oConsciousness, Literature and the Arts
Archive
Volume 2 Number 1, April 2001
_______________________________________________________________
The
Pedagogical and Epistemological Uses of Analogy in Poetry and Mathematics
by
Marcia
Birken and
Anne C. Coon
ABSTRACT:
In
our mathematics and poetry classrooms, we often use analogy to teach new
material by building upon established concepts. Analogy also allows the student
or professor, like the poet or mathematician, to extend the boundaries of what
is known and to find new forms of expression. From problem solving, to exploring
new realms, to imagining a “virtual classroom,” analogy has a vast range of
pedagogical and epistemological uses in mathematics and literature.
“A
common function of poets and mathematicians is to jar us into place, to refocus
our vision so as to order the chaos around us and to connect what is familiar
with what is beyond” (Priestley 17).
With these words, W. M. Priestly brings together several of the assumptions upon
which we have predicated our collaborative work: he asserts quite simply that
there is a “common function” of poets and mathematicians; he describes that
function as having a pedagogical purpose (“to connect what is familiar with
what is beyond”) as well as an epistemological purpose (“to refocus our
vision so as to order the chaos around us”); and in order to describe a
relationship that may not seem immediately clear or obvious, he employs an
analogy (the common function works like a device or an eye).
In our mathematics and poetry classrooms, we often use analogy to teach new
material by building upon established concepts. Analogy also allows the student
or professor, like the poet or mathematician, to extend the boundaries of what
is known and to find new forms of expression. From problem solving, to exploring
new realms, to imagining a “virtual classroom,” analogy has a vast range of
pedagogical and epistemological uses in higher education. Although the study of
analogy has been extended dramatically through research in cognitive science,
psychology, linguistics, and artificial intelligence, for the purposes of this
paper, we will examine the role played by analogy in our respective disciplines
of mathematics and literature.
I.
Using Analogy to Solve Problems and Build on Existing Knowledge
Our collaboration began fifteen years ago when we designed and team-taught a
course in interdisciplinary problem solving at Rochester Institute of
Technology, a highly technical comprehensive university in Rochester, New York.
In this course, we emphasized the cross-disciplinary application of “tools,”
such as analogy, using a variety of assignments in logic, mathematics, and
literature. It was our belief then, and still is today, that students are more
likely to succeed in college when they can see relationships between different
courses and disciplines and when they can apply what they know to new
situations. (For example, students who understand that adding simple fractions
in grade school arithmetic requires finding a common denominator will better
understand how to add more abstract algebraic fractions in an analogous manner.)
Hofstadter reinforces this idea and emphasizes the connection
between analogy-making and high-level perception:
“Analogical
thought is dependent on high-level perception in a very direct way.
When people make analogies, they are perceiving some aspects of the
structures of two situations - the essences
of those situations, in some sense - as identical. These structures, of course, are a product of the process of
high-level perception” (Hofstadter 179).
Indeed,
even standardized college entrance exams emphasize analogy, and intelligence is
often measured by testing analogical reasoning skills.
“Tasks
involving recognition of structural similarity between objects, an important
component of analogical reasoning, figure heavily in tests of human
intelligence; an early study of such tests [. . .] goes so far as to state that
‘intellectual ability can be defined as the ability to reason by analogy from
awareness of relations between experienced characters.’ “(Russell
1).
Furthermore,
using analogy to make connections between the content of different academic
disciplines, in this case poetry and mathematics, may also have pedagogical
value when students are asked to approach new “problems” in one discipline
using knowledge or strategies from the other discipline. For example, one
instructor “encourages his students to use their intuitive knowledge of
geometry to help them to understand a difficult poem” (Rishel qtd. in Growney
“Isolated or Integrated” 65), while another “encourage[s] students to
write a poem after the students have seen hyperreal numbers, reacting as a
Pythagorean might to the discovery of incommensurable magnitudes. . .” (Haack
2)
II.
Using Analogy to Explore the Unknown, Express Abstract
Concepts, and Construct New Knowledge
Analogy has another, different role for college students when it is used to
demonstrate and foster inventive, open-ended exploration and expression. The
history of science is filled with examples of discovery and invention marked by
brilliant analogical leaps, including Archimedes’ “Eureka,” Euler’s
finding the sum of an infinite series, and Newton’s discovery of gravitational
force. And indeed, mathematics
itself has been described as “analogical.”
“One
can see at once the abstract, gestalt-like and analogical character of
mathematics in the way, for example, that one set of equations (Laplace’s)
serves as analogue for the phenomena of change in such diverse fields as
gravitation, electrostatics, electricity, elasticity, and the permanent flow of
liquids” (Leatherdale 28).
Poetry,
too, relies in a fundamental way on the creative use of analogy where it serves
as the basis for much figurative language. The poet Mary
Oliver, when discussing a passage from Milton’s Paradise Lost,
describes the power of figurative language:
“What
becomes apparent in this passage is that the force of figurative language can be
associative and cumulative, as well as astonishing with each individual
figure-that we carry an intended swelling residue with us down the page” (69).
Oliver
continues,
“It
is possible of course to write a poem without figurative language. And
yet...What is poetry but, through whatever particular instance seems believably
to be occurring, a meditation upon something more general and more profound?”
(70).
Whether
we are referring to the “gestalt-like” qualities of mathematics or the
“associative force” of figurative language in poetry, we can encourage
students to look beyond the present and the literal by helping them see the
profound role analogy plays in sparking the creative leap to new knowledge, new
uses of language.
What occurs in this process of creativity? Contemporary research on analogy has
attracted scholars from across the disciplines, and many have addressed the
process involved in the creative use of analogy. Some scholars have described
the conditions in which analogy is employed:
“.
. . [T]he analogical act in so far as it is an illuminating perception of
resemblance is momentary and instantaneous. However, considered in all its
fullness, it should be connected with a whole complex of antecedent behavior and
knowledge and a similar complex of consequent implication and association” (Leatherdale
16).
The powerful nature of cognitive blending has also been well documented: “Human beings routinely - and with the support of cultures and arts - put together things that do not belong together. This creates not confusion, but deep insight” (Turner, UB lecture). However, this “putting together” is not haphazard.
“Although making an analogy requires a leap, that leap need not be blind or random. In all its manifestations, analogy is guided by the basic constraints of similarity, structure, and purpose. . .” (Holyoak and Thagard 7).
Indeed,
psychologists and cognitive scientists have explored the extent to which human
cognition itself is defined by this ability to put things together and
“fundamentally shaped by various poetic or figurative processes” (Gibbs
1). One thing is certain, when creative, curious, intuitive human beings grapple
with new ideas, they often turn to analogy.
And yet there are differences between the use of analogy when proving a
mathematical theorem and when expressing an abstract idea in poetry. Not
surprisingly, students may be much more at ease in analyzing analogy in one
discipline than in the other. Thagard and Holyoak have
made the following distinction:
“Analogy
may be the cognitive basis of metaphor, but, [. . . ] metaphor is often extended
by an associative aura created by metonymy and other figurative devices. Hence
the meaning of a poetic metaphor is not fully captured by the kind of analysis
that would suffice for a more rigorous scientific analogy” (223).
Students
should be encouraged to engage in both kinds of analysis and may be surprised to
see how, sometimes, the two disciplines seem to change roles. Scott Buchanan has
masterfully described the ways in which poetry and mathematics “exchange
disguises.”
“Very
simply, poetry and mathematics are two very successful attempts to deal with
ideas. In this respect they are genuine. Both employ sets of symbols and systems
of notation. In this respect they have very interesting and illuminating
comparisons and contrasts[. . . ]. They exchange disguises so that mathematics,
commonly accepted for its hard-headedness, rigor and accuracy is often poetry
creating a realm of fancy; and poetry, commonly loved for its playful
spontaneity and utter ineffectualness, becomes the mathematical demiurge joining
words and images into a world of hard persuasive fact” (19-20).
An
excellent example of mathematics in its disguise as “poetry creating a realm
of fancy” is the concept of infinity. As early as the sixth century BCE, the
ancient Greeks acknowledged infinity as a central issue in mathematics, but they
did not have a system of notation for dealing with such a novel idea (Maor
3). Prior to this time mathematics dealt with more practical issues of weighing,
measuring, and time. According to Eli Maor, “Infinity
had to wait until mathematics would make the transition from a strictly
practical discipline to an intellectual one, where knowledge for its own sake
became the main goal” (3).
Infinity
is not a fixed number, but rather, infinity refers to the concept of numbers
growing larger and larger without end. Infinity is often linked to the divine or
to a Supreme Being. Infinity has puzzled, confounded, and tantalized
mathematicians, by allowing for this more “poetic” or fanciful way of
thinking. Unlike the solid, indisputable axioms of arithmetic, infinity has
inspired terror as well as wonder. Mathematicians
like Leonhard Euler and Georg Cantor stretched mathematical thought about
infinity. While they certainly used known results from finite mathematics to
think about the infinite, it was their unexpected twists and creative thinking
that opened up the possibility for new ideas about infinity.
Euler was the first to find the answer to the infinite sum shown below:
The
surprise in Euler’s discovery is the unexpected appearance of the number
in the limit of a series involving
only the natural numbers. To this
day, Euler’s series is regarded as one of the most beautiful results in
mathematical analysis. Using
similar methods, Euler was able to find the sum of many other infinite series
involving the natural numbers” (Maor 35).
Euler’s
work, which expanded thinking in finite sums to infinite ones, occurred in the
eighteenth century, but it wasn’t until the nineteenth century that Georg
Cantor showed that there are different classes of infinity, or an actual
hierarchy where some infinities are greater than others. Cantor set up a
correspondence between the set of even integers and the set of all integers,
showing that both of these sets had the same number of elements, despite the
apparent contradiction that the even integers are a subset of all integers.
These two sets are analogous, exactly the same in size, but different in their
respective elements. Further, he showed that there are infinite sets, like the
real numbers, which are so dense they cannot be counted. Such sets are also
infinite in size, but much larger than the infinite set of countable integers.
“Cantor called this type of infinity C,
the infinity of the continuum” (Maor 59). Mathematicians
had to think the absurd (an infinite set of numbers can have a finite sum) or
the impossible (the part is the same size as the whole) before mathematics could
take these major steps forward.
Poets, too, make inspired, unexpected connections. To illustrate the richness of
poetic metaphor based on analogy, we turn to a poem by Wallace Stevens in which
mathematics provides the context for an analogy. This poem offers an opportunity
to engage students in a cross-disciplinary discussion of how analogy is used in
creative expression. Such a discussion would be enlivened by drawing on the
students’ knowledge or preconceptions about poetry or mathematics.
VI
from “Six Significant Landscapes”
Rationalists,
wearing square hats,
Think,
in square rooms,
Looking
at the floor,
Looking
at the ceiling.
They
confine themselves
To
right-angled triangles.
If
they tried rhomboids,
Cones,
waving lines, ellipses--
As,
for example, the ellipse of the half-moon--
Rationalists
would wear sombreros. (Stevens
17)
The
central idea here establishes, by analogy, that “rationalists” dress, and
live, and think in ways that are intellectually and personally confining. The
poet invites the reader to make a leap with him and apply characteristics of
geometric shapes to the ways certain human beings think and live their lives. We
observe the repetitive “square-ness” of the rationalists’ existence, along
with the limitations on their field of vision. We are led to believe the
rationalists lack imagination, even joy, in their lives. The poem goes further,
suggesting that rationalists, if they tried other, less ”squared” ways of
being, again represented analogously by geometric figures that we are now
encouraged to interpret as more flamboyant (rhomboids, cones, waving lines,
ellipses), would “wear sombreros,” the hats of dance. In this relatively
short poem, Stevens is not only eloquent about the privations rationalists
inflict upon themselves, he also comments, indirectly, on the realm of formal
learning, suggesting that there is more pleasure and creativity possible in
forms or ideas that go beyond the “right angles” with which we are most
familiar.
Encouraging students to see how analogy is used creatively in both mathematics
and poetry challenges the students’ assumptions about the “hardheadedness”
of one or the “utter ineffectualness” of the other and may provide students
with increased insight and confidence for approaching both disciplines.
III.
Using Analogy to Change Pedagogy
Pedagogical shifts occur when teachers experiment with different ways of
teaching and learning. Current trends in pedagogy, such as the infusion of
technology into the classroom and the use of collaborative learning, developed
from considering teaching and learning in a new light. By making analogies
between the classroom and the corporate boardroom (team work), between teaching
and entertaining (multimedia), between teacher as autocrat and classroom as
democracy (collaborative learning), and between the linear textbook and the
non-linear brain (hyperlinking, morphing, interactivity of computers), we
created the “new classroom” with its technological emphasis.
Technology is most useful and successful as an educational tool when it allows
students to see or “imagine” differently or to make analogical leaps. When
students are far away from museums, but can experience a virtual tour on the
computer, examining artwork, comparing artists and holding “chats” with
docents, then technology has been successfully integrated into the curriculum.
Similarly, when technology allows us to see mathematics in new ways, by
rotating a three-dimensional object on a computer screen, by changing one
parameter in an equation while simultaneously viewing the changes to its
graphical representation, or by creating animations to show change and growth,
then we are creating paradigm shifts in understanding.
Technology can also make it easier for students to conduct poetic analysis and
understand analogic references. For instance, a poem may contain obscure
allusions, such as those found in “The Merman and the Seraph” by William
Benjamin Smith, a poet and mathematician. JoAnne Growney,
herself a poet and mathematician, offers one explanation, saying,
“[Smith]
sings sadly of the separation between the Merman - perhaps a mathematician,
isolated in his sterile world of thought, and separated from beauty, from
feeling and desire - and an angel or Seraph, who represents the world of
whatsoever is good” (Growney “Isolated or Integrated” 63-64).
The
terms “Merman” and “Seraph” are probably unfamiliar to most
undergraduate students, but a structured computer search could help define these
terms, explain their similarities and differences, and lead to more information
about their historical, societal, and psychological implications.
Continued internet searching on William Benjamin Smith himself would
yield information on the author’s background in philosophy, physics, and
biblical scholarship, as well as the dates he lived.
Collecting and analyzing a range of material, from the etymological to
the biographical, would provide students with insights into the domains or
contexts Smith draws upon to construct the “Merman” and “Seraph”
analogies.
In addition to enhancing students’ ability to conduct research, the multimedia
capabilities and interactive nature of technology also allow students an
unprecedented level of involvement with writers and texts. Students may easily
compare and contrast different versions of a manuscript; write and experiment
within the parameters of a specific poetic form; make visual connections among
rhyme schemes or figurative language by creating links with color and motion;
add sound or animation to a poem; and listen to a poem being read aloud in a
variety of voices.
Whether we use sophisticated searching or graphing methods, visual images, or
sounds, technology has the potential to greatly enhance teaching in the
disciplines of mathematics and poetry. Furthermore, these same elements of
technology may make it easier for students to analyze and create analogies on
their own, as they work on independent projects.
With the support of a Rochester Institute of Technology Provost’s Grant, we
are currently developing an undergraduate course titled “Analogy, Mathematics,
and Poetry.” We believe that by familiarizing our students with the uses of
analogy in mathematics and poetry, we will broaden their understanding of
content material from both areas and will encourage them to apply analogous
thinking in their own fields of study. Beyond that, however, we hope that
students’ intellectual curiosity will be challenged by seeing the connections,
made by analogy, among disparate fields and across different historical periods.
By encouraging students to see their college curriculum not merely as a series
separate, discrete courses, but as a body of knowledge held together by
recurring cognitive links, we are providing them with the tools, vocabulary, and
conceptual framework essential for creative thinking.
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