# Rigorous and Precise Thinking

Teaching writing and mathematics in the same course leaves Ruth Starkman considering the way humanities and mathematics students approach problems.

Everything would have been perfectly ordinary that October morning in my freshman writing course at Stanford University. Bright autumn light reflected up from the Main Quad to our third floor. Unfed, sleepy-eyed freshmen offered ideas about the assigned reading, which I tracked on the board.

As I often do, I drew a doodle to describe a concept in the reading. This doodle — so I thought — demanded less artistry and complexity than my usual sketches of Thomas Hobbes’s "arrant Wolfe," for which I hash out two mangy-looking wolves squinting at each other, or Immanuel Kant’s famous "crooked timber," for which a bent log suffices to get the idea across. Here, I simply tossed up a rectangle with a triangle inside.

My students gasped.

"What’s wrong?" I asked.

“Um … everything." They wagered cautiously.

"Well," I tried. "This is just like the one Lockhart shows in his essay." I was referring to a drawing in Paul Lockhart’s famous 2002 "Lament" about the state of mathematics education. Here it is, precisely as it appears in the essay, not the version I drew in class.

"Sorry … no … not really, well … it’s not even close," they ventured, as if not to hurt my feelings.

My students, mostly young aspiring mathematicians, found themselves so ill at ease here, because their teacher with a humanities doctorate had not bothered to notice that the triangle inside the rectangle touches both corners of the same length and thus forms several other triangles. My doodle — whatever it looked like, I can’t remember — was simply an approximation, a lonely triangloid adrift in a rectangular sea of lopsidedness.

My students had expected greater precision. After all, the course title "Rigorous and Precise Thinking" had suggested as much. Secondly, this was a college writing course, which, as the rumor goes, is supposed to be a smackdown of style, argument and organization, where freshmen quickly learn they must jettison comfortable high school formats and every illusion of their personal literary genius. Expectations for rigor and many other new adventures ran high in this new course, an experimental hybrid college writing/mathematical thinking and proof writing class, one of five liberal arts courses in a new program called Education as Self-Fashioning.

Like the other four ESF classes, this one intended to "engage actively in the types of thinking promoted through these different conceptions of education for life, so as to try those lives on for ourselves ..." and offer students a “chance to shape [their] educational aspirations in dialogue with fellow students and an exciting group of faculty from across a wide range of disciplines — from the humanities and social sciences through the natural sciences and mathematics." I was the writing instructor paired with Professor Ravi Vakil, an American-Canadian mathematician working in algebraic geometry.

Vakil invented the course concept as a rejoinder to C.P. Snow’s "Two Cultures" hypothesis with the hope of showing undergrads, and even the world, that writing in the humanities and writing in math gained force and excellence through similar structures of precise reasoning. Vakil more than delivered on the rigor and precision. His lectures introduced students to proof writing, number theory, set theory, and many other advanced forms of math most academics expect to address only with advanced university students. For my part, I was simply to help students elaborate the readings from Plato, Descartes, Douglas Hofstadter, Bertrand Russell, Paul Lockhart and many others, while teaching writing.

Tellingly, my imprecise doodle proved to be not my first, second, nor even third example of lack of rigor. In fact, the moment seem to demonstrate the deep divide between Snow’s "two cultures," since I evidently betrayed a lack of familiarity with the basic truths of measurement, "mass, or acceleration, pretty much the scientific equivalent of a humanist asking skeptically, Can you read?" Without a doubt, much of that difference proved disciplinary — the very limit this course hoped to transgress.

Yet, we experienced no ordinary rift between the two cultures. The class had read Snow’s famous 1959 Rede Lecture and chuckled at his description of subverbal grunting mathematicians ruining a young humanist’s dinner party experience. My students saw themselves as beyond what old Stanford lingo designates as the split between "fuzzies" and "techies." Interested equally in learning all things humanist and STEM, e.g., Shakespeare and thermodynamics and beyond, these students insisted that math and math culture far surpassed the cartoonish figures of Snow’s dinner party. Nor (my students believed) were humanists so incorrigibly "fuzzy" as to not be able to reproduce a mathematical doodle — or were they?

Had I inadvertently proven Snow’s point, right before the eyes of my epistemologically optimistic students? In fact, both the students and I discovered that many of the clichés about our respective fields proved instructive. I really do need to be more careful in my doodling — and thinking about my doodling — if I am drawing triangles (with mathematical aspirations) and not wolves (no matter how humanistically inclined).

The awkward doodle moment proved not the existence of two never-the-twain-shall-meet cultures, but rather a need for me to look more closely at the other side. Once I recovered from the initial jolt of difference, I began to realize the opportunity for me to reconsider my pedagogy. Not having seen a university math professor teach proof writing before, I witnessed several fascinating interactions while attending Vakil’s sections of our course. Most striking, when Vakil wrote a problem on the board, the room jumped to life with students calling out and frantically waving their arms. He would ask: "How can you prove the square root of 2 is irrational?" and it was as though Vakil were standing at the board waving a bloody steak at a group of famished tigers. Everyone wanted to offer some solution.

Seldom have I been bombarded with solutions or suggestions when I ask students to show me "textual proof" that Sigmund Freud has a Hobbesian view of nature … hint hint … homo homini … wolf sketch, ... *Civilization and Its Discontents,* try page number and reference…Freud 1930a [I929], SE 21:111. That special classroom enthusiasm surely arose from Vakil’s charisma and love of his subject, but the response was new to me because humanities courses that I know at least demand a very different kind of invention. Vakil asked a question and students racked their brains trying to imagine which set of mathematical tools or ideas they might use to solve the problem. Confident that they all share these tools, or at least know of such tools, the students seemed to feel much more at ease trying out different approaches.

In humanities courses, previous knowledge certainly helps, especially with literary references, but at the end of the day, a humanist’s tools remain much more contested and may not be applicable in different contexts. For example, students asked me why I requested they not use the third-person plural perspective "we." I told them writing in the humanities differs from math, where one can simply write in a proof “we assume that *x*=2.” Humanists can neither be sure who that “we” is, nor what to "assume" nor how one can know *x*. All such terms are permanently available for debate.

In contrast, the mathematicians’ particular disciplinary certainty also revealed a fierce loyalty and love of the subject, which produced a very different discourse than I traditionally hear from humanities students who feel a strong affinity with their work. These math students spoke a Russellian language of awe toward the "cold and austere" "supreme beauty" and "elegance" of math. Perhaps other humanists have encountered students who express an emphatic humility before their subjects, but that this for me was as new as the students’ shock at my imprecise drawing. For I learned that day, that my students had not yet adopted a humanistic skepticism toward mathematical precision. For them precision is very real, especially in a world of increasing complexity and Gödelian incompleteness.

For humanists, precision lies elsewhere, side by side with ambiguity, and we pursue it with nuance rather than with proofs. My task therefore became one of translation. I understood little of the doodles and equations that Vakil and the students so hotly debated in his sections, but I knew that I had helped my students articulate arguments within the very different confines of humanistic inquiry. Where they were convinced of certain mathematical truths in the landscape of defined terms, they nevertheless arrived in my class with the classic freshman enormity of themes.

Asked to find “precise” topics in math to write about for their research papers, nearly all 29 students first chose grandiose topics like "the definition of intuition," "the connections between art and math" or "math and humanistic knowledge." With such great ambitions in mind, they also fervently believed in math as a liberal art capable of teaching the exact same virtues of critical (self) reflection as any of the great classical texts I teach from Greek virtue ethics to Rawls.

Most provocatively, they claimed that by practicing mathematical reasoning they were indeed preparing themselves in the fashion of liberal arts education for ethical citizenship. They claimed with confidence their rigorous and precise thinking could lead them to ethical reasoning as equally well as a discussion of the Plato’s “Apology.” For my part, I could not see how debating a triangle or even practicing some form of applied math as statistics would help me lead the "examined life" in a qualitative fashion.

In class, Vakil often reflected on the limits of mathematical reasoning in a mode reminiscent of Greek virtue ethics; that is, perfecting one’s art whether mathematical or literary skill, is surely a virtue, but not one that can replace ethical action. When asked whether excellence in math could prevent one from doing evil, no one doubted the inadequacy of that proposition. History has no shortage of evil uses of math, and the students could quite easily number these. Yet, many of the students persisted in their strong claims for math.

One student asserted a mathematical imperative in times of emergency: "Just imagine it’s war or a crisis: you have a moral obligation to shut up and do the math." By which she meant one is ethically compelled to run a statistical analysis to develop a more concrete understanding of actual dangers. Another student expressed less certainty about quantitative methods. "Statistics aren’t bulletproof, you know; what matters ultimately is thinking clearly, and math trains the mind for such emergencies."

Vakil softened these strong claims for both applied and pure math:

*I'm less certain that this [mathematical reasoning] in any way replaces the approach to the virtues of critical self-reflection through great philosophical texts. I hope that our students will better appreciate the importance of such texts, because of an appreciation of the problems that earlier thinkers were grappling with (and that we should grapple with today). Similarly, I doubt that this is sufficient to lead them to ethical reasoning, although I would make a milder claim that thinking clearly in this way can assist in carrying out ethical reasoning.*

Vakil also elaborated ways in which math could serve ethics, both by providing empirical data and asking Socratic questions about knowledge and decision-making. In the end, we hoped the students finished the course knowing a bit more about practices of rigorous thinking in our respective disciplines, and that they would see these as equally essential and complementary. Could this sprawling, seven-unit course provide a model for future courses? We’re not sure, but are happy to share our data and materials.

## Bio

*Ruth Starkman writes on higher education and teaches college writing, biomedical ethics and social media at Stanford University.*

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Teaching writing and mathematics in the same course leaves Ruth Starkman considering the way humanities and mathematics students approach problems.

Everything would have been perfectly ordinary that October morning in my freshman writing course at Stanford University. Bright autumn light reflected up from the Main Quad to our third floor. Unfed, sleepy-eyed freshmen offered ideas about the assigned reading, which I tracked on the board.

As I often do, I drew a doodle to describe a concept in the reading. This doodle — so I thought — demanded less artistry and complexity than my usual sketches of Thomas Hobbes’s "arrant Wolfe," for which I hash out two mangy-looking wolves squinting at each other, or Immanuel Kant’s famous "crooked timber," for which a bent log suffices to get the idea across. Here, I simply tossed up a rectangle with a triangle inside.

My students gasped.

"What’s wrong?" I asked.

“Um … everything." They wagered cautiously.

"Well," I tried. "This is just like the one Lockhart shows in his essay." I was referring to a drawing in Paul Lockhart’s famous 2002 "Lament" about the state of mathematics education. Here it is, precisely as it appears in the essay, not the version I drew in class.

"Sorry … no … not really, well … it’s not even close," they ventured, as if not to hurt my feelings.

My students, mostly young aspiring mathematicians, found themselves so ill at ease here, because their teacher with a humanities doctorate had not bothered to notice that the triangle inside the rectangle touches both corners of the same length and thus forms several other triangles. My doodle — whatever it looked like, I can’t remember — was simply an approximation, a lonely triangloid adrift in a rectangular sea of lopsidedness.

My students had expected greater precision. After all, the course title "Rigorous and Precise Thinking" had suggested as much. Secondly, this was a college writing course, which, as the rumor goes, is supposed to be a smackdown of style, argument and organization, where freshmen quickly learn they must jettison comfortable high school formats and every illusion of their personal literary genius. Expectations for rigor and many other new adventures ran high in this new course, an experimental hybrid college writing/mathematical thinking and proof writing class, one of five liberal arts courses in a new program called Education as Self-Fashioning.

Like the other four ESF classes, this one intended to "engage actively in the types of thinking promoted through these different conceptions of education for life, so as to try those lives on for ourselves ..." and offer students a “chance to shape [their] educational aspirations in dialogue with fellow students and an exciting group of faculty from across a wide range of disciplines — from the humanities and social sciences through the natural sciences and mathematics." I was the writing instructor paired with Professor Ravi Vakil, an American-Canadian mathematician working in algebraic geometry.

Vakil invented the course concept as a rejoinder to C.P. Snow’s "Two Cultures" hypothesis with the hope of showing undergrads, and even the world, that writing in the humanities and writing in math gained force and excellence through similar structures of precise reasoning. Vakil more than delivered on the rigor and precision. His lectures introduced students to proof writing, number theory, set theory, and many other advanced forms of math most academics expect to address only with advanced university students. For my part, I was simply to help students elaborate the readings from Plato, Descartes, Douglas Hofstadter, Bertrand Russell, Paul Lockhart and many others, while teaching writing.

Tellingly, my imprecise doodle proved to be not my first, second, nor even third example of lack of rigor. In fact, the moment seem to demonstrate the deep divide between Snow’s "two cultures," since I evidently betrayed a lack of familiarity with the basic truths of measurement, "mass, or acceleration, pretty much the scientific equivalent of a humanist asking skeptically, Can you read?" Without a doubt, much of that difference proved disciplinary — the very limit this course hoped to transgress.

Yet, we experienced no ordinary rift between the two cultures. The class had read Snow’s famous 1959 Rede Lecture and chuckled at his description of subverbal grunting mathematicians ruining a young humanist’s dinner party experience. My students saw themselves as beyond what old Stanford lingo designates as the split between "fuzzies" and "techies." Interested equally in learning all things humanist and STEM, e.g., Shakespeare and thermodynamics and beyond, these students insisted that math and math culture far surpassed the cartoonish figures of Snow’s dinner party. Nor (my students believed) were humanists so incorrigibly "fuzzy" as to not be able to reproduce a mathematical doodle — or were they?

Had I inadvertently proven Snow’s point, right before the eyes of my epistemologically optimistic students? In fact, both the students and I discovered that many of the clichés about our respective fields proved instructive. I really do need to be more careful in my doodling — and thinking about my doodling — if I am drawing triangles (with mathematical aspirations) and not wolves (no matter how humanistically inclined).

The awkward doodle moment proved not the existence of two never-the-twain-shall-meet cultures, but rather a need for me to look more closely at the other side. Once I recovered from the initial jolt of difference, I began to realize the opportunity for me to reconsider my pedagogy. Not having seen a university math professor teach proof writing before, I witnessed several fascinating interactions while attending Vakil’s sections of our course. Most striking, when Vakil wrote a problem on the board, the room jumped to life with students calling out and frantically waving their arms. He would ask: "How can you prove the square root of 2 is irrational?" and it was as though Vakil were standing at the board waving a bloody steak at a group of famished tigers. Everyone wanted to offer some solution.

Seldom have I been bombarded with solutions or suggestions when I ask students to show me "textual proof" that Sigmund Freud has a Hobbesian view of nature … hint hint … homo homini … wolf sketch, ...

Civilization and Its Discontents,try page number and reference…Freud 1930a [I929], SE 21:111. That special classroom enthusiasm surely arose from Vakil’s charisma and love of his subject, but the response was new to me because humanities courses that I know at least demand a very different kind of invention. Vakil asked a question and students racked their brains trying to imagine which set of mathematical tools or ideas they might use to solve the problem. Confident that they all share these tools, or at least know of such tools, the students seemed to feel much more at ease trying out different approaches.In humanities courses, previous knowledge certainly helps, especially with literary references, but at the end of the day, a humanist’s tools remain much more contested and may not be applicable in different contexts. For example, students asked me why I requested they not use the third-person plural perspective "we." I told them writing in the humanities differs from math, where one can simply write in a proof “we assume that

x=2.” Humanists can neither be sure who that “we” is, nor what to "assume" nor how one can knowx. All such terms are permanently available for debate.In contrast, the mathematicians’ particular disciplinary certainty also revealed a fierce loyalty and love of the subject, which produced a very different discourse than I traditionally hear from humanities students who feel a strong affinity with their work. These math students spoke a Russellian language of awe toward the "cold and austere" "supreme beauty" and "elegance" of math. Perhaps other humanists have encountered students who express an emphatic humility before their subjects, but that this for me was as new as the students’ shock at my imprecise drawing. For I learned that day, that my students had not yet adopted a humanistic skepticism toward mathematical precision. For them precision is very real, especially in a world of increasing complexity and Gödelian incompleteness.

For humanists, precision lies elsewhere, side by side with ambiguity, and we pursue it with nuance rather than with proofs. My task therefore became one of translation. I understood little of the doodles and equations that Vakil and the students so hotly debated in his sections, but I knew that I had helped my students articulate arguments within the very different confines of humanistic inquiry. Where they were convinced of certain mathematical truths in the landscape of defined terms, they nevertheless arrived in my class with the classic freshman enormity of themes.

Asked to find “precise” topics in math to write about for their research papers, nearly all 29 students first chose grandiose topics like "the definition of intuition," "the connections between art and math" or "math and humanistic knowledge." With such great ambitions in mind, they also fervently believed in math as a liberal art capable of teaching the exact same virtues of critical (self) reflection as any of the great classical texts I teach from Greek virtue ethics to Rawls.

Most provocatively, they claimed that by practicing mathematical reasoning they were indeed preparing themselves in the fashion of liberal arts education for ethical citizenship. They claimed with confidence their rigorous and precise thinking could lead them to ethical reasoning as equally well as a discussion of the Plato’s “Apology.” For my part, I could not see how debating a triangle or even practicing some form of applied math as statistics would help me lead the "examined life" in a qualitative fashion.

In class, Vakil often reflected on the limits of mathematical reasoning in a mode reminiscent of Greek virtue ethics; that is, perfecting one’s art whether mathematical or literary skill, is surely a virtue, but not one that can replace ethical action. When asked whether excellence in math could prevent one from doing evil, no one doubted the inadequacy of that proposition. History has no shortage of evil uses of math, and the students could quite easily number these. Yet, many of the students persisted in their strong claims for math.

One student asserted a mathematical imperative in times of emergency: "Just imagine it’s war or a crisis: you have a moral obligation to shut up and do the math." By which she meant one is ethically compelled to run a statistical analysis to develop a more concrete understanding of actual dangers. Another student expressed less certainty about quantitative methods. "Statistics aren’t bulletproof, you know; what matters ultimately is thinking clearly, and math trains the mind for such emergencies."

Vakil softened these strong claims for both applied and pure math:

I'm less certain that this [mathematical reasoning] in any way replaces the approach to the virtues of critical self-reflection through great philosophical texts. I hope that our students will better appreciate the importance of such texts, because of an appreciation of the problems that earlier thinkers were grappling with (and that we should grapple with today). Similarly, I doubt that this is sufficient to lead them to ethical reasoning, although I would make a milder claim that thinking clearly in this way can assist in carrying out ethical reasoning.Vakil also elaborated ways in which math could serve ethics, both by providing empirical data and asking Socratic questions about knowledge and decision-making. In the end, we hoped the students finished the course knowing a bit more about practices of rigorous thinking in our respective disciplines, and that they would see these as equally essential and complementary. Could this sprawling, seven-unit course provide a model for future courses? We’re not sure, but are happy to share our data and materials.