The greatest shortcoming of the human race is our inability to understand the exponential function.
This is some background to a project I’ve been working on that will heat up very soon.
I’ve liked math for as long as I can remember, but in the past few years I’ve gotten very frustrated with my math classes. In high school, the “trajectory” through a math curriculum is very clear (I could argue this isn’t good, but that’s a different discussion)—do algebra, geometry, calculus. Once you study math in college there is much less of a track. Fields like number theory, graph theory, differential equations, linear algebra, and more all grow organically without a clear path from one to the next. They are all linked and overlapping, yet taught as clearly defined blocks of material.
Yet, the classroom structure keeps from high school. This is already a terrible remnant. Lots of students at little desks all facing the same way, waiting for the professor to deliver content from the blackboard (or worse, projector).
This sucks. We can do better. Also: fuck problem sets.
I know, through things I’ve read, moments of personal insight, and research projects I’ve done, that mathematics is an artistic process. Perhaps especially in “pure”/theoretical maths; my choice of applied math is more akin to engineering. But arts and engineering are both creative fields, best taught through hands-on, open-ended project work. Much of this is rehashing A Mathematician’s Apology.
I feel lucky that I have enjoyed math for so long despite it being presented in such a terrible way. I have numerous amazing professors & teachers to thank for this—shout out to Schuette, Ladd, Fred, and Chun at Cranbrook, Chopp, Berry, Murphy, and Abrams at Northwestern’s McCormick, and Sandstede, Gemmer, Maxey, at Brown’s DAM. These mentors have all been excellent, but only a few (Murphy mostly, Gemmer a bit) have actually pushed back against the traditional math classroom structure. This traditional classroom structure is linked with how math is taught and how it is performed at high levels, and has been perpetuated through time.
How do we escape the traditional structure of curriculum linearity, isolated fields, problem sets, and professor lecturing? Higher maths are a tangle of beautiful ideas, all linked together in incredible ways. We need something that encourages a more flexible strategy for discussing mathematical ideas, one with more cross-disciplinary idea mashing, more peer learning, more space for dumb question asking.
The classes I have had that embrace that (and have been some of my favorites) are studio classes. I had a “studio” experience freshman year at Northwestern through some engineering product design classes. More recently at Brown/RISD in an intro to drawing, a metal studio class, and a web design class. These are terrific and allow for creative thought, sharing, discussion, idea generation, and a balance of abstract theoretical thought merged with craft and hand skills—one of my favorite dualities in life. None of these were perfect… HTML Output perhaps the best, but I learned some things.
The core components of this style of classroom (versus lecture, seminar, et. al.), to me, are:
"Studio Applied Math" is my attempt to bring a studio-style classroom to the Division of Applied Math at Brown. I don’t want to sit through any more lectures. I don’t want to do any more problem sets. I will, unfortunately, have to—but this class will help me find the classroom I’ve been looking for.
SAPMA (“Studio APplied MAth”) will be a project under the umbrella of Brown STEAM but is being led by me and my friend in Applied Math Mara Freilich. It will be a GISP (“Group Independent Study Project”) in spring semester 2015 at Brown (assuming our proposal gets accepted). This means it will be a for-credit class in the Applied Math (APMA) department.
Mara and I share our frustration in the limited scope and variety of higher-level math classes. Many of these fields are very new and still in flux—students should be aware of the people and foundational ideas behind them, if not capable of adding their own thoughts to the field. This is where flexibility and exploration and bravery should be embraced.
SAPMA will (fingers crossed) be a group of 8-10 people who meet once a week for a long span of hours (in true studio fashion) to share what they’ve been working on for the week, have a discussion, perhaps a guest lecture, work, and formulate plans for the coming week. Research and background work is expected to be done outside of valuable group time, during when the emphasis should be on critique and discussion. We will have a class reading list, guest lecturers, and potentially a field trip or two. As for assignments, beyond reading, the minimum will be two “projects” throughout the semester—a midterm and a final. The projects include the presentation of all work done, as an individual, group, or entire class!
What do projects look like in applied mathematics, you ask? Much of it can simply be a literature review—research as an accumulation, combination, articulation, and critique of previous ideas. In can also be programming: much of modern day mathematics is computational, and this is inseparable from our goals in the class. Find a question that interests you and chip away at it—test some of the assumptions, explore some answers, see what other people are saying. This is “math” and can be just as aimless and fraught with dead-ends as the archetypal art-making process. And despite our name (and host department), projects from pure math, physics, engineering, and more are by no means ruled out; in fact, we make for a stronger team by having diverse backgrounds, interests, and fields of work.
SAPMA will also be a way to engage with the quantitative departments at Brown in a new, creative, and high-level way. Our roster is mostly senior or junior students, with connections to professors and research facilities. We hope that by providing a space for free, open collaboration and exploring, we can pique the interest of those around us and continue the conversation of how to best teach and practice mathematics.
We recently set a theme for the class: Complex Dynamics. This includes fields such as machine learning, network theory, nonlinear dynamics, and basically anything with complicated sets of interrelated parts. This broad field is very hot these days, with places like the Santa Fe Institute, NICO at Northwestern, C4 at U Wisconsin, CSCS at U Michigan, and CSDC at ASU, to name a few. It is wide enough to accommodate for our diverse team of mathematicians, computer scientists, physicists, and more—while providing a guiding interest that is relevant in our increasingly complex world.
SAPMA has been a long time coming. Something like it has been bopping around my head abstractly since the end of high school and only became concrete when I talked to Bjorn Sandstede, the chair of the APMA department at Brown, about changing the curriculum in Fall 2013. The opportunity presented itself and I’m going for it. I’ve long been passionate about math, education, and the creative process—this will hopefully be a peak of exploration in all three.
I’ll update on this blog with major updates and on the class blog with more frequent, smaller ones. As the summer closes, we are working on making our reading list and starting our GISP application. Energy is high, though, and I’m ecstatic.
Wikipedia page on vorticity has some awesome GIF animations. Also, I should know this stuff already…
I had a late-night snack with a few friends Wednesday night while in the middle of some problem sets due today.
I brought down with me a particular problem on streamfunctions that had been giving me hell all day. My buddy Dan asked what was keeping me from solving it. Did I have all the necessary tools and techniques? I was pretty sure I did. Was I misreading the question? I didn’t think so. My answer was then simply, “I haven’t spent enough time staring at it.”
The next thing he asked was, “what’s the application?” To this I had no real response. The problem had already been solved and the uses had been worked out; I was just following in the footsteps of some long-dead mathematician or physicist. I was just deriving an expression.
On one hand, this is a bit sad. For much of mathematics, understanding the problem means you are 90% of the way to answering it, but I was struggling so much with just the boring mechanics of it—the last 10%. Algebra. Indicial notation. Variable substitution.
As I’ve been mulling it over since however, I feel more comfortable with this. My friends down the road at RISD spend hours in physical labor: throwing pottery, working the loom, blowing the glass, mixing the paint. Their arts education begins from the ground-up mechanics. Their schoolwork for the first 2 years is mostly gruntwork: project after project that seem irrelevant and annoying and tiring. (I am generalizing here from a few conversations with friends.) Yet at the end of their foundations period, they have a diverse skillset, a nimbleness in their medium of choice, and have often developed philosophical ideas integrated in their craft. This is a very Zen in the Art of Archery idea.
I think that this is the same in math. Math students like myself spend years doing fairly mindless work do develop a dexterity with our medium (scribbles on paper). After all this, we find we can express grand ideas in math language once we graduate to a high enough level. The mechanics become second nature so that you can look beyond them, play with them more flexibly, and begin making more meaningful work. It’s a “you have to know the rules before you can break ‘em” mentality.
So I’ll stare a little bit more at the problem and not worry about any application… I like to think my brain is making the necessary connections and my mastery of medium will come.
To roughly quote my applied math professor today: “Mathematics is a game of twisting problems we don’t know how to solve until they look like easy ones we can solve. Then we prove or pretend they’re the same thing.”
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