NOTES ON THE AESTHETIC DIMENSION OF MATHEMATICS EDUCATION Nathalie ...

NOTES ON THE AESTHETIC DIMENSION OF MATHEMATICS EDUCATION
Nathalie Sinclair
Simon Fraser University

Some of the most important changes in mathematics education over the past century have
evolved in response to perceptions of the unacceptable differences between mathematics,
as conceived of and practiced by research mathematicians, and school mathematics,
which is embodied in textbooks and classrooms. For example, Pólya’s work inspired
many mathematics educators to lessen the gap between mathematics and school
mathematics by infusing the latter with “problem solving,” which Pólya saw as the
defining activity of the research mathematician. A few decades after that, Brown and
Walters persuasively argued that posing problems was just as important–and sometimes
more so—as solving problems, and called for increased opportunities for student problem
posing. Both activities (posing and solving problems) were meant to attenuate
differences, but they also offered an attractive humanistic dimension to the world of the
mathematics learner. Of course, there remain many differences between mathematics and
school mathematics that few would ever consider eliminating—for example, no one has
ever suggested that students should publish new research results or review articles for
publication.

The aesthetic dimension of mathematics has also remained more or less on the side of the
research mathematicians, though with intermittent forays into the school mathematics. In
one sense, it seems natural to leave it in its place, given its tight connection with
published mathematics, that is, with the production of theorems and proofs that
characterize the mathematician’s work. We know, from the writings of a few great
mathematicians, that proofs should be elegant (Gauss, 1865), and that (good)
mathematicians have a special aesthetic sensibility (Poincaré, 1956). We are even told
that the primary goals of the mathematician are aesthetic, and not epistemological (Krull,
1930/1987).

It may be tempting to leave aesthetic with the mathematicians, but it would also be
wrong, and this for two reasons—neither of which has to do with attenuating differences.
First, mathematics education is one of the few disciplines that can (and sometimes does)
play an interpretive role in explaining the nature, purpose and goals of mathematics to the
‘outside’ world, which are infused with aesthetic values. This interpretive role is crucial
to the health of mathematics as a discipline (see Corfield, 2003) and to the monitoring of
the impact of mathematics on society (see Illich, 1994). Second, and more closely related
to immediate pedagogical issues, it can be (and has been) argued that the aesthetic is
deeply connected to learning, and not just to artistic production or appreciation. I
therefore propose a stronger two-way interaction in which the discipline of aesthetics has
much to contribute both to understanding the rationality of mathematics itself, and to
enriching existing theories in mathematics education.

Mathematics Educators as Mathematics Critics
In Lakatos’ (1976) Proofs and Refutations, Gamma and Alpha are discussing the
question of how far they are willing to go in providing a generalization of ‘Euler’s

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formula.’ Gamma notes that it’s a question of what’s interesting or exciting. Alpha
responds that interest is a matter of taste. So Gamma asks “Why not have mathematical
critics just as you have literary critics, to develop mathematical taste by public criticism?”
(p. 98). Indeed, it is difficult to identify the arbiters of taste in mathematics, even though
questions of taste are central to the growth of the discipline. They are central for the
simple reason that mathematicians must choose their objects of study; they are not
somehow dictated by the real world (as is the case with physics). Further, as Steiner
(1998) points out, “There is no objective criterion for a structure to be mathematics—and
not every structure counts as a mathematical structure” (p. 6). Whereas the criteria of the
past may have included utility, contemporary mathematicians have adopted other criteria,
including beauty. Hardy (1940) spoke to this in his famous quotation “Beauty is the first
test: There is no permanent place in the world for ugly mathematics” (p. 85). Indeed
Euler’s formula is a structure worth attending to because of its beauty, that is, its
simplicity and generativity. Csiszar (2003) notes that “aesthetic sensibility becomes a
measure of value itself” (p. 251).

Aesthetics are involved not only in the choice of structures (and, thus, the choice of
mathematical objects, relationships and problems), but also in the communication of
results about these structures. Style is crucial in mathematical communication, as can be
seen in the various guides provided by journals and professional societies (see AMS
(1990)). Mathematicians such as Krull (1930/1987) claim, for example, that the rhetorical
nature of mathematical writing depends strongly on aesthetic devices: “Mathematicians
are not concerned merely with finding and proving theorems, they also want to arrange
and assemble the theorems so that they appear not only correct but evident and
compelling. Such a goal, I feel, is aesthetic rather than epistemological” (p. 49).

Given the importance of aesthetics, Gamma rightly wonders how taste is defined and
communicated in mathematics. In the world of literature, writers produce the works of
art, but literary critics assess and analyse these works in a way that is interpretive,
formative and summative. One might think that mathematicians themselves take on this
role, but they rarely do, and they probably shouldn’t—such questions, which relate to the
image of mathematics in society, cannot be decided within the body of mathematics,
which is what mathematicians concern themselves with (see Corry, 2006). Yet, there are
few people who can fill the role of the mathematical critic in part because mathematical
communication is designed to be exclusive (Burton and Morgan, 2000; Csiszar, 2003).

In the past, philosophers have taken played the role of the mathematics critics, but this is
no longer the case (Corfield, 2003). There’s a sense in which mathematics educators are
perhaps best positioned to replace the philosopher. Unlike philosophers, mathematics
educators have a natural entry into the body of mathematics. But more importantly,
mathematics educators are extremely sensitive to certain aspects of the interface between
body and image: the cultural narratives about mathematics, the historical and
epistemological aspects of mathematics, and the demarcations between mathematics and
school mathematics.

Aesthetic Perspective Informing Mathematics Education
Enzensberger (1998) points out that the mathematics has a certain archaic aesthetic

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sensibility that has changed very little in the past two thousand years, one that speaks
more to the Baroque or the Classical than to any more contemporary or post-modern
sensibilities. Le Lionnais (1948/1986) wouldn’t agree. He sees at least two aesthetic
preferences expressed in mathematics. The classical, on the one hand, which tends
toward harmony and order, and the romantic, on the other, which seeks the bizarre and
the extreme. Whatever the case may be, recent work in a wide variety of disciplines
suggests that the particular tendencies one has are much less important than the fact that
one has aesthetic tendencies at all, and that one uses these pervasively and centrally in
making sense of the world. Using anthropological insights, Dissanayake (1995) writes
convincingly about the satisfactions that come from the successful manifestation of the
basic aesthetic motivation of “making special”. Other scholars interested more in a
cognitive science approach, such as Wilson (1998), argue that humans have predictable,
innate aesthetic preferences they use in making sense of their environments.

Closer to education, and from a more philosophical perspective, Dewey (1933) argues
that the aesthetic is a pervading quality of human reasoning and experience. He begins
his argument by pointing out that the categories used to characterise and communicate
different modes of reasoning and experience (i.e. emotional, intellectual, aesthetic) may
make our descriptions and analyses of them more tractable, but that these categories
impose artificial boundaries and dualisms that do not exist. In other words, not only is it
not sufficient enough to focus on either the cognitive or the affective in our study of
mathematical activity, but one must also strive to understand the way in which the
cognitive, affective and aesthetic intermingle at the most basic level in human inquiry.

The Deweyian approach to aesthetics cares less about whether or not a given theorem is
elegant, or what criteria of mathematical beauty might look like. Instead, such an
approach focuses on the role that the aesthetic plays in human inquiry. As I have
proposed elsewhere (see Sinclair, 2006b), the aesthetic plays a motivational, generative
and evaluative role in mathematical inquiry in the sense that it affects the problems
mathematicians pose, the process through which they go about solving those problems,
and the way they communicate their results. To the extent that school mathematics
engages in the kind of inquiry found in research mathematics, it seems possible that the
aesthetic can play similar roles for students (see Sinclair, 2006a). It thus appears that the
close relationship between aesthetics and inquiry may further inform current theories of
problem solving and problem posing.

However, if we stand back from the role of the aesthetic in research mathematics, and
focus more on mathematics learning, there seem to be some fruitful connections to make
with existing theories and frameworks in mathematics education. One example relates to
Goldin’s (2000) work on the affective domain of mathematics education. As I have tried
to show (see Sinclair, 2004), it is possible to take the elements of the affective domain
(emotions, attitudes, beliefs, and values) as highly intertwined with aesthetic responses.
Indeed, an aesthetic lens helps to illuminate the way in which emotion and cognition
work together in the learning process. Aesthetics thus seems very pertinent to important
questions in mathematics education regarding motivation and anxiety.

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Another arena in which the aesthetic considerations might help inform mathematics
education theories relates to Cobb and Yackel’s (1996) work on the development of
socio-mathematical norms in the classroom. They conceptualize this
development in terms of the ways in which students learn what is
mathematically different, sophisticated, efficient and elegant.
Although they focus primarily on norms around difference, the
norms of efficiency and elegance quite distinctly involve aesthetic
values. If students are going to be comparing solutions, and if the
teacher has to mediate the discussion around this comparison, it
will be difficult to avoid aesthetic values, especially when the
solutions are all correct. At the moment, however, mathematics
educators do not seem well-equipped to handle issues of aesthetics
within school mathematics, whether it’s in terms of students
comparing solutions, students posing problems, teachers posing
problems or teachers assessing student work. One reason for this
lacuna in current theories in mathematics education may be related
to the long-held belief (especially among mathematicians) that
students are unable to appreciate beauty and elegance in
mathematics (see Dreyfus and Eisenberg, 1986). Such beliefs may
unnecessarily constrain both the nature and role of aesthetics in
human thinking.

I conclude by pointing to a number of questions that emerge in
considering the aesthetic dimension of student learning and the
presence of aesthetics in theoretical developments in mathematics
education. These questions span a wide conception of aesthetics that
include the aesthetic as a theme in human experience (as a way that
human organize and derive meaning from everyday situations) and
the aesthetic as a field of study, which includes the nature of
perceptually interesting aspects of phenomena and artifacts.
(1)
How might (and should) aesthetic considerations in
mathematics differ from those in school mathematics?
(2)
How deeply is aesthetic sensibility tied to mathematical
motivation and curiosity?
(3)
Can aesthetic sensibility be taught? If so, how?
(4)
How do theories of embodied cognition relate to
aesthetic perception?
(5)
How can mathematicians help educators gain entry into
the aesthetic values that guide their work?

References
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