Talking...

At another math conference. Talk math all the time or only some of the time?

The traditional Wednesday afternoon break is certainly appreciated; I did no math and instead did some sightseeing and a lot of walking. A good change from sitting and writing for hours.

I have some math friends who can keep talking math endlessly: over dinner, after dinner, over drinks, over breakfast, in conference rooms with no food, during hikes. Other math friends keep changing the topic away from math. Part of this is a question of age and concentration: it's mainly my older or more experienced math friends who can talk endlessly and grad students who can't keep it up -- and I remember grad school and being exhausted by the math talk. Another part is personality. Some people are just more one-track than others.

During the school year, while teaching, I was very appreciative of conferences and the chance to talk serious math. It simply did not happen at my home institution much, unless I was teaching someone about this math. This summer I am more interested in a bit of balance but I still have a heightened appreciation of how rare and important these chances are. I don't think my grad school friends understand that. I hope I am not alienating my grad student collaborator by my endless mathematical conversations with others. This grad student, after all, can return to her home institution and be surrounded by research mathematicians who will chat with her and give her feedback and ideas. Things are not quite the same for me, although next semester I will be back in the research milieu. 

"On Tuesday, the female participants..."

Why have these "women in math" lunches or get-togethers at conferences, in grad school, elsewhere?

Oh, how I hate this tired old discussion. It is just like teaching: you may get older, but freshmen are always freshmen. I may get older and wiser, but there will always, always be some guy who thinks he is super-clever saying, "What about lunch for the men?"

Clever, eh? I bet you never thought of that line!

Current response: well, go have lunch!

Now that we're done with that, to the more educational component of today's post. Why lunch for the women? Because it's nice to meet each other and reassure ourselves that we're not freaks and share some experiences. (Why can't I stop being snarky about this?)

Try again. To share experiences and notes and develop effective ways to deal with the usual non-gender-specific thoughts (my research is never going to succeed! I am sooo dumb! I always forget the statistic on tableau that produces the blah function, so how will I ever prove anything again!) and the more-gender-specific (I don't belong here! All these guys won't talk to me and the guy who knows everything about the KdV equations is scared of girls and scurries into a corner every time I try to ask him about remark four in his recent paper! The senior professor who's lecturing on stacks switches to talking about love and beauty instead of orbifolds every time I come near and it is freaking me out! I want to have a baby, or three! I keep getting nominated for committees, so now they want me to be on the women in math committee and the diversity in sciences committee and the undergraduate curriculum committee and the mentoring committee!!!! I just want to be on the funding committee.)

When we talk, we can figure out some of these things that no one else is going to figure out for us. We can learn techniques for gracefully declining those committee nominations, figure out how to shake up conference speaker lists so that speakers don't just include the male organizer's male friends, get tips on organizing our time between work, travel, family, and the rest of life, learn different ways of seeing the world that might free us from our own prejudices. We can make some important professional connections. We can learn from women ahead of us how they made life as a research mathematician or liberal arts college professor work, with or without kids/aging parents/a demanding Ironman training schedule. These models are important because life and society still do demand different things from men and women, and the model that some senior men present (have stay-at-home wife, move anywhere in world for career, work all the time and have wife take care of kids/parents/Christmas cards) is simply untenable and kicks us right out of the picture.

Guys, we like talking to you. Don't be so gosh-darned sensitive. You're fine. But we need a network of women: for advice, sometimes for validation that we're not crazy, sometimes for a tampon in an emergency. You're not qualified for a number of these things, some for understandable reasons (the last, I hope) and sometimes because you are simply unobservant and/or don't experience the same world as we do. The next time I hear some guy saying, "But he never stares at my chest" I'm going to start screeching like a hyena.

Experimentation in mathematics

As mathematicians, we experiment constantly. Just try some shit for heaven's sake!

I'm talking about several different kinds of experimentation here. I'll look at three kinds below. (This is not exhaustive of course, and there are probably other viewpoints to take).

Just work out a few problems. This is what we'd like our students to do most often when we say "experiment." Try similar types of problems and see what happens. Make links between the ideas. Try solving x^2+3x+4=0, x^2+3x+3=0, x^2+3=0, and so on, and see what different kinds of solutions you get -- before asking for the theorem  ("rule") about what kinds of solutions you get.

Why don't students do this? If math is a series of hazily-understood rules, it's safer to just follow those received rules.

Mess with your assumptions. This is a higher level of sophistication but can still apply even to problems like the quadratic equations above. At some point in school we learn that you can't take the square root of a negative number. This is just a lie adults tell us to protect us from something they believe to be scary. All around the world, though, there's some kid every day who says, "Why not?" That's the right question to ask. Why not?

Sometimes there are good reasons "why not." In mathematics eventually one develops a sense of how things "should be" and it is disturbing when violations are found. This is where interesting things happen. But messing with these assumptions is also very useful. Why can't we take square roots of negative numbers? Well, ... um... in the end, no reason -- so we discover imaginary numbers. Why can't we let time go to infinity in this dynamical system and allow negative populations of gazelles, if only in our minds? Well... no reason -- and then we discover something about stable solutions and that our model actually works for an engineering application. Why can't we take the quotient of a geometric space this way instead of that way? Well.... now we develop Chow quotients, GIT quotients, symplectic quotients, stacks.


Gather data like a scientist. Experimentation by hand or by computer can be deeply valuable. Programming the calculations -- often the only way to gather a lot of data in math -- also forces a different point of view that can be illuminating. (Comparing Sage and Macaulay2's treatment and implementation of Schur functions, for instance, is interesting.) The data you get at the end can be REALLY interesting! You can disprove conjectures quickly by finding counterexamples. You can get a suggestion for a new theorem by noticing a pattern (why are these numbers all even? all 0 mod 4? all prime?). You can discover unexpected connections to other areas of mathematics (the results of the combinatorial experiments gave me formulas that solve this differential equation....?!). You can publish things like "this conjecture has been checked for all n less than 17" or "all n less than 16,092,123".

Pure math involves proof: this is what differentiates it from the other sciences. We should not forget, though, that some of the initial investigatory impulses we have share a lot with the sciences. We shouldn't let our students forget, either.

Atlantic article

Like everyone else in the blogosphere, I feel the need to weigh in on Anne-Marie Slaughter's article on women in the workforce and women in leadership. I'm exactly her subject matter: a female in her early thirties trying to figure out work, family, ambition, and what to do with her PhD.

She mentions that academia made it possible for her to do it all for a long time because of its flexible schedule. I agree, if you can live near your work. When I've lived near my academic job I've enjoyed a lot of freedom and flexibility: I can work really hard and still get a haircut, get groceries, go to the dentist, etc. I've also lived far from my academic job in order to deal with a two-body problem + a mortgage. When I'm commuting a substantial distance, living at home rather than coming home only on weekends, and teaching classes in the morning and attending required committee or department meetings in the late afternoon, I too have felt the stress. Hate it. I hate leaving home by 7 am and coming home at 8 pm. If I have to do it again I will quit -- I learned a lot about work-life balance!

And travel is rough. This summer I am spending five weeks on the road. Sure, it's a choice, and one I've looked at closely. (I believe in making conscious choices to the extent that's possible.) I have considered canceling some of those weeks on the road -- but the conferences seem essential to the progress of my career, if I want to have a career, and the family time seems essential if I want to maintain family connections. On the other hand, time at home with my nuclear family seems pretty important too! I want to see friends and go to cool city events and do a triathlon and weed the garden... when is that going to happen?

I am very fortunate: I get to do work that I find interesting and meaningful while being financially supported for travel to interesting locations. Many people I know find the life, from the outside, almost glamorous (crazy to say about a mathematician's life). On the inside, I don't know. What price am I paying in trying to climb this ladder that in the end seems to have little sawed-off rungs every few steps? It's not like I've got a steady job to rely on...

Learning to learn

I've long been interested in learning to learn -- how people learn -- how excellence and mastery are gained. Somehow I ended up reading the book "Flow" by Mihaly Csikszentmihalyi in high school, for instance. Rereading it more recently I'm surprised I got anything out of it back then. (This reinforces my idea that I used to be smarter.) Two of my other recent favorites are "Talent is Overrated" by Geoff Colvin and "The Art of Learning" by Josh Waitzkin. There are similar ideas in all of these. I love "Talent is Overrated" in particular because of its concrete breakdown of different ways of working toward mastery. It is very clear about what deliberate practice means and involves. It ought to be required reading for college freshmen.

One of the biggest frustrations I encounter with freshmen students in particular is that they don't know how to learn. They feel that doing the homework in a half-assed way in the hour before class ought to be sufficient to get them to real understanding, or that reading the chapter before the test is the best way to study. Deliberate practice is a very new concept for most of them -- but it's an idea that they can apply to sports, learning Arabic, economics, pottery, anything! When I teach again I'd like to bring this idea more explicitly into the classroom. I am very conscious that math is not the real thing I'm teaching in freshman precalc or calc. If I can teach students how to approach problems, then I have succeeded. That statement has many levels.

Freshmen come in with these attitudes for many reasons. The two reasons I see as most relevant are the ridiculous waste of time that high school is for many students and current US attitudes toward learning. High school math seems to spin its wheels for years. Students with a good head start due to good education or high socioeconomic class come into high school ready to zoom through calculus on a superficial level. Students with a poor start come in to high school ready to fail to learn how to add fractions year after year after year.

Our attitudes toward education and learning, too, lead us to believe that downloading information into our brains is the primary activity taking place in learning. Consider what some leading politicians are suggesting with regard to education: that brick-and-mortar campuses will be rendered irrelevant by the ability to download a calculus video from iTunes U. We already have these wonderful sources of information and exercises called books -- they're like videos but written down -- and yet I have met few students who have learned calculus from a book alone! Yet students do believe that skimming a book or watching a video is what learning is. The internal work needed is somehow omitted from our cultural discussions of learning.

So go read about learning and how we do it. It's useful in every endeavor!