participant. It is a choice to look at learning by modeling the social

interactions that take place between people. I will refer to such researchers as social

constructivists and include many working in the Vygotskian tradition. Although social

constructivists do not necessarily reject the individual constructive actions of the learner,

they tend to see language and the social/cultural context as the primary regulative force in

these constructions.

Notice that the distinction between individual and social constructivism is not necessa-

rily based in what phenomena one wants to explain, but rather in the perspective one adopts

as observer in phrasing one's initial research questions. Thus, for example, although Cobb,

Yackel, and Wood (1992) have provided an explanatory framework for social interactions

within elementary classrooms, they have done so from an individual perspective, that is,

imagining what it is like for the individual in the social setting of the classroom. Likewise,

although Vygotsky (1987) had strong interest in explaining intramental (i.e., individual)

higher mental functions, he began from the perspective of what he called the intermental,

i.e., from social interactions. Thus the individual / social constructivist distinction is not

necessarily dependent on one's overall research goals, rather upon how one chooses a point

of entry for understanding the educational process.

As a convenient metaphor for distinguishing these approaches, we might say that for

social constructivists: `̀ Individual constructivist can't see the forest for the trees.''

Whereas for an individual constructivist: `̀ Social constructivist can't see the trees for

the forest.'' In their stronger interpretations, social constructivists may see the trees as

nothing more than a part of the forest, whereas individual constructivists see the forest as

simply and arbitrary grouping of trees. Just as the ecologist and biologist ask different

questions form different perspectives, individual and social constructivist may do the

same, and it may be neither possible nor desirable to reduce their viewpoints to a common

perspective. However, results form one framework may (and should) continually provide a

rich source of new problems from the perspective of the other.

2. INDIVIDUAL AND SOCIAL PERSPECTIVES

ON MATHEMATICS

Recognizing these distinctions in constructivist perspectives leads to differing views of what

counts as mathematics. One version of a distinctly social constructivist version of

mathematics was recently described by Ernest (1991). Ernest's view provides a convenient

place to initiate a comparison of mathematics from a social vs. an individualistic perspective.

2.1. Social Constructivist Mathematics

Although Ernest would not necessarily identify himself with a Vygotskian approach, he

quite clearly takes a social constructivist perspective in his description of `̀ objective

mathematics.'' Objective knowledge, for him, consists of socially accepted forms of

linguistic expressions which evolve over time (thus are fallibilistic) though processes

similar to those described by Lakatos (1976) as `̀ quasi-empiricist.'' He attributes the

CONSTRUCTIVISM AND THE ROLE OF COMPUTERS IN MATHEMATICS EDUCATION 413

certainty we feel for this `̀ objective knowledge'' to linguistic conventionalism, as

described by Wittgenstein (1953). The linguistic rules for `̀ objective mathematics'' are

fairly well-defined and change slowly over time. Ernest describes the knowledge of the

individual as `̀ subjective knowledge'' which he also sees as linguistically based. Thus,

objective knowledge is created when subjective knowledge is publicly expressed and

becomes accepted within a particular temporal and cultural setting.

2.2. Subjective Knowledge and Individual Constructivism

Once one places knowledge within a linguistic domain, it becomes difficult to escape

these bounds when discussing individual knowledge. To place knowledge in language

seems to separate knowledge and knower, thus, it requires some means of explaining how

it is that individuals are knowers of that knowledge. Vygotsky resolves this issue by

describing processes by which the `̀ intermental'' becomes the `̀ intramental'':

Any function in the child's cultural development appears twice, or on two planes. First, it

appears on the social plane, and then on the psychological plane. First, it appears between

people as an interpsychological category, and then within the child as an intrapsycho-

logical category.2

In a sense, Ernest seems to be in agreement with this perspective. For him, all

mathematical knowledge (both objective and subjective) has a linguistic basis. As Ernest

says: `̀ . . . the foundation of mathematical knowledge, both genetic and justificatory, is

acquired with language'' (1990, p. 75). However, because Ernest depends upon the

common linguistic basis to bridge his `̀ objective knowledge'' to individual (`̀ subjective'')

knowledge, he has difficulty tying this knowledge to individual constructivism.3 For

example, when one views operations as based on actions (Piaget, 1970; Steffe, von

Glasersfeld, Richards, & Cobb, 1983; Confrey, 1991), it becomes difficult to reconcile this

with the idea that all mathematical knowledge is linguistically based.

This conflict becomes apparent in the context of remarks made recently by Glasersfeld

and Ernest. In separate discussions held on the same afternoon, Paul Ernest stated that the

mathematical statement, 2 + 2 = 4, is fallible (personal communication, 1992) while

Glasersfeld said that the same claim was true with certainty (personal communication,

1992). Ernest's justification originated in his social constructivist perspective, especially

with its ties to conventionalism, which argues that this statement, like all mathematical

statements is fallible, simply because language use changes and is both culturally and

historically dependent. For Glasersfeld, however, to claim certainty for this mathematical

statement is to claim that one has abstracted the mental operations of isolating two sets of

two units each and has combined these sets into one set which numbers four. For

Glasersfeld, the mental operations create a certainty for which the language, 2 + 2 = 4, is

merely descriptive. If one's use of language changes over time or situation, it will not alter

this certainty. Thus, in articulating a radical constructivist view, Glasersfeld clearly would

not place the basis for all mathematical knowledge in language.

By limiting notions of learning and knowing to language and the use of cultural tools,

social constructivists also seem to limit the possibilities for individual diversity and

cultural change.

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This concern for the observer's status poses a fundamental dilemma for the socio-

constructivist. If the intermental exchanges are what creates and determines the intramental

thinking, then what would lead one to expect diversity in student's interpretation of a

statement, argument or justification? Is it simply accorded to the diversity of social

experiences children have undergone? Giving voice, recognition and encouragement to these

constructive inventions is a primary goal of the constructivist. If classroom interactional

descriptions . . . ascribe a uniformity to students' experience, then the opportunities to build on

the richness and diversity of students approaches will evaporate . . . The pivotal question as I

see it is how do they [socio-constructivists] bring such ideas and experience into coordination

(and, to some extent only, reconciliation or also, potentially, confrontation) with the public

arena of debate and negotiation. (Confrey, 1992, p. 27)

Confrey's concern is not only in the difficulty of reconciling the two positions, but also

that in the particular context of education, the individual constructivist perspective is

essential. Without it, one loses access to the richness of individual constructions that are

at the core of the learning process. Thus, the `̀ viability of the child's model form the

child's perspective'' receives little attention (p. 19) and many `̀ researchers in this

tradition focus so heavily on the social interactions that the rich world of individual

constructs remains relatively unarticulated and discussed in their writings'' (p. 24).

Henderson carries this argument further emphasizing that diversity of background

engenders diversity of mathematics. A viable proof must `̀ answer my why question and

relate my meanings of the concepts involved. The proof that satisfies someone else may

not satisfy me because their meanings and why-questions are different from mine . . .Persons who are different from me (for example, in terms of cultural background, race,

gender) are most likely to have different meanings and thus have different why-

questions and different proofs'' (Henderson, 1992, pp. 3± 4).

If one accepts the validity of separating these two perspectives as I have done,

then a question to social constructivists would be: Given that all education is situated

within social and cultural settings, is that enough to argue that a social constructivist

perspective is sufficient by itself to account for educational activity? The problem

which seems to be central to Confrey's concern, is that with their focus on language

and social interaction, social constructivists may leave out the individual knower

of mathematics.

3. COMPUTERS AND MATHEMATICS

Salomon (1992, pp. 11±12) describes three historical approaches to educational comput-

ing: computer-aided instruction/intelligent tutoring systems (CAI/ITS), programming,

and tools. By tools, he means `̀ cognitive tools'' and includes such software as

`̀ model builders, data sets affording manipulation, conceptual map-makers, simula-

tions and similarly partly or wholly open-ended instruments for the student to operate

and manipulate.'' Salomon claims that neither CAI/ITS nor programming have shown

promise and that the development and use of appropriate software tools provides the

most hopeful route to successful use of computers in schools. An important dis-

tinction for the purposes of this paper between tool software and CAI/ITS software

is the intended goal of the software designer. Although mathematics is, in some

sense, embedded in CAI/ITS software, the intent is to model the learner, providing

CONSTRUCTIVISM AND THE ROLE OF COMPUTERS IN MATHEMATICS EDUCATION 415

prompts and responses to the user that supposedly promote cognitive processes

related to learning. Tool software, on the other hand, is designed to facilitate the

accomplishment of mathematical tasks, whether it is creating representations, aiding

in problem-solving, allowing alternative visual perspectives or whatever. This is not

to say that the designers of tool software do not have strong pedagogical commit-

ments which play out in the design of the software, rather that the software, by

definition, is intended to be a part of user-initiated mathematical activity. As the potential

power of such tools increases with technological advances, software designers must

necessarily face the question of how to distinguish between the capabilities of the

computer and what is important in student-learning. This is a question which will be

addressed below.

First, an example from a classroom is presented which has been chosen to push

at the edges of what it means to learn mathematics in a computer environment with the

intention of raising issues that challenge us to better define our pedagogical commitments.

3.1. Can Computers Do Mathematics?

Three students are engaged as a group with one computer and the multi-representa-

tional software tool, Function Probe (FP; Confrey, 1992) in a mathematics classroom.

They are using the table window to build a table of values for a contextual problem

that can be modeled linearly (distance as a function of years). They have filled one

column in the table window with years by using the `̀ Fill'' command and are

attempting to enter the values in the second column. They have already decided that

the initial distance is 241.5 in. and that this distance decreases by 2.75 in. each year.

Since one option when using the `̀ Fill'' command in FP is to fill a column by declaring

an initial value, a final value, and an iterative arithmetic operation (+,ÿ,*, or ) by a

specified value, one student, Dan, suggests that they can fill the distance column by

starting at 241.5, ending at 0 and subtracting 2.75 each row. He attempts this, but

neglects to change the iterative operation form its default action (plus) to the action he

needs (minus). Thus, FP attempts to fill the column by starting at 241.5 and ending at 0

by adding 2.75. Only one value is entered in the column, 241.5, and the `̀ Fill'' stops.

Dan assumes that he has made an operational mistake in the program. Another student,

Tess, objects:

You have to make an equation don't you understand? Because we want the years to

correspond with the inches.

Since the goal is to create a relationship between years and distance, Tess' argument seems

to be that this correspondence can only be created through an equation. However, Dan

insists that a `̀ Fill'' should work and the third student, Lila, seems to agree. They try

several more times to fill the column, failing each time for the same reason. As they do so,

Tess becomes increasingly upset, attributing their failure to their unwillingness to listen to

her point. Although Dan still expresses certainty about his method, Lila begins to waiver.

Tess states:

I know but . . . you have to have an equation otherwise you have to do each one separately . . .

SMITH416

To which Lila responds:

I think Tess is right. We have to make an equation, otherwise it won't know what to do.

Reluctantly, Dan goes along and the group eventually finds an equation and finishes the

problem.4

Had the teacher reached this group at an appropriate time, he would very likely have

shown them how to use minus in the Fill operation.5 However, an interesting issue would

be how a teacher could have responded to Tess and Lila, had he taken seriously the reasons

they attributed to the failure of `̀ Fill'' for the group. How could he have responded to their

suggestion that the computer in some sense knew that years and distance must correspond

and thus, was unable to carry out an operation which, in their eyes, would fail to make this

correspondence? Although the students were fairly new to FP, they had worked with the

table enough to know that it could algorithmically carry out the indicated operations. Thus,

it would appear that they were attributing, in some sense, a very human quality to the

computer, and ability to create a `̀ problematic'' (Confrey, 1991) within the problem, an

ability for the computer to `̀ do mathematics.''

3.2. Seeking an Appropriate Role

This example is intended to illustrate how computers can undermine the kind of

exploratory constructive processes necessary for meaningful learning. If students do not

distinguish between `̀ meaningful learning'' (Novak and Gowin, 1984) and learning to do

what the computer does, then we have lost a significant part of the educational process.

On the other hand, the program used in the above example is representative of just that

kind of tool software which Salomon sees as the hope of educational computing. Thus,

we must find an appropriate role for these tools which will allow them to reach their

potential. We might ask how our classroom practices allow students to understand what it

is in their own actions that is different form that which a computer does. This issue has

particular relevance in mathematics instruction, particularly in how we view our subject-

matter, mathematics.

It is not inconceivable that in the very near future, a home computer will have the

ability to `̀ listen'' to a student read a typical textbook word problem and then `̀ solve'' the

problem. Computers, in fact, may become very accomplished in Ernest's `̀ objective

mathematics.'' As this process evolves, we will have to keep reformulating our under-

standing of what we call mathematics and our goals in teaching mathematics. This issue is,

of course, already coming up in elementary schools where the debate has begun on

whether mathematical learning is undermined if students are allowed to use calculators.

This issue highlights the importance of the individual/social distinction. Will computers

become mathematicians? What would it mean for a computer to `̀ know'' that 2 + 2 = 4 in

the same sense that Glasersfeld claims to know it?

Wittgenstein (1953, par. 344) points out, `̀ Computers lack that `rest of behavior.''' This

is not something that will be overcome as computers become more sophisticated, for that

`̀ rest of behavior'' is that which makes us distinctly human. If we could imagine a

computer which had the level of caring about others that people do, whose `̀ mental

operation'' was intrinsically tied to its participation in a social network, which had a sense

CONSTRUCTIVISM AND THE ROLE OF COMPUTERS IN MATHEMATICS EDUCATION 417

of humor, and which worried about being unplugged the same way we worry about dying,

we might be able to imagine a computer which could exhibit the `̀ rest of behavior.''

But the question still remains whether there is a distinction between what a computer

can do and mathematics. Mathematically, how important is the `̀ rest of behavior?'' Is it

enough to consider the social world of language and representation?

4. COMPUTERS IN A SOCIAL CONSTRUCTIVIST CLASSROOM

The concerns expressed about relying on a social constructivist perspective in mathematics

education carry over to models of computers in education. In particular, what does one

value in the learning of mathematics if computers are quite good at speaking the language

of `̀ objective'' mathematics? The two social constructivist accounts described in Sections

4.1 and 4.2 may provide the basis for addressing this question.

4.1. Computer as Intellectual Partner

Pea and Salomon proposed the idea of computer as intellectual partner, `̀ a sharing of

cognitive load, the off loading of a cognitive burden unto a tool which partakes in the

handling of the intellectual process'' (Salomon, 1992, p. 17). What makes computers

unique according to Salomon is that they have intelligence. Computers can not only

handle lower level tasks like calculating, drawing, and so on, but allow a new conception

of partnership, where cognitions become `̀ distributed'' in the sense that computer and

partner `̀ think jointly'' (Pea, in press, from Salomon, 1992, p. 17). In this situation,

Salomon argues for what he calls `̀ distributed intelligence,'' where instead of focusing on

individual accomplishment, the emphasis is on the `̀ joint product.'' The computer

becomes a `̀ performance-oriented tool'' (p. 18). As Salomon (1992, p. 18) states:

The common assumption is that intelligence resides in people's heads and that situations can

invite, afford the employment of, or constrain the exercise of this intelligence. The intellectual

partnership with computers, particularly that kind of partnership which is joint-performance-

oriented, suggests the possibility that the resources that enable and shape activity do not reside

in one or another agent but are distributed between persons, situations, and tools. In this sense,

`̀ intelligence is accomplished rather than possessed'' (Pea, in press); it is brought about, or

realized, when humans and tools pool their intelligences and jointly tackle a problem or task.

There are two serious questions that are not directly addressed in this article. First, one

wonders if there is, in their view, no longer a distinction between the roles of computers

and individuals, or if so, how that distinction would be made in the classroom. Second, if

intelligence is `̀ accomplished,'' what is it that we are trying to do in schools? Do we offer

no possibility that students obtain `̀ something'' that they can carry away with them? From

an extreme perspective on situated knowledge, that all knowledge only `̀ exists'' within the

particular social, cultural, and temporal situation in which it is used, this perspective may

begin to make more sense. If problem-solving is nothing more than solving a particular

problem in a particular setting, than the `̀ accomplishment'' of solving that problem does

seem to be paramount and perhaps, it makes little difference how the task was divided

between human and computer. The value of the experience lies only in the ability of the

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participants to be able to repeat under similar conditions. Given the preponderance of

evidence, however, that humans do have the capability to creatively remake old

knowledge in new situations, one must wonder whether the distinctions between the roles

of human and computer should be so diminished.

In spite of these serious reservations, Pea and Salomon have offered valuable insights

on the role of computers. As computers become integrated into school settings, the

nature of the learning activity does change. However, from the perspective of a student,

tools are not simply amplifiers of an independently existing subject-matter (Pea, 1987),

rather tool and problem (or subject-matter) are inherently connected. Thus, we can reject

the idea that representations (including computer interfaces) can be `̀ invisible'' and

begin to seek how our understanding of what it means to learn mathematics changes

when using computers.

4.2. Computer as Cultural Tool

In a more Vygotskian approach, Scott et al. (1992) describe what they call `̀ cultural

constructivism'' which they characterized, in contrast with Piagetian constructivism, as

assuming `̀ not only an active child but an equally active and usually more powerful adult

in interaction. . . . Moreover, cultural constructivism emphasizes that all human activity is

mediated by cultural artifacts, which themselves have been constructed over the course of

human history'' (p. 191). For them, computers play the role of a cultural tool which serves

to mediate human activity:

From this perspective, the historically conditioned forms of activity mediated through

computers must be studied for the qualitatively distinctive forms of interaction that these

artifacts afford and the social arrangements that they help constitute. Moreover, one is

encouraged to seek explanation of current uses of computers in terms of the history of the

technology and the social practices that the technology mediates; one needs to consider the

`effects' of interacting in this medium not only as they are refracted through transfer tests or in

local activity systems (such as classroom lessons) but also in the entire system of social

relations of which they are a part. (p. 192)

There is much in the description of Scott et al. (1992) that facilitates our understanding of

computers in the classroom. The idea that computer-as-tool mediates the activity of

learning supports the idea, also made by Pea (1987), that computers do not simply

`̀ amplify'' our ability to learn but alter the ways in which we learn and, thus, the nature of

what is learned. This suggests that the question of `̀ what'' is learned may be very different

from that of `̀ how much'' is learned and that the questions of `̀ what'' is learned and `̀ how''

that learning takes place may be inseparable. This point is central to the `̀ computer as

intellectual partner'' concept described above, the `̀ computer as tool'' concept of cultural

constructivists, and the individual constructivist model which will be described below.

Scott et al. also point out the importance of developing extended learning contexts

which become embedded in computer cultures, noting several research centers which they

describe as having `̀ an interest in sustaining learning contexts over time and beyond the

confines of their own experimentation and an interest in creating a research agenda (and

therefore an educational practice) in contrast to the technicist tradition in educational

computing'' (p. 242).

CONSTRUCTIVISM AND THE ROLE OF COMPUTERS IN MATHEMATICS EDUCATION 419

There is a broad agreement among researchers of many persuasions in the importance

of developing sustained learning environments in which computers become embedded in

the learning culture. If this does not happen, there seems to be little chance that

computers will have a widespread significant influence in education. However, this can

be a double-edged sword. Just as the benefit of computers may be dependent upon their

becoming culturally embedded, this same process may foster other concerns. How do we

define the goals of a learner in a setting where a computer can `̀ interpret'' and `̀ solve''

mathematical problems?

Although Scott et al. provide little detail about how they would view students'

interactions with computers within a subject-matter context, they do refer to Wertsch's

interpretation of Vygotsky (1985). The idea of a tool `̀ mediating'' experience, in our case

mathematical experience, brings forth the image of an individual and the mathematical

subject-matter with the tool somehow sitting between, `̀ mediating.'' Mathematical content

is separated from the doing of mathematics. This seems to be consistent with the general

Vygotskian view of the individual and physical reality with culture mediating their

interactions. As Wertsch (1985) states, a tool `̀ serves as a conductor of humans' influence

on the object of their activity. . . . [It] changes nothing in the object, it is a means for

psychologically influencing behavior . . .'' (p. 78). In this case, the `̀ object'' is mathe-

matics. One need not postulate the independent existence of mathematics in this view, for

it would seem that Ernest's `̀ objective mathematics,'' a socially constructed view of

mathematics, would be quite consistent with the idea of mediation. However, this also

leads one to the same problem that faced Ernest, reconciling this view with individual

goal-oriented experience. For example, Confrey (1992, p. 18) describes three possible

methods which students might have when working on a specific problem involving

multiplication, each a valid approach to the problem. She states: `̀ I see little in Vygotsky

to help a more knowledgeable other to choose between the options, for each option would

be legitimate to a more knowledgeable other . . . Vygotsky seems to argue that the more

sophisticated method would be the one that relied on semiotic mediation of psychological

tools to accomplish more complex operations.'' Looking back at the student example in

this paper, if one takes the view that finding the equation is the `̀ more sophisticated

method,'' then we must ask if, from a Vygotskian perspective, there would be any

pedagogical concern that Dan was never able to successfully fill the distance column,

since the group did end up finding a correct equation. How could one place a value in

allowing Dan to explore this problem from his own perspective, when his inability to do

so ultimately led to the `̀ more sophisticated'' solution? One might then ask why the

software program should have the capability to fill a column iteratively if the same result

can be obtained with an equation, or, perhaps, why the computer cannot become the

`̀ more knowledgeable other?'' Would the students' goal be to appropriate the knowledge

of the computer?

5. INDIVIDUAL CONSTRUCTIVISM AND THE COMPUTER

Both of the above social constructivist models seem to be consistent with the Vygotskian

idea that intermental processes evolve into intramental processes. Both seem to want to

`̀ pull the computer in'' to make it, in a certain sense, part of the way one thinks and acts on

problems which exist `̀ out there.'' Individual constructivists, on the other hand, would be

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more likely to see the computer as constructed by the individual in conjunction with the

problem. Both are `̀ out there.'' This difference is illustrated in Figs. 1 and 2. In terms of

the previous distinction between social and individual constructivism, one would place the

social constructivist observer within the `̀ culturally situated mathematics'' rectangle,

observing the interaction between individual and computer. The individual constructivist

observer would be sitting inside the rectangle on the bottom left, `̀ observing'' possibilities

for acting upon the experiential world.

It should be pointed out that this model is intended to accentuate ways of viewing the

computer within mathematical situations. Many individual constructivists would argue that

`̀ mathematical situations'' do not exist independently of the actor. Thus, in a sense, the

schematic is misleading. A more complete schematic might include several arrows and

label the whole thing `̀ mathematics,'' emphasizing the interdependence of all the pieces.

5.1. Individual Constructivism Model

Instead of viewing computers as a tool that changes how one does mathematics, an

individual constructivist tends to see the computer as changing the mathematics. In the

previously mentioned example from von Glasersfeld describing the possible mental

actions one might take in constructing the certainty that 2 + 2 = 4, it is clear that he

places mathematics in the mental actions one takes. Those mental actions are created as

tentative viable responses to problematical situations within one's experiential world.

Within the experiential world, `̀ mathematical situations'' and cultural tools are seen as

inseparable, that is one creates problematical situations in relation to how one views

possible ways to resolve those problems. Likewise, objects are created and defined

through actions. Thus, to say that tools do not affect the objects, but only change the

way we act is too weak a role. Statements such as: `̀ I made a graph,'' `̀ I filled a table

column,'' or `̀ I divided two numbers'' become statements about a different mathematical

when one has access to tool software vs. paper and pencil, just as they did when we started

using writing tools.

There is significant evidence of the change in mathematics that evolves as students use

computers and this change grows as the `̀ computer culture'' (Scott et al., 1992) becomes

more firmly established. For example, Rizzuti (1991) documented the power of a

`̀ covariation'' understanding of functions by students working with the fill command in

tables; Afamasaga-Fuata'I (1991) described students construction of quadratic functions in

contextual problems using a process of evaluating net accumulations over time in table

window of FP; Borba (1993) has documented a fundamentally different understanding of

FIGURE 1. Social constructivist model.

CONSTRUCTIVISM AND THE ROLE OF COMPUTERS IN MATHEMATICS EDUCATION 421

functional transformations when working with students who began their explorations

using visually dynamic mouse-driven actions directly on computer graphs; and finally, my

own work suggests that students mathematical language when problem-solving in small

groups may change significantly when their efforts are centered around a problem-solving

tool (Smith, 1993). In all of these cases, the power of the computer became evident to the

extent that it allowed students to instantiate their actions in relation to their own

understanding of the problem. This work along with that of others (Kaput, 1986; Rubin,

1990) has supported the development of an `̀ epistemology of multiple representations''

based on the idea that mathematical concepts are neither singular nor embedded in

individual representational forms, rather are constructed as a process of interweaving

multiple representational forms with the actions we make within problematical situations.

Thus, actions, representations and concepts are inseparable. In the examples cited above,

students created their own mathematics in relation to posed problems where the proble-

matic they constructed was inseparable from their understanding of the actions which were

possible using the available tools (Confrey & Smith, 1991, 1992).

The strongest argument for the individual constructivist approach may be that its goals

are directly aimed at understanding the `̀ rest of behavior,'' that part of people which

makes us human and which can never be part of a computer. This is the point from

which we start, form which we look out, and from which we try to make sense of the

world. In doing so, individual constructivists are not trying to isolate the individual, but

understand how the individual functions within her social, cultural, and physical

environment. As already suggested, this perspective becomes even more important as

computers become more socially adept and as they become more capable of participating

in `̀ objective mathematics.'' It allows us a perspective to distinguish what is important

when a student claims to know that 2 + 2 = 4 beyond her mastery of the conventions of

our language.

6. CONCLUSIONS

What I have hoped to accomplish in this paper is to describe a legitimate tension between

social and individual constructivists approaches to learning, using computers as a vehicle

for describing that debate. The concerns that underlie the social constructivists programs

have led to essential insights into the learning process, many of which have been neglected

or even denied in many individual constructivists programs research findings. Thus, there

has often been a tendency to assume that results from individual studies of students

working on disconnected problems in isolated situations can be generalized to broad

FIGURE 2. Individual constructivist model.

SMITH422

learning environments. In this sense, bringing in the concepts of cultural tool, mediation,

situated learning, etc. have provided an essential framework in which to evaluate all

constructivist research programs. However, the concern expressed in this paper is that

these social concerns should not prevent us from still trying to understand how

individuals make sense of their world. How do social constructivists allow individuals

to create the kinds of culturally rich knowledge that contributes to the continual

evolution of our culture if they do not, in some sense, allow those students to stand

separate from the cultural milieu? If all learning is situational, how do they account for

the inventiveness of children and adults to resolve problems important to them using

methods unseen in their cultural traditions?

The computer is a convenient medium to explore this issue because many of our

societal fears of `̀ technicizing'' education fit well with many individual constructivists'

fears of `̀ culturalizing'' education, making education nothing more than a cultural

exercise. In this sense, I would argue that individual constructivism offers the potential

to lead us out of the culturally isolating educational settings which have been described so

well by Lave (1988) and more recently by Lave and Wenger (1991). Looking back at the

opening quote on this paper, I would not deny the historical/cultural basis for the music of

D. Ford Bailey, but would suggest that, had the Grand Old Opry restricted its membership

to those who could `̀ write down'' their music, Mr. Bailey may never have had the

opportunity to `̀ play the ideas in his head'' on that stage. Creating educational settings that

provide the opportunities for all students to play their ideas should form the core of the

challenge of educational reform.

NOTES

1. This claim admittedly involves a bit of circularity, for it being made in the context of

distinguishing between an individual and social perspective, yet the claim itself is made from an

individual perspective (see Smith, 1993).

2. Wertsch and others have cautioned against interpreting this to mean that for Vygotsky,

thought was nothing more than internalized language. However, it does seem to place the primary

focus on linguistic interaction rather than individual construction.

3. Ernest specifically ties his notion of subjective knowledge to radical constructivism.

4. The two approaches to creating this function suggested by Dan and Tess have been

described as the covariational and correspondence approach, respectively (Confrey and Smith, 1991;

Rizzuti, 1991).

5. It should be noted that this is only the second week that these students have been using FP.

They later become adept at using the Fill command in a variety of situations.

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