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Participationist discourse on mathematics learning

Authors:
Participationist discourse on mathematics learning
Anna Sfard
The University of Haifa & Michigan State University
In the last decade or two, the claim that mathematics learning is a social process can be
heard with such frequency that it became almost a cliché. And yet, those who declare their
belief in the social nature of learning have an important statement to make: They signalize
that in the ongoing debate between cognitivist and sociocultural research communities
they side with the latter. This paper is devoted to explicating theoretical and practical
consequences of this message.
These days, being explicit about what one means while claiming “the social nature
of learning” seems a necessity. In spite of the omnipresence of the word “social” in the
current literature or perhaps just because of it! there is much confusion about how this
term should be understood when applied in conjunction with learning.1 To avoid
undesirable connotations, I use a different terminology. Due to the metaphor for learning
underlying the particular family of sociocultural discourses to be presented on the following
pages, I call these discourses participationist. To bring the special features of the
participationism in fuller relief, I present it against the contrasting background of the more
traditional acquisitionist approach. The origins of participationism can, indeed, be traced to
acquisitionists‟ unsuccessful attempts to deal with certain long-standing dilemmas about
human thinking. After surveying some of these resilient puzzles and presenting basic
participationist tenets, I show how the claim that participationism, if followed in a
disciplined way, leads to the claim that human thinking originates in interpersonal
communication. I finish with a few remarks on the consequences of the participationism for
theory and practice of mathematics education and demonstrate how it helps in dealing with
some of the questions that acquisitionism left unanswered.
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Acquisitionism and its dilemmas
The roots of acquisitionist discourse on learning, which is usually seen as originating in the
work of Piaget, go in fact much deeper. The underlying metaphor of learning as an act of
increasing individual possession - as an acquisition of entities such as concepts,
knowledge, skills, mental schemas comes to this scholarly discourse directly from
everyday expressions, such as acquiring knowledge, forming concepts or constructing
meaning. To get a sense of the impact of the metaphor of acquisition on one‟s
interpretation of human mathematical activities, let me take a look at the following episode,
featuring young children talking with grownups about numbers. The brief scene is the
beginning of a series of conversations about numbers between my colleague Irit Lavi and
two young girls: 4 year old Roni, Irit‟s daughter, and 4 year 7 months old Eyant, Roni‟s
friend. The event took place in Roni‟s house.2
Episode: Comparing boxes with marbles
Speaker
What is said
What is done
1. Mother
I brought you two boxes. Do you
know what is there in the boxes?
Puts two identical closed opaque boxes, A
and B, on the carpet, next to the girls.
2. Roni
Yes, marbles.
3a. Mother
Right, there are marbles in the
boxes.
3b. Mother
I want you to tell me in which box
there are more marbles.
While saying this, points to the box A close to
Eynat, then to box B.
3c. Eynat
Points to box A, which is closer to her.
3d. Roni
Points to box A
4. Mother
In this one? How do you know?
Points to box A
5. Roni
Because this is the biggest than
this one. It is the most.
While saying “than this one” points to box B,
which is close to her
6. Mother
Eynat, how do you know?
7. Eynat
Because… cause it is more huge
than that.
Repeats Roni’s pointing movement to box B
when saying “than that”
8. Mother
Yes? This is more huge than that?
Roni, what do you say?
Repeats Roni‟s pointing movement to box B
when saying “than that”
9. Roni
That this is also more huge than
this.
Repeats Roni’s pointing movement to box B
when saying “than that”
……
……
……
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10a. Mother
Do you want to open and discover?
Let’s open and see what there is
inside. Take a look now.
10b. Roni
Abruptly grabs Box A, which is nearer to
Eyant and which was previously chosen as
the one with more marbles.
11. Roni
1.. 1.. 1.. 2, 3, 4, 5, 6, 7, 8.
Opens box A and counts properly.
12. Eynat
1, 2, 3, 4, 5, 6.
Opens box B and counts properly.
13. Mother
So, what do you say?
14. Roni
6.
15. Mother
Six what? You say 6 what? What
does it mean “six”? Explain.
16. Roni
That this is too many.
17. Mother
That this is too much? Eynat, what
do you say?
18. Eynat
That this too is a little.
19. Mother
That it seems to you a little? Where
do you think there are more
marbles?
20. Roni
I think here.
Points on the box , which is now close to her
(and in which she found 8 marbles)
21. Mother
You think here? And what do you
think, Eynat?
22. Eynat
Also here.
The episode is likely to leave the acquisitionist researcher unimpressed. The girls‟
mastery of counting would just confirm what she knows only too well from previous
studies: 4 and 5 year old children are usually advanced enough in their “acquisition of the
concept of number” to be able to count properly (for a summary of the relevant research
see e.g. Nunes & Bryant, 1996, Dehaene, 1997). Nor will the acquisitionist researcher be
stricken by the fact that in spite of their well developed counting skills, the girls did not
bother to count the marbles or even to open the boxes when asked to compare these
boxes‟ invisible contents. Extensive acquisitionist research on early numerical thinking, in
which young children have been observed implementing different versions of Piagetian
conservation tasks, has shown that at this age, this behavior is quite normal: “Children
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who know how to count may not use counting to compare sets with respect to number”
(Nunes & Bryant, 1994, p. 35).
And yet, knowing what children usually do not do is not enough to account for what
they actually do. An unprejudiced observer, whose analysis is not biased by the sole
interest in the girls‟ ability to “operate with numbers”, is likely to ask questions to which the
acquisitionist researcher may have no answers. Thus, the young interviewees‟ apparently
arbitrary response to the question “Which box has more marbles?” cannot be accounted-
for simply by the reference to „underdeveloped number schemes‟. Similarly, the fact that
the girls agreed in their surprising decisions does not seem to have much to do with
insufficiency of their “conception of number”. Finally, one should rather not count on
acquisitionist explanation while wondering what made the children “justify” their choice in a
seemingly adequate way in spite of the fact that they had no grounds for the comparative
claims, such as “this is the biggest than this one”, “It is the most” ([5]) and “it is more huge
than that” ([7]). If there is little in the past research to help us account for this kind of
phenomena, it is probably because the acquisitionists, while watching their interviewees,
attended to nothing except for those actions which they classified in advance as relevant
to their study. For them, the conversation that preceded opening of the boxes would be
dismissed as a mere „noise‟. The analysis of the remaining half of the event might even
lead them to the claim that the girls had a satisfactory command over numerical
comparisons, although this is not the vision that emerges when the second part of the
episode is analyzed in the context of the first.
Probably the main reason for the shortcomings of acquisitionists‟ accounts is these
researchers‟ belief in the invariability of learning processes across different contexts. In
their research, these scholars are tuned to cross-situational commonalities rather than
differences. For them, individual minds are the principal source of their own development,
whereas the task of the researcher is to discover the universal blueprint of the process. In
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result, acquisitionist discourse is ill equipped to deal not just with inter-personal and cross-
situational differences, but also with those changes in human processes that transcend a
single life span. Indeed, as long as human learning is seen as originating in the individual,
and as long as this process is thought practically impermeable to other influences, notably
those coming from interactions with other individuals, one has no means to account for the
fact that human ways of doing, unlike those of other species, evolve over history. Within
the confines of acquisitionist discourse, there is no cogent explanation for the fact that the
outcomes of the ongoing transformations accumulate from generation to generation,
constantly redefining the nature and extent of the individual growth.
Participationism and its solutions to acquisitionist dilemmas
Although usually traced back to the work Vygotsky and other founders of Activity Theory,3
participationism has, in fact, a more extensive genealogy. As a confluence of ideas coming
from areas as diverse as philosophy, sociology, psychology, anthropology, linguistics, and
more, 4 this relatively new school of thought is a mélange of approaches rather than a
single research discourse. Some of these approaches depart from the acquisitionism only
marginally, in that they merely add social considerations to the traditional individualist
account. Lave (1993) speaks about „cognition plus‟ whenever referring to the talk about the
„social‟ mounted on the top of an acquisitionist discourse. The basic claim that motivates
the more radical form of participationism is that patterned, collective forms of distinctly
human forms of doing are developmentally prior to the activities of the individual. Whereas
acquisitionists view the individual development as proceeding from personal acquisitions
to the participation in collective activities, strong particpationists reverse the picture and
claim that people go from the participation in collectively implemented activities to similar
forms of doing, but which they are now able to perform single-handedly. According to this
vision, learning to speak, to solve mathematical problem or to cook means a gradual
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transition from being able to take a part in collective implementations of a given type of
task to becoming capable of implementing such tasks in their entirety and on one‟s own
accord. Eventually, a person can perform on her own and in her unique way entire
sequences of steps which, so far, she would only execute with others. The tendency for
individualization5 for turning patterned collective doings into activities for an individual
seems to be one of the hallmarks of humanness, and it is made possible by our capacity
for overtaking roles of others.
The difference between the acquisitionist and the participationist versions of human
development is thus not just a matter of "zoom of lens," as it is sometimes presented
(Rogoff, 1995; Lerman, 1998). Above all, it manifests itself in how we understand the
origins and the nature of human uniqueness. For acquisitionist, this uniqueness lies in the
biological makeup of the individual. While participationism does not deny the need for
special biological pre-requisites - such as, for example, the special voice cords and the
ability to discern certain sounds, both of which are the basis for effective human
communication - this approach views all the uniquely human capacities as resulting from
the fundamental fact that humans are social beings, engaged in collective activities from
the day they are born and throughout their lives. In other words, although human biological
givens are what makes this collective form of life possible, it is the collective life that brings
about all the other uniquely human characteristics, with the capacity for individualizing the
collective for individual reenactments of collective activities - being one of the most
important. Human society emerges from the participationist account as a huge fractal-like
entity, every part of which is a society in itself, indistinguishable in its inner structure from
the whole.6
Another notable change that happens in the transition from acquisitionist to
participationist discourse is in the unit of analysis. It is this new unit which I had in mind
while speaking, somewhat ambiguously, about “patterned collective doings”. Other eligible
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candidates for the participationist unit of analysis are form of life, suggested by
Wittgenstein (1953), and activity, the pivotal idea of the Activity Theory. The nowadays
popular term practice is yet another viable option (see e.g. Wenger 1998; Cobb 2002).
Although all these terms are used in the current literature in numerous ways, with the
differences between one use and another not always easy to tell, each of them is good
enough for my present purpose. Indeed, all I want, for now, is to describe participationist
innovation according to those central characteristics which remain basically the same
across different renderings. Whatever name and definition are given to the participationist
unit of analysis and whatever claims about humans are formulated with its help, the
strength of this unit is in the fact that it has both collective and individual „editions.‟
Armed with this flexible analytic focus, participationists have a chance to address
the question of change that exceeds the boundaries of individual life. While speaking
about human development, participationists do not mean a transformation in people, but
rather in forms of human doing. This non-trivial discursive shift is highly consequential, as
it removes the sharp acquisitionist distinction between development of an individual and
the development of collective. The developmental transformations are the result of two
complementary processes, that of individualization of the collective and that of
collectivization of the individual. These two processes are dialectically interrelated and, as
a consequence, both individual and collective forms of doing are in a constant flux,
resulting from inevitable modifications that happen in these bi-directional transitions.
So far, I have shown how participationism deals with the dilemma of the historical
change in human forms of doing. In the rest of this paper I show how it deals with
questions about mathematics learning that acquisitionism left unanswered.
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Consequences of participationism
for the discourse on mathematics thinking and learning
3.1 What is thinking?
Although thinking appears to be an inherently individual activity, there is no reason to
assume that its origins are any different from those of other uniquely human capacities:
like all the others, this special form of human doing could only develop from a patterned
collective activity. This claim is far from intuitively obvious. After all, whatever we call
thinking is usually done by each one of us alone and is generally considered as
inaccessible to others in the direct manner. It is thus not readily evident which „visible‟
human activity might be the collective version of thinking. In fact, one has good reasons to
doubt whether such collective edition exists at all. More than any other human activity,
thinking appears biologically determined and growing „from inside‟ the person. Still,
participationist tenets speak forcefully against this deeply rooted conviction. The next
thesis to explore is that interpersonal communication is the collective activity that morphs
into thinking through the process of individualization.
A powerful, even if indirect, argument comes to mind immediately when one tries to
substantiate this conjecture. The ability to think in the complex way people do is absent in
other species and so is the human highly developed ability to communicate. At a closer
look, communication, like thinking, may be one of the most human of human activities.
This is not to say that the ability to communicate is restricted to people. At least some
animals do seem to engage in activities that one may wish to describe as communication.
And yet, human communication is special, and not just because of its being mainly
linguistic the feature that, in animals, seems to be extremely rare, if not lacking
altogether. It is the role communication plays in human life that seems unique. The ability
to coordinate our activities by means of interpersonal communication is the basis for our
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being social creatures. Our very survival, not to speak about our distinctive forms of living,
depends on our being always a part of a group. And since communication is the glue that
holds human collectives together, even our ability to stay alive is a function of our
communicational capacity. We communicate in order to ascertain the kind of mutuality and
collective doing that provide us with what we need and cannot be attained single-
handedly. The list of human needs that would remain unsatisfied without interpersonal
communication is long and multifarious, and it includes not just the most advanced and
complex cultural needs, but also the most primitive biological ones, of the kind that most
animals are able to provide by themselves, with only marginal collaboration of other
individuals. In the view of all this, it is not surprising that Leont‟ev (1930), one of the
founding fathers of participationism, declared the capacity for communication as the
hallmark of humanness: “[W]e do not meet in the animal world any special forms of action
having as their sole and special end the mastery of the behavior of other individuals by
attracting their attention” (p. 59).
All this, as important as it may sound, is not yet enough to substantiate the claim
that thinking could be defined as a form of communicating. In fact, the current discourses
go directly against this vision when they present thinking and communicating as separate,
even if tightly connected. This, indeed, is how these two activities are pictured in colloquial
forms of talk, through expressions such as „communicating one‟s thoughts‟ or „putting
thoughts in words‟. Our speaking about thoughts as being conveyed (or expressed) in the
act of communication implies two distinct processes, that of thinking and that of
communicating, with the former slightly preceding the latter and constantly feeding into it.
According to this vision, the outcomes of thinking, pictured as entities in their own right, are
supposed to preserve their identity while being “put in other words” or “expressed
somehow differently”.
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Whereas acquisitionists have been working with this dualist vision of human
cognition for centuries, participationists are likely to view the idea of „thought-conveyed-in-
communication” as but a direct result of an unhelpful objectification. With Wittgenstein
(1953), they believe that "Thought is not an incorporeal process which lends life and sense
to speaking, and which it would be possible to detach from speaking" (p. 108). Having
accepted this claim, one can also see that it remains in force when the somewhat limiting
word speaking is replaced with the more general term communicating. Consequently,
thinking stops being a self-sustained process separate from and, in a sense, primary to
any act of communication, and becomes an act of communication in itself, although not
necessarily interpersonal. All this justifies the claim that thinking may be usefully defined
as the individualized form of the activity of communicating, that is, one‟s communication
with oneself. Of course, this self-communication does not have to be in any way audible or
visible, and does not have to be in words.7 In the proposed discourse on thinking, cognitive
processes and processes of inter-personal communicating are thus but different
manifestations of basically the same phenomenon. To stress this fact, I propose to
combine the terms cognitive and communicational into the new adjective commognitive.8
The etymology of this last word will always remind us that whatever is said with its help
refers to these phenomena which are traditionally included in the term cognition, as well as
to those usually associated with interpersonal exchanges.9
To complete the task of defining thinking as an individualized form of
communication, I need yet to explain how this latter term should be understood in the
present context. Since the patterned nature of communication is due to the fact that
different people act in similar ways, communication needs to be considered as a collective
activity, and should thus be described in terms of its global patterns. Restricting the field of
vision to a single node, or to single pair of „sender‟ and „recipient‟, as is done in the
majority of known definitions, would be as unproductive as trying to understand the rules of
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chess from the individual moves of one checker. The following formulation seems to fulfill
this requirement: Communication is a collectively-performed rules-driven activity that
mediates and coordinates other activities of the collective. More specifically, individuals
who participate in the activity of communicating perform actions that are customarily
followed by a certain type of re-action of other individuals. The re-actions may be either
practical actions or other communicational moves. By practical actions, I mean actions
resulting in a change in the physical environment. Opening a window or adding a brick to a
wall while building a house are good examples of practical actions. Communicational
actions are those that affect members of community and have no direct impact on the
environment, although some of them may, in the end, lead to another person's practical
(re)action. In human activities, communicational and practical actions are usually
simultaneously present and inextricably interwoven. Clearly, communication is what
enables inter-person coordination needed for the collective implementation of complex
practically-oriented activities, form preparing foods and garments to building houses,
publishing newspapers, producing films, transporting goods, etc. This said, let me add that
it is also typical of humans to have long chains of purely communicational interactions, in
which every re-action is, in itself, a communicational action bound to entail yet another
communicational re-action. In this process, the participants alternate between the roles of
actors and re-actors, often playing both these parts in one communicational move.
Let me finish this introduction to the participationist discourse on thinking with a
number of remarks. First, the definition of communication speaks about rules that regulate
communication (and thus the commognition in general). It is important to stress that these
rules are to be understood as observer’s constructs, and not as guiding principles,
followed by individual actors in a conscious, deliberate way. Another fact to remember is
that the rules of commognition, are not in any sense “natural” or necessary, as nothing “in
the world” can possibly necessitate the given types of associations between actions and
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re-action. The source of the patterns is in historically established customs. This contingent
nature of communicational patterns is probably the reason why Wittgenstein (1953)
decided to speak about communication as a kind of game.10 Second, because of its being
rules-driven, commognition has dynamics of its own, and it would not be possible without
the natural human tendency for alignment. This said, it is equally important to note that in
commognition, like in any other historically established activity, human players do have
agency. Communicative action almost never determines a re-action. More often than not,
both action and re-action are a matter of construction, to be performed according to rules
that constrain but do not dictate. Third, whereas practical actions are direct actions on
objects, commognitive actions are about objects, that is, they focus interlocutors‟ attention
on an object. Fourth, commognitional actions are performed with the help of mediators,
which can have auditory, visual or even tactile effects on individuals. In humans, language,
which has both vocal and visual (e.g. written) editions, is the principal, although not the
only, form of commognitive mediator.
Finally, just as there is a multitude of games, played with diverse tools and
according to diverse rules, so there are many types of commognition, differing one from
another in their patterns, objects, and the types of mediators used. Like in the case of
games, individuals may be able to participate in certain types of communicational activity
and be unable to take part in some others. The different types of communication that bring
some people together while excluding some others will be called discourses. Given this
definition, any human society may be divided into partially overlapping communities of
discourses. To be members of the same discourse community, individuals do not have to
face one another and do not need to actually communicate. The membership in the wider
community of discourse is won through participation in communicational activities of any
collective that practices this discourse, be this collective as small as it may.
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3.2 What is mathematics?
Given participationist vision of thinking as a form of communication, mathematics can be
seen as a special type of discourse, made distinct, among others, by its objects, mediators
and rules.11 Let me be more specific.
A discourse counts as mathematical if it features mathematical words, such as
those related to quantities and shapes. The conversation between Roni, Eynat and Roni‟s
mother, presented in the beginning of this article, is replete with such mathematical terms
as number-words and comparison-words (e.g. more, bigger), and can thus count as a
case of mathematical discourse. This, however, is just one out of several possible types of
mathematical communication. While many number-related words may appear in non-
specialized, colloquial discourses, mathematical discourses as practiced in schools or in
the academia dictate their own, more disciplined uses of these words. As will be argued
below, neither Roni nor Eynat is using any of the mathematical words the way they are
used by mathematically versed interlocutors (and I do not mean just the grammatical
imperfections of the girls' talk).
Visual mediators to be found in mathematical discourses tend to be quite unlike
those used in many other types of discourses. While colloquial discourses are usually
mediated by images of material things, that is, by concrete objects that are identified or
pointed to with nouns or pronouns and that may be either actually seen or just imagined,
mathematical discourses often involve symbolic artifacts, created specially for the sake of
this particular form of communication. Such symbolic mediation, however, is still absent
from the incipient numerical talk of our young interviewees. Quite understandably, the only
form of visual mediation that can be found in our data is concrete rather than symbolic:
The mathematical task performed by the girls is described in terms of sets of marbles
provided by Roni‟s mother, and is visually (and tangibly) mediated by these sets.
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Endorsed narratives are sets of propositions that are accepted and labeled as true
by the given community. Mathematical narratives, to be endorsed, have to be constructed
and substantiated according to a set of well-defined rules, specific to this discourse. In the
case of scholarly mathematical discourse, these endorsed narratives are known as
mathematical theories, and this includes such discursive constructs as definitions, proofs,
and theorems.6 In addition to the generally endorsed “abstract” narratives such as those
listed above, one can speak about more specific narratives that pertain to concrete objects
and may be endorsed in a given situation. The aim of Roni and Eynat‟s activity, at least in
the eyes of the grownups, is to create such locally endorsable narratives: The girls are
supposed to explore the boxes with marbles and to come up with endorsable statements
that answer the Mother‟s question “Which of the boxes has more marbles”?
Routines are well-defined repetitive patterns characteristic of a given discourse.
Specifically mathematical regularities can be noticed whether one is watching the use of
mathematical words and mediators or follows the process of creating and substantiating
narratives about number. In fact, such repetitive patterns can be seen in almost any aspect
of mathematical discourses: in mathematical forms of categorizing, in mathematical modes
of attending to the environment, in the ways of viewing situations as “the same” or
different, which is crucial for the interlocutors‟ ability to apply mathematical discourse
whenever appropriate; and in production of narratives and their further substantiation.
Routines may be algorithmic, and thus deterministic, or just constraining. The canonic
routine of numerical comparison, which, in our example, the mother expects her daughter
to perform, is an example of algorithmic routine.
3.3 What is mathematics learning?
Learning mathematics may now be defined as individualizing mathematical discourse, that
is, as the process of becoming able to have mathematical communication not only with
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others, but also with oneself. Through the process of individualization, the personal
creativity of the learner comes in.
Let me now go back to the Comparing sets of marbles episode and see whether
this definition helps to make a better sense of children‟s actions. It is now natural to
assume that the observed phenomena are related to the fact that the children have not yet
individualized the numerical discourse they did not yet turn this form of talk into a
discourse for themselves.12 Indeed, there are many signs showing that the girls are
probably at the very beginning of the process. The first evidence can be found in the fact
that they do not use the compare-by-counting procedure on their own accord: The
question “[I]n which of the boxes [are there] more marbles?” ([3b]) is clearly not enough to
get them started, and nothing less than a clear hint by the mother (“Do you want to open
and discover?”, [10a]) would help. Further, the children need mother‟s scaffolding in order
to perform the procedure in its entirety (note, for example, that they stop after having
counted the marbles and they need to be prompted in order to draw the conclusion; see
mother‟s question [15]). It is thus clear that if the girls participate in the numerical
discourse, it is on other people‟s accord and according to other people‟s rules. This can be
summarized in the following way: What for the grownups is the routine of exploration,
geared toward enhancement of one‟s arsenal of “factual knowledge” (endorsed
narratives), for the children is a ritual a game played with others for the sake of the
togetherness that game-playing affords. Note that touching the marbles one by one while
also pronouncing subsequent number words is not unlike incantation of meaningless
rhymes which is often a part of children‟s play. What is now but a ritual, will turn into
exploration in the course of individualization.
The fact that the girls‟ participation in the numerical discourse is ritualized and
undertaken for the sake of connecting with others becomes even more evident when
children‟s actions in the second part of the episode are compared with what they do in the
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first. When the conversation begins, the girls spontaneously respond to the mother‟s
query with pointing to one of the identical boxes. Evidently, the question “[I]n which of the
boxes [are there] more marbles?” when first asked, is not received as a prompt for a
conversation on numbers but rather as an invitation to what the children usually do on
their own accord and willingly: to choosing one of the boxes for themselves. Making
choices, unlike numerical comparisons, is the kind of activity which the girls have already
individualized. It will yet take time until the two types of routines those of choosing and
those of comparing combine one with the other into an individual activity of the child.
It is reasonable to assume that a certain proficiency in a discourse is a prerequisite
for its individualization. Roni and Eynat do not yet exhibit sufficient fluency in numerical
talk. For example, they have yet to change their use of number words. Right now, these
words are for them but a part and parcel of counting. In the future, the words will be used
in many different types of sentences and in multiple roles, as adjectives and as nouns,
among others. Above all, the use of these words will become objectified: More often than
not, expressions such as one, two or two hundred will be used as if they referred to self-
sustained, extra-discursive entities. Similarly, the children‟s use of connectives such as
because will change dramatically. At the moment, this use is clearly ritualized: If the girls
answer mother‟s why questions in a seemingly rational way (see Roni‟s utterance [5] and
Eynat‟s utterance [7], which both begin with the word because), it is obviously due not to
their awareness of the relations between boxes but to their familiarity with the form of talk
which is expected by the grownups in response to this kind of question. At this point, the
girls are already aware of how to talk when answering request for explanation, but are not
yet fully aware of when under which circumstances it is appropriate to apply them. At
this point, the mere appearance of the word why in the interlocutor‟s question may be
enough to prompt an utterance that begins with because and then simply repeats, in a
somewhat modified form, what the question was asking about. It seems reasonable to
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conjecture that in the process of individualization, the awareness of how discursive routine
should be performed usually precedes the ability to tell when such performance would be
appropriate. One may even hypothesize that it is the ability to make independent decisions
about when to apply a given discursive procedure which is the ultimate sign of its
individualization.
The manner in which all these changes in the girls‟ numerical discourse13 are
supposed to happen is implicated in the very claim that learning mathematics is the
process of individualization of mathematical discourse: Discursive change can only
originate in communicating with experienced interlocutors. This vision is quite different
from the one professed by the acquisitionist who assumes, if often only tacitly, that
learning results from the learner‟s attempts to adjust her understanding to the externally
given, mind independent reality. Contradicting the participationist belief in the primacy of
the collective, this latter version implies that learning, at least in theory, could take place
without participation of other people.
Not every mathematical conversation is an opportunity for learning. For a
discursive change to occur, there must be some discrepancy a communicational conflict
between interlocutors. Such conflict arises whenever different participants seem to be
acting according to differing discursive rules. The difference may express itself in a
disparity in the interlocutor‟s uses of words, in the manner they look at visual mediators or
in the ways they match discursive procedures with problems and situations. More often
than not, these differences find their explicit, most salient expression in the fact that the
different participants endorse differing, possibly contradicting, narratives.14 Dissimilarities
between Roni and Eynat‟s numerical discourse and the numerical discourse of the
grownups express themselves in different uses of words and disparate routines, and thus
constitute a good example of communicational conflict, likely to result in a considerable
learning.15
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18
In order to fully individualize numerical discourse Roni and Eynat will have to
overcome this conflict. This is not going to be easy. If the child is to ever use the numerical
discourse in solving her own problems, she must be aware of the advantages of the
relevant discursive procedures. For example, she needs to realize that she may benefit
from choosing according to numbers. And yet, in order to become aware of these
advantages, she has to already use the numerical discourse. The process is thus
inherently circular. The next question to ask is what can possibly motivate the child to
engage in the demanding task of overcoming the circularity.
3.4 Why do we learn mathematics?
The circularity implies that learning mathematics requires readiness to engage in the new
discourse even before one can see its problem-solving potential and inner logic. In other
words, the child needs to be prepared to participate in the numerical discourse in a
ritualized way before she is able to practice the discourse while engaging in self-initiated
explorations. The child‟s motivation for such ritualized action is its immediate social
reward: Roni and Eynat perform the ritual as an act of solidarity with the grownups and in
the attempt to win their approval. Giving the answer that is expected by the interlocutor
may be read as an act of pledging allegiance.
More generally, when the child first engages in mathematics learning, it is because
of her overpowering need for communication, which grows out of the even more
fundamental need for social acceptance. This social concern can clearly be seen all along
the conversations with the girls. The way Roni monitors her mother‟s face, talks to her
and follows her lead clearly indicates that getting the parent‟s attention and approval is
the girl‟s main concern. This wish competes, and is successfully combined, with an
equally strong need to belong with the peer. While making their choices, Roni and Eynat
are careful to stress that their decisions are shared (in the further parts of our transcripts,
Strobl 24 January 2013
19
this need for solidarity with the friend is further evidenced by Roni‟s repetitive use of the
word we, through which she asserts the joint ownership of solutions.)
To sum up, the children have different goals than those envisioned by the
grownups. While counting and comparing, the girls are in fact preoccupied with the
delicate social fabric of their little group, and the conversation on boxes with marbles is, for
them, as good an occasion for inter-personal engineering as any other. While grownups
count in order to get closer to the truth about the world, the children count to get closer to
the grownups. The “exploratory” activities of the young participants are therefore a form of
community-building ritual.
Consequences of participationism
for the practice of mathematics teaching and learning
Our ability to make sense of what we see depends on our uses of words. As illustrated
above, the interpretation of the notion “social” that gave rise to the commognitive
framework made a significant difference in our vision of learning and in this vision‟s
theoretical entailments. In particular, it allowed to account for phenomena that escaped
acquisitionist‟ explanations and it offered alternative explanations for some others. Thus,
for example, what acquisitionists interpreted as showing children‟s unawareness of the
“conservation of number” became, in our interpretation, the result of the simple fact that in
the situation of choice, young learners had no reason to privilege the ritual of counting over
other routines that they had already at their disposal.
Perhaps the most dramatic difference between the acquisitionists‟ and
participationists' visions of mathematical thinking is in their respective messages about the
origins of mathematical learning. Whereas acquisitionists views learning as resulting from
the learners‟ direct efforts to arrive at a coherent vision of the world, participationists sees
learning as arising mainly from one‟s attempt to make sense of other people‟s vision of
Strobl 24 January 2013
20
this world. The former perspective implies that learning, at least in theory, could take place
without participation of other people. In contrast, the idea of mathematics as a form of
discourse entails that individual learning originates in communication with others and is
driven by the need to adjust one‟s discursive ways to those of other people.
Participationism also provokes second thoughts about some common pedagogical
beliefs. For instance, it casts doubt on the current call for “learning with understanding,” at
least insofar as this call is interpreted as the exhortation to never let the student practice
routines which she cannot properly substantiate. According to the present analysis,
students‟ persistent participation in mathematical talk when this kind of communication is
for them but a discourse-for-others seems to be an inevitable stage in learning
mathematics. If learning is to succeed, all the interlocutors must agree to live with the fact
that the new discourse will initially be seen by the newcomers as a game to be played with
others, and that it will be practiced only because of its being a discourse that others use
and appreciate. It is thus now time to rehabilitate the learning that is based on ritualized
action and on thoughtful imitation of the grownups‟ ways with words. Trying to figure out
and then to meet the expert participants‟ expectations is sometimes the only way to initiate
the long process of individualization of discourses. Making sense of other person‟s thinking
is not any less demanding (or respectable!) than the direct attempts to understand reality.
Indeed, entering “foreign” forms of talk (and thus of thought) requires a genuine interest
and a measure of creativity. To turn the discourse-for-others into a discourse-for-oneself,
the student must explore other people‟s reasons for engaging in this discourse.
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21
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1 To illustrate, let me just mention two differing interpretations of the word ‗social‘ to be found in the context
of the famous dichotomy individual vs. social, that lies at the very heart of the current controversies on human
development. At a closer look, those who contrast ―the social‖ with ―the individual‖ may have two different
distinctions in mind. In one of these dichotomies, the term social means that whatever is described with this
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24
adjective has been done or attained by an individual through interaction with others. In this case, the social
could probably be replaced with interactional. The other dichotomy that hides behind the opposition social
versus individual regards not so much the ‗technicalities‘ of individual learning as the nature and origins of
what is being learned. This time, the issue at stake is that of the ontological-epistemological status of
knowledge, with the word individual functioning as almost synonymous with natural or genetically
necessitated, whereas the social is tantamount to human-made. It is this latter, strong interpretation of the
―social‖ that seems to have spurred Vygotsky‘s famous criticism of the Piagetian doctrine (the fact of which
Piaget was likely to be aware only partially, if at all).
2 The study from which the vignette is taken has been reported in Sfard & Lavie (2005).The conversation was
held in Hebrew. While translating to English, I made an effort to preserve the idiosyncrasies of the children‘s
word use.
3 For Activity Theory see, e.g. Leontiev (1947/1981), Nardi (1996), Engeström (1987).
4 In this context, one should mention the significant influence of Wittgenstein, as well as that of two inter-
related, but still distinct schools in sociology: The symbolic interactionism usually associated with Mead
(1934), Goffman (1958), and Blumer (1969); and the ethnomethodological approach initiated by Garfinkel
(1967). Of relevance in this context is also the sociological phenomenology that originated in the
philosophical thought of Husserl‘s and was founded in the first half of 20th century by Schutz (1967). The
direct influence of this latter school of thought on psychology and education can be seen in the work of
German and American researchers see e.g. work by Bauersfeld (1995), Voigt (1985), Krumheuer (1995),
and Cobb and his colleagues (Cobb et al., 1993; Cobb & Bauersfled, 1995). All these schools, be them
diverse as they are, share a number of basic assumptions, which can also be found in most of the current
versions of participationism. They all take the inherently social nature of humans as their point of departure
and agree that actions of the individual cannot be understood unless treated as part and parcel of collective
doings. The patterned collective activities, in turn, are objects of their participants‘ sense-making efforts. The
different schools begin to diverge only when it comes to their respective responses to the question of where
the regularities come from and whether the observed patterns are in any real sense ‗real,‖ as opposed their
lying exclusively in the eyes of sense-making insiders.
5 The terms individualization and collectivization may be viewed as strong participationist versions of what
Vygotsky and Activity Theorists call internalization and externalization. The important advantage of the
present terminology is that it is free of acquisitionist undertones of the traditional vocabulary. In result, the
proposed version of strong participationism does not imply that thinking and behavior are two ontologically
different types of processes but rather promotes the idea that they are two forms of basically the same
phenomenon, which may be termed simply as ‗individual human doing.‘ These two forms differ only in the
degree of their visibility to others.
6 One should not, of course, take this metaphor too far. Not every collective activity can be fully
individualized (reenacted by a single person). Suffices to think about building bridges or performing complex
surgeries. And yet, whatever distinctly human activity has been mastered by a person, the source of this
ability is in this person‘s earlier participation in its collective implementations.
7 This definition resonates well with the conversation metaphor of mind to be found in Ernest (1993, 1994),
Mead (1934), Bakhtin (1981), Holquist (1990) and Marková (2003). See also the idea of discursive
psychology in Harré & Gillett (1995), Edwards (1997).
8 The act of coining my own neologism is certainly rather daring, and I feel I owe an explanation. While
trying to give a name to the just defined discourse on thinking I could, of course, follow the usual practice of
employing a word that already exists in the English language. In fact, after having said that thinking is an
individualized form of communication, I could use the word communication to encompass both categories
that of thinking, and that of interpersonal communication. Indeed, many other human activities that begin
as collective and are liable to individualization do not change their names as a result of individualization: the
individually performed mathematical problem solving is still called problem solving and the task of complex
data processing is called data processing whether it is implemented by a single individual or by a group..
However, calling thinking (individualized form of) communication would require the users to overcome our
deeply entrenched habit of using the words thinking and communicating as denoting different, non-
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25
overlapping types of activities. In introducing the new name I was motivated by the conviction that our view
of communicating as being collective by definition may be too strong to be removed by a mere act of
redefining.
9 At this point, a skeptic can bring yet another argument against the idea of thinking as individualized form of
interpersonal communication. The dilemma of relations between thinking and speech has been stirring one of
the most persistent and encompassing debates in the history of human thought. Considering the fact that no
solution, not even those offered by the most revered of thinkers, managed to bring about a durable consensus,
it may be difficult to understand why the simple statement ―thinking is (can be usefully defined as) a form of
communication‖ should now be accepted as an answer. In response, let me stress two differences between my
present attempt and most of those undertaken in the past. First, what I did has been framed as an act of
defining, not as an attempt to find out what thinking ―really is.‖ Thus, the agreement may be possible
provided I manage to convince others about the usefulness of the proposed thinking = self-communicating
equation. The second difference stems from the fact that the time-honored dilemma which, for centuries, has
been boggling philosophical minds is that of the relations between thinking and language (or speech),
whereas the proposed definition links thinking with communication. The relation between thought and speech
has been, indeed, a leitmotif of philosopher‘s musings about thinking. This is easily explicable, considering
the centrality of verbal communication in specifically human forms of life and the resulting tendency to
equate human communicating with talking. Speech and communication, however, although related, are not
the same: The former is but a special case of the latter. There are numerous non-verbal forms of
communication, and all of them must be considered. Thus, the descriptions of thinking as ―talking to oneself‖
or as ―inner speech‖ are more restrictive than the communicational definition proposed above and as such,
they do not make full justice to the phenomenon we wish to fathom. If the attempts to capture the gist of
human thinking have been invariably deemed futile, it was probably because of the fact that the problem has
been restricted to the issue of relations between thinking and language.
10 More precisely, Wittgenstein (1953) spoke about language games. The metaphor of game, however, is
clearly applicable also to non-verbal forms of communication.
11 Equating mathematics to discourse should not be confused with the time-honored, and often contested,
claim that mathematics is a language. The word language is usually understood as referring to a tool for
representing objects, with this objects being external to, and independent from, the language itself. Therefore,
the statement ―mathematics is a language,‖ unlike its discursive counterpart, could imply that the objects of
mathematics are not a part of mathematics itself. Second, discourses involve many mediators, not just
language.
12 The term discourse-for-oneself is close to Vygotsky‘s idea of speech-for-oneself, introduced to denote a
stage in the development of children‘s language (see e.g. Vygotsky 1987, p.71). Our terms also brings to mind
the Bakhtinian distinction between authoritative discourse, a discourse that ―binds us, quite independently of
any power it might have to persuade us internally‖; and internally persuasive discourse, one that is ―tightly
woven with ‗one‘s own world.‘ (Bakhtin, 1981, pp. 110-111.)
13 Since the only way to actually observe such changes is by watching the child in mathematical conversation
with others rather than with herself, we will need to remember that whatever is found has been informed by
the other participants as well. Still, with an appropriate analyses and the sufficient amount of observations, we
may be able to make conjectures about some general properties of the child‘s participation, as well as of the
individualized form of this child‘s discourse, if any.
14 Since discursive conflict arises in face of differences in meta-discursive rules, a mere difference in
narratives cannot count as a sufficient evidence for such conflict; for example, if one objects to the claim that
―The weather is beautiful today‖, it is indicative of the conflict of opinion, not of discourses
15 The notion of communicational conflict, although reminiscent of the acquisitionist idea of cognitive
conflict, is in fact a different type of theoretical construct: Communicational conflict results from a disparity
between student‘s and teacher‘s discourses rather than from a clash between the learner‘s vision of the world
and the real state of affairs; it is indispensable for learning rather than optional; and it is resolved through
students‘ acceptance and rationalization of the discursive ways of an expert interlocutor and not via their
direct, independent reasoning about the world.
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This article defines communities of practice as an approach for cross-organizational collaboration, highlights several success stories, and provides guidelines on how to cultivate communities of practice to improve performance outcomes. The complexity of today’s challenges and associated performance expectations—in public, private, and non-profit sectors—requires a commensurate capacity for learning, innovation, and collaboration across diverse constituencies. But conventional government bureaucracies are designed to solve stable problems for established constituencies through centrally managed programs and policies. These structures are not sufficient to address the messy problems we face today. Many of our most urgent social problems—in education, community safety, the environment, job creation, affordable housing, healthcare, and more—call for flexible arrangements, constant adaptation, and the savvy blending of expertise and credibility that requires crossing the boundaries of organizations, sectors, and governance levels. One way to integrate efforts across these boundaries is to cultivate “communities of practice” that promote cross-boundary action learning to address national priorities.
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Most previous research on human cognition has focused on problem-solving, and has confined its investigations to the laboratory. As a result, it has been difficult to account for complex mental processes and their place in culture and history. In this startling - indeed, disco in forting - study, Jean Lave moves the analysis of one particular form of cognitive activity, - arithmetic problem-solving - out of the laboratory into the domain of everyday life. In so doing, she shows how mathematics in the 'real world', like all thinking, is shaped by the dynamic encounter between the culturally endowed mind and its total context, a subtle interaction that shapes 1) Both tile human subject and the world within which it acts. The study is focused on mundane daily, activities, such as grocery shopping for 'best buys' in the supermarket, dieting, and so on. Innovative in its method, fascinating in its findings, the research is above all significant in its theoretical contributions. Have offers a cogent critique of conventional cognitive theory, turning for an alternative to recent social theory, and weaving a compelling synthesis from elements of culture theory, theories of practice, and Marxist discourse. The result is a new way of understanding human thought processes, a vision of cognition as the dialectic between persons-acting, and the settings in which their activity is constituted. The book will appeal to anthropologists, for its novel theory of the relation of cognition to culture and context; to cognitive scientists and educational theorists; and to the 'plain folks' who form its subject, and who will recognize themselves in it, a rare accomplishment in the modern social sciences.