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Based on observation of current
practice of proof writing in high school, I concluded
elsewhere (Herbst,
1998, pp. 239, 271) that the two-column proof format works
in solidarity with a clear division of the discursive labor
of student and teacher. The format affords teachers an
implicit criterion to indicate what can be said (hence, it
leaves unquestioned the control over epistemic operators
such as know, need, can say, etc.). The format makes
students accountable for producing an ordered list of
"categorical" statements about a given figure, whereas it
reserves to the teacher the decision as to whether the
produced list of statements constitutes a valid
argument.
Those
observations can explain why the two-column proof format may
be better adapted to the logic or practice of geometry
teaching than other (less regulated) forms of argumentation.
From the perspective of the teacher, this format presents an
efficient tool to ensure and control the production of an
acceptable proof by the student. From the perspective of the
student, the format does not make him or her accountable for
the construction or the explanation of the geometric objects
of knowledge or for the manufacturing of an argument.
Instead, it requires him or her to produce empirical
observations (*)
about the figure at hand.
The two-column
proof format also bounds what can be given to prove in the
sense that it shapes the conceptions of what geometry is.
Two-column proofs work best within a conception of geometry
as the study of figures which have already been constructed
and are always constructible. The study of geometry
conceived in that way is a descriptive study governed by a
general logic that emphasizes the logical connection between
factual properties. To be clear, Euclidean geometry is
indeed all of that but, necessarily, not just that. The
practices unfolded around the two-column proof format seem
to discourage a complementary conception of geometry that
was central in Euclid's Elements and in the notions of Greek
geometry (Caveing,
1990): Geometry is also the study of the conditions that
make the figures constructible. In the study of geometry
conceived in the latter way, the logical links between
statements are pondered with respect to the substantive
strength of the links: What makes a proposition valuable is
not just that a proof exists but also that without a proof
one cannot really know whether the assertion is true
(because there is such a leap from the hypothesis to the
conclusion). As the practice of two-column proofs needs a
given figure and given statements of what is given and what
is to prove, the conception of geometry as the study of
necessary and sufficient conditions is deemphasized (and one
can understand why students may proceed by trial and error
in construction problems after having proved a statement
that entails the construction procedure&emdash;see
Schoenfeld,
1988).
The
two-column proof format bounds the conception of
mathematical proof and of proving in mathematics. By
separating the source of the statements that are formulated
from the arguments that are made about them, this format
emphasizes the role of proof as a certification method,
separated from the search for knowledge or the construction
of the mathematical objects of knowledge. By displacing the
statement from the proof and enabling a policing of the
statements that make up a proof, the two-column proof format
breaks the dialectic between formulation and validation
described by Lakatos
(1976; see also Balacheff,
1991) and its underlying tension between interest to know
and validity of knowledge (*).
As a consequence
of the previous observations, it can also be argued that the
practices associated with the two-column proof format bypass
the epistemological characteristics of mathematical
reasoning. These practices foster a reduction of
mathematical reasoning to an activity involving the
psychology of the reasoning agents and the "natural"
language with which they relate to a "given" mathematical
world: Mathematical reasoning becomes just (general) logical
reasoning applied to mathematical objects, whose conditions
of existence are taken as given. Of course, mathematical
reasoning certainly involves logical reasoning, yet what is
problematic is the exclusion of an epistemological aspect
specific to the mathematical objects being reasoned with and
about. By identifying the logical precedence of "reasons"
with their temporal precedence in the text being studied,
mathematical reasoning becomes more of an activity of
describing a "given" mathematical world than one of
constructing a "possible" mathematical world (or
mathematizing). Some consequences as to the value of
mathematics in general education and to the nature of the
transfer (of the study of mathematics to other situations)
can be envisioned (see Judd,
1928; Skovsmose,
1992; Vygotsky,
1934/1986, pp. 146-209): Mathematical reasoning becomes
suitable to reason about objects that are purely
mathematical or that have already been mathematized
elsewhere (such as the objects of the hard sciences), but is
hardly deemed apt to reason about other objects (such as
those of the soft sciences or ordinary life) beyond the
level of appearances. By fostering the notion of proof as
just logical reasoning about objects that are already
mathematical, the two-column proof format collaborates to
stop the scientific dialectic between empiricism and
rationalism in the construction of mathematical objects of
discourse (see Skovsmose,
1992, p. 6).
 
Note
1. Obviously,
the activity of the student can be one of making empirical
observations even if no physical interaction with the figure
is involved: The observations that can be made are empirical
insofar as they assume an identity between material and
mathematical object. [Back]
2. A good
image of this tension is provided by the rational decision
to circumscribe the definition of polyhedron to having faces
that are simple (i.e., polygons homeomorphic to circles) so
as to achieve a generalization of Euler's theorem based on
the number of holes in the polyhedra (see Lakatos, 1976, p.
66ff). [Back]
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