What is love? baby don't hurt me...

In a conversation with a friend I hadn't seen in a while, the friend said he loved his students and wouldn't trade them for anything. (He's at a community college.) I thought that was interesting -- what does that mean? I suppose it means he's happy working with them; I expect it means he respects them and gets some gratification from helping them. (I feel a little Asperger's trying to parse out what this means...)

Do I love my students? I really enjoy working with some of them and I have very collegial relationships with some. I have friends from both the lower-level classes I've taught and the independent research I've conducted. But I guess I wouldn't say I love my students. This is why I'm pseudonymous, by the way -- it seems absolute heresy to say such things out loud. I don't feel like I can be totally honest without repercussions.

I respect them and try to work with them as adults, while keeping in mind that college students are still developing their adult skills. I enjoy working with them overall. Love? I wonder if we have different meanings for the word "love" -- I know that I don't have the same ideas as everyone else on what romantic love is.

Well, this is part of the largest "soft question" -- I don't love my job. Are we supposed to? Our American ideas of fulfillment through career are a little weird and I don't quite buy into it. Part of the problem as I analyze my job satisfaction is that I get very frustrated at not being able to help all students understand and love mathematics. Like any relationship, this is a two-way street. The students also have to make an effort, and not all do: it's a bad relationship in that respect.

But it would be cool to have a job, and students, I loved. Is it possible?

Girl nerd

This is riffing on the post at mathbabe.org on the making of a girl nerd.

I guess I am a girl nerd. Neither of my parents has a science or math background and my dad did not really do college. Cathy O'Neil says that in her experience a mathematical parent is the greatest predictor of mathematical daughters; in my experience an immigrant parent is the best predictor of a daughter's future mathematical success. I know only a few American-born daughters of American-born parents who are mathematical. They are not unicorns -- they certainly do exist! -- but I don't understand how they got there. An AMS article (pdf) discusses some of these ideas as well so I don't think it's all anecdote. Analysis of female performance in the International Math Olympiad also supports the role of culture in fostering female mathematical talent -- and the evidence is overwhelming that being in the US leads to steadily decreasing female math performance over the generations.

Why? Here I'll put my opinions!

• America does not respect mathematics all that much. Many people and politicians are suspicious of any science, as science does not follow morals in evaluating claims of truth. Same for math.
• In the US junior high girls who participate actively in math are often ostracized. It is not a topic for cool or pretty girls. Marine biologist, though.... you get to play with dolphins, and that's alright.
• Having an immigrant parent insulates you from the above attitudes at home to some extent and home is important. You may acquire other handicapping beliefs, but they're different ones.
• The value of an education is stressed more highly by immigrant parents, in my experience and on average, than by American parents. Part of this is self-selection: anyone motivated enough to move to another country to do something is, by definition, motivated enough to do something. At the same time, immigrant parents may be clueless as to how to obtain an education. Some of my students (usually oldest males from a few local immigrant groups) get calls to come home and take care of the family because of a health emergency, even if it's midterms week, because cultural values clash with the expectations of the US educational system. But the parents do keep saying the education is important in a different way from the way American-for-generations suburban parents say education is important! 
• I think the skill of operating in two cultures (or the similar concept of code-switching) actually gives an advantage to female children of immigrants continuing in mathematics. Mathematical culture in the US is different than mainstream culture, and if you're comfortable switching back and forth and talking with both groups you may be more likely to continue. Men and boys are often excused for being socially clueless if they are mathematically inclined, so they can just keep acting according to the norms of mathematical culture while out at the mall, for instance. Women and girls are not excused but rather judged harshly. For girls in particular to have a comfortable existence in the hell-hole we call junior high in the US they must have some facility in switching between cultures. Transitioning between math club or math circle behavior and girl's volleyball team behavior is important and difficult.

As a young woman I found mathematics very soothing once I got proficient at it. This is apparently a common experience for boys and girls -- the certainty of algebra and trigonometry and geometry are an oasis in a confusing adolescent world. I got involved in some outreach activities and learned about fractals and hyperbolic geometry and that sort of thing, which was undeniably cool. (Fractals are still cool!) Eventually I found a community through math team and my accelerated math classes in high school. There were other girls involved; it was male-dominated but not so much so that I was miserable. (I did take a class one summer in junior high in which I was the only girl -- every other boy in the class refused to talk to me at all. At lunch time I either sat alone or found my female friends in the music class to eat with. Only the teacher would talk to me. That was an exceptional experience, though notable.)

Having parents whose attitudes do not mirror those of the American mainstream is one of the most important supports -- that comes through in both my experiences and Cathy's. Why is the US so toxic to mathematical talent, especially female mathematical talent? What could we do to change it?

Collaboration tools

I have several collaborative projects in progress and I use several tools to make the collaborations easier. What do others use?

• Skype: number one tool. I chat weekly about two projects, and less frequently about another more free-form project. One collaborator lives less than ten miles away from me, but I work in another town and we've both got time constraints that make Skype the easy option for conversation. My other professional collaborators live a thousand miles away, so clearly Skype is best for that communication as well.
• Gchat: I have much less frequently used gmail's chat feature to work with students; somehow they're less shy that way and they don't have as much mathematical material to convey so the lack of equation support is not a big problem.
• Dropbox: best file-sharing around. It's suddenly much easier to keep one version of a working paper up to date, rather than ending up with half-edited this and semi-revised that in about 6 different pdfs. My collaborator groups store our tex files and image/bibliography/supplementary files in a group Dropbox folder and we can compile right in the folder. I love it.
• Google docs: useful for some shared notes or spreadsheets -- I use it for communicating with my students in several classes -- but useless for tex. So annoying to use. Dropbox wins.
• Wunderkit: a new task management program in beta that allows you to add people to a working group, facebook-style. I would love to try using it with a collaborator except that file-sharing might be a bit clunky or non-existent. Perhaps more such functionality will be added as it grows: I'd love to try to use it more fully as a place to leave notes on progress, make to-do lists, etc.

These things make collaboration so much easier and more workable than I found it to be in the past. I think this technology has allowed me to sustain research collaborations despite being at a small liberal arts college in the Midwest surrounded by cornfields or soybean fields or whatever it is they're growing this year. That said, I do love reading the letters between Grothendieck and Serre... wouldn't it be cool if we were still writing letters? I don't know if scraps of my pdfs will be available to future biographers, and there's so much less personal exchange in those latexed files....

Never wasted

As a complement to yesterday's post, I'll offer up one thing about mathematics that has changed my life. I am never bored anymore. There is always a little puzzle to think about. I do some combinatorial research now (it fits well with questions accessible to undergraduates) and so I'm always working on little projects that are advanced by doodling at meetings. I really enjoy mathematics with a visual representation (think ribbon graphs or Young tableaux.). Somehow that visual representation makes it easier to play with concepts while in a faculty meeting or on the bus.

Is this perhaps one reason mathematicians are seen as weird and introverted? This just occurred to me: if we're secretly trying to work out the logical connections between binary strings, Young diagrams, and fixed points of a particular torus action on the Grassmannian we seem far less interested in the company of others than someone who is actually just bored and looking for distraction. Good or bad? Hm!

Wasting Time

This is, I suppose, another rant.

I hate wasting time and I hate wasting my student's time. I want every hour in class to be valuable to them. I want them to use their brains to grapple with the questions of (say) calculus to the fullest extent possible. If you're sitting in a room and showing up, why not do something while you're there?

I do unimportant things ("flick", as we used to say in college). I enjoy reading blogs about fashion, computer programming, fitness, recipes, home-schooling; I do online shopping and window shopping now and then; I read stupid books and listen to low-quality music (Katy Perry? heh -- more likely the video linked below these days). But I do try to enjoy all these things -- I do these things because they give me pleasure. I try to skip events that are not going to give me pleasure or serve some other goal of mine, and if they're serving another goal I try to keep that in mind while I'm there. The first important question of mathematics -- "Why do I care?" -- is one that I ask about other activities I take part in.

My students don't think the same way. They don't have the same urgency or passion. Probably because they're just out of high school, they think that it's normal and "adult" to do useless crap or sit around doing things that don't serve your goals for hours every day. Students who don't really want to do calculus show up anyway, day after day, even if they're failing, because they're "supposed to". Students who do need calculus blow it off even though it's going to be really useful to them next semester. I was no model of foresight; I did not appreciate how important Legendre symbols would be at some point in my life and thought they were kinda dumb. On the other hand, when I thought something was not important I just stopped showing up and did something more important for the moment. Why don't my students do this?

Instead, they show up but some refuse to engage. I say, "Think about this for a moment..." and they say things like, "No, I can't!" It takes incredible coaxing to get some people to think. They seem afraid that it will be physically painful, perhaps give them a rash or a seizure. I need to say, "Think about this for a moment -- I promise it won't hurt -- what do you have to lose? -- and you might figure out the answer." Then they might try it, and often an answer indeed does appear. Astounding! And even if an answer did not appear, they were not harmed by the experience.

Why just sit there sort of ignoring the class around you, wishing you could sleep, when you could actually be in bed? Why sit there staring out the window if you could actually be outside? And if you think it's so important to be there that you stay, why not do something related to the content at hand?

Bait and Switch

One of the things I'm angry about is the US lack of respect for education. It occurs on all levels -- parents who tell elementary school teachers that they can't discipline children for disrupting class, politicians who tell elementary school teachers that they don't deserve pensions or money for Kleenex in the classroom, politicians who tell the rest of us that educators of all stripes are ridiculously overpaid and lazy to boot since they get summers off, the rest of the public who think that I get summers off or only work 11 hours a week (that's the number of hours a week I am explicitly and overtly "teaching," before office hours or class prep or anything else). I get very mad. I feel disrespected and underpaid. If I am really responsible for nurturing and providing life lessons to America's future leaders, aren't I doing something important?

Like all math professors (and many other people), I'm rather ridiculously educated. All that training was in something other than what I'm doing, though. I'm socializing 18-year-olds. I am providing emotional support, math anxiety therapy, career guidance, and life lessons via being a b*tch about homework, academic honesty, and other annoying encumbrances that come along with college. These have nothing to do with my thesis work or any of the other training I got in graduate school.

Between the emotional drain of an introvert doing a job like this -- talking to people 6-8 hours a day -- and the political and ideological climate in the US, I feel like my head is going to explode some days. I want to bring those Republicans who think I'm lazy with me on my long day at work (more or less 7 am to 7 pm), or even my short day when I try to leave to get home by 4 so I can do some research. I want to have them watch a student with severe math anxiety and an absolutely abysmal high school math education try to learn calculus from an online tutorial with no human assistance. Hah. I want them to try to learn calculus from an online tutorial, actually! I want to hear another speech about the lack of trained and educated professionals in the US and the necessity of higher education and then look at our systems of economic reward and punishment. I'm certainly doing my informational interviews with industry these days: I've got academic jobs lined up for the next school year but after that am afloat. There is no reward in academia but the love of the job... and I'm not sure if it's worth the sacrifices.

Failure

I am working in a department with great students overall. One must note, though, that spring semester calculus is often a refuge for students with math anxiety or negative emotions about mathematics. These feelings can often stand in the way of achievement, and I don't know how to deal with them. Optimization -- a particular unit in calculus -- has historically given students real problems (both in spring and fall semesters -- but right now it's spring!).

If you stopped my students on the street and asked them to find the area of a rectangle, they'd do it: no problem. If you ask them on an exam, all that previous knowledge disappears. One year a student got stuck on area of a rectangle because for reasons unknown the student insisted it must have a height. It must be a three-dimensional object, and the height was not given, so the problem was impossible to her. More recently I have avoided such discussions during the exams, but have still gotten A = lw^2 or A=2wl a number of times when asking the classic optimization question involving fencing a rectangular area next to a barn.

This is not so different than physics exams on which the earth has radius larger than the distance to the edge of the solar system, or temperatures occur in negative Kelvins, or the bicyclist goes 16,000 miles in one hour if the speed is 12 mph. I know that students get anxious during exams, in particular. These "brain farts" or --  more chillingly, the real lack of understanding revealed -- lead to poor grades. Students get Ds or Fs. They may fail a class because of their difficulty keeping a clear head or relating concepts in a classroom to real life.

As I'm recovering from grading, though, I'm the one who feels like the real failure. Let's be honest: as teachers, we do measure ourselves by our students' success. If I can't get them to understand or can't get them interested or can't coach them to success, I feel like a failure. Rationally I know that there are many factors involved. If they're in college and can't reliably come up with the area of a rectangle, something went wrong far before I came along. Students who tell me they spend about two or three hours on calculus outside of class are certainly setting themselves up to evade success. As I learned from some wise friends, you can't care about students' grades more than the students themselves do. Students are entirely surprised that the class might be difficult: one student asked me recently if this was supposed to be harder than high school -- she thought she was doing something wrong. There are a lot of things outside of my control here.

It doesn't change my feelings, though. Failure. Whose?

Mastication

Check out the article The bubble within the bubble. It gives a very accurate accounting of the horror that academia can involve, chewing people up and spitting them out. I do not want to live the life depicted. I think that this is a major factor in the "leaky pipeline" that leads many women to leave academia, as well.

Conferences and workshops

I am an "early-career mathematician". I have been going to conferences for six or seven years now; my first conference/workshop was a week at Oberwolfach and it was absolutely wonderful. (Some of the food was odd: I did not know Germans loved curry and pineapples on ham and cheese, for instance. It was surprisingly tasty.) At that first workshop I drank two lattes a day from those coffee machines right outside the lecture hall, and beer in the evening of course. I attended every lecture faithfully and worked in small groups to understand the material. I learned a ton.

My first Joint Math Meetings was quite overwhelming. I was not interested in a particular special session so I ran around from talk to talk -- geometry to combinatorics to pedagogy to mentoring -- and chatted with people occasionally in the halls. I was a graduate student.

As a graduate student, I was a talk-attender: go to the talks to learn the math, right?!

Now I am a people-finder: go to conferences to find the people who might be able to answer that burning question you've got, and go to talks to hear about new fun things if you've got time!

Which do you do these days? What stage of your career are you at?

The absence of evidence...

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Teaching and research are the two main jobs of mathematicians in academia. Research is often valued above teaching, even though I've met a lot of miserable researchers and happy teachers. Both are important: research for its own sake is as valuable as writing symphonies or studying physics, and teaching is good for both future mathematicians and people who will simply use mathematics in their lives in the world. No arguments here about the value of one over the other.

What I'm really interested in here is a strange phenomenon that I see fairly regularly: students and postdocs (and maybe professors?) who seem to feel that since their research is going badly they're meant to be teachers, or whose students hate them and so feel they're meant to be researchers. Utterly mysterious. Why does this feeling have any purchase among graduate students and postdocs? It would make sense to me to feel that you're meant to do the thing you're good at, but to feel that you're meant to do the thing you currently suck less at does not compute. I've seen it at least twice this year, though -- two folks who were intending to do more education-focused math realizing that they're not good teachers and thus deciding they ought to do research. Fine. I don't think it will last long, as it is easier to persist as a mediocre teacher than a mediocre researcher due simply to economics right now.

And why is industry left out of the equation? Maybe you are not good in front of a lecture, and you realize that the life of a researcher is not for you, but you are good at solving problems and doing math. Maybe you're even great at programming or organizing conferences or writing. Look at your strengths, people! Pursue them!

Professors, quit telling the students that aren't so great at research that they are good teachers unless you know they actually are. They are deluded, and upset when they discover the delusion. Even worse is when they don't discover it.

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Having now been in the liberal arts world a while, it's not easier to persist as a mediocre teacher.

Bright star or steady light?

I've long secretly thought that a big ingredient in establishing a career in mathematics is playing a waiting game, especially with the job market as it is. I don't mean a passive waiting game -- if you do not do mathematics, teach, serve, etc you will be pushed out (at least if you don't have a tt job yet!). I mean an active game of patience. Just keep working, being stubborn, contributing in the ways you can.

Plenty of people leave academic math and for many reasons: don't want to do the research to get the PhD, can't manage the persistence or independence to do the PhD, discover they'd rather do something else, manage the PhD and realize academia or math or whatever is not for them, get into the postdoc and realize they'd rather have respect and a nice paycheck or want to live in a particular geographic region or have a sick parent and need to stay someplace or can't find a TT job or ..... This happens at many phases of the career path. I have always known I'm not a stellar mathematician but I'm stubborn, patient, and I love math. I figured I could make it through any of the crap that got thrown my way with patience and stubbornness.

I still think that's true (if I can get a job). I think that I'm growing into my research and making original contributions. I am tackling problems that I'm sure others could think of and solve, but these problems just aren't the next big one on their lists. (I have a list of problems -- a story I want to unravel or a mystery I want to illuminate -- and there are great cool problems I've thought of that don't fit my story that I have not pursued. True for you?) I think I could be a small steady light in the sky of mathematics even though I was never a bright star.

I have to say, too, that many people I knew in grad school and identified as bright stars initially ended up flaming out. How often is that true?

(Some mathematicians have said that math is a young man's game. I think that's true for many bright stars. I certainly don't think it's true for steady lights. There are many men in mathematics who start modestly and continue contributing into old age. I use the gender pronoun intentionally: I have observed as well that many women tend to start more modestly and grow brighter and brighter over time. I think this has to do with American culture in high school and college -- fewer women devote themselves wholeheartedly to mathematics in high school or even college, and thus don't start their cosmic mathematical journeys early enough to burn super-bright in the first month of grad school!)

Do you always care?

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A truth: I do not care about all of mathematics.

Some questions are immediately exciting. I see a paper on the arXiv, get a little tingle when I read the abstract, download it extra-quick, start skimming. Oh, how exciting! That's so cool! I go to a talk and can't stop taking notes. The ideas are so beautiful. I can envision switching to this question and spending hour, weeks, years learning its intricacies.

Some take some introduction: I am cajoled into seeing the coolness by a convincing conversation, a slow unfolding of ideas, a string of relationships. Wow. Now that I spend some time thinking about it, that's awfully cool. Often these are the big ideas -- they take some time to digest.

Some I just can't get into. Why is that so interesting to people? Why do you care about that question? How is that important, other than as a stepping stone to (something actually cool)? Yawn.

I guess I'm happy with being selectively interested. If I got carried away by every current I'd probably never get anywhere. I've noticed that I like certain things -- geometry, dynamics, big pictures -- and am not immediately interested in technicalities, clever tricks, counting things, and algebra that I can't imagine.

While I use my interests to motivate myself and keep awake, the things I don't love turn out to be important. I have to push myself to care about the technicalities and appreciate them for the foundational issues they often turn up. It's ironic how often I need the unloved mathematics to answer the questions that make me happy.

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For instance, I now care (a little bit) about counting things :)

A rant (midterm next week)

One of the things I dislike most about teaching at the small SLAC I'm at is that students are... um... very responsible and grade-conscious...?... but just as competent as any other 18-year-old around (which means "intermittently so"). I assign projects in class with the goals of grappling with open-ended problems, developing critical thinking skills, and learning to communicate mathematical ideas, and they amaze me. (See previous post on ODEs.) On the other hand, I assign regular homework and they amaze me in a bad way.

There is an astounding variety of great excuses for late work. I thought I'd dealt with that by putting three free drops in the homework grade -- three free "I drank too much! My dog got sick! I'm on a volleyball tournament! I forgot I actually have syphilis!" Three free passes, no information or justification needed! One might hope that this would cut down on the number of desperate emails with the above information. Somehow it does not. After all, my students can do some math, and they know that if they get that homework in late and get it score that free drop will be used on the lowest-scoring homework and they will get a few more fractions of a percent in their grade. Must I go to a system in which you only get the drops if you actually refrain from handing in that homework? (That would be amusing in a game-theoretic kind of way.)

Perhaps I should accept that students come in with a Pavlovian response to assigned homework, which involves the ritual of the late assignment as an essential component. Perhaps the ritual of the late assignment is a bonding act, a way of connecting in a comforting way in this scary world of college. It is certainly a way for these students to reassure themselves that they are responsible and doing the right thing: they are trying very hard to get me to let that assignment count.

Of course their priorities are ridiculously misplaced if we simply look at the numbers. With 30-some written assignments in a semester each worth 25 points and normalized to 1/7 of the final grade, each point on a written assignment is about .0013 in the ultimate reckoning. That fourth missed assignment will take you down .03 points. I look for gaps of 20 points to set my grades. To make up that difference you will need 600 assignments, or one really spectacular success or failure on a midterm. Study, young people, and do practice problems -- don't waste your time trying to convince me to accept your late assignment!

A (convenient?) truth

I'm at a SLAC that really values an open-door policy, flexibility in meeting with students, and a lot of attention to undergraduates at all levels. This policy is quite successful in many ways and I agree with many of the values that sustain it.

It is important to learn how to set appropriate boundaries in such a system. If you don't, you'll get eaten alive by thousands of little tiny mouths each taking a bite from your wriggling flesh. Yes, you need to prioritize your service commitments, make time for research, keep up your growing excellence in pedagogy, etc. Today, though, I want to talk about students.

Student A: I know you have office hours on Monday, Tuesday, Thursday, and Friday, but I can only meet Wednesday at 9 am. Can you meet Wednesday at 9 am? I'm really busy with my lab classes but I really want to understand implicit differentiation because I'm very committed to my education.

Professor: (internal sigh as a glance at the calendar reveals weekly squash game. Puts aside squash game because of dedication to accessibility and love of education.) Sure, I can meet at 9 on Wednesday. I'm glad to hear you're interested in understanding implicit differentiation.

Student A: Great! See you Wednesday and I'm really grateful.

Wednesday am: no student. 9:10 no student. 9:20 no student. 11 am no student. Email appears over lunch. "Dear Professor thanks but I figured out how to implicitly differentiate so I didn't come this morning see you in class!"

Repeat with Student B, Stokes' Theorem, with research time rather than squash. You can always do that research later -- this student needs you now!

Repeat with Student C, vertical line test, with faculty coffee hour. Coffee? Who needs coffee?

Repeat with Student D, exponential functions, putting off grading. Well, I don't mind that one (until students start clamoring for the graded work).

Repeat... no, I refuse to repeat.

This year my policy has been that I will not reschedule research time, Skype calls, my trip to the gym, or (most days) my lunch in order to meet with students outside of office hours. I will only meet when I'm genuinely free. I feel less resentful when they don't show up and more present when they do. I go to the gym and faculty coffee hour (or whatever) more often. What's most interesting is that students really don't notice or mind -- they are looking for the most convenient option for themselves but they are not surprised if you have another commitment at a particular time.

Today: "Can't come to office hours have class can I come at 1?" Email sent around midnight. It was the beginning of office hours as I read the email and started responding... and who walked into office hours just minutes later?

Don't think about it too much.

Luke Wolcott's post on ODEs and the meaning of life ponders how philosophical we can be in a math class. I have thought about this too -- has every teacher?

I have taught a lot of calculus, of course. I treat a terminal freshman calculus course as a liberal arts experience as much about philosophy as anything else. The ideas wrapped up in the concepts of limit and infinity are amazing and can blow an 18-year-old's mind. (It's not limited to 18 year olds, but they are the most fun to watch!) As Luke says (although he's talking about modern algebra) we study structure for its own sake. The amazing thing, to me, is that this abstract structure then gives us magical knowledge about the real world. How amazing is that? Why should it work? When does it fail?

I'm very pragmatic in other respects. The terminal freshman calculus course is a life skills course. It's hard, harder than many other courses students take, and students always seem surprised. Just yesterday a student was remarking that it's amazing and disturbing that my class is harder than her AP high school class was. Goodness gracious, I'd certainly hope so! The class is about seeing hard scary things and then doing them in a supportive environment that nevertheless has certain incontrovertible truths. It's like a ropes course except that intellectually you're not wearing a harness and you could actually fall off the rope ladder from one tree to the next. It's like lifting a heavy weight: you can move it, or you can't, but you can always build up to that weight through training and concentration. The math is out there in the world and doesn't care about your feelings. It's like gravity. My students have often come from backgrounds that lead them to believe that trying really does count for everything, and math is a great place for them to learn that trying is only one ingredient of accomplishing something.

The great part of math is that you don't end up with a broken bone if you make that mistake. You just smile and start again. My new students often can't believe it: if you make a mistake, doesn't it hurt? or mean that you're not good at math and so you shouldn't do it?

In a higher-level class like ODEs many students are perfectionists but they aren't so afraid. Math is not such a mystery to them. Here's where my philosophies disagree with Luke's: I found no determinism in the course! I guess I taught ODEs from a "dynamical systems" point of view. We did a lot of modeling. While we did work on finding the recipes for solving various types of ODEs, we spent a lot of time on numerical and qualitative analysis. Every time we encountered a type of ODE we could solve analytically I introduced an ODE that we couldn't solve analytically. In every section we used visualization tools to consider the kinds of behavior these ODEs exhibited. We spent a lot of time on fisheries models...

Sensitive dependence on initial conditions got a lot of attention, and students at the end of the semester even came up with their own chaotic models. We talked about approximation in the computer and looked at nice intro examples like y' = e^t cos t --- this goes nuts pretty fast if you use Euler's method to approximate a solution. You can use the existence and uniqueness theorem to show that your computer estimate is wrong, wrong, wrong. I think this bore some real fruit: one of the end-of-semester projects included a student implementation of a three-body-problem modeling algorithm and the student showed the whole class how rounding error in the computer lead to drastically different outcomes.

This is philosophy, and political science! We talked a lot in a qualitative way about how to think about error and long-term behavior of systems. These have real implications for a lot of topics. I think students gained something beyond mathematics when I made them model a fish population, then add harvesting, then add a disease, then add an oil spill, then change the reproductive rate... I presented the SZR model of the zombie apocalypse and students were picking apart every assumption!

Anyhow. Life lessons (practice, work hard, don't be afraid of challenges) and abstract ideas (limits, infinity, sensitive dependence on initial conditions, strange attractors, the unreasonable effectiveness of mathematics). The idea of truth outside of ourselves that is unmoved by our emotional pleas but will yield to patient and persistent examination. Different kinds of philosophies than Luke's, appearing in some set of classrooms in middle America.

Music to my ears

I like doing math research to music. This drives my significant other absolutely nuts.

Music calms me, allows part of my brain to be distracted while the other part gets to work, helps me enter a flow state. I can only listen to two kinds of music while working and eventually if my thinking gets too deep I do need to turn it off. While computing examples, though, Bach or electronic dance music are really lovely companions. Surprisingly, I've found myself falling for dubstep as well. Best is an hour-long seamless mix without interruptions. iTunes Radio gives a lot of options for this. Maybe I like Bach for similar reasons: the patterns fit that mathematical part of my brain, there are long pieces without interruptions, and there's a certain purity to the music that you just don't get with most pop or country. Bach and techno are about patterns and sounds, rather than giving verbal accounts of tales of romantic woe. I have no mental tolerance for multitasking that involves talking to someone while doing something else that involves writing -- the verbal part of my brain is just not that nimble.

I do confess I wrote much of my dissertation to the sound of Lady Gaga and the Yeah Yeah Yeahs.

And "math rock" never quite fit my math moods...

Music for math? physics? writing as opposed to doing examples? writing as opposed to reading?

Alright... flashbacks to college weekends in the desert....

Writing papers

I'd like to ask other mathematicians about how they write papers. I love doing math for its own beauty and writing papers simply to communicate with the community what has been discovered. I am also a bit concerned about this because I am an "early-career mathematician" and if I don't want to be a "short-career academic mathematician" I need to get some articles out.

I'm working on several projects and each is in a different stage.

One: I wrote some, collaborators wrote some, we have something paper-like which needs major revisions. Collaborators are very-early-career mathematicians so don't know how to write mathematics. But I barely know how to write mathematics well and certainly have no training in guiding beginners through writing for publication, so this is a learning experience for all!

Two: We keep thinking we've got it, almost -- just need to write it up. Have some written notes. I feel far from having a complete paper. Why?

Three: Just started so have such an amorphous pile of small results that it is hard to imagine yet how to put it together.

In post-dissertation-land, I don't generally start with a paper in mind. I start with a problem I want to answer. I work and work on research and reading and throwing away scratch work. I write expository notes to myself and others (students, collaborators). I try to write a fair bit for myself, but haven't channeled most of this material into articles.

Once I have some results that I think are worthy of an article I put them together, with proofs, and start writing the introduction. That's a paper! Time to revise about 17 times.

How do you shepherd projects toward papers? Do you start with a paper in mind or with a problem in mind or with a person you'd like to work with? How intentional are you about keeping that goal of publication in mind?