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Advice for calculus students November 3, 2010

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I am now a TA holding office hours for some calculus classes, and I actually enjoy it a lot. I’m generally impressed with the students. General advice that ought to hold if you’re taking an intro calc class:

1. Don’t freak out because a problem is taking a long time and seems unfairly hard. You’ll say “There must be some mistake!” No, there isn’t. This isn’t high school; you’re being taught by professors, who don’t always know how to gauge the right difficulty level. They do tend to underestimate how long it takes to finish your homework. Also, calculus is actually harder than high school math. If you have to go through a lot of computations and false steps — don’t worry! That’s what math is actually like!

2. I do not have magical TA superpowers of Mathematica. Most of the time, if you ask me for help, I’m going to look at the example on your worksheet and look in the help documentation. You could do that too! The xkcd Tech Support Cheat Sheet is relevant here.

3. L’Hopital’s rule is your friend. Seriously. So is big-O notation. These will save your bacon.

4. 90% of mistakes in multivariable calc result from not drawing pictures. Draw a picture. You are never too cool to draw a picture.

The internet is really hard October 4, 2010

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I had dinner with some computer science students last night that involved this exchange:

CS kid 1: “Oh, I’m no good at math. You math people are too fancy for me.”
Me: “Well, you’re too fancy for me. I don’t even know how the internet works!”
CS kid 1: “Um, well…”
CS kid 2: “The internet is really hard.”
CS kid 1: “Most computer scientists don’t know how it works.”
CS kid 2: “I only know about it because that’s what I research.”
Me: “So I’m not just dumb?”
CS kid 1: “Nope. The internet is really hard.”

It’s been on my mind since I read this Douglas Rushkoff article. It’s a call to arms of sorts, an expression of dismay that we’ve become consumers of electronics that we don’t understand.

For me, however, our inability and refusal to contend with the underlying biases of the programs and networks we all use is less a threat to our military or economic superiority than to our experience and autonomy as people. I can’t think of a time when we seemed so ready to accept such a passive relationship to a medium or technology.

And while machines once replaced and usurped the value of human labor, computers and networks do more than usurp the value of human thought. They not only copy our intellectual processes–our repeatable programs–but they often discourage our more complex processes–our higher order cognition, contemplation, innovation, and meaning making that should be the reward of “outsourcing” our arithmetic to silicon chips in the first place. The more humans become involved in their design, the more humanely inspired these tools will end up behaving.

It hit home, because of course I use lots of devices and programs that I don’t understand. I consume much more than I produce. I can program — I can write a simulation in Matlab any time, and I could probably refresh my memory of C++ if I took a week or two — but I don’t know how the internet works. The techie ideal of having a DIY relationship to the technology I use is inspiring, but daunting. Maybe more daunting than it was in the 70’s, when Rushkoff was a new computer enthusiast; his ideal may have been easier to achieve when technology was simpler. I looked at an old CS textbook written in the 70’s — it wouldn’t have gotten me one-tenth of the way towards the goal of “understand all the technology you use today.”

I did find this non-technical intro to the structure of the internet useful, as a start. But I still wouldn’t say I understand how the internet works.

See Your Group September 23, 2010

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Because I am a geek, and because I love my hometown, I had a T-shirt made.

The inspiration, of course, is the storied Valois Cafeteria in Hyde Park, where you can “see your food.”

It’s a local institution, known for comfort food and more recently famous as an Obama favorite.

Concreteness September 5, 2010

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A friend sent me a quote from Don Zagier:

Even now I am not a modern mathematician and very abstract ideas are unnatural for me. I have of course learned to work with them but I haven’t really internalized it and remain a concrete mathematician. I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its let and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4 , then the sequence continues: 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc. The difference between a mathematician and a nonmathematician is not just being able to discover something like this, but to care about it and to be curious about why it’s true, what it means, and what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrodinger equation of quantum mechanics, and certain models of quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful. Some mathematicians find formulas and special cases less interesting and care only about understanding the deep underlying reasons. Of course that is the final goal, but the examples let you see things for a particular problem differently, and anyway it’s good to have different approaches and different types of mathematicians.

It does seem true that some research mathematicians like concrete examples and some like generalities. I’m not sure how that distinction develops. But as a student I find it much easier to learn beginning with examples, as concrete and elementary as possible, and it’s hard for me to believe that anybody learns well otherwise.

Which is why it frustrates me when an introductory book dealing with representation theory begins by letting G be a group acting on a G-module over a field. It’s not that the generality isn’t useful in the subject. It just seems pedagogically lousy for a first-time reader.

Start instead with groups acting on finite-dimensional vector spaces over the complex numbers — in fact start with SO(3) — and you get a clear sense of the structure and original purpose of representation theory. You can see why it’s fundamentally about symmetry. You can even use (gasp!) polyhedra as an example. It’s so satisfying when a book goes down to the basic level. Call me childish — but I think we all start out as children before we work up to greater sophistication. (The book is Shlomo Sternberg’s Group Theory and Physics and it really is wonderful.)

Bulldog! Bulldog! Bow wow wow! August 29, 2010

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Just a few notes as the week goes by:

1. I’ve got classes picked out now: modern algebra, measure theory and integration, algebraic topology, and harmonic analysis. And the new requirement, “ethical conduct of research” (which I assume means “don’t plagiarize and don’t torture monkeys.”)

2. I love Ikea. I know everybody loves Ikea, but there’s no reason to depart from convention here. Everything is beautiful and cheap and quite a lot of things come in orange.

3. The international students, apparently, were taken aside to learn Yale football chants and associated paraphernalia (including the “Bulldog! Bulldog! Bow wow wow!” cheer.) We Americans were never told anything of the kind. I guess this goes along with the tradition of immigrants knowing the Constitution better than we do.

4. Thanks to a fellow grad student, I’m remembering how much I like geometry and topology. Here’s something I learned (at a pub night, no less!) A hyperbolic manifold is the hyperbolic plane modulo a discrete group of isometries. That means, if you look at the Poincare disc model, some of the points on the circular boundary are limit points of the orbit of some point in the interior under isometries in the group. Consider the set of such points. Apparently: if it’s not the whole circle, it is a Cantor set, and its Hausdorff dimension equals the first eigenvalue of the Laplacian on the manifold. WHOA.

Back to School August 22, 2010

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I just arrived at school, though classes won’t start until the first of September. I have a nice little room; little, but shipshape (at least right now.)

New Haven is far superior to Princeton in terms of having affordable retail within walking distance. You hear a lot about the downsides of going to school in a city, but right now (while setting up the room) I’m really experiencing the upside of not living in a suburban island.

I’ve started meeting the other students — visited that local landmark, the GPSCY (pronounced “the Gypsy”) which is where grad students have social events — so far, people seem friendly.

Packing: it begins August 11, 2010

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I really like cartoonist Lucy Knisely, and one of her latest panels rings pretty true to me right now:

The past few days have been a flurry of inconsistencies. I’m gonna be poor in grad school! No, I’m going to be making more money than I ever have in my life! I’m all grown up! No, I’m an overgrown little kid! I’m a scientist now! But I feel woefully unprepared! The one certainty is that I don’t wanna put huge numbers of textbooks in shipping boxes this morning, but I have to.