October 23, 2012
Mathematics and What It Means to Be Human, Part 3
In May 2009, Michele Osherow, an English professor at the University of Maryland-Baltimore County and resident dramaturg at the Folger Theatre, in Washington, invited her colleague Manil Suri, a mathematician at the university, to act as mathematics consultant for the Folger's production of Tom Stoppard's Arcadia. The play explores the relationship between past and present through the characters' intellectual pursuits, poetic and mathematical. That led to a series of "show and tell" sessions explaining the mathematics behind the play both to cast members and to audiences. In the fall of 2011, the two professors decided to take their collaboration to the classroom and jointly teach a freshman seminar, "Mathematics and What It Means to be Human." Here is the final essay of a three-part series on how the experiment played out. Read Part 1 here and Part 2 here.
Manil Suri: It's reassuring to see that whether in mathematics or the humanities, the last few weeks of a course with a final project follow the same pattern: The jolt of energy that suddenly animates the room when students realize this is it. The grandly ambitious (and hopelessly infeasible) ideas that first emerge. The excitement that turns into consternation when confronted with the impracticality of the studies being proposed. ("And how exactly will you chart all the babies born out of wedlock during the Civil War?" we had to ask one student.)
Most predictable is the panic when the invariable lurking surprises start popping up like the frantic (and usually inaugural) visits during office hours and the 11th-hour abandonment of a doomed project to start another one. I'd forgotten how entertaining it can be to prolong their agony thus—really must remember to give more projects instead of finals!
This time, in addition to the usual pitfalls of research and creative works, students had an extra quagmire to watch out for: the mathematics. It had to be an integral part of what they did (a requirement that some tried, unsuccessfully, to dodge), it had to be correct (a requirement we were more willing to bend), and they had to convincingly show they understood the concepts used. Sadly, that last point overwhelmed one of the most unusual of the projects: a short story based on the Prisoner's Dilemma payoff matrix. Yet another reminder that mathematical constructs can be deeper than they appear.
But we had some sparkling successes, too. One student wrote a story that elegantly finessed chaos theory to imagine a world in which Lincoln might never have become president. Another produced a slick, professional-quality video on beauty and the golden ratio. Mathematically deepest of all was an instrumental composition that musically interpreted what it means to be a sine (or cosine) function—by a humanities scholar inspired to take calculus this semester.
Michele Osherow: Most students who opted for "creative representations" of humanities and math seemed to procrastinate and be far less certain of their plans than the others. Perhaps they thought such an option would be easy, despite our severe and steady warnings. (Had they forgotten Manil was an accomplished novelist?) The repeated bouts of writer's block and artistic despair we encountered during the final weeks were monumental. Everywhere we turned, there was Konstantin with his dead bird.
We had planned from the start to also give students the option to produce a research proposal for quantitatively dealing with any humanities topic. We could see it now: our crew of 13 freshmen submitting completed, original research (involving mathematics, of all things) with their applications to the country's top graduate programs in the humanities.
To nudge them toward real-life research questions, we appealed to their love of cash. The University of Maryland-Baltimore County offers generous grants each year by way of Undergraduate Research Awards. "A real opportunity," we told them. "Once your proposals are crafted, you can parlay them into a URA proposal as a logical next step."
We don't know that any students actually took that step, but then, freshmen often need to work up the courage. One student has continued to develop her project, which attempts to quantify the influence that specific language or word choice may have on survey respondents (thereby aiding statisticians in the elimination of question bias).
In the end, I was more impressed with the students' performance than they were with mine. Our course evaluations were the harshest I've seen in years. I'm not naïve or obsessive about this stuff—I know every class can't work for every student, and I know very high evaluations can indicate a lack of rigor.
Still, the students were critical of areas I'd never before questioned. Some commented that neither of their instructors seemed interested in what the other one had to say. That cut me to the core. I took notes in class, asked questions, confessed my confusion. What does "interest" look like to students nowadays? I wonder, though, if their confidence wasn't shaken when they realized that we two didn't have all the answers, or even all of the questions, but proposed to make these discoveries around the seminar table.
Manil: Although Michele seemed crushed by a few of the student evaluations, I was actually relieved at how good they were.
My experience has been to be severely punished in the evaluations whenever I've tried to do something experimental in the classroom. There was the sparsely attended first-year seminar I taught on mathematics in the media, in which, toward the end, only one student was still showing up to class. (The video talk we created as his class project, on the concept of infinity, has had more than 20,000 YouTube viewers, which makes me feel better.) Another workshop I taught that required participants to use mathematics in a creative endeavor (short story, musical piece, artistic creation) ended with a series of e-mails from an irate student excoriating me for the grading scheme that led to her B.
And then there was the spring 2012 semester. Scheduled to teach the university's usual general-education Math 100 course for nonmajors, and still invigorated by the humanities seminar, I decided to use the most successful aspects from that experience to "perk up" the syllabus (or so I brightly thought) in my intro course. Besides including the videos and computer exercises on fractals and probability, I also employed the basic philosophy of tying everything to real life (for example, having the students read an article on the growth of cities during the chapter on nonlinearity). After all, wasn't this one of the original goals of working on the humanities seminar? To develop materials that could be spread among wider populations, thereby hastening world domination by mathematicians?
It's true that some of the more motivated students genuinely enjoyed my new and improved Math 100. But many in the class simply couldn't be bothered to keep up—as evidenced by their lackadaisical attendance. Which meant they were stumped on tests and couldn't do the homework. Which meant they were in a decidedly ornery mood during the class before the final, when I handed out evaluation questionnaires. My scores were the worst I've ever received in close to 30 years of teaching. Maybe I will show them to Michele to cheer her up.
Perhaps I should have paid more attention to the clues from the humanities seminar. Each math assignment (e.g., on fractal music or chaos in population growth) that I slipped in during the seminar generated so many pockets of anxiety that we had to make them extra-credit. The freshmen seemed fine with understanding underlying math ideas, but, as happens even with math majors, absorbing those ideas with enough precision for applications was a problem—as it certainly was for the Math 100 students. One can probably learn almost anything with reasonable competence, but the time to do so depends on both aptitude and interest (which is why Professor Suri, alas, will never be a singer).
Michele: Well, maybe Professor Suri will never perform live at the Met, but might he be a strong member of the chorus? And couldn't that engagement increase ability, demonstrate skill?
I don't know if it's a difference in our fields or in our persons, but my co-instructor and I ultimately have different views on how much math the students need to understand in order to see the potential for exchange and the possibilities for collaboration between our "two cultures." I know that the small amount of knowledge I gained at Manil's hand as he took me through the math of Arcadia made a difference in my comprehension of the play. But I could not, based on those sessions, apply an iterated algorithm in problem solving. What I could do was understand an iterated system, and apply that understanding successfully to my work as a dramaturg. And to me, that was really something.
Scholars in the humanities are constantly looking for connections between fields. How can we discuss literature without knowledge of history or philosophy or (enter discipline of choice)? The interdisciplinary knowledge may come slowly, but even small pockets have value. I don't think mathematicians are required to make such connections as consistently as we do. Manil may never forgive Howard Moss for his insouciant reference (in his poem "Particular Beauties") to Zeno's paradox, or those countless poets for their inaccurate use of the word "infinite." Such connections, sometimes "yoked by violence together," may not always reveal mysteries, but when they do, it's thrilling.
And that's what seminars foster ideally. They bring together assorted people and perspectives, all focused on examining a subject, so that the participants affect the learning taking place. But I can see now that it was not a setting with which my colleague was as comfortable. There was too much the students did not—could not—know about his topic. Smart and used to speaking up in class as they were, their input appeared initially as a parroting back of our statements, as an indulging in stream-of-consciousness reactions. They participated often, but without necessarily moving things forward. And this drove Manil crazy.
I understood his frustration but knew, from experience, that the students would improve. They'd start to appreciate what meaningful participation looks like. They would recognize that in seminars, participants are responsible for their own and one another's learning. I'm not sure students are up for that responsibility in their first semester, though studies suggest that they learn better from one another than they do from their instructors. (See a 2010 article, "A Case Study of Cooperative Learning and Communication Pedagogy: Does Working in Teams Make a Difference?," in the Journal of the Scholarship of Teaching and Learning.) But wait: I learn from students, too.
Manil: Actually, the imperative of interconnection affects mathematicians as well. Few with gainful employment can operate in a bubble. We are constantly called upon to apply our results to engineering, science, and business (and, very gradually, to humanities as well).
Rather than casting a wide net, what's needed by mathematicians is often more a directed effort, concentrated on the problem at hand. The more pressing problem is one of communication: The volubility of humanities scholars does not necessarily bless those with a more mathematical bent. How to make one's voice heard in a Wall Street team with no other mathematicians? Convince engineers while armed only with a proof's abstractness? Formulate a problem with, or explain its solution to, someone who does not even speak the language?
Those are issues that perhaps a seminar for mathematics students could examine, even if the format falls short in conveying the amount of technical content that must be absorbed in a typical math course. Perhaps it's time for us to plan the sequel to our course: "The Humanities and What it Means to Be a Mathematician."
But coming back to humanities students, what math would I prescribe for them? That's a question I was asked several times while teaching the seminar. Most important (and difficult) would be material that establishes numeracy—in particular the ability to judge whether or not numbers, reported figures, calculated quantities, and so on are plausible in their context. It's a skill that doesn't come automatically to most people, not even math majors.
Closely related would be the ability to recognize what's mathematically relevant. While doing the mathematical exercises, students had a tendency to ramble around, use lots of unnecessary description, without really homing in on the crux of things.
Next would be a course in statistics: The two outside speakers who addressed the class on using math in their research both employed statistical analysis to buttress their conclusions quantitatively, in addition to just qualitatively. We had fun with various numerical facts and figures that one can mine doing searches on Google databases—for instance, charting the historical course of tuberculosis by counting the number of times it appears annually in published works—but students need to be educated on all the misinterpretations and pitfalls possible.
On a more ambitious "wishful thinking" scale would be the math required to model social and historical phenomena. For instance, we read the first few pages of a paper that showed how the differential equations used to track the spread of epidemics could be reformulated to model the spread of ideas. Perhaps that's what the crystal ball shows will gain prominence in the future: the mathematical modeling of sweeping historical movements—a fledgling field called "cliodynamics."
Needless to say, I'd encourage everyone to get enough of an understanding of contemporary mathematical ideas so that they can claim to have a well-rounded education. In return, I pledge to read more Shakespeare.
Michele: Though there are a lot—I won't say infinite, but an impressive batch of natural numbers' worth—of moments I'd redo if I could, I would not have missed out on co-teaching this seminar, tricky as it was. There is a language to mathematics, and I'm devoted to seeing what language produces and provokes. It was fascinating, even if sometimes frightening, to see how different Manil's and my responses were to various texts and ideas. That didn't make one of us "wrong" (though I didn't realize that immediately); it made the possibilities of scholarship more rich. There wasn't a class in which I didn't learn something, and I've given real thought to certain of our topics and ways I might deal with them on my own. It won't be easy, but then, I know where my co-instructor lives.
I admit, too, that I loved entering the department of mathematics and statistics with a purpose. I could never have imagined such a thing. I couldn't have told you what side of campus it was on. I knew no math faculty by sight. I knew only those mathematics course numbers that English majors need to graduate. And I suspect that I was as foreign to those numerical types as they were to me. There was an impressive array of surprised/amused/skeptical/dismayed looks from Manil's colleagues and from mine when they heard about our course. "What do you know about math?" people would ask me. "Not enough," was my answer, "but I'm working on it."
Manil Suri: It's reassuring to see that whether in mathematics or the humanities, the last few weeks of a course with a final project follow the same pattern: The jolt of energy that suddenly animates the room when students realize this is it. The grandly ambitious (and hopelessly infeasible) ideas that first emerge. The excitement that turns into consternation when confronted with the impracticality of the studies being proposed. ("And how exactly will you chart all the babies born out of wedlock during the Civil War?" we had to ask one student.)
Most predictable is the panic when the invariable lurking surprises start popping up like the frantic (and usually inaugural) visits during office hours and the 11th-hour abandonment of a doomed project to start another one. I'd forgotten how entertaining it can be to prolong their agony thus—really must remember to give more projects instead of finals!
This time, in addition to the usual pitfalls of research and creative works, students had an extra quagmire to watch out for: the mathematics. It had to be an integral part of what they did (a requirement that some tried, unsuccessfully, to dodge), it had to be correct (a requirement we were more willing to bend), and they had to convincingly show they understood the concepts used. Sadly, that last point overwhelmed one of the most unusual of the projects: a short story based on the Prisoner's Dilemma payoff matrix. Yet another reminder that mathematical constructs can be deeper than they appear.
But we had some sparkling successes, too. One student wrote a story that elegantly finessed chaos theory to imagine a world in which Lincoln might never have become president. Another produced a slick, professional-quality video on beauty and the golden ratio. Mathematically deepest of all was an instrumental composition that musically interpreted what it means to be a sine (or cosine) function—by a humanities scholar inspired to take calculus this semester.
We had planned from the start to also give students the option to produce a research proposal for quantitatively dealing with any humanities topic. We could see it now: our crew of 13 freshmen submitting completed, original research (involving mathematics, of all things) with their applications to the country's top graduate programs in the humanities.
To nudge them toward real-life research questions, we appealed to their love of cash. The University of Maryland-Baltimore County offers generous grants each year by way of Undergraduate Research Awards. "A real opportunity," we told them. "Once your proposals are crafted, you can parlay them into a URA proposal as a logical next step."
We don't know that any students actually took that step, but then, freshmen often need to work up the courage. One student has continued to develop her project, which attempts to quantify the influence that specific language or word choice may have on survey respondents (thereby aiding statisticians in the elimination of question bias).
In the end, I was more impressed with the students' performance than they were with mine. Our course evaluations were the harshest I've seen in years. I'm not naïve or obsessive about this stuff—I know every class can't work for every student, and I know very high evaluations can indicate a lack of rigor.
Still, the students were critical of areas I'd never before questioned. Some commented that neither of their instructors seemed interested in what the other one had to say. That cut me to the core. I took notes in class, asked questions, confessed my confusion. What does "interest" look like to students nowadays? I wonder, though, if their confidence wasn't shaken when they realized that we two didn't have all the answers, or even all of the questions, but proposed to make these discoveries around the seminar table.
Manil: Although Michele seemed crushed by a few of the student evaluations, I was actually relieved at how good they were.
My experience has been to be severely punished in the evaluations whenever I've tried to do something experimental in the classroom. There was the sparsely attended first-year seminar I taught on mathematics in the media, in which, toward the end, only one student was still showing up to class. (The video talk we created as his class project, on the concept of infinity, has had more than 20,000 YouTube viewers, which makes me feel better.) Another workshop I taught that required participants to use mathematics in a creative endeavor (short story, musical piece, artistic creation) ended with a series of e-mails from an irate student excoriating me for the grading scheme that led to her B.
And then there was the spring 2012 semester. Scheduled to teach the university's usual general-education Math 100 course for nonmajors, and still invigorated by the humanities seminar, I decided to use the most successful aspects from that experience to "perk up" the syllabus (or so I brightly thought) in my intro course. Besides including the videos and computer exercises on fractals and probability, I also employed the basic philosophy of tying everything to real life (for example, having the students read an article on the growth of cities during the chapter on nonlinearity). After all, wasn't this one of the original goals of working on the humanities seminar? To develop materials that could be spread among wider populations, thereby hastening world domination by mathematicians?
It's true that some of the more motivated students genuinely enjoyed my new and improved Math 100. But many in the class simply couldn't be bothered to keep up—as evidenced by their lackadaisical attendance. Which meant they were stumped on tests and couldn't do the homework. Which meant they were in a decidedly ornery mood during the class before the final, when I handed out evaluation questionnaires. My scores were the worst I've ever received in close to 30 years of teaching. Maybe I will show them to Michele to cheer her up.
Perhaps I should have paid more attention to the clues from the humanities seminar. Each math assignment (e.g., on fractal music or chaos in population growth) that I slipped in during the seminar generated so many pockets of anxiety that we had to make them extra-credit. The freshmen seemed fine with understanding underlying math ideas, but, as happens even with math majors, absorbing those ideas with enough precision for applications was a problem—as it certainly was for the Math 100 students. One can probably learn almost anything with reasonable competence, but the time to do so depends on both aptitude and interest (which is why Professor Suri, alas, will never be a singer).
Michele: Well, maybe Professor Suri will never perform live at the Met, but might he be a strong member of the chorus? And couldn't that engagement increase ability, demonstrate skill?
I don't know if it's a difference in our fields or in our persons, but my co-instructor and I ultimately have different views on how much math the students need to understand in order to see the potential for exchange and the possibilities for collaboration between our "two cultures." I know that the small amount of knowledge I gained at Manil's hand as he took me through the math of Arcadia made a difference in my comprehension of the play. But I could not, based on those sessions, apply an iterated algorithm in problem solving. What I could do was understand an iterated system, and apply that understanding successfully to my work as a dramaturg. And to me, that was really something.
Scholars in the humanities are constantly looking for connections between fields. How can we discuss literature without knowledge of history or philosophy or (enter discipline of choice)? The interdisciplinary knowledge may come slowly, but even small pockets have value. I don't think mathematicians are required to make such connections as consistently as we do. Manil may never forgive Howard Moss for his insouciant reference (in his poem "Particular Beauties") to Zeno's paradox, or those countless poets for their inaccurate use of the word "infinite." Such connections, sometimes "yoked by violence together," may not always reveal mysteries, but when they do, it's thrilling.
And that's what seminars foster ideally. They bring together assorted people and perspectives, all focused on examining a subject, so that the participants affect the learning taking place. But I can see now that it was not a setting with which my colleague was as comfortable. There was too much the students did not—could not—know about his topic. Smart and used to speaking up in class as they were, their input appeared initially as a parroting back of our statements, as an indulging in stream-of-consciousness reactions. They participated often, but without necessarily moving things forward. And this drove Manil crazy.
I understood his frustration but knew, from experience, that the students would improve. They'd start to appreciate what meaningful participation looks like. They would recognize that in seminars, participants are responsible for their own and one another's learning. I'm not sure students are up for that responsibility in their first semester, though studies suggest that they learn better from one another than they do from their instructors. (See a 2010 article, "A Case Study of Cooperative Learning and Communication Pedagogy: Does Working in Teams Make a Difference?," in the Journal of the Scholarship of Teaching and Learning.) But wait: I learn from students, too.
Manil: Actually, the imperative of interconnection affects mathematicians as well. Few with gainful employment can operate in a bubble. We are constantly called upon to apply our results to engineering, science, and business (and, very gradually, to humanities as well).
Rather than casting a wide net, what's needed by mathematicians is often more a directed effort, concentrated on the problem at hand. The more pressing problem is one of communication: The volubility of humanities scholars does not necessarily bless those with a more mathematical bent. How to make one's voice heard in a Wall Street team with no other mathematicians? Convince engineers while armed only with a proof's abstractness? Formulate a problem with, or explain its solution to, someone who does not even speak the language?
Those are issues that perhaps a seminar for mathematics students could examine, even if the format falls short in conveying the amount of technical content that must be absorbed in a typical math course. Perhaps it's time for us to plan the sequel to our course: "The Humanities and What it Means to Be a Mathematician."
But coming back to humanities students, what math would I prescribe for them? That's a question I was asked several times while teaching the seminar. Most important (and difficult) would be material that establishes numeracy—in particular the ability to judge whether or not numbers, reported figures, calculated quantities, and so on are plausible in their context. It's a skill that doesn't come automatically to most people, not even math majors.
Closely related would be the ability to recognize what's mathematically relevant. While doing the mathematical exercises, students had a tendency to ramble around, use lots of unnecessary description, without really homing in on the crux of things.
Next would be a course in statistics: The two outside speakers who addressed the class on using math in their research both employed statistical analysis to buttress their conclusions quantitatively, in addition to just qualitatively. We had fun with various numerical facts and figures that one can mine doing searches on Google databases—for instance, charting the historical course of tuberculosis by counting the number of times it appears annually in published works—but students need to be educated on all the misinterpretations and pitfalls possible.
On a more ambitious "wishful thinking" scale would be the math required to model social and historical phenomena. For instance, we read the first few pages of a paper that showed how the differential equations used to track the spread of epidemics could be reformulated to model the spread of ideas. Perhaps that's what the crystal ball shows will gain prominence in the future: the mathematical modeling of sweeping historical movements—a fledgling field called "cliodynamics."
Needless to say, I'd encourage everyone to get enough of an understanding of contemporary mathematical ideas so that they can claim to have a well-rounded education. In return, I pledge to read more Shakespeare.
Michele: Though there are a lot—I won't say infinite, but an impressive batch of natural numbers' worth—of moments I'd redo if I could, I would not have missed out on co-teaching this seminar, tricky as it was. There is a language to mathematics, and I'm devoted to seeing what language produces and provokes. It was fascinating, even if sometimes frightening, to see how different Manil's and my responses were to various texts and ideas. That didn't make one of us "wrong" (though I didn't realize that immediately); it made the possibilities of scholarship more rich. There wasn't a class in which I didn't learn something, and I've given real thought to certain of our topics and ways I might deal with them on my own. It won't be easy, but then, I know where my co-instructor lives.
I admit, too, that I loved entering the department of mathematics and statistics with a purpose. I could never have imagined such a thing. I couldn't have told you what side of campus it was on. I knew no math faculty by sight. I knew only those mathematics course numbers that English majors need to graduate. And I suspect that I was as foreign to those numerical types as they were to me. There was an impressive array of surprised/amused/skeptical/dismayed looks from Manil's colleagues and from mine when they heard about our course. "What do you know about math?" people would ask me. "Not enough," was my answer, "but I'm working on it."
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