Archive for the ‘background & analysis’ Category

(Final?) Reflections on the blog

March 15, 2013

I’ve decided to take a (permanent?) hiatus from writing this blog, and wanted to conclude with some reflections and analysis on the posts to date.

A look at the Table of Contents shows that there are 25 flavors and 7 seasons, and a few posts on background and analysis.  I think that these posts capture the main recurring mathematical experiences that I’ve had as a graduate student and now a postdoc.  In a strange way, the blog feels complete, not to say done.  When I reflect on my day-to-day and month-to-month experience doing research math, I mostly find that I’m revisiting experiences, not undertaking new ones.  Am I approaching a comprehensive list?  I’m sure it can’t be so, and look forward to the unknown.

The process of writing the blog has, I think, accomplished the goals I set out with.  Writing posts has forced me to be more self-reflective about how I do math.  This self-reflection has helped me to tweak my process, to be more efficient, productive, and enjoyable.  It has also brought meaning and depth to my academic research life.  Writing posts has also helped me articulate the recurring aspects of my experience, and confirmed for me the stability and intersubjectivity of these experiences.  Although no one else’s voice appears on this blog, in conversations and other mathematicians’ writings I’ve seen parallel experiences described.  I think that other mathematicians would find a lot (of course not 100%!) to agree with in my posts.

If I were to do it all again, I would probably use the same procedure and the same questions for flavors and seasons.  Except I’d change flavor question #6 to “what experience does it resolve to, after how much time?”, and maybe rephrase the questions as actual grammatical sentences.

Where to go from here?  

This blog aimed to articulate recurring math research experiences.  I left out quite a few singular experiences, and a careful description of these might also be interesting.  For example, the night I submitted my first paper to a journal, then spent hours walking the streets in an intense rain and wind storm.  Or when I found out my first paper got published.  But maybe this becomes too subjective and less relevant.

This blog avoided discussing the experience of specific mathematical ideas (what does it feel like to think about a Bousfield lattice? what do you see in your mind? how?).  Doing so could be interesting.

My next step, however, will be learning more about first-person methods for understanding experience.  I’m obviously not the first one to try to carefully describe his or her lived experience, and lately I’ve been finding out about a body of research that establishes specific techniques for doing so, uses these techniques to examine certain experiences and draw conclusions, and analyzes the limitations and pitfalls of these techniques.  I wrote about these first-person methods in a paper I wrote for the Journal of Humanistic Mathematics about contemplation in mathematics.  So I’ve been reading, and will continue reading, these books and related material: On Becoming Aware, The View From Within, and Ten Years of Viewing From Within.  With more theory under my belt, I’m excited to think about how I could integrate this sort of intentioned self-reflection into a classroom setting (as a kind of metacognition), to get students thinking about (and consequently improving) their learning process.


Contemplative Education

August 7, 2011

I recently attended a regional meeting of the Association for Contemplative Mind in Higher Education.  Simply put, it’s an organization of college professors that incorporate contemplative practice in their classes.  The conference was very diverse – with people from the arts, social sciences, activist programs, and even a few scientists.  Some talks were detailing success stories, others were participatory and demonstrated specific contemplative practices to use, still others were theoretical and visionary.

Let me back up and explain what I’ve learned about this thing: contemplative education.  First, a quote:

“The faculty of voluntarily bringing back a wandering attention, over and over again, is the very root of judgment, character, and will. . . An education which should improve this faculty would be the education par excellence” 

-William James, 1890

If you believe this, then maybe you’ll believe a second one:

“Universities have forgotten their larger educational role for college students. They succeed, better than ever, as creators and repositories of knowledge. But they have forgotten that the fundamental job of undergraduate education is… to help them grow up, to learn who they are, to search for a larger purpose for their lives, and to leave college as better human beings. So totally has the goal of scholarly excellence overshadowed universities’ educational role that they have forgotten that the two need not be in conflict.”

-Harry R. Lewis, former dean of Harvard College

If you believe the first and second quotes, and you care about teaching, then you might start to wonder if there’s a way to develop “the faculty of voluntarily bringing back a wandering attention,” for example.  There’s a word for this.  Focus.   And there’s a way to become more focused.  Meditate.

If a technique for becoming more “focused” seems like a good idea to you, something worth looking into, but maybe needing some scientific grounding to be more appealing, well, guess what?  You’re living in the right decade.  In recent years, there have been dozens of scientific studies – neuroscience, psychology, health care, education – more or less confirming the claims of meditators.  These (secular and non-secular) meditators base their techniques in millenia-old wisdom traditions from around the world.  Here is one link to a list of research articles and books.  Here is a link to a review of research that pertains specifically to the benefits that meditation brings to a college classroom.

This is not sketchy science.  This is, for example, recurring 5-day workshops hosted in India by the Dalai Lama, bringing together Buddhist scholars and Nobel-prize-winning scientists like (current US Secretary of Energy) Steven Chu.  This is, for example, panel discussions at MIT and Stanford, with thousands of academics attending.  If you believe in global warming, you should believe in the benefits of a meditation practice.

Increased focus is just one of these “proven” benefits of meditation.  Prior to the act of refocusing attention is the act of noticing when it wanders – this is called mindfulness.  There are meditation techniques that improve mindfulness, and then there are meditation techniques that use that mindfulness to improve focus and concentration.  The same techniques will help you to be aware of and in control of emotions (like stress).  And then there are techniques to go deeper into the objects of attention, and for example cultivate curiosity, creativity, open-mindedness.  And there are techniques to foster concern and compassion for those around you.

The teachers at the conference had been using meditation in their classes, with some subset of these “goals” in mind.  As I said, there were many success stories told – coming from chemists, physicists, law teachers, as well as art and social science teachers.  (Here are some syllabi used, in a range of disciplines.)  But many of them have also experimented with other “contemplative practices” – things like contemplative movement or deep listening.  Here is a page with a helpful diagram of the diversity of contemplative practices, and some info about many of them

The post up until now has been discussing a teaching pedagogy, one that I think is fascinating and holds a lot of potential, and I’ll be experimenting with in the near future.  But this blog is supposed to be about research, not teaching.

All the benefits of meditation – greater awareness, focus, balance of mind, insight, creativity, interpersonal communication (and more!) – are yours for the taking, IF you’re willing to establish a personal contemplative practice.  I say this from personal experience, and with the above research articles as empirical evidence.  If you don’t like sitting meditation, then look into one of the other contemplative practices.

I’ve heard two nice analogies for the role meditation might play in a balanced life.

One is hygienic.  You keep your body clean, so you should keep your mind clean.  You nourish and exercise your body, so you should nourish and exercise your mind.  Meditation is a way of clearing out the clutter, of giving wholesome food to your mind and letting it go for a quiet walk outside.

The second is more scientific.  In order to perform experiments, a chemist needs a lab with the right tools.  The untrained mind is unwieldy – easily distracted, prone to dullness, never still but always jittery and burdened.  Meditation cultivates your mind as a tool – steadies it, sharpens it, gives you practice in controlling it.  Of course, traditionally the purpose of this was to allow meditators to go deeper into the nature of reality, in order to find the most universal truths and embrace the world with the most expansive compassion.  But you can use it to do better math, too.

Common Ground

April 28, 2011

The University of Washington’s Mathematics Department has an annual departmental potluck and musical event – a chance for faculty and grad students to socialize and show off some of their musical and performance talent. Each year I’ve tried to contribute something unique; last year in 2010 I wanted to entertain but also provoke thought. The opportunity to perform in front of a captive audience of my colleagues and mentors is too tempting to resist.

My goal was to capture what it’s like to be a mathematician, in an entertaining yet self-reflective way. My hypothesis is that, across our different fields, the combinatorialist, analyst, and algebraic topologist, etc, share many day-to-day and week-to-week experiences. Of course there is a diversity of experiences, but there are also similarities. It’s rare for the department to come together, and I wanted to use the occasion as an opportunity to express and address these commonalities.

For example, there’s the common, yet somewhat contradictory, philosophical stance of the practicing mathematician – “a formalist during the week and a Platonist on Sunday” (Hersh and Davis). There’s the nonlinear feedback loop and co-evolution of definitions, hypotheses, conjectures, and theorems (as in Lakatos’ Proofs and Refutations). There’s getting into the one-pointed concentration zone, where you feel like a mere conduit for the mathematical flow to express itself. There’s the use of machinery to break problems into smaller, more manageable pieces, slowly extracting small results that bit by bit add together. There’s getting completely stuck. There’s the paperwork and grading, and the extremely difficult casual conversation with the non-mathematician about your work. There are the times that distance and leisure suggest new insights or new perspectives. There’s the creating of new perspectives, and the discovery of what must be so. There’s that moment when the last piece falls into place, and certainty resonates within you (à la Poincaré). There’s the disappointment of finding a counterexample, and having to throw out a week (or month) of work.

My performance tried to express all these, in under ten minutes. It was part theater, part performance art, with a dynamic soundtrack and lots of bizarre props. There’s a video of this performance on my website, (well, any day now).

In a sense, this blog is an attempt at continuing the programme set out in that performance: to express some of the common experiences encountered along the mathematical path.

After the show, the feedback among my classmates and the faculty was better than I hoped. “Luke, you hit the nail on the head!”, “We were all just talking about it together… that’s exactly what it’s like to do math!” “The music you chose was perfect!”

I’ve tried to engage other mathematicians, in more or less formal settings, in this level of meta-mathematics. A continuing issue is that most mathematicians feel that any discussion involving math must proceed with utmost rigor and objectivity. I strongly disagree with this sentiment.  My performance at the math musicale succeeded, I think, because the medium – absurdist performance art – made it clear: we are contemplating mathematics, but in a looser, more subjective way.

So, as you read the blog, please withhold your insistence on rigor. It is necessary to math, but it is not all of math. Try to be mindful of how it feels to surrender rigor – how the subjectivity floods in, and in the flood what are the branches that we can hold on to? These are the threads of intersubjectivity – shared experience.

An everyday metaphor

When you go to the movies to see a romantic drama, the plot usually pulls from a database of common themes.  Likewise when you listen to the lyrics of songs on the radio.  If you’re watching a romance or listening to the radio, you sort of know what to expect.  The themes can be trite and arranged uninterestingly, as in your average romantic comedy.  They can also be presented artfully and uniquely, as in a Leonard Cohen ballad.  These themes are, in part, normative scripts that we are socialized to (e.g. hetero- versus homo-relationship paradigms), but they also express universals that appear across all or most human cultures.

So if this blog were about partnerships rather than mathematics, some flavors might be: the second date, meeting the family, cooking a meal together.  Some seasons might be: getting to know each other, traveling together, long-distance.  Thank Goodness I’m not making a blog about those things.

sticky post: Flavors and What?

April 27, 2011

The purpose of this blog is to document the flavors and seasons of the mathematical experience.  I’d like to dedicate the blog to the discoverer/inventer of projective geometry, Girard Desargues.

His 1639 masterpiece, “Rough draft for an essay on the results of taking plane sections of a cone,” with only 50 copies published, was ignored for about 200 years. According to  Raymond Wilder, there are two possible explanations for why his work, “one of the most unsuccessful great works ever published,” went unappreciated for so long.

1.  Perhaps it wasn’t until the mid 1800’s that the mathematical community was prepared for the revolutionary idea that Euclidean geometry was just one of many equally valid and self-consistent geometries.  (In the 1830’s, the incomparable Gauss developed his own non-Euclidean ideas, but chose to carry them to his grave (1855), because, as he wrote to a friend, “I fear the ‘clamor of the Boeotians.'”)

2. Or maybe it’s because Desargues’ work was “couched in a strange terminology, much of which was borrowed from botany.”

For more info about this project, go to the About the project page.

Find a list of all flavors and seasons in the Table of Contents.  Recent posts are below.  Please add comments.