The articulation and structuring of
conceptions
in the mathematics class:
argument and public knowledge
by
Patricio Herbst
The University of Michigan, USA
If the mathematics class is considered as a system of
public knowledge, what are the possible uses of argument in
the articulation and structuring of knowledge within this
system. The term public knowledge is used here to
indicate two simultaneous distinctions. The mathematics of
the class as public knowledge is distinguished (1) from the
mathematics (prescribed) to be transmitted and acquired (the
official knowledge), and (2) from the mathematics (known or
developed) by the individuals that play roles of learners or
teachers (their personal knowledge). These distinctions are
analytical; they mean not to discredit or neglect official
or personal knowledge. Rather, these distinctions serve to
identify a challenge to undertake: That of fleshing out a
notion of public knowledge which is dialectically tied to
the other two notions.
We need to find ways of talking about the
relation between argument and mathematical knowledge within
the complexity of the mathematics class. The problems with
proof in the mathematics class have often been framed in
terms of the logical structure of mathematical discourse or
the personal conviction of the participants. A focus on the
mathematics of the class aims at developing a way of talking
about proof more closely related to the class's construction
of mathematical knowledge.
Recent examples of mathematics classroom
research (e.g., Yackel & Cobb, 1996; Ball & Bass,
forthcoming) have encouraged thinking about the mathematics
class as a place where a specific kind of mathematical
knowing exists and functions. Ball's example of Sean's
Numbers (numbers that are odd multiples of two--such as 6 or
10) illustrates very clearly that such specificity means not
just an indigenous way of doing, but it involves indigenous
objects (1)
of that doing (Ball, 1993). I suggest that observations like
that support the hypothesis that the mathematics class is a
cognitive locus: A sort of ecosystem that produces its own
objects of knowledge and, one would expect, its own
communication codes, substantial forms of argumentation,
indigenous theories, and inscription and memory devices.
According to the hypothesis that the mathematics class is a
cognitive locus, the mathematics (of the) class is enabled
(and constrained) as much by the cognition of the
individuals involved as by the structure of the knowledge to
be taught, but is irreducible to either of those factors.
Studying such system of public knowledge may help us
understand more systematically what mathematics might be
known and done in schools. Whereas knowledge about how
individuals develop specific mathematical ideas or about
ways to organize the knowledge to be learned provide
motivation and warrant for suggesting that changes be made
in school mathematics, they are not enough to estimate their
cost or even demonstrate that they are possible. We need to
know more about the logic that underlies the knowledge
practices of school mathematics in order to envision the
conditions of possibility for those changes.
The mainstream perspectives of mathematics
education research on proof can be correlated with the ways
of conceiving of the mathematics class outlined in the
previous paragraph. Some researchers interested in the
individual development of notions of proof have defined
proof as those strategies by which people convince
themselves or persuade others of the validity of their
mathematical assertions (see for example Harel & Sowder,
1998). In contrast, scholars interested in determining the
import of mathematical proof (as a general practice of the
mathematical sciences) to the practice of mathematics
education have considered proof as the generic ways of
validating and explaining truth in mathematics (see for
example Fawcett, 1938; Hanna, 1995). Those perspectives may
be useful to study the development of the notion of proof by
individuals or to monitor the fit between what is called
proof in the mathematics class and in mathematics. However,
the conceptualization of the class as a system of public
knowledge recommends one to avoid a reduction of proof to
conviction or to logic. Instead, the conceptualization of
the class as a system of public knowledge recommends to take
up other sorts of problems that seem to be critical for an
investigation of a classroom-based notion of proof. Those
problems concern the possible roles of argument in
constituting, articulating, and structuring a system of
public knowledge.
The problems of constitution,
articulation, and structuring of public knowledge involve,
at least, tensions between knowledge as product (as public
wealth) and production of knowledge (as public work), both
at a global level (accumulation and growth) and at a local
level (creation and trade). At a global level one can
envision a tension between the structure of knowledge (as a
product) and the structuring of knowledge (as part of its
production). These tensions can be located in the history of
mathematics as well as in the individual development of
mathematical ideas. The following reflection on mathematical
practices and their history should help make the point that
there is an intimate connection between a substantive,
practice-based, notion of argument (or proof) and the
problems of public knowledge presented above.
The creation and trade of knowledge in
mathematical practice concerns fundamentally the
articulation of definition of objects and proof of theorems
(if-then statements about those objects). There is always a
tension between the interest to claim a conclusion and the
possibility to establish such a claim in continuity with
what is known: This tension relates the conclusion (stating
a categorical fact) and the proposition (providing a
hypothetical statement). Inasmuch as one can load more or
less the conditions under the "if," a proof will more or
less likely be achieved and, thus, warrant claiming the
conclusion. But clearly, those conditions can be so
demanding or the proof so trivial that the hypothetical
statement may become unimportant. Mathematicians do not just
strive to provide the most impeccable proof that a
conclusion is warranted, they also search for the
(relatively) minimal conditions that enable an acceptable
proof to exist&emdash;and this naturally raises the
expectation of the ingenuity in the proof itself
(2).
The work of proof in this local aspect of producing
mathematical knowledge cannot be reduced to the logical form
of the argument (as it involves substantive knowledge and
values about knowledge). The work of proof in this local
aspect of producing mathematical knowledge cannot be reduced
to the personal knowledge of the participants either (as it
involves a synthesis between epistemological positions that
usually coexist in a person).
The accumulation and growth of
mathematical knowledge concerns fundamentally the
structuring of mathematical theories, memoirs, discourses,
or texts (which could be understood as intellectual
architectures that integrate a body of objects and proved
propositions, and suggest problems for future work). There
is always a tension between the capacity of theories to
house large number of results within a coherent structure
(the accumulation aspect) and the capacity of theories to
enable the actual formulation of problems and generation of
solutions (the growth aspect). This tension affects the
constraints to which a proof submits. If a theory is going
to enable the community of researchers to foresee what to
do, it has to allow insights that are not formalized or
axiomatized, concentrate the rigor where it is needed, avoid
handcuffing intuition. Reciprocally, if a theory will give a
coherent account of what a community knows, it needs tools
to avoid the circularity of arguments or the ambiguity of
information. (3)
The work of proof in this global aspect of production of
mathematical knowledge cannot be reduced to the building of
a logical architecture (as it involves not just the
existence of a logical structure but also a substantive,
prospective, rationale for it). The work of proof in this
global aspect of production of mathematical knowledge cannot
be reduced to the sharing of the personal knowledge of the
participants (as theories involve the reconstruction of
fictional but common pasts and enable the envisioning of
common futures).
The search for a classroom-based notion of
proof may, therefore, benefit from addressing analogous
questions of articulation and structuring of the mathematics
(of the class). This proposal thus intends to attack the
problem of proof in the classroom by circumventing it.
Instead of abstracting a generic notion of proof in
mathematics as a discipline or aggregating the ways in which
classroom participants are individually convinced, the
metaphors of economic production help give a working
definition of proof from the perspective of public
knowledge. The provisional definition I suggest includes and
extends the one proposed by Balacheff (1987, p. 147):
Proof is an explanation [of the truth
character of a proposition] accepted by a given
community at a given moment of time. The decision to
accept it can be the object of a debate whose principal
objective is to determine a common system of validation
for the speakers.
I suggest that proof can initially be conceived of as a
prospective and retrospective tool for the public control of
knowledge production: What counts as proof depends on what
is being (and to be) done and known at various levels of
cognitive activity (problems, propositions, models,
theories).
A focus on the objects of public knowledge
dealt with in the class makes one aware of the fact that to
recognize a claim being formulated needs an appraisal of the
normative ways of representing and enunciating that are
specific of the objects dealt with. The recognition of the
conditions under which a claim is formulated and of the
validity of its "proof" depend on the conceptions being
handled and those being kept away. What things count as a
claim or as a proof in the mathematics of the class are not
understandable just in terms of an indigenous but generic
logic or an indigenous but generic language. A refined
search for understanding about proof in the mathematics (of
the) classroom could use working inside specific conceptual
fields rather than in mathematics in general.
There is an important relationship between
what may possibly count as "proof" and the way conceptions
are articulated. The various conceptions related to a
mathematical notion are not likely to be available at any
given time (even if students have used them in the past).
This seems to fulfill the purpose of enabling some work to
be done as well as placing bounds on what arguments and
counterexamples can be offered. What are the arguments by
which various conceptions associated with a single
mathematical notion are integrated into public knowledge? If
various conceptions are reconciled into a unified theory,
how are they brought to "argue" against each other, what do
they "say," and how do they develop a "normal argument"? If
such reconciliation does not happen in fact, what are the
ways in which parallel and disconnected conceptions maintain
their own senses of "normal argument"? These questions are
formulated metaphorically because an important part of the
work to do involves developing theory that can formulate
them in more precise terms.
Nicolas Balacheff (largely inspired by
Gerard Vergnaud's theory of conceptual fields and by Gaston
Bachelard's notion of epistemological obstacle) has provided
an abstract characterization of conceptions that
permits to describe substantive examples of argument.
Mathematical conceptions are defined as coherent domains of
practice and modeled as a quadruplet of four interrelated
systems, namely:
the problems that can
normally be attacked,
the operators that normatively handle
those problems,
the systems of representation that
permit to express and attack those problems, and
the structures of control (including
the kinds of proof) that permit to determine whether
a
problem has been satisfactorily solved.
I have suggested that the question of what proof might
look like is intimately related to the articulation and
structuring of student conceptions: What counts as proof and
what works as proof depend on what the class knows and could
know. Whereas such conceptualization of proof may respond to
a vision of school mathematics more closely related to what
mathematicians do, it also upgrades the stakes of the
teacher. In such a vision of proof in school mathematics,
what are the tools available for the teacher to engage
students in argument and to ensure that such arguments are
mathematical?
The study of how conceptions are
articulated and structured provides a base to understand the
rationality behind current uses or lack of use of argument.
One will then be in better position to understand more
precisely what are the specific challenges that teachers
face in trying to make the practices of school mathematics
more similar to those of working mathematicians. Further,
one may be able to work toward the development of technical
support to meet those challenges.
Notes
(1)
The fact that Sean's numbers admit of a precise mathematical
definition does not make them objects of the mathematics of
mathematicians (by themselves they are relatively
uninteresting to be created). Insofar as there is no
cultural imperative to transmit and learn Sean's numbers,
but these numbers nevertheless emerge as mathematical
objects in Ball's class, they can be an example of an
indigenous object. [Back]
(2) For example, it is relatively easy to
prove Euler's theorem (that vertices plus faces minus edges
equals 2) for the case of a convex polyhedron, using
projective geometry. The observation that this invariant
holds for some other (but not all) polyhedra imposed the
need to define the largest set of objects for which this
theorem was true (hence, the definition of simple
polyhedron) and to search for new tools that enabled a proof
to exist. The resulting argument was still compelling, yet
rather more metaphorical than the projective argument (see
Frechet & Fan, Lakatos ). What counted as an acceptable
proof was an emergent of the best synthesis between the
interest in claiming the conclusion and the importance of
the proposition stated. [Back]
(3) It seems to me that a prime example of
this equilibrium is found in Euclid's Elements (exactly with
all its merits and faults). [Back]
References
Balacheff N. (1987). Processus de preuve et
situations de validation [Processes of proof and
situations of validation]. Educational Studies in
Mathematics, 18, 147-176.
Ball D. L. (1993). With an eye on the mathematical
horizon: Dilemmas of teaching elementary school mathematics.
The Elementary School Journal, 93, 373-397
Ball D., Bass H. (in press). Making believe:
The collective construction of public mathematical knowledge
in the elementary classroom. In D. Phillips (Ed.),
Constructivism in education: Yearbook of the National
Society for the Study of Education. Chicago: University of
Chicago Press.
Fawcett H. (1938). The nature of proof&emdash;The
National Council of Teachers of Mathematics Thirtheenth
Yearbook. New York: Bureau of Publications of Teachers
College, Columbia University.
Hanna G. (1995). Challenges to the importance of
proof. For the Learning of Mathematics, 15(3), 42-49.
Harel G., Sowder L. (1998) Students' proof
schemes: Results from exploratory studies. In: A. Schonfeld,
J. Kaput J., and E. Dubinsky (eds.) Research in collegiate
mathematics education III. (Issues in Mathematics Education,
Volume 7, pp. 234-282). American Mathematical Society.
Yackel E., Cobb P. (1996). Sociomathematical
norms, argumentation, and autonomy in mathematics. Journal
for Research in Mathematics Education, 27, 458-477.
Reactions?
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The reactions to the contribution of Patricio Herbst
will be published in the May/June 2000 Proof Newsletter
©
Patricio Herbst
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