Monday, March 27, 2017

College Physics Taught Without Problem Solving? Preposterous!

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In the most recent issue of Physics Today (March, p. 10), a 2nd year student in astrophysics at University College,  London  saw his roughly 2-page letter ('How to teach me physics: Tradition is not always a virtue') published. While  acknowledging "physics is the most exciting endeavor I can imagine", Ricardo Heras also wrote:

"The basic courses of my first two years were disappointing. They didn't really give me the opportunity to join that great adventure. Most of my lecturers followed traditional teaching approaches based heavily on solving standard problems and learning by rote, with no hint of free inquiry or discussion. They seemed to be convinced we would understand physics through that method. I was not enthusiastic. "

Mr. Heras then went on to complain that while he and fellow students "spent a lot of time and effort solving textbook-style problems" they didn't really understand physics by doing so and stated that he was "mainly trained to use problem solving techniques."  He then quoted Richard Feynman (of the Feynman lectures fame)  who wrote:

"I don't know what's the matter with people, they don't learn by understanding , they learn by some other way - by rote or something"

Let's first note that the Feynman Lectures in Physics (a  3-volume work) definitely  exemplifies the author’s unconventional approach to physics teaching. But even today most physicists I know look at it as an interesting experiment but only use the texts as  supplemental material to their undergrad courses (whether in QM, Electricity and Magnetism or Thermal Physics) but not as a standalone text.  This is understandable because Feynman drifted all over the place, and didn't follow the usual trajectory for teaching physics, e.g. mechanics, heat, wave motion, optics, electrostatics, E&M, atomic physics, and maybe some quantum physics.

Interestingly, Feynman explored some intriguing problems, such as finding how maser states vary when a maser cavity frequency is nearly - but not exactly- equal to the resonance frequency, w o (Cf. Vol. III,  Sec. 9.5 ' Transitions off resonance')  Despite numerous such examples scattered over 3 volumes, there was no supplemental problem set or booklet to accompany the lectures. Many have opined they might write such a set eventually, but co-authors Babcock and Leighton never did. .  In any case, it appeared Feynman himself didn't regard having such problems as being of paramount import, as this student Heras doesn't. Indeed, Heras even quotes David Goodstein from a Feb., 1989 Physics Today piece on Feynman:

"If his purpose in giving them was to prepare classes of adolescent boys to solve examination problems in physics, he may not have succeeded particularly well… . If, however, his purpose was to illustrate, by example, how to think and reason about physics, then, by all indications, he was brilliantly successful."

Heras' own frustrations are evident when he writes:

"The aspects of physics I have understood best so far are those I have studied for pleasure. I understood special relativity better when I derived the Lorentz transformations in a different form. This task was much more exciting than the usual assignment of calculating the length contraction of a rod."

But, of course it would be!   The key aspect as well is that when the student studies physics on his own he can apply the creativity and free inquiry he so often finds absent in the class-lecture setting. But this should not be mystifying. Check any university course catalog - even I suspect University College, London - and you will see course listings by credit hours. These give an indication of the time allocated in class for lectures each week. Also labs may be listed separately with their credit hours assigned (Often 1 cr. hr. but the student is actually in lab for 3 hrs.). The whole point is that the university schedule conforms to a specified time frame.  Administrators, for understandable reasons, want to make sure each undergrad - for example - can matriculate in 4-5 years, not take 10 or 15, which Feynman's "create and think it all out" rubric might require.

So it was easy for Feynman to write (as Heras quotes him):

"The best teaching can be done only when there is a direct individual relationship between a student and a good teacher—a situation in which the student discusses the ideas, thinks about the things, and talks about the things."

But let's face it, Feynman is talking here about one -on -one tutorials!  Of course that's the best teaching! But how the heck are you going to apply that to a class of say 300 first year calculus physics undergrads - and that's for one section sitting in an auditorium?

Let's also not fool ourselves that problem-solving in physics is not critically important and often discloses how a student is able to think his way through a problem based on using known principles (not "rote").  Conducted with the most interesting, thought absorbing problems, problem solving can be a boon to inculcating physics principles. The object then is not to move away from problems (given they are used at every stage to gauge whether the student can advance - see the end of this post) but to craft better problems!

One such problem  I've given as part of a 2- year General Physics course, is shown below:
A group of 4 astronauts lands on Mars with solar radiation collection material of total area 2000 m 2 . If the efficiency of the material is 30%, and the ambient night time temperature on Mars (for their base location at Isidis Planitia) is -40 C (10C day time), will they have adequate collecting material if the solar constant on Mars is 620 W/m2 ? (Assume insulating material with a thermal conductivity of 0.08 W/mC, and a need to keep the inside area of their domecile at least at 10 C, requiring solar radiant energy collected of at least 1,200 W per minute for an area of 10 m x 10 m.)

Estimate the thickness of insulating material they're likely to need in order to make it work. Comment on whether this expedition is even feasible given the limits of their materials, and that no more than 100 m
3  of insulating material can be taken.
The preceding problem clearly makes use of physics principles which the student needs to know to arrive to the solution.  Also, it is clear the student can't just solve the problem by "rote", or by "finding some appropriate equations, putting them together, manipulating them algebraically."  In other words, problem solving need not be mutually exclusive with free inquiry. In fact, a homework problem can afford the opportunity for such inquiry that the limits of course and class structure don't.   Hence, I usually blame physics lecturers for offering uninspired problems for homework and tests, as opposed to creative ones that force the student to go beyond the rote or plug-in paradigm.

While one can sympathize with Heras’ poignant pleas for more “creativity” in physics teaching (especially at the undergrad level), the fact remains that the entire current structure of physics education is founded on mastery of content, as reflected in tests taken at various stages. These determine whether the student is qualified for promotion and even admission to the gateway for ultimate passage (the Ph.D.) which depends on passing a series of comprehensive examination.

To modify this didactic structure in favor of creative in- class learning simply wouldn’t accomplish the goals of physics departments as they are presently structured. For one thing, the time consumed for such learning would surely be much greater  than for the current lecture-lab format. Of course, one could assign projects such as I have during my physics teaching career in the 1980s- early 90s, but this is outside of class time. Hence, it does not facilitate learning by supporting independent student creativity in class.
What I have done, to a limited degree, is allow students - such as in general physics, calculus physics or space physics classes I've taught - to design some of their own labs. The design can be presented as a kind of "thought" experiment in the first instance, and then followed up by providing the specific apparatus that would be technically needed to carry it out.

For example, consider the design of an experiment to allow the student to simulate a "subflare", for which I have used the following:

Some solar flare models are based on 'equivalent inductive circuits' in which the circuit is suddenly interrupted or broken when the switch is opened, e.g.

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 When the circuit is broken the collapsing flux through the coil tends to maintain the current  I o hence will generate a spark at the gap if the switch is opened. Suppose the current is rising in simple circuit with a coil, a source of emf, a switch and an inductance, L. Let the current in the simple circuit rise at the rate dI/dt per second. If L is the circuit inductance then the back emf is:

Eb = L (dI/ dt)

The rate at which work is done vs. the back emf is:

 Eb I = LI (dI/ dt)

How might you use this to design an actual  circuit to illustrate how a solar flare occurs via sudden "circuit breaking"? List all the components needed and the specifications.  How would you estimate the magnitude of energy released? How would you re-design the circuit to prevent sparking , i.e. original energy stored in magnetic field of coil now stored in electrostatic field of capacitor? (This latter would be analogous to a double layer in a solar coronal loop which stores excess magnetic free energy. Thus, if its capacitance is large enough  the potential difference across it - and hence across the switch - never rises high enough to cause a spark, or flare in the case of the loop circuit configuration). 

Followup problem: In your lab experiment design, suppose a 1 A current is to be broken (without sparking) in a circuit with self -inductance 1 henry. Find the maximum threshold p.d. across the capacitor, beyond, i.e. which cannot be exceeded. Thereby find the least capacitance that can effectively connected across the switch.

This exercise not only tests the student's creativity and free inquiry skills in simulating a subflare in an inductive circuit, but also how such a flare can be "stifled" under the appropriate physical conditions.  It also addresses Heras' complaint that:
"Traditional teaching methods urge us to perform standard calculations that rarely spark our creativity. Being immersed in such teaching, I feel trapped in a labyrinth whose exit can only be found by solving a ton of mostly uninteresting textbook problems."
Perhaps Heras would have been more at home being challenged in my space physics labs, where free inquiry was given plenty of leeway. In space physics the student is introduced early on to the importance of the Earth's magnetic field, and in particular as the basis for the magnetosphere - on which the aurora depends. In another variant of the earlier experiment, I will set out the following materials  for a space physics lab with no specific instructions for assembly or application: a rectangular coil of at least N = 100 turns mounted on a light wooden frame, a square wooden base 10 cm x 10 cm, two set screws, additional wire, wooden stops, other assorted screws, pins, selected apparatus including flat needle pointer.  If assembled correctly it will appear in finished form below:
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The student is then required to construct a working ballistic galvanometer which he or she will then use to find: a) the magnetic flux φ  linking the coil of N turns, the angle of 'dip' and c) the ratio of the vertical component ( B v  ) of Earth's magnetic field to the horizontal component  ( B H ).   An illustration of the quantities in terms of the coil orientation is shown below, where d  denotes the angle of dip:
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The preceding examples are intended to illustrate that there is scope to interject inquiry and creative aspects into labs as well as homework problems. However, I believe it is unrealistic to expect entire classes to be devoted to free inquiry or creative learning via exclusively "first principles" understanding - by which I mean the student is responsible for all or most of the creative effort to learn all his physics "first hand" as it were - with zero outside input.  If we had all the time in the world, or at least more than 4 hours per week for lectures, 3 for labs, plus seven or eight years for physics students to graduate - that might be fine.

But seriously, what physics department today could even remotely entertain such a class or mode of subject delivery?  It would require vastly more time and resources, as well as a totally radical rethinking of physics pedagogy and would come up against the existing system for promotion and qualification, not to mention how we integrate students into the formal university course system. I am not saying it could never work, only that free inquiry and creativity have only limited scope as current physics departments are designed.

Perhaps the optimal time for such exclusive student pursuit of free inquiry is when Heras proceeds on to the pursuit of his Ph.D.  Then, by selecting a problem of inherent appeal, he can develop lines for  original expression of his curiosity, creativity and inquiry not readily available in standard courses. But to expect beginning physics students to do this in more than limited and controlled settings is, frankly, ludicrous.

Of course, to reach that ultimate Ph.D. inquiry point he will have to pass his Ph.D. comprehensive examinations, and that will entail solving a lot of  difficult “traditional” problems! Generally, there will be six exams in six subject areas: classical mechanics, thermodynamics, statistical mechanics, mathematical physics, quantum physics, and electromagnetic theory. These will generally be four hours each.

One of the problems (of five) for the classical mechanics comp given at Univ. of Alaska Fairbanks is shown below:
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Even before crossing that threshold, he will need to get through the physics GRE subject exam, which consists of a solid 100 problems to be done in 3 hours. See e.g.

True, the problems are multiple choice (5 options) but that doesn't mean one can race through them. Consider the example shown below:

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The diagram (Fig. 1) shows a resting cylinder of weight W. The coefficient for static friction for all surfaces is 1/3.  The applied force for P = 2W
61)The distance d for which the counterclockwise motion is initiated by P is:
A) d = r/3    B) d = d/2   C) d = r/ 4  D) d = 2d/ 5   E) d = 3r/ 5

62) The vertical reaction force at point A is:

A)    0.3 W  …B)0.5W…..C) 0.8W…. D) 0.9 W…..E) 1.5W

63) The vertical reaction force at point B is:

A) 3W…..B) 2.7 W……C) 1.5 W…..D) 2.1 W….E) 2.5 W

64) The horizontal reaction force at point A is:
A) 1.5 W…..B) 2.1 W….  C) 0.9 W……D) 1.2 W……E) 1.8 W

65) The horizontal reaction force at point B is:

A) 1.5 W…..B) 02.1 W…..C)  0.9 W….D) 1.2 W…..E) 1.8 W

My point in showing the above? Problems form the core for determining physics advancement at various stages. If you don't like working problems, consider them "too traditional", or whatnot, then physics may not be the "endeavor" for you.

Final note: The Scientific American blog also discussed Heras' letter but framed it in terms of "individualists" vs. "collectivists", i.e. in terms of research bent. Thus, the "individualist" does his own research and publishes his own paper, while the "collectivist" is part of an ensemble of authors - maybe 5 to 15 - who each contribute part of the overall work. But this is the wrong emphasis. No where does Heras even mention research. His complaints are with undergrad physics education and excessive use of traditional problems. His preference is for more "free inquiry" than problem solving, not realizing they need not be mutually exclusive if the problems are designed properly.  As far as research goes, free inquiry (and problem solving) are part and parcel of the process whether one is part of a team ("collective") or on his own ("individualist"). Hence, the SciAm blog mixes apples and oranges in trying to parse Heras' complaints.

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