What would math class look like if it were a fine art?
Wahoo Public High School in Wahoo, Nebraska is by all accounts a school with ample opportunities for their students. They have the standard array of English, history, and science courses, with enough money left over for band, drama, fine arts, many sports teams, and a couple of very surprising electives (zoology, web design, and fashion design to name a few, see their current curriculum guide for the whole list).
And Wahoo HS has given some truly unique opportunities for their students to do things like watch a live heart surgery and visit the video control room of the Ralston Arena event center. Wahoo seems to treat its teachers right, too: in 2008 they sent a group of teachers to DC to learn more about its history.
I don’t bring all this up just to praise Wahoo HS for its quality (though they clearly deserve praise), but also to use it as a platform for a thought experiment: what would math class look like if it were considered a fine art instead of a science? The claim that this even makes sense is a common one from mathematicians. They say that doing mathematics involves a lot more creativity than is present in the majority of high schools, and it’s a disservice to students to misguide them as to what mathematics is all about by focusing on memorizing facts. A good school like Wahoo gives us a nice platform to explore the possibilities.
Let’s start by comparing two Art and Math courses at WHS by their respective syllabi. The first is Pre-calculus with trigonometry, and the second is Art 1. These syllabi seem to be standard as far as course syllabi go at WHS.
Though the syllabus is a minuscule part of the learning experience in a class, they do give insight into the attitudes of the administration and/or the teachers. I want to highlight the attitudes of the art class, which is more or less what one expects from an art class. The syllabus states a student only needs a pencil, eraser, and a creative mind (sounds familiar to mathematicians). The stated “philosophy” of the course is to be “free and creative” while having the opportunity to try a variety of media and understand the principles of design. Here are the grading standards for Art 1:
Note the key words: originality, creativity, and “piece of work.” It’s clear that this class is about creating something that displays your understanding of craft and technique.
The math class, on the other hand, begins with this list of expectations:
“Be prepared,” “demonstrate proficiency,” and “respond appropriately.” The effect is clear: math is careening toward you and you should brace for impact. The syllabus continues by outlining an exact day count to spend on each topic throughout the year, ranging from “analytic trigonometry” (15 days) to “practice for ACT and NeSA test” (10 days). No opportunities, no exploration, no originality or creativity. You must memorize and practice to meet standards and prepare for a college course you are unlikely to need or use.
This course follows the same path that most high school trigonometry courses follow. And if it is taught in the same way that mine was (and I went to a comparably well-off public school), then even the best students leave pre-calculus with plenty of rapid-fire mechanical skill but without doing much mathematics, if any. This is what many think develops critical thinking skills, but it is unfortunately the mathematical analogue of memorizing dates in history (useful, but far from the core of historical analysis). In fact, most mathematicians have relatively poor mechanical abilities. Fine details are something that we accept takes time and deliberation to get right, and I personally save it for the quiet days where I can devote a few hours to crossing all my t’s (and after recording the details, I promptly forget them).
Before I go on I want to make something clear: I’m not blaming the teacher of this course. I imagine whoever teaches this pre-calculus course busts his or her ass off six days a week to get these kids to learn anything. The teacher is probably so beleaguered by students, administrators, tests, and standards, that we can expect little else. No, my problem is not with the dedicated teacher, but with the content and organization that has been undoubtedly imposed on them. It’s not hard to find specific examples. Lacking a directive to force feed students information about secant, cosecant, and cotangent, every sane math teacher would completely obliterate them from the menu. They were invented as a shorthand for navigation books hundreds of years in the past and have yet to find a reason to be in our classrooms.
What constitutes a good mathematical curriculum is a whole can of worms I don’t want to open right now, because it would detract form the purpose of this thought experiment: to see what math class would be like if it were treated as a fine art. Indeed, I posit that most issues with dense, overwhelming, pointless curricula would be completely bypassed in the scenario I am about to propose!
But before I do, the reader needs to understand that mathematics is not about factoring polynomials or computing derivatives. Mathematics is first and foremost about crafting arguments to explain why patterns behave the way they do. This is also known as the process of conjecture and proof, though the word “proof” in high school mathematics has been largely hijacked to mean something different. If we as a society value “mathematical thinking skills” and want to teach that to our students, then we absolutely must make conjecture and proof the primary focus of secondary school mathematics education. And there is no precise formula one can follow to find beautiful proofs. They must be discovered (often with a lot of random guessing and mistakes), and revised until they are as pristine and logically airtight as one can make them. Mathematicians understand this at a deep level; it is a personal journey that doubles as weightlifting for your brain. As Paul Lockhart writes addressing the beginning mathematics student,
What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty. Yes, your ideas won’t work. Yes, your intuition will be flawed. Welcome to the club! I have a dozen bad ideas a day and so does every other mathematician. … The real difference between you and more experienced mathematicians is that we’ve seen a lot more ways that we can fool ourselves. … As you do more mathematics, you will literally get smarter. Your logical reasoning will become tighter, and you will begin to develop a mathematical “nose.” You will learn to be suspicious, to sense that some important details have been glossed over. So let that happen.
So with that understanding: how might we write a syllabus (and in doing so, organize a course) if it were modeled on WHS’s Art 1 rather than its current model? Here’s my attempt:
Syllabus for High School Math 1 #
Purpose: High School Math 1 is a course that provides an introduction to mathematical reasoning through a multi-topic experience. Students will learn and apply the principles of argument to produce creative, aesthetically pleasing proofs that reflect their understanding of mathematical patterns.
Materials: You will only need a pencil, eraser, and your creative mind.
Daily Expectations: My expectations of students are that they use their class time wisely and work on the project at hand. They should be open to the critiques and ideas of their fellow classmates, with whom they will discuss their proofs and conjectures.
Grading: Your grade in math will come from the following:
- Your finished pieces of work (portfolio) and how they reflect the principles of argument
- Originality of your work (collaboration allowed)
- Creativity of your work
- Being neat and careful
- Using your class time wisely
- Going above and beyond the questions presented in class (forming your own questions)
- Using your abilities to the fullest
- Provide the opportunity to learn and understand the principles of beautiful arguments.
- Provide the opportunity to explore a variety of mathematical topics.
- Provide the opportunity to critique and be critiqued in a constructive manner.
- Provide an atmosphere where mistakes are common, and over time develop a sense of neatness and carefulness (make a mess, and revise, revise, revise!).
- Provide the opportunity to be free and creative.
As you can see, I did little beside change the art-specific words to math-specific ones and add some words on discussion, critique, and learning from mistakes.
As wonderful as this sounds to me, I can already hear the protests. How can we just let students sit around and daydream about patterns? There are some basic things they just need to know and be taught!
Well, Art 1 is not without lecture or instruction. The syllabus has a long list of topics relating to line, texture, shape, perspective, color, that are driven by example (drawing a paper bag for texture, etc.). Math teachers will produce their own curriculum in much the same way: showcasing beautiful arguments and using them as a platform to discuss technique that students can use in their own compositions. After all, a teacher who has studied mathematics for at least eight years must certainly have a whole treasure trove of puzzles, paradoxes, and favorite proofs to share. If not (if a teacher has little or no experience reading, critiquing, or creating original mathematics), then I question whether they’re fit to teach. Was there ever a high school art teacher without a few dozen favorite works of art or who has never painted an original piece? I’m not talking about important mathematics, mind you. It can be totally insignificant! But a teacher who has never even come up with their own question and tried to answer it cannot expect to teach such modes of thinking. Just as a music teacher who has never hummed an original tune cannot expect to teach improvisational jazz.
It doesn’t take years of experience to start doing original mathematics, and I’ve seen many students go from a general disinterest in mathematics to conjecturing and attempting proofs over the course of an hour. How is it that I see so many bright young students get interested in mathematics so quickly? It’s not a bugle trumpeting my own teaching accomplishments but a testament to the universal appeal of mathematical inquiry.
I can further imagine the argument that mathematical ability is too variable to have everyone go through such a course. But even if you really mean that with respect to the sort of mathematical ability I describe (conjecture and proof, to which again I’d argue almost no students have experience), then it’s still no different from any other subject. Some students unavoidably have more experience in writing or dance or science and that’s okay. The same problem even plagues Art 1 (if you believe “some people just can’t draw”), and art teachers readily compensate for it! Why can’t we do the same thing for mathematics?
One might question: how can we expect mere students who are supposed to enter knowing nothing about mathematics, to craft deep logical arguments, to debate intelligently, and to discover beauty in mathematics? Again, we already do exactly this for almost every single subject! Students are regularly expected to fashion arguments in essays, to comprehend some depth in literature and art, and to understand and debate the politics of governments throughout history. But more to the point, a student’s first few years of proofs are guaranteed be trivial to a seasoned mathematician just as a high school debate team is nothing in comparison to the Supreme Court. But to constrain their mathematical education from a blanket assumption that important mathematics is beyond their abilities is as insulting and illogical as refusing to teach them to write because they will not become a Hemingway. The point of a mathematics education is to learn to think critically and reason, not to prepare you for a Fields Medal.
And what about standards? And preparing the students for college calculus? Every math professor in college already knows that the students coming into freshman calculus can hardly do algebra. As a teacher of these same students, I would trade their shaky hold on algebra for even a single year of developing critical thinking skills. The truth is that you can learn algebra and calculus exponentially faster once you have the critical thinking part chugging away. And maybe in a world like I propose we wouldn’t have to make college calculus so gruelingly awful either.
I’m not demanding a sweeping reform to make all mathematics education like this. That’s not feasible and would likely be implemented just as poorly as current mathematics standards are. I understand that we need to be sensitive to the students and the learning environment.
No, I did this thought experiment so that readers might question the assumptions of the current establishment. Such experiments are exactly what mathematically-minded folks do with any problem: assume some hypothesis is true (mathematics is treated like a fine art), and poke it to see what can happen. I like the outcome, and even if I (a hypothetical high school math teacher using the standard pre-calculus syllabus) couldn’t make mathematics a fine art officially, I would definitely reconsider my course organization. Because that is the kind of education I want for my kids, where they’re not fearful and loathsome of something that isn’t math, but rather have enough control of their own mathematical destiny to grow and truly learn.
Have other ideas (critiques, suggestions, counterexamples) you think I should hear? Find me on Twitter @MathProgramming.