Consciousness, Literature and the Arts

Archive

Volume 4 Number 1, April 2003

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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.

 

I. Introduction

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.

 

III. Exploring Creativity and Fixed Form

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.

Stanza I                                               Stanza IV

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

Stanza II                                             Stanza V

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

 

Stanza III                                            Stanza VI

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.

 

IV. Creativity in the Computer Age

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." 

 

VI. Conclusion

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

Birken, Marcia and Anne C. Coon. "Analogy, Mathematics, and Poetry"  (course website)

Rochester Institute of Technology. http://www.rit.edu/~mkbsma/analogy.

Birken, Marcia and Anne C. Coon. "The Pedagogical and Epistemological

Uses of Analogy in Poetry and Mathematics”. Consciousness, Literature and the Arts, University of Wales Aberystwyth (March 2001), Vol. 2, No.1. http://www.aber.ac.uk/~drawww/journal/editorialboard.html

Gibbs, Raymond W., Jr. The Poetics of Mind: Figurative Thought, Language, and

Understanding. Cambridge: Cambridge University Press, 1994.

Glazier, Loss Pequeño. Electronic Poetry Center. SUNY Buffalo, 2002,

http://wings.buffalo.edu/epc/.

Hardaker, Wes and Gervais Chapuis. Escher Web Sketch. The University of Lausanne,

Switzerland. http://www-sphys.unil.ch/escher/.

Holyoak, Keith J. and Paul Thagard. Mental Leaps: Analogy in Creative Thought. Cambridge, MA and London: The MIT Press, 1995.

Lakoff, George and Mark Johnson. Metaphors We Live By. Chicago and London: The University of Chicago Press, 1980.

Leatherdale, W. H. The Role of Analogy, Model and Metaphor in Science. Amsterdam and Oxford: North-Holland Publishing, 1974.

Lee, Kevin D. TesselMania. MECC, 1995.

Pugh, Sharon. L. Jean Wolph Hicks and Marcia Davis. Metaphorical Ways of

Knowing: The Imaginative Nature of Thought and Expression. Urbana, Ill.:

National Council of Teachers of English, 1997.

Phillips, Mark, Nina Amenta and Jeff Weeks. JAVA Kali. JAVA Gallery of Interactive

Geometry. http://www.geom.umn.edu/java/Kali/.  

Ruef, Kerry. The Private Eye: Looking/Thinking By Analogy. Seattle. Wash.: The

Private Eye Project, 1992.

Shekerjian, Denise. Uncommon Genius: How Great Ideas Are Born, Teaching the

Creative Impulse with Forty Winners of the MacArthur Award. New York:

Viking, 1990.

Taylor, Paul. OULIPO. 2001, http://www.nous.org.uk/oulipo.html.

Turner, Mark. The Literary Mind: The Origins of Thought and Language. New York and

Oxford: Oxford University Press, 1996.