A Teacher’s Life – Four Stages in the Practice of Physics
J’aime mieux forger mon âme que la meubler.
Michel Eyquem de Montaigne.
A willingness to be puzzled is a valuable trait to cultivate, from childhood to advanced inquiry.
One of the joys of teaching at St Stephen’s College is that our old students return to us (winter is their chosen season), and when they come back they bring to us new ways of looking at the world, in particular the practice of physics. On one such visit, which I particularly remember, I asked the visiting old student, whom I knew to be a very thoughtful person, to say a few words to the students present, requesting him especially to reflect on what he had learnt at various stages of his career as a student. (He was then doing his PhD.)
He said, “At school I learnt how to solve a problem in one way; at St Stephen’s I learnt how to solve the same problem in many ways; at Cambridge I learnt how to work on my own; at Princeton I learnt how to discover a problem.” This has become one of my stories. I tell it – usually in the classroom but also in conversation with colleagues – to illustrate various aspects of the process of learning. I tell it, for example, to students looking satisfied at having solved a problem, to emphasize how important it is for them not to get stuck in a single approach. I tell it to let students know that the spoon-feeding to which they became accustomed at school, and of which they are not freed at College, could become a serious handicap later. But I tell it most often to drive home a point that is easy to miss in the classroom and the lab – that the knowledge of physics is not enough for the practice of physics.
The Practice of Physics
If I had to present my understanding, somewhat idealized, of the practice of physics, I would say that it has the following four stages: discovery, formulation, solution, and return. Stage 1, discovery, consists in sensing the contours of a scientific problem in a phenomenon. In stage 2, formulation, the scientific problem is removed from its physical moorings and placed within the fecund matrix of physics. Experiments are designed and performed, to isolate the parameters that describe the physical situation. It is described by a mathematical model that fits within the framework of theoretical physics (sometimes, at the highest level of practice, the framework is modified or enlarged in this process). In stage 3, solution, a path is opened up from the general mathematical model to particular solutions that can describe physical situations in terms of measurable parameters. This stage often involves the use of a suite of very sophisticated mathematical and computational techniques, which can take several years to learn. In stage 4, physicists return from the abstractions of the mathematical model to the physical situation in which it originated, and at the same time a whole family of other possibilities, including perhaps some not dreamt of in their philosophy. Experiments are performed to corroborate and test what the model predicts, and to explore what it hints at. The four stages do not necessarily unfold in sequence: stages 1 and 2 do usually precede the others, but stages 3 and 4 usually happen together (and are separated here only for the purposes of this essay). Furthermore, the later stages feed back into the earlier ones, and the process keeps correcting itself. Theoreticians often start at the junction of stages 2 and 3, and, after going to the end of the process, try to imagine what the beginning might have looked like.
A Thought Experiment
A good example of the practice of physics can be seen in a pedagogic exercise that was carried out by a teacher in classroom full of 10-year-olds. (This is a true example that I heard about in a talk on Youtube.) She handed out a number of weights attached to strings and asked the kids to play with them. Then she asked them to try to figure out what the time period of oscillation depended on. There was wild debate in the class, with various factions taking different positions – the weight! the length of the string! the shape of the weight! the amplitude of oscillation! gravity! the wind! At this point the teacher calmed the children down and asked them to do a series of experiments to test, one at a time, the parameters that the time period really depended on. Very soon the kids arrived at the astonishing result that the period of the pendulum does not depend on the weight or the amplitude of the oscillation (two parameters that seem so obviously connected with the motion), that in fact it depends only on the length of the pendulum, with longer pendulums having longer periods. What we see is the unfolding of stage 1 and a little of stage 2 in the practice of physics. But the 10-year-olds, while being full of the qualities that make discovery possible – curiosity and enthusiasm and freedom from regimentation – lack the technical skills necessary to go further.
Let us imagine that one of the 10-year-olds, say Chirag, has a sister five years older (let’s call her Gargi). They talk about the pendulum experiment over dinner, and Gargi takes the idea back to school with her. (This happens long before the era of the Internet – so she doesn’t automatically google.) Their teacher suggests that they repeat the experiments done by the 10-year-olds. The students are a little reluctant to repeat what’s been done by their juniors, but the teacher points out that they can do a much better job of understanding the phenomenon, because they’re more mature and know certain techniques that the 10-year-olds don’t, in particular something they have just learnt – graphing. They can characterize the dependence of the time period of oscillation (T) on various parameters, such as the length of the pendulum (l). It’s clear that T increases with l, as discovered by the 10-year-olds, but what the 15-year-olds discover is that the T-l curve is not straight. What kind of curve is it? They argue over it for a while. Then Lilavati, who’s been quietly watching the others, especially the boys, shouting at each other, points out that the longest pendulum in their sample, which is about four times as long the shortest, has about double the time period; could it be, she asks, that T depends on the square root of l? If so, shouldn’t they be plotting the square of T versus l if they want a straight line? There is some initial resistance. Why should we plot T2 when what we’re observing is T? many ask. But after a small intervention by the teacher, the kids decide to try out Lilavati’s suggestion – and, sure enough, T2-l is a straight line. So Lilavati was right – T depends on √l, not on l. They’re ready to move on.
Now they try to understand why T seems to be independent of weight. The teacher briefly explains that weight and mass aren’t the same thing – that weight is mass times the acceleration due to gravity (g). Could it be, she asks, that T depends on g even if it does not depend on mass? If so, what experiments could they perform? Was there any place in school where their weight might suddenly change? She asks them to imagine standing on weighing scales. Under what conditions would the reading change? What if the floor suddenly fell away under the scales? A lift! A lift! shouts a little boy that everyone calls Tiny. The class runs to the physics lab, borrows a weighing machine, and Tiny and a couple of others rush to the school lifts and keep punching all the buttons until one door finally opens. Of course they’ve forgotten to bring a pendulum, but Miss has followed them with a smile and pendulum. Tiny is given the honour of standing on the weighing scales while holding the pendulum. Gargi times the oscillations. They take the lift all the way to the fourth floor, then down to the ground floor, and back up to the second, where the entire class is waiting impatiently for them. What do the find? It turns out to be a little difficult to do the observations, since the reading on the scales changes only as the lift starts and stops, but still, it looks like T increases when Tiny’s weight decreases, i.e. when the lift starts downward, and decreases when his weight increases. By this time, the principal of the school has arrived to deal with the tamasha by the lift. He sends the students and their teacher back to the classroom. So they decide to test other things.
They still can’t quite believe that the time period could be independent of the amplitude; so they embark on a series of carefully-timed experiments in which the time period is measured for a range of amplitudes. They discover that there indeed is a very slight increase with amplitude; they need to repeat their experiments several times, and get better stop-watches to determine this increase, and in the end they’re convinced that it’s there. But, try as they might, they are unable to figure out how exactly T depends on amplitude; characterizing this dependence is not easy; even Lilavati is unable to suggest anything this time.
So they move on to the relation between T and m, the mass of the bob. It just doesn’t seem possible that T should be independent of mass. So they try a large range of weights, and they begin to see some strange and interesting results. So long as the weight is small in size, much smaller than the length of the string, its magnitude makes absolutely no difference to T, but if it happens to be large in size, then it does seem to make a difference. Once again, the dependence of T on the distribution of mass is highly non-obvious. How does one even talk about distribution of mass? – how is a ball different from a rod of the same mass? At this point, Meghnad, another quiet student, says – In that case, let’s just see what happens if we use a metre rod – at least it looks simple. How are you going to hang it from different points? – it doesn’t have holes. Hmm, maybe I can just hold it between my thumb and forefinger and let it oscillate. Of course there’ll be lots of friction, but it should still oscillate for a while. They do the experiment as Meghnad suggests. And this time they get a crazy, utterly incomprehensible result: T is the same at four points on this pendulum! They try a few other solid objects, and the results become even more complicated. They’re lost; they tell Meghnad that his crazy ideas have only confused them – that they’ll never listen to him again. We’re still at stage 2.
To move up to stage 3, let us enlarge the conversation at the dining table – let us add an 18-year-old cousin, say Tathagata, who recently joined St Stephen’s College to study physics. He and many of his classmates dream of becoming great physicists. They enjoy the idea of physics. They know about Newton’s laws of motion, about the inverse-square law of gravity, about differential equations, and a lot more. They bring together the law of gravity and the second law of motion. They see that the differential equation is non-linear and therefore outside their ken, but their teacher points out that for small angles sin ϑ = ϑ. On trying this out, they find that the resulting differential equation, which should hold for small-amplitude oscillations, is linear. They know how to solve this equation; they have learnt the technique in their mathematical-physics course. They see that it has sinusoidal solutions with angular frequency √(g/l): thus they show that the time period of a pendulum is 2π √(l/g). What they’d found about the dependence of T on l was right; and now they see how T would have been found to depend on g if they’d been able to do the necessary experiments. They extend their theory to include a damping term. It becomes possible to determine how the time period changes as result of damping. And they see that if damping is sufficiently large, there comes a critical point beyond which the motion of the system is not oscillatory any more – something they hadn’t even imagined. Later their teacher shows them how to handle the case of large-amplitude oscillations as a term-by-term series expansion beginning with 2π √(l/g). It turns out, furthermore, that there is a way to characterize the distribution of mass in pendulums that are not simple – thus the mystery of Meghnad’s pendulum, and of all pendulums whatever their shape or size, is also solved. And oscillating pendulums, they finally realize, are just examples of systems hovering around equilibrium, all of which behave in the same way. Then their teacher shows them how, when two or more pendulums are connected by springs, they can find hidden simple harmonic oscillators behind the now not-so-simple systems; and when he finishes dramatically by repeating the claim in some fancy book, that physics is the subset of all problems reducible to coupled oscillators, they feel very chuffed indeed.
With a single mathematical model of a simple phenomenon – drawing on a universal law of physics that describes all classical motion, another universal law of physics that describes gravity, and an adequate representation of damping – these 18-year-olds have been able build a model that produces, for the behaviour of any pendulum, indeed for any system slightly out of equilibrium, a coherent, convincing, comprehensive, and profound explanation, and one that is, in addition, not just beautiful (in an austere way, no doubt) but simple (if one looks it right), and that leaves them at the threshold of a universe of possibilities. They, and their collaborators – Lilavati, Tiny, Gargi, and Chirag, and all their classmates – have gone through the four stages in the practice of physics: they have returned where they came from, but they have risen above it.
At the end of their first year in college, some of the students – the cousin at the dinner table among them – go to various research institutes to do summer projects. One of them works at NCBS, the National Centre for Biological Sciences, with a physicist turned biologist. The first question she asks him is whether he understands the damped harmonic oscillator. It’s very important in biological systems, she says. Of course! he says, I know it well. Do you understand where all the terms in the equation come from? Yes, I’m pretty sure I do. What about the damping term? Yes, that comes from the velocity-dependent force. Tchch, I mean what’s the physical origin of the velocity-dependent force? Just saying it’s velocity-dependent is not saying much. Well, it’s because of the collision between the air molecules and the object, or because of viscosity if the ambient medium is a liquid. All right, we’ll examine that more closely later; but tell me this, young man: if the oscillator is damped, shouldn’t it come to a halt after enough time has passed? Yes, it must – it does. No, it doesn’t? No? – then what happens? It comes into thermal equilibrium with its surroundings. Thermal equilibrium? Yes, the time averages of its potential energy and its kinetic energy approach kBT/2 each. That’s not obvious in a large pendulum because the corresponding motion is too small to observe, but in a microscopic pendulum you would see it very clearly. Really? – Wow! So the equation we wrote was not right? Well, it was incomplete – you needed to add a noise term: it should be a stochastic differential equation. A stochastic differential equation – what’s that? Come, you’re going to find out, and then you’re going to help me solve this problem I’ve been struggling with….
Does it happen like this? No, of course not. Most 10-15-year-old kids in India aren’t lucky enough to have teachers who can guide them in a process of discovery of the kind described above. (The first story here is true, but it didn’t happen here.) Most dinner-table conversations don’t bring together the buoyant, unfettered enthusiasm of a 10-year-old, the somewhat cooler eagerness of a 15-year-old, and the technical skill of an undergraduate. Furthermore, the structure of the narrative above gives us the impression that the enthusiasm and eagerness of the discoverers in this story, the 10- and 15-year-olds, has somehow been communicated to the solvers, the undergraduates. In fact that does not happen most of the time. We see the possibility of that in the summer project experience recounted above, but the college classroom is quite different.
For the vast majority of those studying physics, the burthen of undergraduate education, and indeed also a significant part of graduate school, consists of two parts: first, an introduction to the great paradigms of physics – usually removed from the tortured history of their discovery, and instead recast for greater elegance and ease of comprehension –, and, second, training in the vast panoply of experimental and theoretical methods that stages 2 and 3 in the practice of physics require, again almost always through examples carefully refined to highlight method.
Why is this so? There are good reasons. The repertoire of the practising physicist is so enormous that it would be impossible to teach everything as if it were being discovered for the first time. When it comes to the great paradigms, the way they came into being for the first time is often so fraught with false starts, futile detours, dashed hopes, and sustained misunderstandings, that most students learning physics historically would be filled with despair, and perhaps never really understand the meaning and ramifications of the paradigms, which is what matters most when they are being used. When it comes to methods, removal from context happens almost naturally when one tries bring out the features and power of a method: since many of these methods are intricate and complex, and since a real situation may require the use of several such methods at the same time, it makes sense to remove them from reality and embed them instead in artificial situations that highlight one method at a time with the greatest clarity and force.
However, I don’t think that is the entire truth. It is also true that it is often more satisfying for teachers to teach, and for students to learn, ideas that have been cleaned up and pared down. Teaching and learning become easier in the short term, but the long-term consequences of an excessively hygienic exposure to physics may be quite different. It may produce masters of the entire spectrum of known techniques – who top every exam, are highly regarded by their peers and teachers, gain admission to the most prestigious universities –, masters who, faced with the unforeseen and unknown, lack the conviction and courage to find their way through it. Their beautifully trained systems – like immune systems allowed to grow without adequate exposure to germs – may fail in a crisis or, worse still, leave them ill-equipped to embrace confusion and error, the stuff of real discovery.
The question therefore arises: Should the undergraduate training of a physicist include more than just the imparting of ideas and methods? Is it necessary – would it be useful? – to include a psychological preparation for the uncertainty that appears to be a necessary part of the act of creation and thus a necessary prelude to discovery? I believe it is.
I am sure there are there are young men and women who are naturally endowed with the self-confidence needed to leap into the unknown. But I am equally certain that there are others who can be helped to develop that self-confidence. One tends to regard self-confidence as innate, one of those things – like “intelligence” – that cannot be improved. I myself have a strong tendency to think like that. Yet, twenty years of teaching convince me otherwise – that, given the right conditions many otherwise unwilling to risk the unknown will take the first steps into it, and, once they find their way through, break free. I know that this view is contrary to that of many scientists; having had to fight for their ideas, they believe that those who don’t have the guts don’t deserve to be helped. I too believe that – except that I believe that guts is not innate and immutable: it can be discovered and helped to grow.
I watch a programme on the Internet called Île en Île, in which Francophone writers from all over the world are asked about their writing lives. Most of them are men and women of colour, often belonging to former French colonies, and brought up on French classics written by dead White men. In an interview that I watched recently, the writer in question said, “I didn’t think it – writing, telling our stories – was possible for us: I grew up thinking this was something that they did, not us.” I am sure she speaks not just for coloured, Francophone writers but for the entire world of those dis-advantaged by pre-existing perceptions of their possibilities, which are shaped not just by their innate qualities but equally by an ecosystem of history, culture, and politics over which they have little control.
What this writer discovered in her world I will assume in mine – that there are students out there who, given the right conditions, can develop the courage to be creative.
Some Lessons and Suggestions
The search for clarity and order is certainly one of the driving forces of physics, but it is not easy for undergraduates, or even beginning graduate students, to appreciate that this order, if it is to be found by them, must lie on the other side of disorder. A story Steven Weinberg tells in his Four Golden Lessons for aspiring scientists illustrates this point well: “Another lesson to be learned … is that while you are swimming and not sinking you should aim for rough water. When I was teaching at the Massachusetts Institute of Technology in the late 1960s, a student told me that he wanted to go into general relativity rather than the area I was working on, elementary particle physics, because the principles of the former were well known, while the latter seemed like a mess to him. It struck me that he had just given a perfectly good reason for doing the opposite. Particle physics was an area where creative work could still be done. It really was a mess in the 1960s, but since that time the work of many theoretical and experimental physicists has been able to sort it out, and put everything (well, almost everything) together in a beautiful theory known as the standard model. My advice is to go for the messes – that’s where the action is.”
The question for those of us involved with physics education is – How do we inculcate in our students this readiness to dive into the mess? Would it help to present the paradigms of physics always in their own history, thus revealing the difficult paths that led to them? Probably not: it is important not just to know how something was discovered but also to re-imagine it. Furthermore, physics re-imagined is always clearer than physics discovered, and helps faster and clearer learning of the basics – which is important because of the sheer amount of material that an average physics students must learn to reach the stage of research. Nevertheless, a certain exposure to the history of physics – the difficulties, surprises, and serendipities encountered in making discoveries – is, I think, an essential part of physics education. Here is what Weinberg says on this, in the above-mentioned article: “Finally, learn something about the history of science, or at a minimum the history of your own branch of science.” He then adds: “The least important reason for this is that the history may actually be of some use to you in your own scientific work”. While a knowledge of history may not be directly useful, it does, I think, provide a psychological preparedness for the reality of research that is indeed useful.
Knowledge of history, however, is by itself far from enough. The fact that discovery involves falling into confusion before emerging from it needs to be communicated more effectively and through methods that allow students to experience some of that confusion themselves: the textbook, the classroom, the lab must all become arenas for discovery rather than just the learning of paradigms and methods.
The vast majority of problems in undergraduate physics textbooks, even very good ones, are toy problems, which may very effectively teach certain skills but do a poor job of communicating what real physics is like. I am not suggesting that they should all be replaced by discovery-oriented problems: the vast majority of the toy problems are indeed very useful, and necessary, but I think it might be useful for students, after they have done twenty such problems, to try one or two in which they need to find a problem in the fog of a physical situation, formulate it, solve it, and then return to reality. This can happen very naturally if opportunities for serious research by undergraduates exist in an institution. (We have been seeing this for example at St Stephen’s in the last couple of years.) Unfortunately, such opportunities are usually restricted to a few students. For the others, i.e. the majority, it may be useful to assign small projects that are not excessively well-defined and thus give them some taste of the various stages in the practice of physics, and that take their minds away from the regimentation of the classroom.
The centre of interaction between teacher and student, however, remains the classroom. This space is undergoing a revolution in some parts of the world: peer instruction, the inverted classroom, MOOCs, all of these are removing the teacher from his position of authority and making learning more self-driven and democratic. The inverted classroom in particular – where students watch their lectures at home, and in the classroom solve problems, work through difficulties, pose questions, etc with the help of one another and the teacher – promises to be a long-overdue transformation, one that will finally make students the prime movers in education. This will undoubtedly give them the confidence necessary for discovery. (It is not clear that the inverted classroom will come to India any time soon, except perhaps in institutions like the IITs.) None of these changes are by themselves necessarily discovery-oriented, but some of them, like peer instruction and the inverted classroom, are more likely to build the willingness to take risks that is necessary for discovery. Even the traditional classroom leaves plenty of space for independent thinking and exploration if the teacher refuses to play the role of an omniscient and unimpeachable guru who must answer every question posed and clear every doubt that arises.
In physics we have not just the classroom but also the laboratory. The lab can easily be turned into a space for discovery, so long as the only concern of teachers and students is not simply to get the experiments done. The lab offers – far more naturally than the classroom – opportunities for making mistakes and suffering the ensuing confusion, and thus for eventually finding one’s way out of it. Unless lab work is completely vitiated by “perfect” set-ups requiring no more of the students than the pushing of buttons, it automatically helps them develop the very valuable skill of trouble-shooting. The lab, if effectively used, can become a matrix not just for experimental skills but also, interestingly, for theoretical skills. In India, for example, physics labs generally don’t have enough equipment for all the students to do the same experiment at the same time. This makes it difficult to ensure that every student already has the theoretical background to do the experiment allotted to him or her. This is usually seen as a disadvantage, and indeed for most students it is; however, it offers an advantage which our best students have often profited from: being unfamiliar with the theoretical background to the experiment allotted to them, and finding the “theory” provided in most Indian manuals incoherent and woefully inadequate, they are forced to figure out a lot of things by themselves. Just recently, in fact, a very theoretically-oriented student told me how he and a few others had had to work out the theory behind a particular experiment almost entirely on their own, and how they’d learnt greatly from this process.
Another forum in which the idea of discovery can be very effectively brought out is something like a physics society, a student-driven body that organizes talks and encourages discussions. Former students doing research and current students who’ve done summer research projects can be encouraged to talk about their experiences at a forum like the Feynman Club of the Physics Society of St Stephen’s, bringing back into the predictable world of undergraduate physics a whiff of the spontaneity that is a part of all discovery.
Of course, for all these things to happen – for labs, classrooms, physics societies, textbooks and other spaces in a typical physics department to turn into zones of discovery – there has to an acceptance by those in authority – teachers, administrators, scientists guiding summer projects – of the idea that confusion is as important to pedagogy as clarity. It is not an easy to get this point across. Several years ago, I asked a number of students who had just finished their first year in college what their experience so far had been. One of them said something that I’ve never forgotten: “Sir, when I got into St Stephen’s I thought this place would be great: I thought the teachers would be wonderful – I thought that I would just sit back in class, and understand everything.” I was stupefied – not by his thinking of us teachers as less than wonderful, but by this excessively clear statement of what he thought learning meant. Unfortunately, his understanding of learning as a completely passive, utterly confusion-free process is very widespread, and not just among students. There is very little in the Indian schooling system that allows students to imagine that they can make discoveries, and that to do so they must be willing to enter into confusion before emerging from it. I am not sure how school teaching in India can be changed to inculcate the qualities that that discovery requires – curiosity and willingness to take risks, above all. But even college is not, I think, too late. It thus becomes incumbent upon those of us who teach in undergraduate colleges to allow this to become a possibility, instead of imagining that our task is restricted to imparting knowledge and clearing doubts. If students are to develop the independence, resolution, and patience to walk by themselves through darkness into light, teachers must develop the wisdom, maturity, and patience to allow the process to unfold – not always, certainly, but some of the time. And it is equally incumbent upon research scientists who take on undergraduate students for summer projects to allow them to struggle through some of the uncertainty that they know precedes discovery, instead of taking the easy way out and leading them through everything, or, worse still, just helping them learn another technique or method that they haven’t learnt in college.
The Heart of the Matter
All of us have a strong tendency to run away from confusion, to stay close to what is clear and certain. We associate confusion with error, and see error as evidence of foolishness, which we are taught from early childhood to avoid, especially in public. It is difficult for our brains to accept confusion and error as the necessary and inevitable foreshadowing of insight and creativity. As a result, we imagine that creative geniuses are creatures who never make mistakes. But that is very far from the truth. Error is not just inevitable, it is essential. One of the clearest statements on this that I have come across is the following passage from Récoltes et Sémailles, by the revolutionary mathematician Alexandre Grothendieck:
La découverte de l’erreur est un des moments cruciaux, un moment créateur entre tous, dans tout travail de découverte, …. Craindre l’erreur et craindre la vérité est une seule et même chose. Celui qui craint de se tromper est impuissant à découvrir. C’est quand nous craignons de nous tromper que l’erreur qui est en nous se fait immuable comme un roc. Car dans notre peur, nous nous accrochons à ce que nous avons décrété “vrai” un jour, ou à ce qui depuis toujours nous a été présenté comme tel. Quand nous sommes mûs, non par la peur de voir s’évanouir une illusoire sécurité, mais par une soif de connaître, alors l’erreur, comme la souffrance ou la tristesse, nous traverse sans se ﬁger jamais, et la trace de son passage est une connaissance renouvelée.
(The discovery of error is one of the crucial moments, a moment uniquely creative, in every work of discovery, …. To fear error and to fear the truth are one and the same thing. He who is afraid of fooling himself is powerless to discover. It is when we fear fooling ourselves that the error that is in us becomes immovable as a rock. For, in our fear, we attach ourselves to that which we have one day decided is “true”, or that which was always presented to us as such. When we are moved not by the fear of seeing an illusory security collapse but by a thirst to know, then error, like suffering and sadness, goes through us without ever hardening, and the trace of its passage is a renewed knowing.”)
The most important word in that passage is fear. We fear making mistakes: we fear doing things that make us look unsure, confused, ridiculous, that make us feel as if the ground beneath our feet were crumbling. Do creative people, those like Grothendieck, not know – do they not feel – this fear? Here is what the great American artist Georgia O’Keefe had to say about this: “I have been absolutely terrified every moment of my life and I have never let it keep me from doing a single thing that I wanted to do.”
- Bikram Phookun
25th August, 2016.