The problem with maths and physics teaching

[cosmic onion]

Please don't take this too seriously -I do still like my maths and physics but I have a gripe about how I have been taught some of it, best give some definite examples.

1 - Maxwells electromagnetics. In his original work there were 2 parts. Part 1 was a touchy feel thought experiment where he imagined an ether existed and proceeded to describe the workings of this ether by means of cogs and wheels. An intuitive approach. In part 2 he put skin and bones on it with a mathematical description of what an imaginary ether would be. Finally after another 30 years Heavside re wrote Maxwells equations in the modern form we have today. My gripe is that we now learn Electromagnetic without the intuition that Maxwell used. Its like someone building a skyscraper then taking down all the scaffolding and being left left with this beatiful edifice but no comprehension of how it was built

2 - Euler Lagrange equation in calculus of variations. Eulers derivation was a localised derivation which was intuitively simple and gives a great and easy intuitive way of looking at it. Lagrange derivation and the one accepted and taught now is less intuitive but more analytic and precise. But I bet my house that Lagrange used the same intuitive approach as a first attempt at a derivation before he did his more analytic derivation. Again creating academic skyscrapers and hiding the dirty scaffolding.

3 - Engineering students learning Laplace transforms. I was taught formulas and processes and when I looked at it it appeared that an astounding genius must have thought it up, 'approximate a function using a family of decreasing and increasing rotating phasers , spirals. But it turns out inow order to understand it properly you need to think of Fourier and laplace transforms in the context of vector spaces and metrics within these spaces , linear algebra. Then all becomes apparent. Again a case of building skyscrapers then throwing away the inconvenient and ghastly looking scaffolding.

So in summary why can we not learn from intuition when of value 1,2 and real maths when of value 3.

 

[S.G. Janssens]

So in summary why can we not learn from intuition when of value 1,2 and real maths when of value 3.

Perhaps it is because people learn differently. For example, in case (1) I would personally tend to disagree: Just give me the system of PDEs, I don't care so much about the experimental history. However, in case (3) I would tend to agree, but that is probably because "scaffolding" here seems to have a different meaning: It actually refers to the mathematical formalism. (That is why in this I case find it beautiful and not "ghastly looking".)

It is my experiences that looking at "scaffolding" (in the narrow sense of studying the historical path that lead to a result or theory in its present form) can be very useful, but rarely so when studying a subject for the first time. Perhaps for other people the opposite is true.

 

[cosmic onion]

Thanks for the reply. I have never taught so I bow to your experience. When learning I would have thought original intuitive methods would be good way to introduce a subject. If these are in there historical context then it would add a sense of gravity to what was being taught (example we learn about energy directly proportional to v squared in a matter of minutes but historically it to 100 years for this to be widely accepted by scientific community ). If I knew this as a student I might have thought longer and harder about this and other basic maths and physics. As you say there is no one best way of learning.

 

[S.G. Janssens]

I have never taught so I bow to your experience.

No need for bowing, and my experience is still limited.

If I knew this as a student I might have thought longer and harder about this and other basic maths and physics.

Indeed, I think awareness of the historical context (as well as the existence or many misses and dead ends) is important. It is humbling and helps to understand that the acquisition of knowledge and insight is not a straightforward, linear process. However, for me when I learn something new in mathematics, it works best to first encounter a presentation that is clean and succinct, and this seems harder to achieve with a lot of scaffolding present.

 

[cosmic onion]

I also like to live in my subspace. Interesting conversation.

 

[Dr.D]

Sadly, I think that another side of this issue is the argumentative / litigious way we live today. Regarding Maxwell's ether, someone will shout, "You idiot! There is no ether. I don't believe a word of what you have said. Rigorous proof is the only way!" Most teachers don't want to deal with that, so they would prefer to present an unassailable proof, even if it is clear as mud. Then there will be no argument.

 

[cosmic onion]

I've never contemplated the awswer in those terms. Very interesting

 

[Andy Resnick]

Please don't take this too seriously -I do still like my maths and physics but I have a gripe about how I have been taught some of it, best give some definite examples.

1 - Maxwells electromagnetics. In his original work there were 2 parts. Part 1 was a touchy feel thought experiment where he imagined an ether existed and proceeded to describe the workings of this ether by means of cogs and wheels. An intuitive approach. In part 2 he put skin and bones on it with a mathematical description of what an imaginary ether would be. Finally after another 30 years Heavside re wrote Maxwells equations in the modern form we have today. My gripe is that we now learn Electromagnetic without the intuition that Maxwell used. Its like someone building a skyscraper then taking down all the scaffolding and being left left with this beatiful edifice but no comprehension of how it was built

2 - Euler Lagrange equation in calculus of variations. Eulers derivation was a localised derivation which was intuitively simple and gives a great and easy intuitive way of looking at it. Lagrange derivation and the one accepted and taught now is less intuitive but more analytic and precise. But I bet my house that Lagrange used the same intuitive approach as a first attempt at a derivation before he did his more analytic derivation. Again creating academic skyscrapers and hiding the dirty scaffolding.

3 - Engineering students learning Laplace transforms. I was taught formulas and processes and when I looked at it it appeared that an astounding genius must have thought it up, 'approximate a function using a family of decreasing and increasing rotating phasers , spirals. But it turns out inow order to understand it properly you need to think of Fourier and laplace transforms in the context of vector spaces and metrics within these spaces , linear algebra. Then all becomes apparent. Again a case of building skyscrapers then throwing away the inconvenient and ghastly looking scaffolding.

So in summary why can we not learn from intuition when of value 1,2 and real maths when of value 3.

I've been thinking about this post for a while, because you make some excellent points. The problem is simply this: not enough time. There isn't enough time in a single course to cover both the subject matter (say, E&M I or II) and in addition, cover the historical development of that same subject.

To be sure, choices can be made: this is why I cover special relativity in my Intro II course rather than Intro I; the historical development of SR derives from considering the Lorentz Force law.

 

[jasonRF]

Interesting post. I agree that sometimes not presenting the original intuition is unfortunate. However, there are also many instances in which the historical evolution of the concepts and theories is not helpful, at least for a first exposure to a subject.

Regarding 1, I have only flipped through Maxwell's treatise and found it very hard reading indeed. Interestingly, one of the (many) important contributions of Maxwell is the addition of the displacement current to electromagnetic theory. However, at least according to Nahin (Oliver Heaviside, Johns Hopkins University Press, 2002) Maxwell envisioned it as arising from polarization or displacement of charges in dielectrics. At the time there was no experimental evidence that the displacement current even existed. Of course, now we know it exists in free space so any intuition based on dielectric polarization is suspect at best. I think the modern "derivation" that appeals to the continuity equation is more intuitive and only requires a few lines of math. One can always read up on the original thinking and history later; as indicated by Andy Resnick, modern day courses are already packed so getting to the point efficiently has its benefits.

For 3, I completely agree. I think I was lucky that my TA for "advance engineering analysis" explained Laplace transforms as an inner product, using the language and concepts of linear algebra and pointing out how complex exponentials were eigenfunctions of a linear constant coefficient differential operator. My TA spent at most 10 minutes on this, and I remember it clearly even though it was 25 years ago. This also highlights the centrality of linear algebra in applied math - at least for linear problems! We had to take a semester of linear algebra first semester sophomore year, before any of the courses that covered differential equations, Fourier series and boundary value problems, integral transforms, etc. When I took linear algebra I was convinced it was abstract for no good reason, but practically every semester after that my classes used the language and concepts of linear algebra so I couldn't help but see how incredibly useful the abstraction is. When I look at the transcripts of engineers that we interview these days, I almost never see a case where they took linear algebra before ODEs or other classes in which they might learn about many of these topics. In some cases they never took linear algebra at all, which is a mistake. This is clearly a choice made by curriculum committees, and certainly reflects the fact that modern graduates seem to be expected to know more topics (but in less depth) than I did when I graduated.

By the way, the problem of eliminating the scaffolding seems worse when I read refereed journals. Sometimes I think that authors (even engineers!) "dress up" their results to try and make them look more impressive, when a straightforward exposition would be better for the vast majority of the audience that would actually use their results.

Jason

 

[Dr. Courtney]

In addition to the time squeeze mentioned above, there is the tendency to teach toward the middle of the student abilities in the classroom.

I was one of the stronger students in my undergrad Physics and Math courses at LSU, and I don't think I would have grasped the approaches you outlined above. They would have been very confusing for me. I may have grasped them when I retook the undergrad courses at MIT, but this was my second pass through the standard undergrad courses, AND I had done a lot of outside work improving my insights and skills.

Different people learn in different ways. My learning is more experiential and hands-on and much less about what the theorist was thinking at the time. I do better connecting it to the experiments giving rise to it as well as later experiments most directly validating it after the fact. The mind of the theorist is not an essential part of how I learn or process science. I don't care. I want to know the predictions of the theory, how to make them, and how they are confirmed by experimental reality.

 

[David Lewis]

There are teachers who would love to make learning more enjoyable and interesting, and provide a deeper, more intuitive level of comprehension, and they know how to do it, but it's expensive.

 

[Stephen Tashi]

It's an interesting question whether instructors teach subjects the way they themselves understand them. I think the ones that do understand generally teach that way. However, I think that experts can become so familiar with certain topics that they forget how they originally managed to figure them out. Some teachers keep the struggle to understand fresh in their minds and they communicate it well - for example in the Feynman lectures on physics or the video lectures by Leonard Susskind.

 

[haushofer]

I have this experience with general relativity. It's very comforting to see how Einstein struggled with the whole notion of general covariance and coordinate transformations; see e.g. his "hole argument". I've encountered more than one who was thoroughly familiar with GR and all its algebraic calculations, but was seriously puzzled by an explicit example of the hole argument. Formal math and algebra can cover up a lot of intuition and conceptual understanding.

 

[David Lewis]

That is a good point. When John Bell first proposed his Spaceship Paradox, he informally surveyed colleagues, and most of them got the wrong answer. Yet if you gave them a quiz on SR, they'd get an A+. They know the equations backwards and forwards. They can solve any quantitative riddle you throw at them. The problem is they don't understand what the math says is actually going on.

 

[symbolipoint]

This is the important part:

...

Different people learn in different ways. My learning is more experiential and hands-on and much less about what the theorist was thinking at the time. I do better connecting it to the experiments giving rise to it as well as later experiments most directly validating it after the fact. The mind of the theorist is not an essential part of how I learn or process science. I don't care. I want to know the predictions of the theory, how to make them, and how they are confirmed by experimental reality.

The discussion of the whole topic in most of the posts here seems slightly unfocused, maybe because the topic is hard to discuss, but the quote there seems good as what can be done to help best in learning.

 

[mathwonk]

I agree with Andy. We are given a "syllabus" or list of topics to cover, and doing justice to the background and motivation for each topic as well as explaining the details, would take several semesters, not the one we have. the one class in my life (intro to number theory) when i was given no syllabus but just told not to bury the students, I went so slow I could not believe it (we spent the whole first hour just discussing primes, and writing down the prime numbers up to 100, and discussing possible patterns), but every one loved it including the advanced graduate students taking an undergraduate class. the 3rd quarter i shifted into high gear and proved dirichlets theorem on primes in arithmetic progressions, with many sounds of suffering and wailing from the class.

the first time i taught college geometry i assumed everyone knew high school geometry and studied different axiom systems and various exotic models. for them. it blew the class away. finally i got the nerve to just spend the whole semester teaching high school geometry from scratch, literally from euclid, and it was the best class i ever gave. i myself finally learned that euclid is the absolute best geometry book in existence and far more subtle, deep, and rigorous than i had always heard. I also learned that the best part of the earlier course on models had actually been lifted directly from euclid without comment, by the modern authors.

moral: GO TO OFFICE HOURS! we have no syllabus there and can answer any question in detail.

 

[David Roberts]

Great discussion. I fall on the side of the line that is "why not have both?" I know that time is of the essence and most times we can't educate a growing mind quick enough to pump the information into it before it hardens, like quick dry cement.