Fusing Modern Physics With a Dated Curriculum

Enrico Uva

An "open letter" in the form of a youtube video to the president regarding physics education complained that most of the physics taught in high school is at least 150 years old. It does score some good points in arguing why modern physics should be included in a basic course. How can students accept the notion of a dense nucleus consisting of positively charged particles without having any knowledge of the strong force? How does carbon 14's beta decay make any sense without weak interactions? And twin paradoxes aside, without relativity we have no idea why gold has its color, why thallium is poisonous and why mercury is liquid.

But as Chad Orzel points out, it's unfair to paint the older physics as "boring", especially when a Newtonian approach is still very useful at the macro level. He also points out that getting people all fired up about "cool modern physics" à la Feynman or Tyson is a different business from showing them how to actually do and use physics.

In Britain, a couple of physics professors from Nottingham, while pleased to see some inclusion of modern physics in the prerequisite A-level courses, would like to see more. At the same time they realize that there's the risk of packing too much content into two years, not allowing students to properly grasp concepts. They mainly complain about how physics freshmen seem to struggle with math, partly as a result of physics-prep courses shying away from the use of calculus to derive basic physics concepts.

From Quebec to Japan, there has been an unfortunate downplay of algebra in high school math courses. I'm honestly clueless about what to do with a math curriculum for non-science bound students, but for those with an aptitude for math, science and engineering, a derivational approach with a barrel full of algebra leading to calculus should be the norm.

It's ill-advised to expose those students to a mish-mash of topics in middle school and to a full grade 10 chapter of obscurities like the step-function.

The modern world is full of choices, spinning adolescent-heads in every direction. Ideally if there would be more young people who knew what they liked at an earlier age, they could take less courses and focus more on specific subjects. As for the danger of not creating well-rounded individuals, well that's what summers, winter vacations and spring break could be used for.

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As for the danger of not creating well-rounded individuals...

That danger has long since passed and become the norm. The average individual graduating high school can't write very well, their spelling is atrocious [even with spellcheckers], and their mathematical skills generally terminate at the fourth grade level [basic fractions].

We're not likely to make up that deficit with a few humanities or general arts classes at the university level.

Gerhard Adam | 01/21/13 | 17:41 PM

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The average individual graduating high school can't write very well, their spelling is atrocious [even with spellcheckers], and their mathematical skills generally terminate at the fourth grade level [basic fractions].

That sounds pretty balanced to me.

derek_potter (not verified) | 01/26/13 | 06:13 AM

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If I had to give a one and a half hour motivational lecture to high school students with a modicum of calculus (knowing the basic idea of derivative and integral) how would I do this? I'd start of with the problem of a lifeguard who can run faster on the beach than he or she could swim to a drowning person placed diagonally away in the water. Calculus would come to the rescue with the optimal distance to run along the beach and the optimal distance to swim to reach the drowning person in the least time. Then I'd discuss the generalization of this idea by something called the calculus of variations. If I have a pole of height A and a pole of height B less than A, than what would be the shape of a rope suspended between the top of A and B? The answer is some unique function, just a soap bubbles figure out from the edges of arbitrary, closed wire loops. I'd do a proof using this tool showing that the shortest distance between two points is a straight line, and then I'd tell them about the so-called Lagrangian formulation of physics. I'd show them the Lagrangian for the harmonic oscillator, how the calculus of variations then supplies the equations of motion which admit oscillating solutions--plug in a sine wave do the work in a minute. Then I'd tell show them how to turn their equations of motion into the Schrodinger equation, one historical path to quantum physics from humble classical physics, and I'd tell them that to solve these kinds of equations they'd need to learn differential equations. If I had the time left (and I would try hard to) I'd even tell them that with differential equations come symmetries that underlie the particles and fields that underlie our understanding of the universe. Then I'd had them a forty page syllabus I wrote up taking a reader class by class from college freshman to PhD in either math or physics. I would hope that I'd given them a path with vision of the "why" and the "how come", and that it is important to study the unsexy stuff of potential and kinetic energy and rotational physics, etc., because, as with my harmonic oscillator example, this stuff underlies the deep, sexy stuff like atomic and nuclear physics.

Alas my experience is not with high school students, but rather in teaching material from college freshman through 1st year graduate courses in both mathematics and physics. Students in these fields are currently subjected to dry, disconnected prescriptions devoid of the "why" and the "because" of what they are learning.
Like everyone else, I wanted to understand "big picture" theoretical physics, and so I took a lot of mathematics (a 36 hour non-thesis MS in math) on my way to a PhD in physics. It was mostly a monumental waste of time. I took a lot of undergraduate and graduate courses in (abstract) algebra, differential equations, analysis, topology, etc., and I had no notion that these specialized fields had anything but superficial relations to each other because none of my text books, or the references in the libraries, or any of the my professors ever connected these fundamental subjects. Why not? Because most professors can't, being themselves parrots of crap they don't truly understand. So I fell back on history. Why was (abstract) algebra developed? Because there were problems that needed to be solved, quite often physical problems in crystallography, in differential equations of all types, linear, nonlinear, ordinary and partial (look up Sophus Lie). The PhDs I hung out with at Los Alamos National Laboratory might have read some paragraph about this history at some point in their past, but like myself they had no depth beyond this trivial level of awareness if at all.

At the conclusion of my studies I figured that I understood about half of what I had supposedly mastered in a true, fundamental sense. The other half was prescription knowledge that may as well have been witch doctor incantations. If you see a differential equation of type X, then apply method Y. Why? Because it works. This is not understanding. It's aping, having faith, and just plain deceit.

It doesn't have to be this way. I went back to history and over a period of a decade I was able to create a set of notes tying (abstract) algebra, topology, and differential equations to the physics problems that motivated these and other branches of mathematics supplying not just the how, but the "why" and the "how come" without which students can't have a basis of understanding. The notes take you from the sophomore level with a year of calculus and calculus based physics, step-by-step, example by example to the foundations that mathematicians and physicists need to get through graduate school, e.g., the various paths to quantum physics, quantum field theory and general relativity from classical physics and the concomitant developments in algebra and topology by Sophus Lie and others to the fundamental treatment of the equations of physics, and the extraction of their symmetries. It is more untrue than true that math and physics are too developed to allow for a strong, general understanding--I say it's a lie, and physics and math is killing itself with it.

Alex Alaniz

The notes are at http://www.physicsforums.com/blog.php?b=4342

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A Alaniz

Alex Alaniz | 01/22/13 | 01:52 AM

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If you see a differential equation of type X, then apply method Y. Why? Because it works. This is not understanding. It's aping, having faith, and just plain deceit.
It doesn't have to be this way....
It is more untrue than true that math and physics are too developed to allow for a strong, general understanding--I say it's a lie, and physics and math is killing itself with it.

Very good points, among many others. Thanks for your thorough comment.

Enrico Uva | 01/22/13 | 09:26 AM

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Much to agree with there, but on one point, I must stand and defend:

Why was (abstract) algebra developed? Because there were problems that needed to be solved, quite often physical problems in crystallography, in differential equations of all types, linear, nonlinear, ordinary and partial (look up Sophus Lie).

I have taught history of mathematics, and I would say that more often than not the mathematics, especially abstract algebra, was developed for its own sake. The time of Euler is my favourite, but the 19th century is also well on my map. Abstract algebra (groups, rings, etc) was developed by mathematicians without particular physical applications in mind. The infinite groups brought to light by Sophus Lie are so fundamental to particle physics, but Lie himself seems to have had little do with physics, and his major work was done before particles even started to be discovered.

One exception to the general rule is vector analysis of the type introduced by Gibbs and Heaviside to describe electromagnetism in particular. Differential equations are the strongest exception, and these were very much developed in response to physical problems, such as that of the vibrating string: also Fourier developed his series with reference to thermal conduction.

Back to the mathematics for its own sake – matrices were developed by Arthur Cayley (1821–1895), who in 1858 published “A Memoir on the Theory of Matrices”, but towards the end of the 19th century they effectively went into dormancy. It was the field of quantum mechanics in which they emerged again. Here is how Max Born, in “The Restless Universe” (1935) explaining, in an extremely modest way, how he came to realise that Heisenberg’s quantum mechanics was represented by matrices:

In 1925 Heisenberg put forward a decisive idea; this was seized on by Jordan and myself, who worked out the appropriate mathematics, the so-called matrix mechanics. You may wonder how this came about. A student occasionally goes to lectures about abstruse subjects just for fun and speedily forgets all about them This is what happened to me with a lecture on higher algebra, of which I recollected little more than the word “matrix” and a few simple theorems about these matrices. But that sufficed. A little playing about with Heisenberg’s physical formula showed the connection. Then it was an easy matter to refresh my memory and apply the results. This form of quantum mechanics, which was also brought to a high degree of perfection by Dirac quite independently, is not only the earliest form of quantum mechanics, but perhaps the most fundamental; but it is so mathematical and abstract that it cannot be made intelligible without the use of mathematics.

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Robert H. Olley Quondam Physics Department University of Reading England

Robert H Olley | 01/26/13 | 16:09 PM

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One of my favorite mathematicians is the late Morris Kline. I especially liked his book: Mathematics, The Loss of Certainty. Maybe I'm being too abrupt, be he thought that mathematicians who claimed to be developing mathematics in a vacuum (for the sake of abstraction) were mental masturbators, ignoring the physical nature that inspired practically all of their mathematical musings.

This is a sidebar argument however. What is more fundamental? Physics or math? Who is more important? Who cares? (I'll let M. Kline 'talk' to those mathematicians who erroneously thought their math was pure.)

The net effect of today's teaching is essentially: math majors learn abstract math with no connection to underlying physics, and hence with great loss of context. Physicists learn math tricks and prescriptive procedures with great loss of foundational maturity.

My MS was 36 hours pure math: algebra, topology, analysis, etc.
My PhD was in particle physics. There was no connection between these two degrees. This is nuts! This is what's wrong with Math/Physics pedagogy.

I knew things about monoids, groups, subgroups, normal subgroups, towers of subgroups. Didn't know how they related to differential equations (ordinary, partial, linear or nonlinear) via Sophus LIe. I didn't know of connections between all of my algebra to the differential equations of physics, their reduciblity, their topology, the connection with matrix groups, the extraction of eigenspectra. Know that I have both (thanks to my own self-education) do I see the full richness of mathematical physics.

I knew things about sigma algebras, Borel sets, etc. Didn't know how any of this related to the relationship between probability and stochastic differential equations when I got into financial physics. My background in statistical physics was useful, i.e., Monte Carlo methods. Slowly did I connect the dots of Rudin and Royden to probability and stochastic differential equations. By the way, discretizing stochastics (diffusion) differential equations for numerical methods gives abstract linear algebra a basis in concrete intuition.

I knew things about point set topology and global topology, but without connection to applications whatsoever, eg. to differential equations--Lie again.

My PhD math friends/acquaintances at Los Alamos have very weak physical intuition. My Los Alamos PhD friends think math is voodoo and are happy to follow prescriptions.

(I typed this up in a rush...should get back to it with more time; bottom line: Math more than physics is killing itself with its current pedagogical methods).

Alex Alaniz (not verified) | 01/26/13 | 19:37 PM

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I’ve just just around to watching that video. I prefer transcripts myself — trying to follow that interview is like trying to keep up with a chicken, a duck and a turkey in conversation (雞同鴨講).

When I talk to someone in the business (in one of our learned bodies) he says that talking to University Physicists is often difficult, because they are particularly interested in selecting out potential university physicists, so that the A-level course (years 11 and 12) is not well suited to the majority of those who would otherwise like to study at A-level.

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Robert H. Olley Quondam Physics Department University of Reading England

Robert H Olley | 01/25/13 | 15:51 PM

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One of my favorite mathematicians and historians of mathematics is the late Morris Kline. I especially liked his book, “Mathematics, The Loss of Certainty”. Putting my interpretation of his thesis in my own words, Morris Kline thought that mathematicians who claimed to be developing mathematics in a vacuum (for the sake of abstraction, free of physical muck) were mental masturbators, ignoring the physical nature just one or two steps removed that inspired the lion’s share of their “pure” mathematical musings.

Not counting my undergraduate courses in “pure” mathematics, I earned a 36 hour, non-thesis MS in mathematics. My analysis training was based on texts by Rudin and Royden, my algebra background based on Fraleigh and Lang. I forget the authors in probability, functional analysis, topology. How much of this pure mathematics could I connect in some tangible way to my undergraduate degree in physics? Not at all. How much of this pure mathematics could I connect to my PhD work in physics? Not at all! Yet my physics readings were filled with talk of symmetries and topologies and Hopf bundles and so on that was nothing more to me than babble. But the situation was actually worse. I thought I had a good notion about ordinary and partial differential equations based on physical grounds. But I had absolutely no notion of how deeply algebra and topology touched these differential equations.

How was knowing about group theory, and normal subgroups at all connected with differential equations? I couldn’t tell you after an MS in math and PhD in physics. Nor could every PhD in math or physics I’ve ever run into. Nor could I tell what connection normal subgroups had to do with the topological concept of the connected component of some domain within mathematics itself, forgetting about physics. This is a criminal state of affairs isn’t it?

I persevered through decades of confusion, piles of textbooks, mountains of literature, before I began to see the unity of mathematics within itself and with physics, and physics began to become unified in my mind as well.

Here I was, a guy with a BS physics, MS pure math, PhD in particles and fields feeling like a charlatan until I ran into the first four chapters of “Lie Groups, Lie Algebras, and some of Their Applications,” R. Gilmore, Dover 2002 AND “Symmetry Methods for Differential Equations, A Beginner’s Guide”, P. E. Hydon, Cambridge 2000. So all this dry, meaningless, symbology in Lang and Fraleigh and in my texts in local and global topology suddenly had contact with physics: how we took different paths (Lagrangian, Hamiltonian, Poisson Bracket, etc.) to quantum mechanics and quantum fields and more in terms of infinitesimal generators, their algebraic structures and topological natures, but let me slow down and bring this into greater relief.

Why is ordinary differential equations taught as botany? Recognize the species of ODE and apply method Y(X). Crap! Robbery! Ditto for partial differential equations. One thing I got out of classes such as these was that occasionally super geniuses get born who somehow see fantastic tricks to solving differential equations. I can ape their methods, but never hope to produce methods of my own. I am dumb. I cannot be a mathematician. I had a sophomore level course in abstract algebra (with the book by Durbin I believe). Why not connect abstract algebra and topology to ordinary and partial differential equations (linear and nonlinear) between the sophomore and junior level for math, physics and even engineering majors? Do you want charlatans, as I was after my MS/PhD, for the next generation of mathematicians, physicists and engineers? Today’s physicists and engineers are pretenders who know how to apply tricks they understand very poorly if at all. Today’s mathematicians don’t know that algebra is tied to topology, and together that these are tied to differential equations. This is not okay.

Teach applied and pure math together with physics—it’s not that much, and it’s highly unified—and you’ll have stronger and more creative physicists, engineers and mathematicians.

Did I ever understand Galois’ work with respect to the insolvability of the quintic? Not really. Not with all that division ring crap. Neither has any other mathematician I’ve ever run into. Then one day I ran into the first chapter of “Lie Groups, Physics, and Geometry” R. Gilmore, Cambridge, 2008 AND the first five chapters of “Groups, Representations and Physics,” H. F. Jones, Institute of Physics, 1998 and for the first time I understood BOTH: Galois theory and the construction of character tables for finite groups with applications to physics. Then from readings of mathematics and its history (and what we need to teach STEM majors) I understood the nature and motivation of synthesizing new, “pure” mathematics: There are limits to what the Greeks could do with straight edge and compass, so add tools, and there are limits to what can be done by radical extension, so add tools, e.g., elliptic functions, and there are limits to the vector potential formulation of field theories, so add connections over principle bundles.,…,etc.

There are dividends not just for the undergraduate. In graduate school my introductory book into quantum fields talked about the parallel nature of the covariant derivative and parallel transport with internal symmetries and geometric theories like general relativity, all suddenly far better understood thanks to Hydon and Gilmore (2002) long after I finished school.

There are dividends in the practical world as well. I hated Rudin and Royden. I liked statistical physics. I got Monte Carlo methods. I got hired as a financial physicist (a quant). While Monte Carlo was great, I suddenly wanted to understand Black-Scholes-Merton theory and martingales. Oh, so that’s what all of that analysis was about—applications to probability theory connected to stochastic partial differential equations! Pretty good stuff to know if you’re a nuclear weapons physicist working on neutron and radiation transport. And all that linear algebra starts to become useful in discretizing differential equations for numerical quadrature, and for finding principle components in high dimensional problems.

My MS was based on 36 hours of pure math. My PhD was in particle physics. There was no connection between these two degrees. This is nuts! It doesn’t have to be this way. My 323 page notes I recently posted are my attempt at this combined with a syllabus to better navigate traditional math and physics pedagogy. I think if a mathematician wants to practice abstract math, he or she is far better served with a more cohesive, fundamental grounding in physics as Kline believed and with a better understanding of the relationships of the various mathematical branches.

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A Alaniz

Alex Alaniz | 01/27/13 | 02:08 AM

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Why is ordinary differential equations taught as botany?

Don't be hard on botany! :) To verify whether classifications based on morphology are valid, a botanist nowadays has to venture in DNA sequencing. Even prior to such an approach, botanists looked for other biochemical evidence. And there are evolutionary aspects, paleo-botanical ones etc.

Enrico Uva | 01/27/13 | 04:51 AM

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I’ve read Kline’s magnum opus (Mathematical Thought from Ancient to Modern Times), and I too have a high regard for him.

Certainly there was a complete disconnect between the maths as taught in our Maths Dept and what was taught to students in our quondam Physics Department.

History of Mathematics went a long way to clearing up things that had stumped me for decades. Simply reading about Mersenne cleared up a misconception about the vibrating string that had trapped me for years, so I could then take on board how the differential equation was developed.

I had a look on Amazon (UK) at the first few pages of Gilmore book you recommended. Certainly it brought home to me that Sophus Lie was trying, so to speak, to extend the work of Galois to differential equations. However, this customer review seems apt:

I just bought this book, and was very enthusiastic to read it as the author seems to have some very good ideas about topics to be covered. His approach to using Galois theory (solution of polynomial equations using group theory) is a great way to motivate the study of Lie Groups. Further, the sections on using the EXP map to recover a Lie Group from its Lie Algebra is covered after a very clear explantion of the real problems that occur and how to get around them. However, the author has really confused himself over the need to teach basic Group Theory first. The definitions of cosets, group conjugacy, character tables are very imprecise and confusing. If you do not have previous exposure to these concepts, all this book succeeds in doing is confusing you. Most of these problems belong to the first chapter. However, as this is a book on Group Theory for Physicists, Chemists and Engineers, this really de-rates the book in my opinion. Also, when the author talks about operations over a complex field, come on, you are teaching group theory to the uninitiated but you refer to a field. This is just off-putting. There is no need to use these terms. This book suffers from the problem that most books on mathematics for physicists at the Graduate Level used to suffer from; the most important concepts are imprecise and the reader is left wondering if she/he is stupid or is missing a big piece of background, which could be easily and concisely provided. It is fair to say that most modern books on subjects such as these are better, particularly those combining classical mathematical methods with more modern mathematics, such as the beautiful work of Stone and Goldbart. It was a shame this book is weak on proper definition because I really like the subjects it attempts to cover in the order it covers them. If you do want to read this book and you have not got a background in mathematics, I recommend reading an elementary book on Group Theory and a book on Group Representations. Ledermann's books on these subjects are still very good, although a little dated. Ian Stewart's book on Galois Theory is very nice reading too.

I have just got Gilmore’s earlier book (ISBN 0471301795) out of the library, and it’s somewhat better in this regard.

What I’ve been after is this: how exactly does group theory relate to particle physics? I know, for example, about double cover, and use this example of the Real Projective Plane RP2 to show how, in one more dimension, SU(2) is the double cover of SO(3). (Note to Brits: even New Zealand is not our antipodes!)

Here is roughly what I said in the lecture:

This is because of the Pauli exclusion principle, which keeps the electrons in pairs in the orbitals of the atoms which make up all of us.

And while SO(3) is the group of rotations in real space, SU(2) is the group of rotations of the space the electron lives in. So if you hear someone talking about an electron having to rotate twice to come round to its original position, that’s what they’re talking about.

This might hit the target with a few physics students, but by then there were only maths ones. But I myself am someone mystified by what exactly is the connection, and so I was really talking as a historian.

Have you come across Yearning for the Impossible: The Surprising Truths of Mathematics by John Stillwell?

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Robert H. Olley Quondam Physics Department University of Reading England

Robert H Olley | 02/01/13 | 07:19 AM

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I stumbled onto "Mathematical Thought from Ancient to Modern Times" by Kline in 1987 or 1988. I was in trouble in physics graduate school the first time around. My undergraduate physics work was done at a very weak school and I was young and naive. I had no notion that the quality of an undergraduate degree varied so much from school to school. I had skipped high school and I had worked and supported myself and eventually a young wife. Now I was failing, loosing my minority fellowship, and wondering if I was a physics idiot. I was, however, doing well in my pure math classes at the University of Texas at Austin. Unfortunately, my fellowship was for physics not for mathematics. Thankfully I was accepted by the United States Air Force and ended up at the then Phillips Research Laboratory in Albuquerque NM, whereupon I spent four years fixing my undergraduate deficiencies surrounded by civilian PhDs. Kline's history of math book made me realize that I wasn't stupid by making me realize that mathematics and physics progress by fits and starts, and that there are logical, traceable paths to current math and physics. I didn't know if I could ever go back to graduate school, but thanks to Kline, I did know that through retracing history there was nothing out there that I couldn't eventually understand and master, and that this was true for all other lost, baffled and mystified students.

Robert, I agree with the Amazon critic of Gilmore 2008. I coupled the first four chapters of the earlier (Dover) book by Gilmore to H F Jones 2nd ed text on physics and group theory. Years later when I discovered Gilmore's new 2008 book, I was excited but very quickly disappointed. What I did get out of Gilmore 2008 was coupling its chapter 1 with the first four chapters of H F Jones to arrive at, for the first time, a fundamental, sound understanding of the insolvability of the quintic by radical extension (Galois theory). I had never felt I had understood Galois theory in the dry algebraic language of writers such as Lang.

Over the last few years I weaved a set of notes designed to give either a PhD student in physics or math the kind of history based, step-by-step guide to algebra, topology, differential equations, the calculus of variations to Lagrangian, Hamiltonian and Poisson formulations of classical physics, and how these formulations relate to Sophus Lie's EXP operating on infinitesimal generators, and ultimately how this work then served as the foundation of for building quantum physics and quantum field theories.

Gilmore's Dover book, chapters 3 and 4 on the geometric meaning of the commutator blend in perfectly with "Quantum Field Theory" 2nd ed. L. H. Ryder's chapter that shows the great similarity between quantum fields and general relativity in terms of parallel transport and the covariant derivative.

My notes, written in airports and in airplanes while I've traveled for the USAF as civilian this time, have typos I'm sure, but I think they supply that which still eludes you: what is the connection between particle physics and group theory. I had three audiences in mind: the beginner who can penetrate the notes a little further as his/her background increases in traditional math/physics courses, the math/physics BS graduate/graduate math/physics student, and the postgraduate mathematician and/or physicist. I attached a "syllabus" to these notes to supply the list of math/physics courses that should be mastered (with suggested reading material), and a discussion of math as expounded by M. Kline in his "End of Math" book, as well as my understanding of the limits of physics. I suggest to the reader many additional sources, and in particular the 2nd ed. book on "Geometry, Topology and Physics" by M. Nakahara as motivated by the illegible book by Naber on guage fields.

Between "A First Course in String Theory" by B. Zwiebach and "Loop Quantum Gravity" by Gambini and Pullin, LQG does seem to be better grounded with respect to the background independence of general relativity, and for this I appreciate the background in connections over bundles by Nakahara.

Alex Alaniz
AFOTEC HQ, Kirtland AFB NM

My notes have been posted by John Baez on his webpage, but the most updated version is on the Physics Forums blog.

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A Alaniz

Alex Alaniz | 02/03/13 | 01:52 AM

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they are particularly interested in selecting out potential university physicists, so that the A-level course (years 11 and 12) is not well suited to the majority of those who would otherwise like to study at A-level.

For something like physics it might be best to make an exception and actually have a pair of courses of varying difficulty, even within the A- level. We had something like that once upon a time. Either course allowed students into science programs, but if an entrance exam deemed the student weak in physics, he would have to take a bridge course in his first year of college.

Enrico Uva | 01/25/13 | 23:01 PM

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