Analogic Thinking in Science and Math

Humanistic Mathematics Network Journal, Sep 2018

Martha D. Patton

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Analogic Thinking in Science and Math

Humanistic Mathematics Network Journal Analogic Think ing in Science and Math Martha D. Patton 0 Recommended Citation 0 University of Missouri , Columbia , USA Follow this and additional works at; http; //scholarship; claremont; edu/hmnj Analogic Thinking in Science and Math Martha D. Patton Campus Writing Program 325 General Classroom Building University of Missouri-Columbia Columbia, Missouri 65211 (573) 884-6221 One of the great pleasures of my position in a nationally known writing-across-the curriculum program is discovering in many scientists a deep appreciation for humanistic thinking. The science wars wage on in the background, and I do find plenty of evidence of rifts between the “two cultures” that C. P. Snow described a half century ago. Nonetheless, there are on our campus many mathematicians and scientists who not only harbor all sorts of artistic talents, but also call upon their students to use language and to think imaginatively. Long before Professor Dennis Sentilles’ calculus course was formally designated “writing intensive,” he had asked his students to write. That is, Sentilles recognized the power of language to help students conceptualize the mathematical procedures they were working through. His most noteworthy assignment, now a staple in his writing intensive sections, asks students to compare differential calculus to a videotaped tennis game. Students use the extended metaphor (see “A Leitmotif for Differential Calculus,” facing page) to explain the nature and measurement of time and motion and their representations from practical, cognitive, scientific, and mathematical points of view. Intrigued by this professor’s assignment, I wanted to review other scientists’ use of analogic thinking and to investigate, however informally, some students’ responses to analogic thinking. Following is a brief tour through the history of analogic thinking in science as well as a discussion of analogic thinking as reported by six students, three from Sentilles’ calculus classes and three from a writing-intensive genetics class that also foregrounds language and imaginative thinking. ous, expressive, personal, or persuasive. It certainly does not favor metaphor. This prejudice against “literary language” was strong in 1660, when members of the first British society of scientists denounced “all amplifications, digressions, and swellings of style” and called for a return to a “primitive purity” of language. Instead of the “superfluity of talking” that has “overwhelm’d most other Arts and Professions,” the new sciences demanded a “naked, natural way of speaking; positive expressions; clear senses; a native easiness” (as quoted by Locke 4 and Bizzell 642). Three hundred years after the founding of the British Royal Society, many scientific style manuals still pan any use of metaphor or figurative language. Sentilles’ use of an extended analogy in calculus might be suspect except for its heuristic or pedagogical value. Analogies might be useful for communicating something to a broad or popular audience, but many scientists would still argue that analogies have little place in discovery or in communicating to a specialized audience. A look at the history of science suggests otherwise, though: analogic thinking has been important both in the discovery and the communication of knowledge, as well as in the more obvious role of teaching. An informal protocol/interview analysis of six students suggests that analogic thinking may be valuable, not so much because it bridges old and new concepts and expedites learning, but because in many cases it disrupts and slows down learning. A Leitmotif for Differential Calculus Imagine being out on the tennis court with the ball rising toward you. Differential calculus is the mathematics that describes and measures change in such an ever changing “time-ball” system. One can use this easily imagined setting to elucidate a leitmotif for differential calculus along the following conceptual theme line, where f(t) is the height of the tennis ball at time t: Computation < ——-— Abstraction < ———— Life/Reality —-— > Cognition Geometry of Graph Math Model Videotape External Reality Cognition Graph of f The function f The whole videotape The path of the ball Image on frame taken at time t Length of time spent taping (# frames) All frames on the videotape Change between images Sense of change between images Where the ball is at time t How long the ball is in flight All the different positions of the ball How much the ball rises between two moments How much time changes How quickly the ball appears to rise between two separated moments One point on the graph: (t, f(t)) Horizontal axis Vertical axis f(t) Domain of f Range of f Vertical increment in graph Difference in function values: f(t+h)-f(t) Slope of secant (two point) line to the graph between (t, f(t)) and (t+h, f(t+h)) Slope of tangent line at the point (t, f(t)) All tangents to the graph of f Average rate of change: f (t + h) − f (t ) h f’(t)= h → 0 lim f (t + h) − f (t ) h The derivative of f: f’ as a new function Horizontal shift or increment Difference in underly- Time between frames ing variable: (t+h)-t=h Video motion seen at time t How fast the ball is rising at time t Viewing the video The flight of the ball The time-ball “system” State at time t Duration of events All individual states of the system Change-of-state Time span Average rate of change (rise): one’s sense of motion over a span of time Rate, or change-of state, at the moment t The motion, or flow, of the system serted that “logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention,” and he credited analogy with being the guide to mathematical invention. The Italian rhetorician Giambattista Vico made similar claims two hundred years earlier. Many scientists, including Humphry Davy, Robert Hooke, Johannes Kepler, Antoine Lavoisier, and Robert Oppenheimer, have also acknowledged the role of analogy in discovery or in intuition (Leatherdale). Perhaps the most famous scientific analogy is Friedrich August Kekule’s account of dreaming about a serpent biting its own tail just prior to his discovering the structure of the benzene ring: analogy about which scientists are still undecided. This distinction is similar to one made by Mike in the discussion below: students might be irritated by the false elements of an analogy, but constructively troubled by a “third element,” the part that slows them down and causes them to mull over the concept. Few scientists or philosophers of science deny that analogies offer a heuristic value—in the classroom or in the profession, but there is less consensus about the necessity of analogy for scientific explanations. Hesse, among others, argues that analogy is necessary for scientific argument. Philosophers have asked parallel questions about the role of analogy in language. Friedrich Nietzsche’s radiDuring my stay in Ghent, Belgium, I lived in a cal assertion that all language is metaphoric (and, fine room on the main street. I sat in this room therefore, analogic) has become commonplace in the and wrote on my textbook, but could make twentieth century. Postmodernists have largely disno progress—my mind was on other things. I missed the cautionary hedge in I. A. Richards’ comturned my chair to the fire and sank into a ment, “Even in the rigid language of the settled scidoze. Again the atoms were gamboling before ences we do not eliminate or prevent [metaphor] withmy eyes. Little groups kept modestly in the out great difficulty” (92). I contend that we in the late background. My mind’s eye, trained by the 1990s need to revisit Richards who, on one hand, deobservation of similar forms, could now dis- nounced “the one and only one meaning superstition” tinguish more complex structures of various and boldly asserted that “metaphor is the omnipreskinds. Long chains here and there were firmly ent principle of language” and, on the other hand, joined; all winding and turning with snake- recognized greater rigidity and stability in the lanlike motion. Suddenly, one of the serpents guage of science. caught its own tail and the ring thus formed whirled exasperatingly before my eyes. I woke To the degree that philosophers and scientists agree as by lightning and spent the rest of the night that metaphor and analogy play a vital role in science, working out the logical consequences of my they aren’t entirely celebratory. Turbayne cautions that hypothesis (qtd. by Leatherdale 20). victims of metaphor are trapped unwittingly by prevailing metaphors, much as Thomas Kuhn argues that Astonishing as some accounts of analogic thinking are the prevailing metaphors in a given paradigm both for scientific discovery, they are less controversial than shape and limit scientists’ thinking. the accounts of analogy in scientific argument, particularly in scientific induction. While Aristotle cau- However, the “problem” areas of analogies might be tioned against argument by analogy (as many logi- prime sites for “disequilibrium,” Jean Piaget’s term cians have since), Francis Bacon recognized the im- for the tension between the known and unknown that portance of analogy to scientific argument. John motivates leaming—in this case, learning about sciMaynard Keynes further credits Bacon with distin- ence and learning about language. This is the concept guishing between positive and false analogies. Twen- that Robert Mayer builds upon in “The Instructive tieth-century philosopher of science Mary Hesse Metaphor: Metaphoric Aids to Students’ Understandmodifies Bacon’s distinction between positive and ing of Science” (1993). false analogies by examining the positive and false elements within any given analogy. Within any one ANALOGIC THINKING IN NOVICE SCIENTISTS analogy are both positive and negative components. Mayer is not the first to think about the role of a parThe predictive power of analogic thinking comes, ac- ticular kind of analogy, metaphor, in learning. The cording to Hesse, from a third element, the part of the 1970’s marked the “cognitive turn” in psychology and in “metaphorology,” a time in which psychologists, journal written for experts in neurophysiology. My cognitive linguists, anthropologists, and literary theo- initial question, to what degree do student readers of rists widely accepted the premise that metaphor is not science think analogically, developed into the followjust a marker of deviance (genius), as Aristotle be- ing six questions as the interviews took place: lieved, but is common (by degree) to all thought. Cognitive linguists explored not only linguistic structures in a text, but also the ways in which a reader processed them. In the following quarter of a century, sociolinguists have called attention to the importance of context and social relations in discourse. • Do these six students recognize as potential analogies the same linguistic structures identified by experts? • Do these six students process potential analogies analogically? (That is, are potential analogies read literally or figuratively, and, if figuratively, in what ways?) • Do these six students find analogies helpful in un derstanding the content? • What happens when an analogy breaks down, as most analogies eventually do? • To the degree some analogies bridge new and old information, how does the bridging work? • Does analogic thinking lead to greater insight into the nature of language? In a 1994 study of student processing of metaphor, Understanding Metaphor in Literature, Gerard Steen analyzes the ways in which students of literature process both potential metaphors (linguistic structures identified by experts as metaphor) and realized metaphors (cognitive reconstructions of potential metaphors). As a discourse analyst working in the realm of pragmatics, Steen assumes that the reader, the text, and the context are all constituents in the study, but that the reader is at the center of the investigation. He assumes that a reader’s goals are partly socially de- In general, there was a wide discrepancy between termined and that discourse communities share cer- potential and realized analogies. Most of the students tain regularities and conventions. In this study Steen only realized or reconstructed the potential analogies attempts to move reception theory from the text to when they talked or wrote about them. These students the reader as a locus of discourse analysis. Steen first found some analogies much more helpful than others, asked students to underline metaphors in a text to see but all of the students interviewed affirmed the poif students recognized as potential metaphors the same tential instructional value of using analogic thinking linguistic structures as those identified by expert read- in the sciences and of having qualitative learning preers, in this case a panel of literature professors. cede quantitative learning. And, for most students, the interview project led them to think about the language of science in ways that had never before occurred to them. The not-so-literal dimension of language is more pervasive than most students had realized. This awareness, in turn, did lead a few students to think about the ways in which science is “made” in new ways, but it did not cause them to question the value of science. These conclusions, along with the success of the written assignments foregrounding analogy in the calculus course, point to the value of making deliberate use of carefully selected analogies in the sciences. Sharing scholarly debts to pragmatics and discourse analysis, I wish to explore the relevance of Steen’s inquiry to science literacy studies. I broadened and altered the scope from a study of literature students’ processing of metaphor to a study of science students’ processing of analogy. Over a dozen students participated in this study, but I focused on six, three from the calculus course described above and three from a writing-intensive genetics course. Slightly modifying Steen’s methodology, I asked students first to underline analogies in each of three texts and explain them in a taped interview afterwards, and secondly to “think aloud” as they orally read three different passages. In each of the two sessions, the underlining/ explanation session and the think-aloud/explanation session, students responded to an excerpt from a work of popular science written for a general audience, an excerpt from a science text (Human Genetics) written for college students, and an excerpt from a science POTENTIAL ANALOGIES With one exception, students preferred the popular science genre to either the text excerpts or to the technical academic articles, and they attributed their preference to the abundant images and comparisons in the popular science writing. Students made comments such as “It got you interested” and “That was helpful; I probably wouldn’t’ve understood it [the article] CONSTRUCTED ANALOGIES without them [the analogies]” — or “It gave me some- Although Steve was the most active reader—the most thing concrete to hold in my mind.” In the first popu- ranging in his connections beyond those presented in lar science excerpt, entropy was compared both to an the text—all six students constructed analogies when engine running out of gas (a conventional analogy given an opportunity to write or talk about them. used in most physics courses) and to a casino closing Some analogies were grounded in “potential analodown (a novel and productive analogy for all six read- gies,” those text structures that expert readers would ers). Even when students claimed to enjoy excerpts identify as analogies; other analogies were compariwith many “potential analogies,” though, they didn’t sons between seemingly-literal information in the text always identify the analogies as such. Five of the stu- and something in the students’ experience. For exdents rarely identified as analogies anything other ample, few students read “blind watchmaker” as than similes or phrases that were announced by tags much other than a placeholder for an unfamiliar idea such as “...is like.” None of the students, for example, in the think-aloud interviews, but the more they identified “cDNA library,” “transcription,” “editing,” talked, the more they began to make sense of “blind” or “palindrome” as part of an extended linguistic anal- and to sort through similarities and differences in the ogy, even though they could readily identify more sonar capacity of bats (a result of chance and evoluterms in the same group once the extended analogy tion) and the sonar capacity of machines (purposehad been pointed out to fully designed by engithem. The distinction made neers). agnieds i“scaofnfisrtmruecdtebdy”thaenastluo-- ...if you force people to❝ make analogies or have hTahde tahlsroeejucsatlccuolmusplsetuteddetnhtes here between “potential” dents’ “monovalent” read- an analogy set up for them, the fundamental parts course in which they were ing of many conventional of calculus won’t be glossed over so much, but asked to explain in writing analogies (by a literal read- will be used and understood. --Steve an extended analogy of a ing of a conventional anal- videotape of a tennis game. ogy). With the exception of As indicated earlier, the the same student (Steve), the undergraduates disliked professor of this course, Dennis Sentilles, compares the technical article, which made little blatant use of calculus to an ever-changing “time-ball system.” A analogy. function is compared to the whole videotape of the path of a tennis ball, and the domain of f is compared Steve, the most advanced of the calculus students, to the length of time the ball is in flight (the number expressed decided appreciation of analogies, but con- of frames on the videotape) and the range of f is comstructed his own analogies with or without the prompt pared to all the different positions of the ball (or all of the “potential analogy.” As he put it, “I’d almost the frames on the videotape). All three students found say that any time I see something that I’m familiar this approach to calculus revolutionary and construcwith, the whole index [of mind and memory] is tive. Steve, who claimed to have an intuitive underopened up, and I can pull out my file card and say, standing of the equations, found himself re-defining ah, here’s one!” The lack of “potential analogies” in and clarifying ideas that had already made some sense the technical article did not bother him because he to him. He found the videotape analogy indispenswas rifling his own mental files, including many “re- able and, when asked if he would teach in the manceived” analogies from other texts and lectures. He ner of Sentilles, responded, “most definitely...I think liked the technical article precisely because it was the if you force people to make analogies or have an analmost foreign to him, because it challenged him the ogy set up for them, the fundamental parts of calcumost to construct his own analogies or to recall analo- lus won’t be glossed over so much, but will be used gies from memory. In the genetics text excerpts, also, and understood.” That, unfortunately, is what Tom one phrase after another would elicit an analogy not described as being the case in previous courses: “Bepresent in the text structure but present in Steve’s fore I went in there, I just got pushed through calcumemory from a drawing on the blackboard in a pre- lus, and I didn’t really learn to conceptualize it.” vious course or from a picture in an old textbook. Learning to think qualitatively and not just quantitawords that have a lot of different meanings, or could have...and you have to think...and it makes you mull it over.” In other words, the “meaningful calculus” sought by professors such as Sentilles or the “instructive metaphor” sought by learning theorists such as Richard Mayer comes about by gaps that motivate new learning. tively proved to be revolutionary for him in other courses as well. “After this course, after I started in this course, my grades shot up because I would sit back and look at something and say, ‘Okay, I can’t get this, why? What are we doing?’ And after a while, I’d say, ‘ah, that’s why!’” Mike, the third calculus student, also said that “for me, it was the analogies that made my understanding. I couldn’t just throw those out.” All three of them, though, identified the writing process as the place where the received analogy started to make full sense, where the “potential analogy” created by Sentilles became a “constructed” or “reconstructed” analogy in their own minds. Two tentative conclusions might be drawn from this: first, potential analogies might not offer much if students aren’t asked to play with them, if students aren’t given the time and resources to reconstruct them. Second, all analogies are not equally valuable: Sentilles had experimented with many other potential analogies before settling on the time-ball analogy that proved to be powerful for a wide variety of math students. BRIDGING OLD AND NEW INFORMATION All six students were quick to credit analogies with getting them interested in new material. Only one of them expressed much interest in entropy, but all of them found the article about entropy quite interesting. It appears, though, that analogies function even more effectively by breaking up a bridge, by creating a hurdle, or slowing down a train of thought. In both cases, the analogy often functioned as a placeholder, a space for a concept that would become better understood in time. Several of Mike’s comments pointed to still another function of analogy: a bridge not from the unfamiliar to the familiar, but from the now-thoroughly-understood to memory. In other words, analoANALOGIC BREAKDOWN gies can function as a way of compressing and repackSteve identified “negative analogy” or inaccurate com- aging already-understood concepts for long-term storparisons in both the genetic text and popular science age. excerpts. He conceded that these limitations could lead to misunderstanding. Instead of ending the anal- INSIGHT INTO LANGUAGE ogy, though, he felt readers should keep extending The more I gave these six students an opportunity to the analogy: “Keep re-defining, keep talking.” Here, I explain themselves, the more they realized that they would like to distinguish between those analogies that were dependent by degree on analogies. Put another fail because the student already possesses a more re- way, the more they tried to remove themselves from fined understanding of the concept and those analo- analogic thinking, the more they realized they couldn’t gies that are troubling, provocative. Most students do it. They began to realize that words and concepts who found some analogies too simplistic simply are born and grow and change, and most found it skipped over them. In the Human Genetics text, nu- impossible to express scientific concepts in absolutely merous similes were used, and some simply failed for value-free language. Many had never before thought students who had considerable coursework in biol- about the etymology of conventional vocabulary such ogy. For example, most students found useful a com- as “bacteria” (little staffs). Several returned to the secparison of Vitamin D and a faulty receptor to a ferry ond interview with examples of analogies they had unable to dock, but advanced biology students found found in other science texts. Although students gained other similes limited. However, if an analogy was trou- insight into language, their insight was a byproduct bling (not simplistic, but troubling), most students of their learning about something—entropy or natufound even the troubling part stimulating. Mike and ral selection or transcription or neural transplants. Steve both objected to some of the emotional implica- They weren’t bashing science, but were gaining intions of the casino analogy for entropy; neither wanted sight into it. to view entropy as something “bad” or something to lament. Mike didn’t fault the writer, though, for an CONCLUSION imperfect analogy is thought-provoking, slows a Writing across the curriculum (WAC) programs, such reader down: “Because if he didn’t have words in here as the one in which I work, have largely substituted a like that you’d just read it and go on...but then he used belief in linguistic positivism (which treats language as if it were a transparent medium and writing skills Locke, David. Science as Writing. New Haven: Yale UP, as if they were generalizable across all contexts) with 1992. a belief that language can never be completely “clear,” can never be completely rid of analogy, and, even if it Mayer, Richard E. “The Instructive Metaphor: Metacould, it shouldn’t. 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Martha D. Patton. Analogic Thinking in Science and Math, Humanistic Mathematics Network Journal, 2018,