Consciousness, Literature and the Arts
Archive
Volume 4 Number 1, April 2003
_______________________________________________________________
Developing
Analogical Thinking and Creativity in University Students
by
Anne C. Coon and Marcia Birken
Key
words: analogy, creativity, university students, poetry, mathematics,
interdisciplinary
Abstract:
A professor of literature and a professor of mathematics discuss how their
interdisciplinary course "Analogy, Mathematics, and Poetry" helps
students see links between poetry and mathematics and encourages the students'
analogical thinking and creativity. The paper discusses many aspects of the
course: establishing connections between poetry and mathematics; homework,
classroom and lab activities that foster analogical thinking; the interplay
between fixed form and creativity; how creativity may be affected by the advent
of the computer; and the range of interdisciplinary research conducted by the
students.
As
faculty members from two different colleges of Rochester Institute of Technology
(RIT), we have found many opportunities during the past seventeen years to bring
together our disciplines of mathematics and literature. We have written
elsewhere about our recent research into the pedagogical and epistemological
uses of analogy in poetry and mathematics (Birken and Coon, 2001). That research
was fundamental to developing and team-teaching an interdisciplinary course,
“Analogy, Mathematics, and Poetry,” that we offered to RIT students for the
first time in the spring of 2001. The course attracted students from
engineering, computer science, and mathematics, as well as from graphic arts and
film studies. We hoped that in addition to meeting other content-based goals we
established for the course, students would learn to do the following:
·
see connections between the liberal arts and science
·
understand the roles played by analogy in poetry and mathematics
·
recognize pattern and form in both disciplines
·
explore how both disciplines address abstract topics such as infinity and
paradox
·
and develop their own analogical thinking and
creativity.
In
this paper, we will examine the methods by which we tried to accomplish these
goals, specifically focusing on elements of the course that emphasized making
interdisciplinary connections and developing analogical thinking and creativity.
We will discuss several in-class activities, homework assignments, and lab
exercises and present a brief summary of the interdisciplinary research
conducted by the students.
II.
Developing Analogical Thinking in an Interdisciplinary Setting
Beginning
with the title of the course, "Analogy, Mathematics, and Poetry," we
emphasized that we would be talking about mathematics
and poetry together and that
analogy would have an important role in establishing the connection between the
two disciplines. We recognized that we faced two challenges in developing the
course. First, we were linking two disciplines that are not often seen as
natural partners. Most cross-disciplinary efforts at the university level have
brought together closely aligned disciplines. For example, creation of
multi-disciplinary courses is common within the sciences and technology (courses
in bioinformatics and physical chemistry); among the sciences, technology, and
engineering (courses in genetic engineering and microelectronic engineering);
and within the arts, humanities, and social sciences (courses in philosophy of
religion and art history). While there are some university courses that cross
the boundaries of science, technology, and the liberal arts, such as history of
science, biomedical photography, or art and mathematics, there are few role
models for a collaborative course between literature and mathematics and even
fewer that specifically link poetry and mathematics. The second challenge we
faced was to find ways to develop analogical thinking within a broad
interdisciplinary context at the university level.
Although
others have written about strategies for developing creative analogical thinking
in younger students (Pugh, Hicks & Davis, 1997; and Ruef, 1992), we were
challenged to develop curriculum, homework and research assignments, and lab and
classroom activities that would encourage analogical thinking in upper-division
university students enrolled in a wide range of academic majors. What we
discovered was that analogy could play a critical role in making the
interdisciplinary connections we sought. To bring the content areas of the
course together, that is, to link ideas as disparate as the use of metaphor in
poetry and the representation of ratio and proportion in mathematics, or to show
how developments in mathematical understanding of infinity are reflected in
figurative language poets use to describe infinity, we often focused on how each
discipline relies upon analogy. For the purposes of this course, we used
“analogy” to encompass a variety of heuristic devices, cognitive processes,
and forms of figurative language employed for explaining the unknown, for making
leaps of intuition or discovery, and for creative expression. To introduce
students to basic concepts from contemporary research on analogy, we drew upon
work from cognitive science and linguistics, especially that of Turner (1996),
Holyoak and Thagard (1995), Gibbs (1994), and Lakoff and Johnson (1980).
Thus,
along with content material from both poetry and mathematics, we wove the
concept of analogy throughout the course. After introducing analogy and examples
of its use in both poetry and mathematics on the first day of class, we asked
students to read the parable of the "Prodigal Son" and consider how
analogy was being used. We then assigned the students to work in pairs and write
a parable representing an observation about human behavior. Despite its
simplicity, this early in-class assignment effectively engaged students in
thinking and using written language analogously.
We
believed that it would be important to help students become more aware of their
own analogical thinking if they were to move beyond simply observing or
describing analogical relationships, so we devised several ways for the students
to put their ideas about analogy into practice. For example, after a lecture and
discussion on how analogy is employed to teach new material -- by relating the
unfamiliar to what is already known -- we asked the students to become teachers.
Their assignment was to write a paper demonstrating the use of analogy in
teaching a mathematical principle; the paper was to take the form of a teaching
module written for students in a high school or college mathematics course.
Students were encouraged to draw upon their own experiences in learning
mathematics, to begin by stating the concept being taught and then show how
constructing an analogy to something already known could be used to explain the
new concept. For example, one student explained how knowledge of the real number
system can be used -- by analogy -- to teach complex numbers.
Later,
in a hands-on classroom exercise, students used analogy to explore the unknown
and make their own leaps of discovery. We gave groups of students objects whose
origin and purpose were unknown to them and asked the students to compare the
objects they had been given to other things with which they were familiar; to
try and imagine where the objects had come from and what their use and age might
be. By this point in the course, students had become familiar with some basic
concepts and terminology from cognitive science, so we encouraged them to apply
those concepts to their inquiry by asking, "What is the source domain or
familiar body of knowledge you are using in examining this object? How do the
three constraints on analogic thinking described by Holyoak and Thagard
(similarity, structure, and purpose) help you with this task?" ( 5-6).
Examples
of some of the objects we used are shown below.
As
they struggled to find language to describe the objects, speculated about the
objects' origins and use, and invented a new use for the objects, the
students’ analogies became increasingly inventive. This activity helped
students make links between their understanding of basic concepts and their own
experience, while it also encouraged them to be creative, to test their ideas,
and to look at their familiar world with new eyes.
In
a more complex assignment given later in the term, we brought together
mathematics, denotative language, and analogical thinking. After lecturing on
the Dutch graphic artist M.C. Escher and discussing many examples of his work
with tessellations (that is, the repeated tiling of a plane), we asked students
to create tessellations of their own, by hand or by using special computer
software. The follow-up writing assignment encouraged students to consider their
work from three distinct perspectives, specifically, to identify the types of
symmetries that were displayed in their tessellations; to describe their designs
using concrete language; and to describe what the images suggested to them,
using figurative language. This assignment – in its various parts – became
for us an almost perfect blending of mathematical elements, analogical thinking,
and written expression.
Another
important goal of this course was to develop students' appreciation for
creativity and to foster their inventiveness both within and outside their
majors. We were working with a self-selected group of students who had an
interest in both math and poetry, but the more technical students often did not
trust their creative instincts beyond using them to solve engineering or
computer problems. At the same time, the more artistic students, although quite
imaginative, had explored their creativity primarily within a specific design
area or medium, such as graphic design, film, or photography.
We wanted to broaden all students' use of creative thinking across the
disciplines. One way we tried to
accomplish this goal was by repeatedly examining the interaction between fixed
form and creative expression. We manipulated fixed forms from both poetry and
mathematics, using the computer to move words around on the page or to create
new number sequences from old, showing how "creative play" is
initiated. Then we showed instances where poets or mathematicians had worked
within a fixed form to create something new.
One
example of a fixed form that intrigued the class is the sestina. Attributed to
the 12th century Provençal poet Arnaut Daniel, the sestina relies on
a pattern of six repeating end words, rather than rhyming sounds, to create its
structure. Within the first six stanzas of the sestina, the poet can develop a
narrative through the repetition of the six end words; the final three-line
stanza brings together all six repeating words. Some of the ways that writers
can stretch the limits of this formulaic pattern include using the repeating
words as different parts of speech, carrying an idea over from one line to
another, and ending a thought in the middle of a line. All of these strategies
can be used to ensure that the repeating end words appear in the correct order
while exploiting the narrative possibilities of the form. Although the pattern
of repetition is formal and highly structured, the effect on the poem is subtle:
the repeating words help create a story that is tightly wound with recurring
ideas.
Mathematically,
we can describe the pattern in the first six stanzas of a sestina as a permutation of six end words. Numbering these final words in each
line as 1, 2, . . . 6, the mathematician would describe the permutation as:
(1, 2, 4, 5, 3, 6). Using
the “skeleton” of a sestina below, we can follow the permutation.
Word 1 falls at the end of line 1 in Stanza I, then passes to line 2 in
Stanza II, line 4 in Stanza III, line 5 in Stanza IV, line 3 in Stanza V and
finally resides in line 6 of Stanza VI.
Line
ending in word 1
Line ending in word 5
Line
ending in word 2
Line ending in word 3
Line
ending in word 3
Line ending in word 2
Line
ending in word 4
Line ending in word 6
Line
ending in word 5
Line ending in word 1
Line
ending in word 6
Line ending in word 4
Line
ending in word 6
Line ending in word 4
Line
ending in word 1
Line ending in word 5
Line
ending in word 5
Line ending in word 1
Line
ending in word 2
Line ending in word 3
Line
ending in word 4
Line ending in word 6
Line
ending in word 3
Line ending in word 2
Line
ending in word 3
Line ending in word 2
Line
ending in word 6
Line ending in word 4
Line
ending in word 4
Line ending in word 6
Line
ending in word 1
Line ending in word 5
Line
ending in word 2
Line ending in word 3
Line
ending in word 5
Line ending in word 1
In
a homework exercise for the course, students composed their own sestinas, often
with remarkably creative results. It was the first time some of the students had
tried to write poetry, and the highly structured form of the sestina provided a
pattern within which they were happy to experiment. Rather than seeing the
conventions of the sestina as confining, most students were intrigued by the
challenge of creating a poem that could fulfill those conventions. After the
members of the class had completed this assignment, they were much more
interested and prepared to discuss the limitations and problems inherent in this
fixed poetic form than they would have been before writing a sestina themselves.
In
mathematics, we returned to M. C. Escher, using his work to show the tension and
interaction between fixed form and creativity. When Escher began to create his
famous tessellations, he relied upon the three basic symmetries of a single
shape: translation, rotation, and reflection. As he became more intrigued with
the form, Escher's tessellations became increasingly complex, combining
different symmetries with multiple tessellating shapes. In addition to symmetry
in the Euclidean plane, his later work illustrated other mathematical concepts
such as positive and negative space, infinity of scale, transformation of shape,
and non-Euclidean geometry. Escher’s early use of a single fixed form
completely filling the plane gradually metamorphosed into elaborate, intricate
designs similar to those found in higher order mathematics and the natural
sciences.
Students
enjoyed studying the powerful visual messages of Escher and spent much time in
the computer lab experimenting with software that was designed to help them
create tessellations, including TesselMania, Escher Web Sketch,
and Kali. A few students preferred to
create their tessellations by hand, in the manner of Escher, but most felt the
computer assisted them in ways they could not have explored manually.
This difference in approach led to one of the class's many discussions on
the role of the computer in fostering creativity.
Difficult
questions about the relationship between creativity and technology arose while
we were developing the curriculum for the course. Ultimately we brought many of
the questions to our students, just as they presented us with questions of their
own.
Typical
queries included:
·
Does the computer stifle or encourage
creativity?
·
Is writing with a word processor more or less
creative than using pen and pencil?
·
Do computers assist the creative process in
problem solving?
·
Can a computer generate poetry?
·
Does a computer algorithm "do"
mathematics?
·
Is a proof done using complicated computer
algorithms (such as the proof of Fermat's Last Theorem) valid in the
same sense as a traditional proof?
·
To what extent do technological advances that
further our understanding of mathematics also influence the figurative language
or form used by poets?
As
professors we enjoyed exploring these ideas with exceptional students who felt
equally at home with literature and technology. The engineers read (and
sometimes wrote) poetry; the computer science majors sought beauty, as well as
relevance, in the complexity of fractals and chaos; the math majors knew that
Einstein's Theory of Relativity and Andrew Wile’s solution of Fermat’s Last
Theorem had been celebrated in poetry; and the fine arts majors appreciated the
mathematical patterns found in tessellations. These students were more open to
suggestion, willing to explore, and eager to see relationships than most
students at our institution. Their approach to answering the questions listed
above was often more "technical" and computer-oriented than the
professors' more cognitive, philosophical approach, but certainly no less valid.
Although these questions did not have simple answers, and in some cases led to
more questions rather than answers, the in-class discussions were themselves
most creative.
Opinions
on the role of computers in poetic and mathematical creativity fluctuated
throughout the term, but there was some consensus on how computers affect
mathematical inquiry. Those
familiar with using technology in math classes felt that graphing calculators,
computer algebra systems, and statistical software packages allowed them to
“see” and understand mathematics in new ways. They used these devices to
visualize in three dimensions, manipulate data, experiment with changes in
graphs, and test hypotheses, all of which enhanced their ability to think beyond
the given problem. Students felt freer to explore and make connections when
computers helped them to answer the difficult “what if” questions:
What if I cubed this variable instead of squaring it?
What if I limit all the scalars to being negative in this function?
What if I increased/decreased the scale of this graph?
What if I tried to integrate all functions that have this form?
What if I change the matrix slightly and try to find the inverse?
What if the data includes 500,000 points instead of 500 points?
On
the other hand, there was less consensus on how computers affect poetic form and
thought. Early in the course we introduced the students to a number of web-based
poetry sites, such as the Electronic Poetry Center (Glazier, 2002) that houses a
vast collection of electronic poetry resources. Other sites we presented
demonstrate how mathematicians and writers have experimented together with
poetic form; for example, a site devoted to the Oulipo movement describes a
system where "each substantive or noun in a given text, such as a poem, is
systematically replaced by the noun to be found seven places away in a chosen
dictionary" (Taylor, 2001) -- an experimental strategy that can easily be
undertaken using a computer. Although some students were resistant to the idea
that computers could be used to enhance creativity in poetry, others found many
ways to use computers, from creating "concrete poetry," where the
words are arranged in a specific shape or design, to locating rhyming words
through on-line resources. The question as to whether a computer can write
poetry persisted in student debate throughout the course. The devotees of
artificial intelligence insisted that computers are indeed creative, while other
students took the position that machines cannot express emotion or beauty. We
were pleased that some students chose to continue their inquiry into the subject
of computers and creativity in their final research project.
V. Student
Research: Extending the Inquiry
The
students' final projects exemplified the connections we made in the course. We
encouraged students to go beyond the course material and to pursue projects that
reflected their own interests while drawing connections between or among our
three areas of inquiry: analogy, mathematics, and poetry. We presented the
students with five broad topic categories and asked them to develop their own
area of focus within one of those categories. The following examples from each
area will demonstrate the range of work done by the students in their projects,
the results of which were presented orally and in a formal paper:
·
Topic A.
Analogy in Poetry
Student research on "The Use of Poetry to Describe the Sublime" and "Poetry and Analogies for Death"
·
Topic B.
Analogy in Mathematical Thought
Student research on "Analogy and Invention of the
Rubik’s Cube"
·
Topic C.
Individuals Whose Work Linked Mathematics & Poetry
Student research on Archimedes, Isaac Newton, Benoit Mandelbrot, and Madeleine L’Engle
·
Topic D.
Technology and a “Problem” from Poetry or Mathematics
Student research on "Computer-Generated Poetry" and "Genetic Algorithms and Neural Networks”
·
Topic E.
New Ways of Seeing
Student
research on "Concurrent Developments in Modern Science and
Modern
Poetry," "Mathematics and Contemporary Music," and "Gertrude
Stein and Cubist Art."
As
we consider how we are educating students for life in the computer age, life in
the 21st century, we are concerned that students not view their
academic studies and their own work or future careers in isolation. We want
their relationship with technology to be one that enhances mental activity,
rather than draining it. We feel compelled to demonstrate the links between fields of
study and bodies of knowledge, and to demonstrate the power of analogy as it is
used in the teaching, technology, creative expression, and discovery that link
human beings from generation to generation.
Finally, we believe it is critical that students become actively involved
in class assignments and research where they are encouraged to be creative, to
experiment, and to engage in intellectual inquiry that goes beyond their own
academic discipline.
References
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Marcia and Anne C. Coon. "The Pedagogical and Epistemological
Uses
of Analogy in Poetry and Mathematics”. Consciousness,
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2, No.1. http://www.aber.ac.uk/~drawww/journal/editorialboard.html
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Raymond W., Jr.
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Understanding.
Cambridge: Cambridge University Press, 1994.
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SUNY Buffalo, 2002,
http://wings.buffalo.edu/epc/.
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