(There’s a printer- and footnote-friendly pdf version—with links still working!—here if you prefer.)
This fall, I’m teaching a freshman writing course at the University of Michigan.1 I’ll be writing a lot about this course and related topics over the coming months, so I thought I’d take a moment to introduce the course and talk about what I’m planning on doing.
Before I go into the details, let me give the quick (and simplified) version of my plans.
We’ll begin with an introduction to pure math: the beauty of proofs and abstraction, not the rote calculations of high school. Then, we’ll segue into areas of theoretical linguistics (syntax, semantics, and pragmatics) that overlap significantly with pure math (especially logic, set theory, and theoretical computer science). Finally, we’ll use our knowledge of math and theoretical linguistics to think about writing itself, in particular how to write (and argue, and think) effectively. Throughout, we’ll be motivated by the premise that mathematical thinking is crucial in understanding much of the world, not only in the linguistics we’ll be focusing on, but also in economics, computer science, music, medicine, sports, political science, and many other areas.2
I’m writing this with the dual goals of publicizing the course and soliciting feedback and advice. I’m eager to publicize this, since I think this experiment—the course is very much an experiment—offers a different and potentially valuable approach to math and writing education. Because it is new and experimental, I’m eager to hear suggestions as I finish designing and preparing the course over the summer. (As I mention below, I’m especially looking for suggestions of readings to assign.) Please let me know what you think, and feel free to forward this on to others who might be interested.
Also, over the summer and fall I plan on posting online much of the material I’m going to cover, so if you’re interested in following along, subscribe to this blog for updates.
My audiences for this post
Before I go into the details, I have a few words directed to the different audiences for this post (as well as future posts on the course):
- readers without any specialized background
- mathematicians and math educators
- linguists (and computer scientists, philosophers, and logicians, who fit somewhere between the math and linguistics camps3)
- English professors and others who teach writing
- academics and educators more broadly
- any UM freshmen thinking of taking the course
Feel free to go to the section relevant to you and skim the other parts, or jump straight to the details of the course, although you might find some of the other sections of interest, especially the penultimate one.
To readers without specialized backgrounds
I’m teaching this course because I think the material I’ll be discussing is important and underappreciated. Over the summer, I’m going to write a series of posts that will introduce some of the ideas I plan on teaching in the fall, topics like Gricean implicature, recursion, and set theory. Since I’ll be teaching this material to 18-year-olds who don’t have backgrounds in math, linguistics, or logic, I very much want to know what in my expositions is and isn’t clear to readers without specialized backgrounds. So please read these posts and offer feedback. (I’m going to direct parts of this post to more specialized audiences. So I apologize if I won’t have space to explain in this post what these terms mean. But most of this post should still be accessible.)
To mathematicians and math educators
We are doing a tremendous disservice to the world by focusing our introductory math course offerings in high schools and colleges on applications-based precalculus and calculus courses. Instead, we should be teaching students what pure math is actually about: abstraction, proof,4 and beauty. Many others have said this, and have offered introductory courses with this goal.5 I hope my course, with its unusual focus on combining pure math with writing, and with its focus on “applications” of math outside the traditional areas of physics and engineering, might be of interest.
A challenge for the course will be teaching pure math—proofs, abstract definitions, and logic—to students without much math background (and, in most cases, no background in pure math). I strongly believe both that this can be done and that it should be done, but I’m aware that this might be difficult. (I’m particularly aware of this given the time limitations of the course; because of the focus on writing, I’m not going to be able to assign many homework exercises or proofs.) Suggestions from mathematicians who have taught such courses before are welcome!
Update: See this post I wrote about the Potsdam Miracle, which suggests that this can be done.
To linguists (and philosophers, logicians, and computer scientists)
What I’m suggesting—using theoretical linguistics in a systematic way to help students understand writing—seems to be unusual. My impression of theoretical linguists6 is that when they do think about the connection between linguistics and writing, it’s mostly in the context of specific quirks of language or discussions of prescriptivism.7 (This is the sort of stuff one sees on Language Log.)
Back in college, I went to a talk where Mark Liberman, the noted Penn professor and founder of Language Log, decried the low enrollments of introductory linguistics courses, especially compared to the burgeoning field of psychology. From his (provocative) abstract8:
About ten years ago, a publisher’s representative told me that introductory linguistics courses in the U.S. enroll 50,000 students per year, while introductory psychology courses enroll about 1,500,000, or 30 times more. The current number of Google hits for “linguistics department” is 60,900, while “psychology department” has 1,010,000, or 14 times more. The Linguistic Society of America has about 4,000 members, while the American Psychological Association has more than 150,000 members, or about 38 times more. Comparisons between linguistics and fields like history or chemistry give similar results.
It’s easy to accept this state of affairs as natural, but in fact it’s bizarre, both historically and logically. Furthermore, it’s part of a larger and much more serious problem. Those who are resigned to the fate of our academic discipline should still be disturbed that contemporary intellectuals are taught almost no skills for analyzing the form and content of speech and text, or that reading instruction is so widely based on false or nonsensical ideas that a quarter of all students have difficulties serious enough to interfere with the rest of their education.
The type of course I’m designing (or a variant where everything that’s called “math” is relabeled as “linguistics”) could help solve Liberman’s problem.9 After all, there’s an even bigger market for freshman writing courses than there is for intro pysch.
So I hope linguists will think about what I’m suggesting. Obviously, this has to be done in the right way, with the right approach to questions of prescriptivism/style and an appropriately agnostic attitude to the ideological battles within linguistics. What follows below only gives an outline of how I plan to tie linguistics to writing instruction. The course materials I’ll post over the summer and fall will show much more.
I’d love to hear feedback and suggestions from theoretical linguists, especially (a) advice in teaching ideas from the technically challenging areas of syntax/semantics/pragmatics to freshmen10; (b) any experiences with explicitly unifying pure math and linguistics as a pedagogical ploy11; and (c) any experiences systematically using theoretical linguistics to teach writing. (To any linguists at the LSA Summer Institute at UM this month, I’m in Ann Arbor, so I’d be happy to meet in person to discuss these ideas or hear suggestions.)
To English professors and others teaching writing
By teaching this course, I want to show that there’s a place for writing courses centered around a careful analysis and study of language that includes advanced topics in linguistics like syntax, semantics, and pragmatics.
In part, this is simply joining the fight against naive prescriptivism, while maintaining that this doesn’t mean that we have to adopt a relativism that ignores stylistic and grammatical solecisms “so long as they don’t impede comprehension.” Usually, this more nuanced approach to (prescriptive) grammar is taught by emphasizing the historical and social context of prescriptive grammatical rules. I agree wholeheartedly that these aspects are important; students should learn that a dialect is just a language without an army and that African American Vernacular English (as “Ebonics” is often called) is as logical and expressive as standard English. If, however, we want students to understand the norms that do exist, we can’t merely give them a list of rules when they have at best a superficial understanding of how language works. Instead, we can and should teach them some actual linguistics. I hope this experiment will show that such an approach can work—not just by helping students learn about grammar and mechanics but also by giving them a linguistic sophistication that will help them read, understand, and write texts throughout their lives.
As I prepare and teach the course over the summer and fall, I’ll be putting much of my material online; I hope you’ll come back and take a look to see how these ideas can be implemented. I would love feedback about my plans for the course or any of these ideas. (I’d especially like to hear suggestions from anyone who has assigned students extramural reading, as I’m planning on doing.)
To academics and educators more broadly
I think we need to fundamentally rethink education, in the following way. When I said above that we’d be working with the premise that mathematical thinking is crucial to much of the world, what I really meant was that there is a core set of analytical and deductive tools—among them logic, probability, algorithmic thinking, economic reasoning, and linguistic analysis—that underlie this thinking. You can call this “math” if you want, but it’s much broader. Students (and educated adults) might get bits and pieces of this here and there if they’re lucky—algorithmic thinking and economic reasoning being the most common, thanks to the popularity of computer science and economics courses—but few get the whole perspective. (I was lucky enough to be introduced to many of these ideas through the wonderful Symbolic Systems major at Stanford, which I pursued along with linguistics before switching to math.)
This should change. This course is just a start, and I’m only going to be able to focus on a subset of these tools. (I hope to encourage students to explore some of the other tools in their projects.) I see this course as part of a much broader educational program that incorporates all of these perspectives. At some point soon, I hope to write a manifesto of sorts explaining these ideas in more depth.12 For now, though, I hope that academics and educators will be interested in this course as a case study in this broader educational philosophy.
To any UM freshmen
If there are any UM freshmen reading this who are thinking about taking the course, please feel free to contact me for more information. I’m incredibly excited to teach this course, and I’m eager to have students who are intrigued by what I’m planning on doing. I hope this gives you a sense of what the course will be about. (Of course everything here is tentative; we’ll have an official syllabus by the fall.)
A quick outline of what I have below: First, in the content section, I’ll offer a more detailed narrative of what I’m planning to cover, with some comments and explanation. That narrative greatly oversimplifies the order in which I’ll be covering things; math, linguistics, and writing will actually be intertwined throughout the semester. I have an extremely rough version of a calendar, which shows how I plan on fitting all of these topics into my 27 scheduled classes; you might want to take a look at that after my narrative. Then I’ll discuss the assignments and readings. The calendar shows roughly how these fit in (including a few more details about possible assignments). Finally, I’ll conclude with a short discussion about student backgrounds and some future steps.
I should emphasize that all of the specifics are very tentative. I’d very much appreciate feedback as I finalize this.
Now let me offer some more details on the content I’m going to cover. (Some of the words I use here will be complicated to non-experts; skip over those parts, and keep going. As I post more about the course over the summer and fall, I’ll be explaining many of these terms.)
First I’m going to introduce students to pure mathematics: abstraction, logic, proof, and so on. In a sense, one can be agnostic about the specific material taught; number theory, combinatorics, or logic all could be used to teach this. The key will be trying to give a students a sense of both the importance and the beauty/wonder/joy of real mathematics, and also to give them some practice in engaging in pure math so they can succeed in later parts of the course.
Then I’m going to shift the pure math content from traditional first-proofs-course topics like number theory into topics relevant to writing, especially logic and linguistics. Even though these aren’t traditionally taught in the math curriculum—the rare student who sees them will do so in linguistics, philosophy, or computer science courses—the content is very mathematical, and so follows nicely from the introduction. Some of the things I might talk about include:
- Recursion, Trees, and Formal Grammars. Natural language syntax is based on these mathematical concepts. Once I explain13 these concepts—I’ve got some thinking to do about the best ways to do this14—I’ll talk about how we can parse natural language using these ideas. Once students understand constituency, for example, they’ll see the logic that determines most punctuation in the English language.15 More generally, studying syntax will help students develop linguistic awareness and sophistication that will help them throughout their writing.
Logic (propositional and predicate calculus). This will be important for several reasons. First, it will help students understand mathematical proof. Second, it will help students understand and think about the nature of argument; this naturally leads into discussions of fallacious reasoning, a common topic in freshman composition courses. (I’ll also probably have a day about probability, misunderstandings of which are similarly a source of fallacious reasoning in the world.) Third, this will help us with understanding semantics and pragmatics.
Algorithms. Many of the “rules” in writing—be they punctuation or bibliographies—are straightforward recipes, i.e., explicit algorithms. Of course, there are interesting discussions to be had (and we will have them) about why these rules are the way they are, but students still need to be able to follow these instructions. Rather than holding students’ hands in teaching each step, say, of writing a bibliography, I’m instead going to focus on how the rules provide an explicit algorithm that students should be able to follow and teach themselves. After all, in the real world, they’ll be expected to learn complicated tasks all the time; they need to develop the ability to teach themselves. (In addition, of course, algorithms are fundamental mathematically; we’ll talk about that, too.)
Set Theory, Semantics and Pragmatics. Now we get to some of the most exciting stuff.
If syntax is a machine that determines what sentences are grammatical16—i.e., it’s a function that takes in a string of words, and returns a boolean (i.e., true or false) for whether the sentence is grammatical (along with a parsing of the sentence)—then semantics is a machine that takes in a grammatical sentence (with a parsing) and returns a boolean for whether that sentence is true in the world. [Update: I should have been a bit more careful with my language here. To be more precise: semantics takes the grammatical sentence and a possible world (such as, although not necessarily, the actual world we live in), and returns whether the sentence is true in that world.] For example, if I say, “Blue the is sky,” the syntax machine will say the sentence is ungrammatical; if I say, “The sky is green,” the syntax machine will say the sentence is grammatical, but the semantics machine will say the sentence is false (because the sky isn’t green); and if I say, “The sky is blue,” the syntax machine will say the sentence is grammatical, and the semantics machine will say the sentence is true. [Update: That holds if we give the semantics machine our actual world. If we give the semantics machine a world with green skies, these last two would switch.]
Then there’s a third step. It turns out that the truth-conditional meaning the semantics machine tells us—that is, the determination whether or not the sentence is true in this world—isn’t precisely the same as the actual meaning as we interpret it. Instead, there’s a third process, the pragmatics machine, that takes the truth-conditional meaning of a sentence and interprets it based on the context it was said in. For example, if I say, “Barack Obama has one daughter,” the best way to analyze things is to say that the sentence is “literally” (i.e., truth-conditionally) true, because he does have a daughter. In fact, of course, he has two daughters. Therefore, any reasonable speaker would say instead the sentence “Barack Obama has two daughters.” The pragmatics machine, knowing this fact about the reasonableness of human speakers, therefore interprets the sentence “Barack Obama has one daughter” as meaning “Barack Obama has one daughter and does not have more than one daughter.”17 Semanticists (this term subsumes those who study pragmatics) apply similar reasoning to a wide range of linguistic constructions. For example, it turns out that the best way to analyze the meaning of “or” is to treat its truth-conditional meaning as the inclusive “or”, and use the pragmatics machinery to add the exclusive sense in certain contexts. (Thus, a dose of education in pragmatics would I hope cure mathematicians—precocious seven-year-olds I fear are helpless—of jokes depending exclusively on literal interpretations of truth-conditions.)
Semantics is essentially applied (elementary) set theory, which is a natural topic to cover in the intro-to-pure-math section.18 I’m particularly excited to teach enough of this to be able to explain the difference between restrictive and non-restrictive clauses.
Pragmatics has rather more subtle prerequisites: one needs both (a) an intellectual sophistication (in understanding abstraction and the scientific process19) to understand the division of labor between semantics and pragmatics (note how useful it was for me to describe things in terms of boolean functions,20 which I’ll make sure to explain); and (b) an understanding of the human reasoning that underlies Gricean logic. Prerequisite (a) is going to be hard—I might have to resort to my professorly ethos to convince them, rather than use logos and teach them about this scientific process, although I’ll try—but I think prerequisite (b) is feasible. This Gricean logic is beautiful and powerful21; it’s changed the way I think about almost everything I read and write. I hope this perspective can help students not just in their writing but in other areas as well; for example, there are similarities between explicitly Gricean reasoning and the logic of literary criticism (cf. Aristotle on plausibility) and the law (what would a reasonable person do?).
Of course, there are challenges in teaching both semantics and pragmatics. I’ll only be able to teach a small part of these theories,22 and I’ll have to think carefully about how I can present it without offering too much of the formal apparatus. I’d love to hear suggestions about this.
Information theory and efficiency. There’s a lot of interesting stuff about the various ways in which a language can be “efficient.” I’ll go from information theory in mathematics to ambiguity in linguistics (e.g., with garden path sentences) and finally to discussions about how we can communicate most effectively as writers. (For example, I might have students take a look at the work of Jose Saramago to think about the role punctuation plays in writing. I might also look at some excerpts from the Oulipo school; please let me know if you have suggestions!)
Meanwhile, we’ll be talking more specifically about writing. (I want to assure any of the powers-that-be who are sanctioning this course: don’t worry, this is still very much a writing course!) In addition to the topics above (the boundary between this set of bullets and the previous one is fluid), some things I’ll be talking about are:
- Norms and Prescriptivism. Obviously, a discussion about prescriptivism is necessary whenever the term “grammar” is being used. At some, probably very early on, I’ll have this discussion, where I’ll debunk Strunk and White and other naive prescriptivists, explain what descriptive grammar is, and offer a reasonable middle ground between naive prescriptivism and a grammatical relativism that ignores the norms of standard written English. A key goal is that students develop a linguistic awareness, so that they can choose which rules to follow and which to break, depending on the context.
- Exposition. Throughout the course, a lot of our focus will be on what makes good exposition and explanation. In the readings, I’m going to introduce students to many examples of good exposition, and I’m going to design many of the writing assignments so that they will require students to explain mathematical and other concepts clearly. (Note, of course, that writing good exposition—just like teaching effectively or giving a good talk—requires an understanding of one’s audience. I’ll definitely be talking a lot about that.)
- Cadence. One of the challenges in writing is to move from the local structure of the sentence to the “semi-local” structure of the paragraph: how do you string sentences together so that they flow naturally? Perhaps the only way to learn to do this is to read widely and develop an ear for the cadence of writing. At the very least, I’ll try to help students develop this ear by encouraging them to read widely. I hope, though, that I might be able to offer some suggestions along these lines. (Advice welcome!)
- Parallelism and Structure. Both in thinking about cadence, and in thinking about how to organize the larger scale of a paper, parallelism and structure are crucial. I’m not sure how exactly I’m going to do this, but I’d like to have some discussion of this, ideally drawing on some of our mathematical concepts (e.g., hierarchies, graph theory).
- Rhetoric. I’m definitely going to talk about ethos, logos, and pathos, one of my favorite topics from my own freshman composition course many years ago. (I’ll point out how it even applies in a field like math; good math exposition is not just a matter of logos.)
Obviously, there’s a lot here. Only partially tongue-in-cheek, I’d say that I’ll be happy if I can get my students just to know where to put their commas. (I can say this because if they fully understand this they’ve understood recursion with its connection to parsing and set theory with its connection to semantics, two of the most difficult concepts in the course.23)
The actual schedule won’t go in this precise order—I’ll be mixing math, linguistics, and writing throughout the semester, and there will be a few additional topics that don’t fit perfectly into the typology I’ve suggested.24 I’ve started working on the syllabus, trying to fit all I want to cover within the 27 scheduled classes, and I have an extremely tentative version of the schedule here if you’re curious. (Feedback from people who’ve taught these things before is welcome!)
I’m going to give a lot of writing assignments, both informal and formal. There will be a focus on summary, exposition, explanation, and analysis. I’ll start with assignments where students try to explain basic ideas; for example, after I introduce the concept of mathematical proof, I think I’ll have students try to write out an exposition of a proof they’ve mastered. At some point, after they’ve been exposed to logic (and perhaps probability), I’ll have them write an article pointing out fallacious reasoning in something they’ve read. (They might, for example, respond to an article that draws inappropriate conclusions about nutrition by conflating correlation with causation.) This more advanced assignment requires students to summarize someone else’s work, explain the mathematical concepts necessary to expose the error, and then (ideally) analyze the issue and discuss broader implications.
The final project in the class will be a paper about an application of mathematical thinking to the real world, based on the students’ own interests.25 For example, a student with a music background might write about how tonal harmony is based on the same basic mathematical ideas (formal grammars) that underpin the linguistics that we’ve studied. First, they’d learn about and research the topic on their own (I would of course be there to guide them). Then, as an initial writing project, they’d write up an exposition of these ideas for a generally-educated audience. Finally, they’d expand upon their exposition to write a longer paper where they’d discuss broader implications. (For example, the student writing about music theory might discuss how music theory could be used to teach mathematical concepts in high school, or discuss what these concepts mean for computer-aided composition.) After this main project is done, in the tail end of the course, I think I’ll also have them present their work in a few variations: an elevator pitch of their idea, a short presentation of their ideas to the class, and perhaps some form of shorter writing (maybe an op-ed discussing the social implications of their project, which would require much more concise exposition than their papers).
I have a lot of ideas for topics students could choose for the final project.26 (I’d welcome more suggestions from readers.) Of course, I hope students will explore things and find topics of their own interest. Some of the topics I can think of are:
- Benford’s law (with implications, e.g., in forensic accounting)
- Social Choice Theory and Mechanism Design (with many implications in politics, sports, and other areas)
- Music Theory
- Machine Learning (with applications to Netflix, Facebook, and machine-grading of essays, and so many other areas)
- Economic Theory (with so many applications in so many areas)
- Game Theory (with applications to many areas not just in economics but in sports and elsewhere in society)
- Probability (with applications to medicine, science, and so many other areas)
- Information Theory
- Finance (e.g., the corporate asset pricing model, the efficient market, arguments for index funds, etc.)
- Computability theorems in logic (e.g., P vs NP, Church’s thesis, Godel’s Theorems, all with many implications, etc.)
- Natural Language Processing (e.g., how an iPhone autocorrects spelling, or machine translation)
- Modal Logic
- “Digital Humanities” corpus-based approaches to studying literature
- More topics in linguistics (e.g., language change, dialect variation)
- Math Education (e.g., discussing the content in secondary curricula)
- Debates about our national debt (the controversy over Reinhart-Rogoff?)
- Big Data (in computer science, in biology)
- Law (e.g., the logic of legal reasoning, the use of probability in the courtroom)
- Statistics and Education (debates over psychometrics, testing in schools, etc.)
- Probability/Statistics and Grades (e.g., what does grade inflation mean—for one take see Ellenberg; perhaps tie in Spence’s theory of education and signaling?)
The only way to learn to write is to read many examples of good writing.27 The philosophy behind my reading selections will be that students should see examples of the types of writing I expect them to produce. Therefore, we’ll be reading plenty of expositions of math (of course), as well as more analytical articles that discuss implications of mathematical concepts in the world.
In addition, for the more technical material in linguistics and math (as well as discussions of the application of these areas to writing), I’ll be writing lecture notes. This is in part because there probably don’t exist good references for the specific topics I’m covering that (a) assume minimal background; (b) take this course’s philosophical approach of mixing math, linguistics, and writing; and (c) work within the narrow scope and time frame I have for many of these topics. (If you have suggestions, please do let me know.) But it’s also because I hope eventually, if this experiment works, to write a textbook based on this course. I’ll be posting many these of these notes here over the summer and fall.
Finally, I’m going to require that students do extramural reading outside of the assigned readings for the course, again because they need to see as many examples of good writing as possible. My dream would be that each student regularly reads, say, the New Yorker, the Economist, the New York Times, and Slate. Obviously, I can’t require that. What I am going to do is require them to read a certain amount of high-quality prose each week outside the syllabus, and send me an email describing what they’ve read and sharing any thoughts they had. Sometimes, I’ll then have them incorporate what we’ve learned in class in their responses; for example, after we discuss rhetoric, I might ask them to observe some rhetorical techniques in their readings. In the beginning weeks, I’ll probably guide students to sources of excellent prose, including those mentioned above,28 but I hope that they’ll eventually explore based on their own interests.29 (Ideally, this might help them discover topics for their papers—both for the paper where they have to critique fallacious reasoning and for their final project.) If anyone has tried something like this in the past and has any suggestions—there are the obvious issues of how to assign a reasonable amount and have students do it without it becoming a burden—I would love to hear. (One thing I’m considering doing—advice welcome—is allowing some, though not all, of this extra reading to be from their other classes.)
I would love suggestions for readings. I already have some ideas, but I won’t prime you by listing them. For all of these, I’m looking for things that aren’t too long, ideally in the range 0-15 pages. Here’s a list of some types of readings I’m looking for (the very tentative schedule is here if you want to see how these might fit in):
- Above all, I’d love to have as many examples as I can of expositions long and short of the sort of “Math and the World” topics I’m thinking about—applications of mathematical thinking to other areas (e.g., social choice theory, probability, the math behind linguistics, the math behind finance, etc.). I’d like examples both of expositions that simply explain these concepts and also more analytical articles that discuss social or other implications. (Has anyone looked at Nate Silver’s book? Is there a chapter there that would be good? Or from similar other books?)
- essays on the beauty of math/introducing pure math
- self-contained expositions of accessible and beautiful proofs
- self-contained introductions to propositional logic (truth tables) and predicate logic (syllogisms?), each of which I’m planning on spending only one day on
- examples of articles that commit logical fallacies (ideally prominent or socially significant ones), to give students examples they can write critiques of
- articles that point out logical fallacies in arguments other people have made
- an introduction to logos, ethos, and pathos from rhetoric
- reasonable takedowns of naive prescriptivism that still acknowledge a role for the conventions of standard written English
- resources (not just articles—also books that I can put on reserve or websites) that offer good advice on style and mechanics. (Obviously, I’m looking things that are as far from terribly problematic Strunk and White as possible.)
- self-contained introductions to basic probability theory
- self-contained introductions to elementary set theory
- are there self-contained introductions to pragmatics? (Perhaps I can assign Grice’s original article?)
- examples of great short writing tout court. On one day, I’m going to talk about style. There’s more to great writing than just following rules. I want to show them some fun pieces by writers with a wide variety of styles. (For example, I might show them Anthony Lane, HL Mencken, Joan Didion, and William Deresiewicz.) If you have suggestions, let me know!
- a short introduction to information theory
- accessible introductions to legal reasoning
- accessible introductions to economic reasoning
- an accessible analytic philosophy paper that makes a clear argument from a set of axioms/premises. (That is, a short paper that’s really explicit in its use of basic logic and reasoning, as is often discussed in intro philosophy courses, where students then are asked to reconstruct the argument.)
The syllabus is far from finalized, so there will be readings that don’t fit into these categories. If there’s some article that you think might work well in this course, even if it doesn’t fit into this list, please let me know!
Student Backgrounds: A Challenge and an Opportunity
As I mentioned towards the beginning, I strongly believe that pure math can and should be taught to a wide range of students, not just those with strong secondary math backgrounds. Indeed, I think many students’ distaste for math comes quite reasonably from their boredom and frustration at the stultifying exercises of high school. I imagine I’ll get students from a range of backgrounds in this course. Because the math content has few prerequisites, I hope it will be accessible to students with less math experience, and because the content is different from most undergraduate math courses, I think it will still engage students with more math experience. Indeed, I hope the variety of backgrounds can be an advantage. I’m going to encourage students to work together not just in editing their writing (as is common in writing courses), but also in helping each other understand the technical material; more advanced students will benefit tremendously from helping explain concepts to students who might be struggling.
I also hope that this course might introduce math as a major and career path to students who would not otherwise have considered it. We don’t do a good job in math of attracting a wide range of students. Much of the problem has to do with students never being exposed to the beauties of pure math in the first place. I hope my course can be a very small step in helping this. But even those who get such exposure often become disenchanted. One reason for this, I think, is the intimidating nature of the first proofs-based class that students see in college.30 The collaborative environment of this class, where evaluation is based on writing and not tests or homework and where the material and emphasis is unusual, will I hope create an environment where students can feel comfortable enough to discover a love for math.
Finally, I hope that my perspective on grammar and writing will encourage students to feel more comfortable with writing and the humanities. I remember being frustrated by grammar days in middle school English—by restrictive clauses taught without set theory—because it didn’t make any sense. Perhaps students, especially those with a more mathematical bent, will appreciate the more technical approach I’ll adopt to these linguistic and grammatical issues.
Even though I’m a mathematician, I think the humanities are the crown of human achievement. We do an injustice to our students if we do not try and instill an appreciation if not a love for the great works of literature. In my course, I won’t be able to do anything directly besides exhorting my students with every spare breath to read Tolstoy and Dostoevsky. But I hope at least that by introducing students to some of the beauties of language and math I might inspire a few at least to drink from this great well of human creation.
Conclusion and Future Steps
As I said at the beginning, this is an experiment; we’ll see how it turns out. In December, after the course is finished, I’ll follow up here with some thoughts on the experience. Meanwhile, as I prepare the course, I’d welcome any suggestions you might have—especially over the next few weeks, as I finalize the readings and syllabus. Through the summer and fall, I’ll be posting here about some of the topics I’m going to cover; comments about these expositions would be very helpful, so I can improve them before I show them to my students.
I’m graduating (fingers crossed!) this year, so I’ll probably only be able to teach this course once at Michigan, but I hope to continue working on these educational ideas. (As I promised above, at some point I’m going to write a manifesto about this broader educational philosophy.) I’d love to hear from other educators who share or are interested in these ideas. Please be in touch—I promise I’ll keep adding things here!
- Why as a math Ph.D. student can I do this? I’m doing this as part of a program offered through the University of Michigan Sweetland Writing Center that allows Ph.D. students in any field to teach freshman writing courses about a topic related to their discipline. (To any grad students at UM reading this who are interested in learning more: feel free to contact me for more about the program.) The official course title is “English 125: Language, Logic and Information: Using Mathematics to Understand Writing, Communication and Argument.” (I rather like the title “Math, Writing, and the World” that I’ve chosen for this blog post, though.) I’ve posted the course description I sent to the university on the homepage I’ve made for the course, but the description I’m giving here is much more in-depth. ↩
- Some might note that I haven’t referred to the obvious applications of math in the physical sciences and engineering. Such applications—usually based on calculus—have long been widely appreciated in our educational system. (Indeed, they determine our introductory math curricula to an unhealthy degree. Without such pressures, perhaps more math departments would offer proofs-based introductions to pure math as their primary introductory offerings, instead of applications-based calculus.) The sorts of things I’m thinking about tend to have a much different flavor—discrete math rather than calculus—than physics-based applications; are much more accessible without significant math background; and, I would argue, are much more important for the average educated adult to be conversant in. (Lest anyone accuse me of bias against calculus or physics, I should note that my research area is PDE of water wave equations. I love mathematical analysis, but I don’t think it’s what we should be teaching the average student. Do we need our doctors, congressmen, and journalists to understand mechanics or probability more? Ideally, we would expose our students to some calculus, but the focus would be different—the goal would be understanding the difference between the national debt and the national deficit, or the intuition behind continuous probability, not how to expand a Taylor series or calculate the work to lift an object.) [Update: I should make clear that teaching scientific thinking and the scientific method is crucially important, too, and physics certainly can and should be a part of this. What I’m saying is that, given very finite constraints of student time, we need to think more about what content we’re focusing on.] ↩
- In fact, philosophy and computer science deserve credit for picking up the slack where mathematics has failed, by offering accessible intro-proofs courses (logic in philosophy and discrete mathematics in computer science). ↩
- As I noted in a previous post, pure math isn’t just about proof: less rigorous heuristics and intuitions are crucial. To a first approximation, though, we can focus on proof (along with abstraction and beauty) as the salient difference between pure math and what’s taught in schools. (It would be interesting, of course, to think about incorporating these more heuristic mathematical processes into math education.) ↩
- There is a long history of this. In the 1960s at Dartmouth, my father took an Introduction to Finite Mathematics course designed for students with minimal backgrounds. (The course was taught by the logician John Kemeny, who is also known for introducing the BASIC language for teaching computer science and programming and went on to become Dartmouth’s president.) A more recent example is Paul Lockhart, whose essay A Mathematician’s Lament should be required reading. (Lockhart has since expanded the essay into a book, which I haven’t had a chance to read yet.) ↩
Specifically, I’m referring to theoretical linguists from the broadly Chomskyan program I was brought up in. I say “broadly” because Stanford, where I was originally a linguistics major before shifting to math, had a syntax faculty consisting mostly of former students of Chomsky who had shifted away from his specific program but still did work within the broad generative program that he had envisioned. For more on this, see Mark C. Baker, “Syntax,” in The Handbook of Linguistics, p. 267:
The broadly Chomskyan approach to syntax is often called formal or generative: “formal” because it uses mathematical style formalisms and definitions in presenting its analysis, such as the recursive rules above; “generative” because it seeks to explicate how fluent speakers can generate new well-formed expressions of a language. Many current syntacticians still accept these labels for their work, while others have reacted against them and what they see as some deficiencies and excesses of this approach. However, it seems fair to say that even those that react most loudly against the Chomskyan approach have been positively influenced by its most basic ethos and by some of what has been learned from that perspective.
By focusing on the broadly Chomskyan tradition within theoretical linguistics, I’m ignoring functionalist approaches like M.A.K. Halliday’s Systemic Functional Grammar. I know very little about the functionalist tradition, and would love to learn more. From the little I do know (based on some very cursory reading in the past week), it seems that functionalist approaches would complement the sorts of things I’m planning on doing. The core ideas that I want to teach my students are the concepts of recursion/generativity, parsing, ambiguity, constituency, truth-conditional semantics, Gricean implicature, etc. These naturally fall out of the broadly Chomskyan program—and they are naturally linked with the math content I’ll be teaching. The functionalist perspective could be very useful, especially in thinking about style and discourse. But, given my background and the time constraints of the course, I probably won’t be able to include much from this approach.
I should note, also, that I know little about the applied linguistics world, which I imagine has been more interested in these issues. Furthermore, when I’m talking about (broadly Chomskyan) theoretical linguistics, my focus is on syntax, semantics, and pragmatics. This focus reflects my interests and background; I don’t mean to neglect or disrespect the other important subfields! ↩
- It is of course important to keep speaking out against the more foolish aspects of prescriptivism. Linguists (for example, Geoffrey Pullum) have done well to make many of these points. (See also Anne Curzan’s article Says Who in PMLA (behind paywall, though accessible through most university libraries) for a good take from the English Department world.) I think one has to have the appropriate, nuanced view. I don’t have space to address that here, but I want to make sure my students get it, so I will be writing my take at some point over the summer. (For a start, see my (slightly ranting) recent blog post on John McPhee and the Irregular Restrictive Which, especially some of the footnotes.) ↩
- See his abstract, his slides, and his blog post about the talk. ↩
- In fact, I would suggest an even more expansive program than just using linguistics to teach freshman writing. If you’re going to teach linguistics in order to teach writing, then why not also use it to teach students about analytical and scientific thinking? The sort of thinking one does, say, in examining minimal pairs is precisely the sort of analysis we should be teaching students. And it doesn’t have to start in college, either! I was lucky enough to take a theoretical linguistics course during a summer program when I was in high school (with the wonderful Gulsat Aygen), and I’ve thought ever since that this should be part of secondary education. There were some studies in the 1980s by Fabb (Linguistics for Ten Year Olds) and by Honda and O’Neil (Triggering Science-Forming Capacity Through Linguistic Inquiry) from MIT about teaching theoretical linguistics in middle and high schools, but I’m not aware of much else. (At least I wasn’t when I looked into it in 2004; perhaps there have been developments since then. It seems that Honda and O’Neil now have a book on this.) ↩
- This has been done before, in a freshman seminar Steven Abney teaches here at Michigan. ↩
- As should be clear by now, I think it’s mostly a matter of labels and framing whether you characterize what I’m doing as “linguistics” or “math.” Of course, the choice of label and frame matters, and there are reasons why I’m choosing to frame things mostly in terms of “math.” (Part of it has to do with my position—I’m in a math department (although teaching a course in an English department), so it makes sense to call the course something that coincides with my academic bona fides. Part of it has to do with the perception I want students to have; I’m trying to make “grammar” sound less scary by phrasing it as math. (Of course, one could do the reverse, too.) And part of it is because of my goal to show that math is much more than what most people think it is.) ↩
- I can also now explain the cryptic blog category Gamut at the top of this and other posts: this is the term I’ve used to describe this program. The name is, in part, an homage to LTF Gamut, the collective pseudonym of a group of Dutch linguists and logicians who authored the wonderful two-volume textbook Language, Logic, and Meaning. (The astute reader will also observe homage in the official title I chose for the course. In fact, I’d misremembered the title of Gamut’s book; only now do I see that the third noun wasn’t “information.”) It also evokes both the breadth of the disciplines it touches and, by its etymology, music theory—a paradigmatic example of the “Gamutian” philosophy, since tonal harmony is a formal grammar just like natural language. Both this program and the name grew out of many discussions about these ideas with my good friend Alex Romanczuk, to whom I owe a tremendous debt (and who deserves none of the blame for anything I have written here). And so, I promise you, dear reader, the “Gamut Manifesto,” sometime soon, I hope. ↩
- This language might suggest that what I’ll be doing is simply lecturing to students. I’ll continue to use such language, but I’d like to assure readers that almost all of the teaching will be interactive. (There’s of course much more to say on interactive classrooms, flipped classrooms, etc. But my course is experimental enough in content that I won’t—intentionally at least—be doing many experiments in form.) ↩
- Perhaps games are the most natural way to introduce some of these ideas? The advantage of using games is that they avoid the bad associations people have with both math and grammar. That might not be ideal for this course, but in other situations—for example, in applying these ideas in secondary education—this could be very useful. ↩
- Obviously, there are a lot of subtleties. But, to a first approximation, the comma in contemporary written (American) English is used primarily as a recursive/parenthetical marker, with the notable exception of lists. ↩
- I’m using this in the non-prescriptive sense. I will of course discuss these matters. ↩
- To any semanticists out there: It’s been a while since I’ve studied this, so I might be a bit rusty on some of the details, and I’m definitely not up with the latest theoretical debates. I’m looking forward to brushing up on my semantics and pragmatics this summer as I prepare the course. ↩
- The details are in the rather formally imposing Montague grammar. It’s been a while since I’ve looked at this (indeed, I studied it before I had any of that mystical quantity “mathematical maturity”). Obviously, I won’t include all of that apparatus here, and will only be able to do a small part of this. ↩
- By this I mean the sort of thinking that allows one to say that the “best” (i.e., most scientifically plausible, by criteria including parsimony, elegance, and other things) theory is one that subdivides things into semantics and pragmatics. ↩
- Math Education Aside: Note, too, how I used the machine metaphor for functions. It would be wonderful if students could learn about functions from this perspective, rather than just thinking of functions in terms of polynomials, etc. After all, if you see things this way, concepts like inversion become so much simpler. Whenever I teach or tutor, I make sure to emphasize the function-as-machine metaphor, drawing it out. I’m always surprised by how few students learned this in high school. ↩
- It has a beautiful self-evidence that reminds me of Darwin’s natural selection and Adam Smith’s invisible hand. ↩
- The subtleties of how pragmatics interacts with restrictiveness are beautiful, but I’m aware that this might be a bit too much to include in this course. ↩
- Cf. this (informal) study by a math professor who found out that understanding the chain rule was by far the best predictor of performance in a calculus class. The point of my tongue-in-cheek remark above is that if students can understand comma placement they’ll have understood a whole lot of stuff in my class. ↩
- For example, I’m planning to have a class on economic and legal reasoning. Far too few students—or educated adults—understand economic reasoning, which explains so much of our world. At least there are econ classes that many undergrads take; almost no one except law students is exposed to legal reasoning, which is a tremendous shame. (This state of affairs obviously is in the legal profession’s economic self-interest.) Sometime soon (I hope!), I’ll write a blog post “Why Everyone Should Study Law—Contract Law” explaining these views. For now, my advice is to do what I did a few years ago, when I was curious about how legal reasoning works: buy a used contract law book (they’re cheap!) and read it. (The one I used was Frier and White.) ↩
- I will have a generous interpretation of the “real world”; if I have a student with a lot of background in pure math who wants to write about the abc conjecture, I’ll probably be fine with it, so long as their project will call upon all the same skills. The point is for students to write about something that excites them. ↩
- Indeed, many (though not all) of these I’ve thought about writing myself, for this blog. A short example that I did write is my post about judgment aggregation. ↩
I’ve always loved this blog post by Jordan Ellenberg, where he discusses an experiment of trying random bad novels to see if he could learn anything from them. It’s interesting to think about learning to write as a formal learning problem (in the sense of machine learning). How much is this supervised learning versus unsupervised learning? What sort input do we give—only good writing, or also bad writing?
Something I’ve often wondered is whether one can create a linguistically sophisticated theory of good writing. Let me explain what I mean. Theoretical linguistics has produced a powerful theory of spoken natural language; the syntax machine can take a string of spoken words and say “yes this is English” or “no this is (syntactic) gobbledygook.” Is there a somewhat formal theory that can explain—at least in part—what “good writing” is? After all, if some humans can learn the patterns that separate good from bad writing, in theory we ought to be able to model what they are doing in their minds. For example, if you move beyond Strunk and White’s idiotic proscription against the passive voice, you’ll see that in specific situations there are often good reasons to use or not to use the passive; perhaps these reasons can be formalized by thinking about things like topicality? Such a theory would require a careful understanding of a lot of things, among them: discourse structure; the pragmatics and semantics of connectives like “and”, “but”, “although”, etc.; metrical and phonological issues as they manifest themselves in writing and reading; punctuation; how we parse complicated sentences; and the historical and idiomatic associations we have with many specific constructions and styles. I don’t know of much work in these issues, at least from the Chomskyan side of linguistics; I’ve heard there is work by people from more functional perspectives, which I will have to check out. (In the seminar I took last semester preparing to teach this course, we read a paper by George Gopen and Judith Swan where they took such an approach. I’m not sure I agree fully with their analysis—I haven’t had the time to grapple with their theories carefully—but this is the sort of thing I’m wondering about.)
Even if there is such a theory, there remains the pedagogical question of whether it’s best to teach it to students, rather than have them discover their own internal theory about what makes good writing, having done lots of reading by themselves. (Indeed, perhaps the reason it was hard for me to read the details of Gopen and Swan’s article was that I’ve already developed my own internal theory for the issues they discussed.) ↩
Some of my high-minded friends have questioned my inclusion of Slate on this august list of periodicals. I’m consistently impressed by the high level of intellectual discourse on Slate. For the purposes of this writing class, and for most (though not all) nonfiction, I subscribe to the “good prose is like a windowpane” theory. Slate meets my standards for good prose. (Indeed, comparing it with the New Republic, which likewise impresses me with its intellectual seriousness, I find myself cringing at the New Republic‘s uneven prose a good deal more often. Perhaps, though, this has something to do with my reading The New Republic on paper and Slate online.) The articles on Slate are a perfect model for the type of writing I want my students to do: clear explanation and exposition coupled with analysis and argument. Besides, unlike these other publications, Slate is free and therefore easy for my students to access.
My one qualm is the slightly left-leaning slant of the publications I’ve mentioned, the Economist excepted. I think it’s important that instructors not impose their views on students, and that they actively encourage their students to hear from new perspectives. (I hasten to add that the left-leaning slant of the publications mentioned does not necessarily represent my political views; I hope that statement is sufficiently vague that nothing can be inferred from it about my actual views, which I’ll continue to obscure.) The reality of arts and letters in America is that much of the best writing is in publications that are predominantly leftist, so it’s not surprising that many of the publications I’m most familiar with lean this way. I’ll definitely add the Wall Street Journal to the list I give my students. The other more right-leaning publications I sometimes read—e.g., Commentary, Andrew Sullivan’s Dish (if you can call that conservative), or Reason—have a more narrow focus. I’d welcome suggestions of high-quality general interest publications that offer a wider range of ideological perspectives. (Yes, of course one can separate the reporting in the Times or the New Yorker from the editorial stance. But when every other New Yorker opens with a Hendrik Hertzberg column, and when Paul Krugman or Maureen Dowd are seen as the face of the NYT, and David Brooks as the token conservative, it sets a certain tone. Of course the same holds for the Journal, which has excellent reporting, not to mention increasingly good arts and culture coverage.) ↩
- In encouraging students to read widely based on their interests, reading that they will mostly do on the internet, I face the challenge of ensuring that they’re looking at carefully written and edited prose. This matters a lot, because I want them to see models of proper mechanics. Unfortunately, the demise of copy-editing makes this a bit of a challenge. I’d be happy if a student interested in sports reads Grantland, but I’d be worried if they read the much sloppier stuff on parts of EPSN. I find a lot that’s well-written on Gawker, but it’s not always perfect. Even on Slate, the terrific Matt Yglesias sometimes could use copy-editing (although perhaps these are quickly corrected and I’m just checking my RSS feed too often). I think what I’ll do is encourage students to read from a variety of sources, and nudge them away from ones that might not be good models for their prose. ↩
- My own college experience confirms this. I had a strong math background from an excellent public high school, yet I felt intimidated when I tried taking Stanford’s honors math sequence, because it seemed like everyone in the class was a “genius” who had spent all their high school summers doing math camps. (The course’s reputation for requiring weekly all-nighters only added to this sensation.) I ended up taking a more circuitous route to a math major, via much less intimidating logic courses, and then some wonderful upper-level courses in the math department that were accessible without the honors sequence. In the end, many of the “geniuses” in the honors sequence dropped out of math, and students like me who took a more unusual route ended up being the ones who succeeded in the major and went on to grad school. ↩