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The first premise of this paper is that proof must be
part of any mathematics curriculum that aims, as it should,
to reflect mathematics itself and the important role of
proof within it. The second is that the most significant
potential contribution of proof in the classroom is in the
promotion of mathematical understanding, a role that it
plays in mathematical practice as well (Thurston, 1994).
Some educators, proceeding from these premises, have
considered ways to make effective use of proof in teaching,
and especially in the last twenty years there has been a
significant reorientation towards intuition in the teaching
of proof (Dörfler and Fischer, 1979). Wittmann and
Müller (1988) speak of "intuitive proof"
("inhaltlich-anschaulicher Beweis"), and Hanna (1990) and
Dreyfus and Hadas (1996) draw on the distinction between
explanatory and non-explanatory proofs.
These educators, however, have
concentrated on the internal aspect of proof, focussing in
the main on its function within mathematics (Hanna and
Jahnke, 1993; 1996). This paper seeks to redress this
imbalance somewhat by investigating proof primarily from the
external viewpoint, with a focus on one of its important
external aspects: the relationship between physics and
mathematical proof. Jahnke (1978) and Winter (1983) have
already argued that the usual opposition between "intuitive"
and "deductive" is unacceptable, and that mathematical proof
should not be seen as a turning away from observation and
measurement, but rather as a guide to an intelligent
exploration of phenomena. The specific question the paper
poses is twofold: What is the possible role of arguments
from physics within mathematical proof, and how should this
role be reflected in the classroom?
Previous scholarly
work
The first part of this question has to do with
mathematics itself. The close cooperation between
mathematicians and theoretical physicists has led to a
heightened awareness of the many benefits that mathematics
derives from physics (Jaffe and Quinn, 1993). Jaffe (1997)
points out that physics has traditionally been a source of
important problems for mathematics, contributing in this way
to its progress, and that in turn mathematical results have
helped in the solution of difficult problems in physics.
In a paper on the phenomenology of proof,
Rota (1997) maintains that the benefits of this close
association are to be seen in mathematical proof in
particular. Mathematicians often remain dissatisfied with
proofs that, though they establish without a doubt that a
theorem is true, provide no insight as to why it is true.
Physical concepts and models can make an important
contribution to understanding in such cases, and can even
help mathematicians devise purely mathematical proofs of a
more explanatory nature. In addition, however, an argument
from physics may form an integral part of a mathematical
proof.
The second part of our question relates to mathematics
education. In approaching this issue, we have taken our cue
in large part from two publications that deal directly with
the role of arguments from physics in the classroom: Winter
(1978) and Polya (1981). The ideas contained in these works
would be a good point of departure for the suggested
investigation. References to physical laws do appear in
other educational publications, but only as remarks in
passing. Castelnuovo (1971), for example, introduces
projections and shadows when treating the notion of
similarity. From a theoretical viewpoint, Struve (1990)
discusses geometry as an empirical science in contrast to
geometry as a theoretical system. In Bender and Schreiber
(1985) one finds a different conception of the relation of
empirical and theoretical geometry, based on the ideas of H.
Dingler. The following paragraphs discuss work published on
closely related issues.
Some recent publications describe various
approaches to making proof meaningful in the classroom with
the help of empirical arguments. Dreyfus and Hadas (1996)
showed that an empirical approach to teaching geometry with
dynamic software can bring students to see that proof is
required to explain results that are unexpected or
counterintuitive. De Villiers (1995), Mariotti (1995), and
Mason (1993) discuss several dynamic geometry constructions,
illustrate problem-solving methods not possible with pencil
and paper and advocate the use of dynamic software for
fostering new insights into traditional geometry theorems.
Greer (1996), as well, describes the use of empirical
arguments for proving.
Among mathematics educators there are also
those who advocate basing the teaching of mathematics upon
its various applications. This movement includes suggestions
for a closer relationship between mathematics and the other
sciences (OECD, 1991), a theoretical framework for what is
called Realistic Mathematics Education (RME), which
advocates using reality as a source for mathematization
(Freudenthal, 1983; Streefland, 1991), and other projects
that in various ways seek to strengthen the role of
applications in mathematics teaching. For the higher grades
of school teaching, one must also take into consideration
the publications of the ISTRON group (Blum, 1993). None of
these proposals deals explicitly with the teaching of proof,
however.
What is meant by "arguments from
physics within mathematical proofs"?
To explain better the concept behind the proposed
investigation, we would like to draw a clear dividing line
between using arguments from physics within mathematical
proofs and merely using physical representations or
illustrations of mathematical concepts or theorems. An
example of the latter is the representation of the laws for
natural numbers by geometrical configurations of pebbles.
The underlying idea we suggest, on the other hand, is to
apply in a mathematical proof a complex law of physics as if
it were a mathematical theorem. For this there are
historical as well as educational examples.
To begin with the former, the application
of physics to mathematics has a long history. When a purely
mathematical proof of a theorem proves elusive or awkward,
mathematicians have often found that the introduction of
concepts and arguments from physics yields a straightforward
proof. A famous example is Archimedes' use of the law of the
lever for determining volumes and areas (compare his work on
"the method"). Another equally famous example, from the
calculus of variations, is the so-called Dirichlet
principle, which asserts the existence of certain minimal
surfaces as solutions of certain boundary value problems. In
the 19th century, Dirichlet and Riemann took this theorem as
obvious for physical reasons. Weierstraß later
criticized its use, however, forcing mathematicians to look
for a purely mathematical proof of the principle. This was
quite hard to achieve, but in the end the effort led to
considerable progress in the calculus of variations (Monna,
1975).
As an example of the application of laws
of physics to mathematical proofs in education, we might
first mention the construction of the Fermat point of a
triangle, where the most elegant method is to model the
triangle by a physical system consisting of a perforated
plate and weighted ropes (Polya, 1981). Some theorems of
elementary geometry, such as the Varignon theorem that the
midpoints of the sides of a quadrangle are the vertices of a
parallelogram, can be proved most easily by applying the
laws of the lever or the notion of centre of gravity. For
these and numerous other examples see Winter (1978). A final
example is the mean value theorem of differential calculus.
If we interpret the derivative of a function as the velocity
at a given instant, then the mean value theorem follows
directly from the observation that a car going from A to B
must have had, at least at one point, the mean velocity as
its actual velocity.
Such applications of physics do much more
than illustrate a theorem. By introducing productive
concepts, they make possible a more satisfactory proof of
the theorem, and one that, on the basis of an isomorphism
between the mathematical and the physical constructs, is no
less rigorous. (In this they differ from much of what has
come to be referred to as "experimental mathematics," which
in its essence consists of generalizations from
instances.)
For the mathematician, indeed, the use of
concepts and arguments from physics is primarily a way to
achieve a more elegant proof. But such a proof may also be
illuminating, in different ways. It may reveal the essential
features of a complex mathematical structure, provide a
proof that can be grasped in its entirety (we call this the
holistic version), as opposed to an elaborate and almost
incomprehensible mathematical argument, or point out more
clearly the relevance of a theorem to other areas of
mathematics or to other scientific disciplines.
These broader benefits are invaluable even to the
practising mathematician, so they clearly have great
potential for promoting understanding among students.
Unfortunately this potential is not being exploited,
however, because concepts and arguments from physics have
not been integrated into the classroom teaching of proof to
any great extent and certainly not in any organized way.
This is not surprising, since there is no body of research
work on this topic that might provide guidance and tools for
teachers and curriculum developers.
Prerequisites for success in using
arguments from physics
The educational aspect of our question actually comprises
two tightly linked issues: How can the actual role of
arguments from physics in mathematical practice best be
reflected in the curriculum, and how can such arguments best
be used to promote understanding?
To address the first issue, one would have to examine the
epistemology of mathematics implied by much of present
classroom practice and compare it with accounts of the
nature of mathematics implied by the practice of mathematics
itself or espoused by mathematicians and philosophers of
mathematics.
Implicit differences of epistemology are
important. For example, students are often taught that the
angle sum theorem for triangles is true in general just
because it has been proven mathematically. Ignoring the fact
that measurements have shown this relationship to hold true
for real triangles as well, this practice implies a very
specific and limited view of the nature of mathematics and
its relationship to the outside world. Students do not share
this view, however, bringing to the classroom the belief
that geometry has something to say about the triangles they
find around them. In this they may unwittingly be closer
than the curriculum to the broader view of the nature of
mathematics held by most practising mathematicians. For this
reason it should come as no surprise to educators when
students are taken aback, misinterpreting the assertion that
mathematical proof is sufficient in geometry to mean that
empirical truth can be arrived at by pure deduction.
It would seem that educators themselves
need to come to the classroom with a more satisfactory
understanding of the nature of mathematics, one that
encompasses its relationship with the empirical sciences and
everyday human experience. Of course the curriculum itself
should be informed by the same understanding.
The second issue is the use of proof for the promotion of
understanding. Students being introduced to mathematical
proof come to the classroom with preconceived notions and
complex epistemological uncertainty. Educators need to
understand both much better than they do today. When
confronted with the proof of a theorem, for example,
students quite often say that they have understood the
proof, but still ask for additional empirical testing. From
a purely mathematical viewpoint such a request seems quite
unreasonable, and teachers usually take it as an indication
that the students did not really understand what a
mathematical proof is. From the viewpoint of a theoretical
physicist, however, the same request would seem quite
natural; no physicist would accept a fact as true simply on
the basis of a theoretical deduction. Thus a consideration
of the role of mathematical proof in theoretical physics may
well shed light on the way in which students view proof.
Keeping in mind the viewpoint of the
theoretical physicist is useful when analysing how students
approach proof when using dynamic geometry software such
Cabri Geometry or the Sketchpad, which allow explorative
work similar to experimental physics. Comparing students
with theoretical physicists also promises to be of help in
understanding how teachers might best cope with the
questions that may be created in students' minds by the use
of concepts and arguments from physics in mathematical
proofs.
Broader educational
aims
There are also broader educational reasons for studying
the use of arguments from physics within mathematical
proofs. We will sum up these up in four statements.
* As already mentioned, there is a trend in all
Western countries away from using proof in the classroom.
In our view this development will undercut the
educational value of mathematics teaching and should be
countered by fresh approaches to the teaching of proof.
* Of course the growing trend to experimental
mathematics should be reflected by an increased emphasis
on experimental mathematics in the schools. But
experimental mathematics in the schools should not be
just "mathematics with computers." From an educational
point of view, this would be a dangerous development.
Rather, one should be guided in the use of experimental
mathematics by the question of how it contributes to our
understanding of the world around us.
* The holistic aspect which many arguments from
physics can bring to mathematical proofs (see above) is
an important part of mathematical competence that is
frequently underestimated. Instead, there is a
predominance of step-by-step procedures. Seeking good
examples of instances where arguments from physics are
useful to mathematics proofs will contribute to
developing a way of teaching and learning mathematics
which is more balanced in this regard.
* In Western countries physics is less and less a
required subject. Physics is the discipline nearest to
mathematics, however, and to maintain meaningful and
interdisciplinary mathematics teaching it will therefore
become necessary to include some elementary physics in
the mathematics curriculum. In Germany there are already
proposals in this direction, as a consequence of the
country's poor results in the Third International
Mathematics and Science Study (TIMSS), and the "Konferenz
der Kultusminister" intends to decide upon a large scale
project exploring new types of interdisciplinary
teaching.
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