Thursday, March 29, 2007

Risky Business

Background: Liberal Arts Colleges in American Higher Education: Challenges and Opportunities, report of a symposium held on the Williams College campus in 2003.

It's hard to answer the question "Is Physics a Liberal Art?" when you're not exactly sure how to define "the liberal arts." But somehow this was the embarrassing position in which I found myself last fall while writing my lecture. (How did this happen? Wasn't I a Williams College graduate? Hadn't I just spent three years teaching the liberal arts??)

At one point in my research, I went to the websites of Williams College, Amherst College, and Swarthmore College to see what I could learn about the liberal arts from their mission statements. You can click the links yourself, but to summarize, Williams and Amherst offer quotations from past Presidents. Swarthmore doesn't seem to have a mission statement, but gives a few paragraphs about the College. With all three colleges, it's part philosophy, part marketing. Swarthmore works hardest on the marketing angle, presumably because its reputation as "The College with a Conscience" causes parents to worry about their children's ability to support them in their old age. "Just don't you worry," the text seems to say; "Swarthmoreans get to have it both ways. They can study broadly AND push the boundaries of plasma physics; they can be investment bankers AND care about alternative energy sources."

Language like this helps us approve of the liberal arts, but that's different from telling us what they are. So, looking for something more concrete, I simply googled "what are the liberal arts". None of the top fifty liberal arts colleges showed up in the first page of results, but result #4 was from the website of Kauai Community College. There I read the following two paragraphs:

What Are the Liberal Arts?

The liberal arts are the studies that develop general intellectual capacities, such as reason or judgement, rather than specific professional, vocational, or technical capacities. These studies encourage students to think clearly and creatively, to seek and assess information, to communicate effectively, to take pleasure in learning, to learn to adapt to change, and to live more consciously, responsibly, and humanely.

Why Study the Liberal Arts?

What is crucial in today's world of rapid change is not specialized training but the ability to think critically and make sound judgements. A student's best career preparation may be one that emphasizes general understanding and intellectual curiosity. The liberal arts develop these abilities and qualities of thought.

I was struck by the marvelous clarity of these words, and by the contrast they made when placed next to the flowery and equivocal language of Willams, Amherst, and Swarthmore.

One of the most important parts of KCC's text is the place where it says that the best career preparation may be one that emphasizes general understanding and intellectual curiosity. I like the honesty of this, because it seems to me that a liberal arts education is always a bit of a risk: a willingness to wait and figure life out when you get there. Years ago, a friend of mine was told by his undergraduate advisor: "A bachelor's degree in physics is just a license to think on your feet." Any liberal arts degree has that quality.

High school seniors from all over the country come to Bennington on college tours, and from time to time one of these young people will visit me in my office. The question I get is usually something like this: "What I really want is to be an engineer. Can I be an engineer if I come here?" I say yes, because the answer is yes; but I'll also add that coming to Bennington is clearly not the way to maximize your chances of being an engineer. Clearly, the way to do that would be to go the university route. The point is that a liberal arts education doesn't seek to maximize specific outcomes like "being an engineer;" and from what I can tell so far, the process works better when the people involved aren't asking for that kind of security. This is one way in which the reputations and the have-it-all strategies of Williams, Amherst, and Swarthmore might work against them. Making these schools the very safest choices possible for students and faculty might inevitably involve sacrificing the vitality of the risk-taking element of a liberal arts education. But, as the private colleges are finding out, it's hard to sell risk at $40,000 a year.

Thursday, March 22, 2007

Civic Life in the Information Age

Background: Civic Life in the Information Age, by Stefanie Sanford, Ph.D., Deputy Director for National Initiatives for the Bill & Melinda Gates Foundation.

Civic Life in the Information Age is a striking new book that questions much of the conventional wisdom about Generation X. The book describes the results of Sanford's recent study involving in-depth interviews with 40 individuals aged 22-41, all of them tech-savvy residents of Austin, Texas. The interviews focused on the respondents' views about citizenship and politics. Sanford divides her respondents into four subgroups: cyberdemocrats (working in the policy and government sphere), wireheads (programmers, web designers, and other members of a new middle class), high-tech elites (entrepreneurs, rebounding failed entrepreneurs), and "trailing Xers" (22-26 year olds, many in college). The differences between these groups are the concern of most of the book, but by the end Sanford brings the strands together to paint a picture of Generation X with respect to the ideals of citizenship.

A key finding in the book is that instead of failing to meet the challenges of citizenship (as Boomer Generation academics would have it), Generation X has actually reprioritized the classical duties of citizenship in its own way. For example, while traditional theory states that voting is the defining act of citizenship, the respondents in Sanford's study were likely to view voting as meaningless or irresponsible without a sound knowledge of the issues; likely to view the media as standing in the way of their gaining that knowledge; likely to view politicians as inefficient ways to get things done anyway; and likely to see voting itself as altogether less important than the need to respond to crises and challenges in specific cases (Hurricane Katrina, local neighborhood cleanups).

This "just-in-time citizenship" was to me one of the most interesting aspects of the respondents' take on civic life. And while we probably should castigate Generation X for not voting, we should also figure out a way to take advantage of their "just-in-time" responsiveness by combining it with what Sanford characterizes as a strong work ethic and a pragmatic obsession with results. But maybe that's just my pragmatic obsession with results talking.

Indeed, speaking as one member of Generation X, I at least can say that Sanford has arrestingly captured the mindset of my own social and work circles. "We" GenXers emerge from the interviews as a prickly group with an intense work ethic, a mania for effectiveness and efficiency, a hatred of talk and meetings, a pragmatic wish to find out what works, a corresponding impatience with ideology, and a risk-taking and entrepreneurial spirit. Though originally Sanford may have set out to write a fairly academic book about society's changing definitions of citizenship, I think she has actually ended up with something much bigger: a book that just might help an entire generation to understand both itself and its potential.

Sanford ends by making a number of detailed recommendations, and by considering the implications of her work for Education, Entitlements, and Health Care. I was particularly interested in the implications for education, because I have noticed that while many of my friends in Generation X have gone for advanced degrees, they have finally eschewed academia in favor of writing and consulting; and many have passed on advanced degrees altogether so that they could become entrepreneurs or start new organizations. In my own case, I chose a college without a tenure system so that I, and not a committee of my elders, could direct my scholarship and teaching. The days of climbing the ladder at work and paying one's dues seem to be over. Will my generation's risk-takers, who value effectiveness and efficiency, and who want to be rewarded for their performance, ever find a home in the seniority-based teachers' unions or the gasifying tenured faculties of the colleges? Or will my generation remake those institutions as soon as we get the chance?

Sunday, March 18, 2007

The Wisdom of Solomon

Last Friday in my Rediscovering Math class, we talked about exponential growth. This was not an easy subject to approach for my audience, and sometime soon I hope I can share what we did, because it was an interesting session. But something that has remained with me since then is what happened when I went to the Web the night before to see if there were any good materials I could cherry-pick. When I googled "exponential growth", I was surprised to see this awful, awful page ranked #5 (out of about 1,000,000):


This is probably a relic page, dating as it does from an environmental studies class given way back in 1999; but the fact that the Web still ranks it so highly makes me wonder about the state of this art. (And until we get something better, is there such a thing as a linkectomy? I'm doing my part, by not using a hyperlink.)

The monotone awfulness of this thing assaults the senses, but in particular I wonder about its take on "the two principal problems in energy management." Can these ever really have been, as this professor suggested, "Failure for policy makers to understand the concept of exponential growth" and "Failure for legislation to be formulated and passed to give us a long term energy strategy"? Is it really possible that all of the failure belongs this conveniently to others, while none of it belongs to us, the educators and the technically adept?

When the Intergovernmental Panel on Climate Change released its 2007 report, lead author Susan Solomon was asked by a reporter "to sum up what kind of urgency this sort of report should convey to policy makers" (article
). She answered, "I can only give you something that’s going to disappoint you, sir, and that is that it’s my personal scientific approach to say it’s not my role to try to communicate what should be done. I believe that is a societal choice. I believe science is one input to that choice, and I also believe that science can best serve society by refraining from going beyond its expertise. In my view, that’s what the I.P.C.C. also is all about, namely not trying to make policy-prescriptive statements, but policy-relevant statements."

In this response Dr. Solomon was making an important point, namely that the report in itself is not and was never intended to be a policy document, but rather a statement of scientific consensus. Consequently, her role in leading the panel was not to argue for what should be done about the problem. So far, so good. But what worried me was the possible suggestion in these remarks that Dr. Solomon sees her role in this process as having concluded. I sincerely hope that will not the case. Given all that she has accomplished already, and given the years she has spent bringing this report to light, it is hard to ask that Dr. Solomon give even more. But how can the one person who best understands this report not be at the center of the discussion of what to do about it?

Perhaps the most hopeful view we can take of Dr. Solomon's remarks is that by preserving her objectivity in this way - by presenting herself as a dispassionate master of the facts of the case - she may actually be enhancing her ability to affect energy policy in the United States over the coming years. But that will only happen if she decides to take the next step.

Saturday, March 17, 2007

So Far, So Good?

Background: the Monty Hall paradox (here are some of my tips for understanding and explaining it). Optional: David Sklansky, Theory of Poker.

I was at my neighborhood pub last night when I overheard a snippet of conversation that nearly made me pick my first bar fight. (I've been in plenty of bar fights, just never picked one before.)

WOMAN: "Do you think you made the right decision?"

MAN: "So far."

So far? I shook my head and took a drink. This guy sounded just like all the lousy poker players I've ever known: people who think drawing to an inside straight is fine as long as they hit their hand. They confuse the payoff with the play. His decision was right, I fumed, if it was arrived at in the right way, not if it does or doesn't pay off in the actual event. (Many a fisticuff has arisen over this important point of principle.)

This very idea had just come up in my Rediscovering Math class. We were playing the Monty Hall game, with me as the host, and with an imaginary jeep as the prize behind one of the doors. The student who was the game's contestant said, "OK, I'll switch because I believe the math, but if the jeep is behind my first guess, I'll be pissed." In reply I said, "But would you be pissed, really?" And here another student said, "Personally I think it would be OK even if I lost, because by switching my guess, I at least put myself in a position to win." That's how you keep the payoff separate from the play.

When the woman asked, "Did you make the right decision?", why is it so much to ask that the man should answer, "Well, I made the best decision I could based on the information I had at the time; it's working out so far, but win or lose, I think I made a sound decision on this one." More head shaking, another sip of my drink.

But as I sat there longer, I continued to reflect on the man's "so far" answer. My hackles went back down as I registered the helplessness behind his words. What decision had he been facing, and how did it feel to be still waiting for the other shoe to drop, wondering if everything would turn out all right? It struck me that there is a cruelty in this common view of decisions, this picture of accounts that remain open until the final score is tallied. But when you don't know how to assign yourself points for making rational and careful decisions, what other way is there to keep score?


Rational and careful is not synonymous with mathematical or robotic, and in fact, being rational demands that we pay attention to emotional factors. Good decision-making doesn't ask us to quantify emotions and values, nor does it ask us to ignore them in favor of "objective" considerations. But it does demand that we gain some distance from these emotions and values by naming them, and by investigating them as factors in our decision-making process. This is not something we often do naturally, but it is, I hope, something we can be educated to do, and something we (I) can improve at throughout the course of our lives. Another thing I hope to teach my students in Rediscovering Math is that a momentous decision with real consequences for ourselves and for others is not only a process of sorting through emotions and values, but also a process of discovering crucial facts, of being creative about options, and of bringing our clearest sort of thinking to bear. One of my students today pointed out that it is especially in a problem fraught with emotion that thinking systematically can help us cope.

If I could reclaim the last point for a mathematician: Poker is a poor metaphor for life, but it illustrates one essential feature of the rational life, which is that even the major life decisions are about (as my student put it) "putting yourself in a position to win", not winning or losing in the event. If this message helps us not to feel regret when the world inevitably fails to give us what we ask for, then there is a way in which the rational is in fact the most human option, because the most merciful.

So Advanced, It's Simple

Keeping it simple can be a bad bargain when it means failing to explain, failing to convince, or failing to support us as we move up to the next level of complexity. A handful of examples along these lines:

1. Kinematics

The curriculum for Newton's Laws traditionally begins with 1-D kinematics, and traditionally emphasizes the reduction of 2-D problems to 1-D problems via component analysis. It is an important feature of the world that phenomena can be sliced up and viewed one component at a time. But just because you can work one component at a time, that doesn't mean you always have to.

Over the years I've come to feel that trying to build a 2-D understanding on a 1-D foundation is a losing strategy. It's sort of like trying to build a rectangle out of line segments: it works fine, as long as you have an eternity to spend on it. In my book manuscript, I've taken the approach of presenting kinematics in two dimensions from the very beginning. We look at all kinds of curvilinear motions and draw lots of pictures, as we learn what the position, velocity, and acceleration vectors mean. The vectors themselves are allowed to spend a little time in the limelight, instead of being chopped up and used for components. Circular motion becomes relatively easy and loses its ad hoc character.

This approach does require that we learn some "vector calculus," but I have found that the essential facts are relatively few and surprisingly easy even for nonmajors to grasp - perhaps because working with vectors and trajectories in space allows you to bring your visual imagination to bear in ways that sketching a_x-t graphs does not.

2. Walking

To zeroth order, you could posit a human being walking in a straight line at a constant speed. But this is not so much walking as it is floating - and it doesn't feel right. The next order of complexity seems crucial for understanding the process. It seems to me that over the cycle of each stride, your center of mass speeds up during the "launch" phase, glides ahead for a moment, and then slows down during the "heel strike" phase. A graph of the acceleration component over time leads to a graph of the net force exerted by the floor on your feet; this allows you to visualize the friction-force vector flipping back and forth, maintaining the net force at zero in a time-averaged sense. But now it's a "dynamic zero" that we're seeing, not a "floating zero" of the sort that we would associate with a pebble drifting through space.

This is certainly a more complicated picture than the simple proposition of "walking in a straight line at a constant speed." But here we may have a case in which the zeroth-order approximation is so far from the reality and the felt experience that it's actually confusing.

3. Steady state flows

The book I'm using right now for introductory physics confines its treatment of heat conduction to the steady state. The idea behind restricting to the steady state seems to be one of shielding students from the complexity of the full dynamics of diffusive heat transfer. In a way this makes sense, but the trouble is that it doesn't acknowledge the fact that the steady state viewed in isolation is highly counterintuitive. If there's construction on the highway and two lanes constrict down to a single lane, have you ever in your life observed the traffic speed to double at the constriction point? Of course not; but in the steady state, this is what would happen. (Once I did actually take part in a steady-state traffic jam, though it wasn't on the interstate. It was at the Blue Water Bridge in Port Huron, Michigan: one lane bridge, seven lane toll plaza; roughly two minutes to advance each car length on the bridge, roughly fifteen minutes to advance each car length on the plaza. Multiplying the number of lanes by seven divided the speed by about seven!)

Frustrated by the troubles my students were having with conduction problems and the steady state, I sat down one night last week and did some numerical work on the temperature diffusion equation, funneling the output into a handful of little movies. One of them is embedded below; for a clearer version, click here; some explanation is attached to the YouTube video. Feel free to send me an e-mail and I'll send you .mov files and/or Mathematica code. (This was some quickie programming the night before class....)

We discussed this movie and the accompanying T(x,t) animation for quite some time. Far from being intimidated or confused by the complexity of the diffusion-equation behavior they were seeing, the class seemed to gain a much clearer idea of just what I had been asking them to do by focusing on the steady state in the first place.

Given the physicist's overwhelming desire to simplify (which I described earlier in Is Physics a Liberal Art?), it's worthwhile to keep in mind Einstein's famous corrective: "Everything should be made as simple as possible, but not simpler."

Wednesday, March 14, 2007

The Wittgenstein Within

When you toss a golf ball straight up into the air, and you say to the class, "Today I thought we could talk about the acceleration of the golf ball when it's at the peak of the trajectory," it's like the moment when a psychoanalyst says, "Today I thought we could talk about your mother." The key thing is to maintain a pleasant neutral expression no matter what you hear next.

Many years ago while living in Oxford, I went to see an art film about Ludwig Wittgenstein (let's call him "W" for short). One memorable scene takes place during the years when W was still living in Vienna; it gives a hilarious impression of what a terrible teacher he was supposed to be. As I recall the scene, W is tutoring a youngster in math. We see the child sitting at a desk, W standing by her side, and a blackboard behind both of them, its surface virtually painted white with chalked expressions from the propositional calculus. The student is all of twelve years old, and looks terrified. W says to her: "So. What is" (gesturing vaguely at the board behind him) "this." The child puts her head down and gives no answer. After a second or two of this, W loses control completely, yelling at her and yanking her earlobe painfully. There's a quick cutaway before we see the melee go any further....

I don't support that kind of student-teacher interaction, but I have to admit that there are times in office hours when I do feel a little like this guy. I can feel my patience beginning to wear thin like Ken Doral's hair, and I can actually almost hear his voice in my head saying "Obviously energy is the strategy to use here because THAT'S what this CHAPTER is about!"

In general, it is the subtlety of Newton's Laws that impresses me. But then again, sometimes I catch myself thinking that the laws are so simple! Why can't people just get it? I wish I could remember more clearly what I myself was like at the end of my first college physics course. Maybe if I could see how limited my own understanding was, it would help me to make sure that I never end up yanking somebody's earlobe.

(Side question: Are any experiments such as diSessa's classic study being done longitudinally, so that we can see retrospectively what future Ph.D.s' freshman understanding of mechanics was really like?)

Stuart Crampton, the Barclay Jermain Professor of Natural Philosophy at Williams College, taught me mechanics in the spring semester of 1988, when I was eighteen years old. I had arrived at Williams to find myself far behind many of my classmates in terms of preparation, and I was finding Physics 104 so difficult and so intimidating that before the first midterm I went to Stuart's office and asked him if getting a C in his class would mean that I couldn't be a physicist someday. To Stuart's great credit, he was very kind to me while also giving me a straight answer to the question I'd put to him.

That midterm proved to be one of the memorable turning points in my life. I still remember two of the problems with complete clarity. One of these was to compute the rate of energy dissipation in a ball bearing dropped through a cylinder of oil. Another was to imagine trying to stop an oncoming tank by throwing thousands of mudballs at it. (The mudballs stick to the tank.) At first I was completely nonplussed by the tank problem, but suddenly I realized that this was essentially a rocket in reverse: whereas the rocket dispels exhaust and consequently speeds up, the tank gains mud and consequently slows down. So I wrote down the rocket equation, fiddled with a couple things in it, and answered the question.

The test came back with an A, or maybe even an A+. Coming from the place I'd come from, I saw this grade and the first thing I thought was that I could make it here, that I'd found a subject and a society to which I belonged.

But no one sails through physics. I remember in that same class struggling mightily with the problem of "popping a wheelie" on a motorcycle. The point of the problem was to understand why the rider must continually speed up in order to maintain the bike in a wheelie position. (Ever notice how brief these "wheelie" events are? It's because you can only keep speeding up for so long before you decide you're going fast enough!)

In fact, I still have a hard time thinking about this problem unless I move to an accelerating reference frame - which is a move that I enjoy trying to avoid. So I have work to do. I'll take another crack at it sometime soon. It'll help me remember that we all struggle.

Monday, March 12, 2007

Is Physics a Liberal Art?

At Bennington College there is a wonderful, Oxbridge-esque tradition called the Tuesday Night Supper Club. Three or four times a semester, on a series of Tuesday nights, fifteen or twenty faculty members choose to stay on campus after a long day of teaching. At five-thirty in the evening (evenings start early in Vermont) everyone gathers in a small dining room in Commons. The room is part of the students' dining hall during the day, but on Supper Club nights the swinging doors are kept closed, and the evening is dressed up with wine and tablecloths. Faculty members sneak back from the foodservice line with plates full of food, trying to avoid their students' questions about the assignment due tomorrow. Before long people are filtering back and forth for coffee and dessert, and the chosen speaker walks to the front of the room to address his or her colleagues.

Usually these presentations are a way for the faculty to get to know one another's research. But when I was invited to give a presentation at the Supper Club last fall, frankly I despaired of talking about my research. My audience, I knew, would consist almost entirely of artists, writers, and performers. Not only no physicists, but no physical scientists at all. Why in the world would I go in front of this audience with a specialist's colloquium of the type that I might give to a physics department?

But if I wasn't going to talk about my latest paper, or my next paper, then what was I going to talk about?

Finally I decided to do something along different lines. The title of my presentation was "Is Physics a Liberal Art?"

I thought of this lecture earlier tonight, when a friend had this to say about my earlier post on the place of Newton's Laws in the physics curriculum: "As you hint at, a lot of this comes down to why students should learn intro physics (or calculus) in the first place. So much of our curriculum requirements are in place simply for historical, departmental turf-protecting reasons. So: why should the liberal arts student need to learn physics at all?" Not long ago, an iconoclastic computer scientist at Bennington asked me a similar question: "Why does a liberal arts college even need a physicist?" (It was a friendly question, although that fact is somewhat hard to convey here!)

Despite the title of today's post (and despite the neater-than-it-should-have-been conclusion to my talk, an ending which I felt driven to by the requirements of the genre) I don't claim to have the answer to these questions. But in light of my friend's comments, I thought I'd share the presentation.

Here is a link to the lecture in Word document format: "Is Physics a Liberal Art?". The Word document contains the images used in the PowerPoint presentation that accompanied my lecture, but if you want better images, then the PowerPoint file itself is also available here: TNSCPresentation-02.ppt.


By the way, one lucky winner identified the title of the last post (PUUPh ("puff") Give) as an allusion to the line "puff, puff, give" from the movie Friday! For proving himself seriously smart, this reader wins a deluxe copy of my vanity-published puzzle book, Word Puzzles for the Seriously Smart. Enjoy!

PUUPh ("puff") Give

Background: Tough Choices or Tough Times, the December 2006 report from the National Center on Education and the Economy (NCEE)'s Commission on the Skills of the American Workforce - these are the folks who brought you A Nation at Risk in 1983 and kicked off the standards movement; Taking Science to School, the 2006 report from the National Academy of Science's Committee on Science Learning, K-8; LiPing Ma, Knowing and Teaching Elementary Mathematics (1999).

Mainstream K-12 reform takes it pretty much as given that America's math and science teachers aren't good enough at math and science, and that this deficit is affecting our students' learning outcomes. For example, one of the key reforms advocated in Tough Choices or Tough Times is to move somehow from today's world, characterized as one in which most teachers come from the bottom fifth of college graduates, to a world of tomorrow in which most teachers come from the top fifth.

One of the key texts to have in the back of your mind in this discussion is LiPing Ma's must-read book, Knowing and Teaching Elementary Mathematics. Ma coined the term PUFM (pronounced "puff-em"), which stands for Profound Understanding of Fundamental Mathematics. If you have PUFM, then you probably know a slew of different algorithms for long division, half a dozen ways to explain why you need a common denominator to add fractions, and just why it is that dividing by a fraction is the same as multiplying by its reciprocal. The book argues that while elementary math teachers in China can do all this stuff, elementary math teachers in America are fuzzy on it - with consequences for student outcomes.

Sometimes I think that the standards people may have confused the need for teachers to have PUFM with the need for six-year-olds to have PUFM, but that's a post for another day. In the meantime, what seems to be clear is that you need PUFM to teach -FM.

Elementary math is not intellectually trivial stuff, by the way. The base-10 place value system is a towering achievement in human intellectual history - not to mention relatively recent in the scheme of things. According to my old copy of The World of Mathematics, the standard algorithm for long division dates from as recently as 1491!

Although the teacher-quality critique is pretty standard talk in K-12, I have often wondered at the apparent lack of any analogous concerns in the higher-ed setting. I guess one reason would be that it is academics who define what expertise is - and, being pretty satisfied with themselves, they have decided that expertise means a Ph.D. But of course, having a Ph.D. in physics mostly means that you spent an intense five or six years in a laboratory (or, for the theorists, in a cafe) pushing the frontiers of human knowledge or what have you. It would seem that this is poor training for teaching freshman physics.

Graduate students in physics nowadays take teaching seminars, but to order of magnitude, this activity amounts to roughly 1% of the integrated mental experience of a graduate program. 0%+1% is infinitely larger than 0%, and the results show it. But if you were serious about creating teachers of undergraduate physics, wouldn't you spend more like, say, 50% towards that end?

Why do the people who teach physics majors the core curriculum have to have research Ph.D.'s, anyway?

Or let me put it another way: Why are physics majors systematically denied the opportunity to learn physics from people with substantial training in the teaching and learning of physics?

Just brainstorming for a minute here, I am seeing two kinds of academic appointments in a higher-ed setting: an instructorship appointment, in which teaching craft and teaching effectiveness really matter, and a research appointment. To hold the research appointment, you have to do the usual Ph.D. To hold the teaching appointment, you have to do a teaching-intensive graduate program. Students majoring in physics would take courses from the instructors, but also take topic seminars with the researchers. In the topic seminars, researchers are allowed to do what they do best, which is to give inspirational and completely incomprehensible PowerPoint presentations.

Instructors will also, by the way, hold part-time research appointments to work in the research labs and contribute to frontier projects, though they won't lead these projects the way researchers do.

So, what do the instructors learn in graduate school? You guessed it: they learn PUUPh! (pronounced "puff"). Profound Understanding of Undergraduate Physics. The curriculum for this writes itself: intensive work on the force concept; common difficulties with the field concept; multiple ways of looking at the derivative [as a slope, as a rate, as a marginal return, as a functional] and the integral [as an area, as an accumulated change, as a functional]; dimensional analysis; lots of ways to explain line and surface integrals; intensive work on the conceptual foundations of stat mech and its relationship to phenomenological thermodynamics; linear algebra with special attention to vectors-vs.-components; special relativity with lots of discussion of the paradoxes; quantum theory with attention to identical particles, the meaning of the wavefunction; making arguments of a tensorial character; etc.

The vast majority of practicing physicists don't have PUUPh, any more than practicing mathematicians have PUFM. Unless you're invited to slow down and ponder the intricacies of some of these rich ideas, you get by perfectly well without doing so. I learned a lot about F=ma by writing a book about it, long after I'd become able to apply it to my professional work. And I leave aside the issue of pedagogical content knowledge - not just knowing what a flux integral is, but knowing five ways to explain it and a dozen excellent problems to assign for it.

In addition to doing a better job with the basic content for the physics major, with the right training instructors would also be able to do much more interesting things for non-majors. I'm thinking along the lines of the recommendations in the recent National Academy of Sciences report Taking Science to School, another worthwhile read. This report's recommendations are for K-8, but frankly even the colleges are utterly failing to do this work. Doing better here will require college science teachers to learn much more about the kinds of misconceptions and naive theories that 18-year-olds come to college with. A subject for another post....

P.S., a prize for whoever identifies the allusion in the title!

Saturday, March 10, 2007

Style, pedagogy, and Purcell

This is not an idea yet. But I've been thinking lately about the relationship of curriculum development and teaching effectiveness to issues of pure style and aesthetics. Coming from the physics education perspective, I have been accustomed to thinking of curriculum development as a science. And I do think this is the best way to frame it. We need to know what works, and there are facts about this, in the way that there are facts about which therapies work for illnesses. And the main innovations in my teaching of Newton's Laws (well, at least I'd call them that) have come from the hard work of gathering data over a span of years. I might have done it more formally, made the data more quantitative, but the fact is that I asked, and asked, and asked, and listened, and changed my methods radically, and noted the results and changed my methods again.

But then again it might not be so accurate, this picture of me in a classroom with a white coat and a clipboard. The way it usually happened was in a sunny office with big windows, and two people with trust and warm feelings for each other. One of them maybe even beautiful, and one of them unintimidating and perhaps not so ugly either. (I guess if I was really serious about the aesthetic, I'd go back and correct that Bogart-esque rhythm.) In thinking about teaching and curriculum, we overlook questions of personalities and the personal for good reasons. But when you have a subject like physics that is so terrifying to people, I wonder if effectiveness might depend significantly on emotional and psychological factors. At any rate, I feel that my own effectiveness depends in part on these things.

I first thought of this general topic when I read the responses to a questionnaire that I gave to my students after we'd used a draft of the book as our primary text in physics class. One of the students wrote, "I'll always remember the sentence 'Forces are evanescent things.'" At first I skimmed past this, and then it sank in: he wasn't saying he'd remember the fact that forces are evanescent things, but rather the sentence that told him this. It struck me that something interesting had happened here: in crafting a successful sentence, I may have actually accomplished something concrete on the pedagogical front. (And of course the rhythm of the surrounding writing must also have contributed somewhat towards making this key sentence "pop.")

Students complain about the "dry" writing in the standard physics texts; can lively writing and a discernible authorial voice contribute significantly to learning outcomes? I think it is possible, perhaps especially with respect to a deep idea like the Laws of Motion. A friend wrote in response to this post, "From your comments below, I wonder if part of the distinction you're
making is a personal relationship you can have to the ideas in Newton's
Laws - the way you can to Hamlet - as opposed to the prosaic
relationship with the lenses. So in the right book about Newton's Laws
we'll want to hear the author's voice."

There are some other loose associations between "purely aesthetic" considerations and "purely intellectual" ones - such as the way certain qualities of verse make poetry amazingly easy to remember, or even the fact that in a published study (available here), professors' teaching evaluations were significantly dependent on the physical attractiveness of the professor. (See especially Figure 1 on page 23 of the PDF; every scientific paper should have a graph whose horizontal axis reads "Percentile of Beauty.") The authors are careful in stating their conclusions, but they do entertain the notion that the results may have been less about discrimination than about effectiveness. (p. 14 of the PDF) I wonder if, when you disaggregate their data by teaching department, subjects like physics get a "terror bump" in the strength of the effect.

I remember the sensual pleasure I felt during my undergraduate years when I regarded the proportions, the layout, the illustrations, and even the smell of Purcell's masterpiece Electricity and Magnetism. Didn't all these pleasures have something to do with how eager I was to apply myself to this difficult text? (Click the link, and notice that fifteen years after I graduated from college, this book is still sold in the same hardcover format, with its peculiarly square height-width ratio.)

Here is the first Amazon review of Purcell's book, by Professor Henrique Fleming of Sao Paolo, Brazil. The emotion is unmistakable, the book and the reviewer virtually characters in a romance:

"This book must have been a work of love. The reader of it who fails to fall in love with electromagnetism would better change his direction of study, as he will not find anything better, including the marvellous Feynman's "Lectures on Physics". Following a more-or-less historical approach, except for the early use of relativity, the author strives to get the results from a full understanding of the physical situation. This is obtained by the use of very clever intuitive models. After that comes the mathematics, rendered natural and welcome. An outstanding example is the treatment of polarization of a dielectric sphere, where most of the physics is derived from a drawing! Another feature, to be found only in books written by great physicists, is the ability of stretching the argument up to its limit, getting results we wouldn't think possible with so little formalism. Problems are extremely good and real. The drawings, done by the author himself (so I read some! where) are very beautiful and helpful. Some of the exercises are of numerical character, motivating the use of computers. After meeting this book I could never teach introductory electromagnetism from another text. The author, Edward Purcell, is a Nobel prize winner who discovered, among many other things, nuclear magnetic resonance."

It is instructive to compare this with the second review, by a Kevin Costello (perhaps this person?):

"This is a (very heavily mathematical) introduction to the physics of Electricity and Magnetism. Although it has some strong points (It takes the time to explain the math behind div, curl, etc. and some of the problems are rather neat), it also has its weaknesses. First and foremost, the problems are almost universally without solution, except for a select few whose numerical answer are written below the problem. Combine this with the almost total dearth of examples, and you have a pretty serious problem for anyone trying to learn the material on their own. The sections themselves are also sometimes rather poorly explained. Chapters 5(explaining magnetism as the relativistic effect of moving charge) and 10 (dielectrics) are both fairly confusing and hard to understand. At times I found myself begging Purcell to include one, just one little measly example that could possibly make more sense than his pages and pages of writing. Once or twice I found myself not noticing that I had a fundamental misunderstanding of a facet of the material, just because no example or solution in the book provided a counterexample to my way of thinking."

It will come as no surprise if I say I'm solidly with Henrique Fleming in this argument, but the second review does raise some interesting issues. I do actually remember Purcell's book as brutal when it came to the problem sets, and yes it was precisely because there were no example problems in the chapters. Purcell must have taken the view that the only way to learn these things is to struggle over them; it can't be shown, it must be discovered. This is actually not far from the currently accepted wisdom in some ways. I don't take it as far as Purcell did - my book has plenty of worked examples - but compared with contemporary texts, my book probably looks a lot more like Purcell than most of what's out there today does. Connell's last point is more interesting, to do with Purcell's failure to anticipate and prevent common misconceptions. Here physics education research has made leaps and bounds since Purcell's day.

It may after all be reasonable for Connell to ask for more in a text, but many students seem to want the impossible, which is for physics to be made effortless. And the publishers will certainly promise it to them. Not long ago I read the back cover of Physics Demystified, where it says it's going to make physics "fun, effective, and totally painless." I guess if you are talking about 'commodity physics' (haven't got a better way to characterize it yet), this is possible. But ask the most successful physics students if physics has been painless for them, and all you will get is bitter, bitter laughter.

The place of Newton's Laws in the physics curriculum

As you may know, I've just written a manuscript for a book about Newton's Laws of Motion. The inspiration for today's post was my deepening realization that in one sense at least, this project sails against the prevailing winds of postsecondary science education. By this I mean that the book does not attempt in any way to connect the core ideas to issues of importance in the world. Everyone in physics is trying to do that today, not just in the liberal arts colleges but also in universities, with courses on Physics for Future Presidents and so on. Let me say that I think the prevailing winds are blowing in a good direction, and I foresee myself doing a lot of work that goes with the flow. But let me also try to draw a distinction here. There is 'commodity physics' and then there is 'deep physics.' Commodity physics includes topics like lenses, heat conduction, circuits, and even to some extent energy. This stuff is a commodity in the sense that just about anyone can learn it, and just about anyone can teach it. And, since it doesn't take long to learn, having learned it you are in a position to apply it all over the place. This is where the textbooks are going.

But the Laws of Motion are not a commodity; they're precious metal. They are buried deep, hard to reach; those who catch a glimpse of them experience awe, perhaps even mania.

There are very different reasons to study physics "as a liberal art." One reason can be for the citizenship stuff, and in that case you end up pushing ideas such as problem-solving and rates of change; and you end up linking topics such as heat conduction to applications like R-values for home insulation. Great; worth doing. But the other side of the liberal arts is the great ideas, and the stretching that we do to reach them, and the way they stay with us for life. Newton's Laws of Motion are a masterpiece. Teaching them so that you can connect them to social problems is like teaching Hamlet so that your graduates can be impressive at cocktail parties.

A clue to the distinction I'm trying to make between Newton's Laws and other parts of physics may be found in the fact that it is not very easy to invent good examples of applying the Laws of Motion in a technology or industry setting, whereas for topics like energy or circuits it is effortless to do so. It's as if one were to try to write an essay question along the lines of, "How would Hamlet have handled the Cuban Missile Crisis?"

I wish I didn't have to make this kind of distinction between commodity physics and deep physics, but the difficulty and subtlety of the Laws of Motion force you to make a choice. The views are from the top, and it takes a long time to get there.

Lately I've been having the radical thought that Newton's Laws should be removed from most introductory physics courses. You're probably laughing out loud right now, but bear with me for a moment. If you were to suggest this idea to a physicist, I bet he or she would object in the following way: "Are you crazy? It's all *based* on Newton's Laws for God's sake!" But I say, (A) If the Laws are so important, then why aren't you more upset that so few of your students are learning them? (B) If it's really "all based on" Newton's Laws, then why don't students' difficulties with the Laws doom them for the rest of the introductory year? (C) Energy and momentum have central roles in contemporary physical theories, but force does not; so can't we just get over the historical blip that was 1687-1926? (D) Remember, I'm not touching the curriculum for physics majors or engineers, so try to stay calm. (E) Let's take a careful look at the curriculum and see who actually has to know what. Yes, doctors have to know about forces because of structural anatomy and muscles; but when they do this work do you really think they're using what *you* taught them? They forgot all that a long time ago. They're just using everyday intuitions about force; and everyday intuitions are fine for everyday purposes.

Once you remove Sir Isaac from the non-physics-major curriculum, you can offer non-majors a course specifically about Newton's Laws. This is more like a course on Hamlet, in that students take it not for what they can do with it, but for what it adds to their inner lives. And notice that with a whole semester to spend, we can make another radical move, which is to really teach non-physics-majors the Laws: not a cartoon version of them but the full vector dynamics, the way it's done in my book. I can report that this works, because I actually taught Physics 1 last fall to a class of mostly non-majors using a draft of my book, and it was great. I had more success in getting the core ideas across than I ever did before, and that includes my experiences teaching Physics 131 at Grinnell College and TA'ing Physics 7A and 8A at UC Berkeley. But the pace was radically slow by the usual standards; and even at that slow pace, if I'm honest I have to admit that the end result was that the students' understanding was fragile. After such herculean efforts on everyone's part, I could still trip them up at will with certain conceptual questions. It was the failure mixed with the success that led me to wonder if more would be gained than would be lost by removing this material from the introductory physics curriculum altogether.

Expert practice, expert tutelage, and the role of homework in college physics courses

Background references: Dr. Anders Ericsson's faculty webpage; The Cambridge Handbook of Expert Performance; the 5/7/06 Sunday NYT Magazine article "A Star is Made" ; and the Sunday 3/14/07 NYT Play Magazine article "How to Grow a Super-Athlete".

Now that we have a handle on the kinds of practice that do and don't lead to continued improvement over long timescales, perhaps now we can go on to ask about the kinds of student-teacher relationships that seem reliably to foster high achievement. For example, if we take a look at experts and the teachers they've had over the course of their education and careers, what patterns can we see in their student-teacher relationships? Does any teaching arrangement other than intensive 1-on-1 tutelage reliably support the effective work that Ericsson calls "deliberate practice"? What is the feedback cycle we see happening between a teacher and a student engaging in deliberate practice? How does the teacher's own domain expertise play a role, and how does the answer to that question depend on the discipline of study?

I have a feeling these questions would become acutely important to the design of a school based on the concept of deliberate practice.

Just a couple of informal observations on the logistics of expert tutelage: In music, the elite teaching structure seems to be 1-on-1: you have a personal voice coach, or the piano teacher sits on the bench with you. The same is true of many sports, such as tennis and golf. Even in team sports, where there may be just one head coach for an entire team, it still seems to be the case that the actual teaching moments are 1-on-1. This is partly because there are different roles on any team. For example, in basketball you have the point guard, the center, the small forward, etc. The coach doesn't say, "Today we will practice shooting the basketball." Instead, somebody from the coaching staff will take the center aside and work on the drop step; somebody else will take the guard aside and work on defensive stance; etc. Maybe these moments are 1-on-2,3,4 instead of 1-on-1. But they are certainly not 1-on-10.

This line of thinking has actually led to a new idea for how to handle homework in my college classes, which I am experimenting with right now in my second-semester introductory physics course at Bennington. The problems I'm trying to address here involve the role of homework. For one thing grading homework is a huge hassle for me, and the feedback comes to the students too late, and they have little incentive to absorb the careful solutions. My colleagues in the humanities tell a similar story - some of them barely read the papers they assign. Everyone knows it's not working. In the research university setting, where you have introductory physics classes with hundreds of students enrolled, this problem is being addressed by putting all of the homework assignments on the web. The students submit their answers and get rapid feedback, without the professor even being involved. It is ironic that the professors are so grateful to be spared any contact with the homework cycle, when research shows that it's almost solely by the doing of problems that students actually learn physics.

Here's the system I'm trying out. The cycle begins on Fridays with me distributing a problem set in the usual way. The students have a week to work on it. The following Friday, instead of collecting the homework, I give out careful solutions, and the students have about a week to study them. Then, the following Thursday, I have each student in to my office for a fifteen minute 1-on-1 session, like a mini oral exam. I choose a homework problem at random from the assignment, perhaps with a small numerical change, and the student solves the problem at the board. I see how the student is thinking about it all. There are some follow-up questions to pinpoint trouble.

The student gets an instant pass-fail grade. To compute the student's final semester grade, I will subtract the number of Fails from the number of Passes, add 1-to-3 points for the final lab project, and then simply rank the students bottom to top. Cut points are introduced into the scale based on whatever the usual considerations are.

We just had our first day of oral exams, and it was amazing. The students were nervous, but most of them did great. It was clear that they had worked on the problems, studied the solutions, re-worked the problems, and practiced the solutions. It is amazing what can happen when the work gets done! On the way out of my office after the exam, several students told me how excited they were by this system. They love that the expectations are so clear. One said, "I actually thought about the homework, which I've never done in a science class before."

For this system to make sense, you can't treat homework problems like a commodity, the way the publishing industry and many instructors treat them. The problems have to be so valuable that you are comfortable making the entire accountability system focused on solving them. The problems also have to be, to use Gauss's phrase, "few but ripe," because the students can't master very many problems in a short time. People give lip service to reducing the pace of introductory courses, but I am actually doing it, and the results are fascinating so far.

When I unveiled this strategy to the students on the first day of class this term, I could tell they were scared of having to come in and perform at the board in front of me. I made sure to say, "Look, this system is not about punishment. It's really about sincerity. I sincerely want you to understand and be able to do these problems. So let's work with a system that won't let us fail in that." After Week 1, they are on board.