The coming revolution in physics education

[Dale]

I give up on selling you my scientific programming course.

I am very curious about why you are so stuck on your course as is with no willingness to consider improvements?

Personally, I like the idea of using available computational tools in physics (and statistics and math) curricula. So I am naturally predisposed to support the concept. If you can’t even make the effort to win over someone like me then you are going to have a really hard time winning over others that may not be predisposed in favor of the concept.

Joke #2 - A student walks into Dale's Culinary Arts class, and asks, 'Where's the kitchen, you know, stove, pots, and pans?' And Dale answers - 'All you need is your cellphone, we'll be ordering in.'

Hahaha!

 

[Will Flannery]

I am very curious about why you are so stuck on your course as is with no willingness to consider improvements?

Personally, I like the idea of using available computational tools in physics curricula. So I am naturally predisposed to support the concept. If you can’t even make the effort to win over someone like me then you are going to have a really hard time winning over others that may not be predisposed in favor of the concept.

I've been trying to 'win over' people now for quite some time, NSF reviewers, journal editors, etc. So, I've given this a lot of thought. Until this thread not one person, (1), has acknowledged the basic idea of the course, much less been won over .. but in this thread we read ...

Twenty-one years ago, I taught high school for one year. During one of the semesters, I was assigned a high school computer programming course to teach (as well as math and physics courses). For one of the coding projects, I taught the students Euler's method, and had them use this to analyse orbital motion. Their programs produced "real-time" animations in two spatial dimensions of the the motion.

I kind of like Euler's method as a teaching tool and have used it to teach high school students to compute rocket trajectories as well as to compute forces from videos of experiments. For systems where the forces are changing slowly, it is not bad, and it can be implemented in a spreadsheet. Students can learn it even before they have a clue about calculus with the idea that for sufficiently short times, the assumption of constant acceleration is "good enough" for sufficiently short time steps. Then students can shorten the time step and see if their calculations change much. Sure, the spreadsheets might have 1000-10000 rows, but they are all cut and pasted from the first few, and this allows students to easily see how the motion is progressing.

and even ...

When I was first exposed to numerical methods as an undergraduate, it was a real eye opening experience. The challenges of tough integrals, derivatives, and differential equations melted away, and I realized that the real challenge of physics was reducing the problem to an equation that could then be solved on the computer. Before then - I saw reducing the problem to an equation as about the first 30% of the problem and then beating that equation into submission with the traditional analytical (pencil and paper) techniques as the last 70% of the task at hand. Physics was mostly math, and I was not good at math, and I did not like math. About the same time, I also felt cheated by realizing that all the prior problems had been cherry picked to be solvable with traditional pencil and paper methods - most problems cannot be solved with pencil and paper. These revelations should really occur before a physics major is a junior in college.

So, I'm making progress.

I began trying to sell - 'Mathematical Modeling and Computational Calculus'
Then - 'Clearing the First Hurdle in Mathematical Physics, Unsolvable Differential Equations'
Finally - 'The Coming Revolution ... '

Having failed to win over anyone for some time I began trying to 'sell the sizzle, not the steak', and started including the graph of the Apollo trajectory in every communication, calling it fantastic, etc. Which it is. But to date not one person has acknowledged it. The results in the paper are not only extraordinary even undreamed of in high school, they would be extraordinary in college.

And ... we haven't even gotten to partial differential equations ... !

The difficulties of analytic calculus completely dominate the university classical physics curriculum, and once you take that away, or at least put it off to the side, the landscape is entirely different and you get to see where the difficulties (apart from analytic calculus) in mathematical physics lie.

Difficulty arises going from 2-d to 3-d rigid body modeling as you need Euler angles/quaternions and the moment of inertial tensor. Note that dynamics is not even in the USF physics program.

When you get to stress and strain in materials, again missing from the USF curriculum, you need tensors big time, and if you think stress and strain in materials is outdated, the stress and strain tensors are great preparation for the gravity tensor in general relativity.

Etc.

... I could go on forever ...

Well .. I'm going on for one more little thing - computational calculus makes the physics and STEM curriculums easy at least until difficulties like tensors are encountered. So, it's like I'm trying to sell space heaters to Eskimos and no one is buying ! Millions spent by NSF on improving STEM education and they're funding things like, no kidding, 'Culturally Authentic Practice to Advance Computational Thinking in Youth' and 'Integrating Computational Thinking into Mathematics Instruction in Rural and Urban Preschools', these are real programs funded with millions.

 

[Dale]

The results in the paper are not only extraordinary even undreamed of in high school, they would be extraordinary in college.

This is substantial exaggeration. My sophomore year in college (mid '90s) we did double pendulum numerical simulations with ordinary students in our standard differential equations course. So this is standard college sophomore level stuff from 20+ years ago. Two decades later for such exceptional high-school students to accomplish something similar with a 2/1 student/teacher ratio is good, but not a particularly strong demonstration of the curriculum. I would expect such extraordinary students, with that level of individual attention, to be able to produce results ~3 years ahead of their peers regardless of the curriculum. Claiming this as "extraordinary in college" really diminishes your credibility.

A much more convincing test of the curriculum would be a physics test taken after your class and after a traditional class, with standard students in both. Does being able to simulate physics give typical students a better grasp of the underlying physics?

 

[Will Flannery]

This is substantial exaggeration. My sophomore year in college (mid '90s) we did double pendulum numerical simulations with ordinary students in our standard differential equations course.

You're comparing apples and oranges. Every mechanics book covers pendulums because that is one of the few non-trivial systems where the DEs can be solved. So, I pick up 'Mechanics' (the only mechanics book I've got) by Symon, and it has 7 pages on the simple and compound pendulum. You won't be able to find any textbook on classical mechanics that computes orbits (radius as a function of time), much less rocket trajectories, as the DEs are unsolvable analytically.

 

[Dale]

You're comparing apples and oranges. Every mechanics book covers pendulums because that is one of the few non-trivial systems where the DEs can be solved.

Double pendulums cannot; there is no analytical solution. It is an apples to apples comparison.

 

[Will Flannery]

Double pendulums cannot; there is no analytical solution. It is an apples to apples comparison.

Well, you're right. See https://www.myphysicslab.com/pendulum/double-pendulum-en.html

I'm surprised to see such a complex sim in a traditional DE class. In fact, I'm surprised to see any sim in a traditional DE class. There's nothing like it in the USF physics DE class http://ewald.cas.usf.edu/teaching/2015F/3113/syllabus.pdf

And still, no orbits, no rockets.

 

[Dale]

I'm surprised to see such a complex sim in a traditional DE class. In fact, I'm surprised to see any sim in a traditional DE class.

Well, it was our standard required course in the mid 90's for all engineering majors at a typical big engineering college.

But still, no orbits, no rockets.

So what? Were some of the proposed names of your course "orbital mechanics" or "rocket science"? No, those were just examples of analytically unsolvable DE's. A double pendulum is also one, so it is clearly and obviously an apples to apples comparison.

The point stands that this is not "extraordinary in college" success, this is "sophomore level in college" success. That is still a good accomplishment, but not nearly living up to your overselling. A more moderate approach would be more credible.

 

[Will Flannery]

No, those were just examples of analytically unsolvable DE's.

Not at all. These aren't 'random examples'. These are representative core problems in 3 branches of physics. The three branches that are characterized by ODEs as opposed to PDEs. Hey, I should have pointed that out in the beginning.

The orbits/rockets analyses represent the core of central force motion. And the only similar results at USF are obtained not in the classical dynamics (physics dept) class but in the elective senior year course in computational physics.

The electric circuit examples represent the core of electric circuit analysis, and the methods used in the industry standard SPICE (simulation program with integrated circuit emphasis) program which EEs use to analyze newly designed circuits.

The rigid body examples are representative of the analysis of rigid body dynamics, another huge branch of physics. One of the rigid body analyses described in the paper is a 2-d rigid body rocket launch to orbit. There is a rigid body rocket problem analyzed in Classical Dynamics PHY 3220, the text (Classical Dynamics, Thornton) reads pg. 374 ...

The actual motion of a rocket attempting to leave earth’s gravitational field is quite complicated. For analytical purposes, we begin by making several assumptions. The rocket will have only vertical motion, with no horizontal component. We neglect air resistance and assume that the acceleration of gravity is constant with height. We also assume that the burn rate of fuel is constant.

It takes a couple of pages of calculus to solve the equations of motion this problem which has been simplified to point of irrelevance.

The text continues:

All these factors that are neglected can reasonably be included with a numerical analysis by computer.

Which is what the example in the paper does. So, our trajectory approaches realism, whereas the PHY 3220 analysis is totally unrealistic.

 

[Dale]

Not at all. These aren't 'random examples'.

I never said the examples were random. But the accomplishment isn’t the specific problems solved, it is using a numerical method to solve an ODE with no analytical solution. So, again, the accomplishment isn’t super stupendous amazing wonderful fantastic even for college, it is solid sophomore level work.

 

[Will Flannery]

So, again, the accomplishment isn’t super stupendous amazing wonderful fantastic even for college,

Let's just say fantastic, even at the college level.

A (physics) student at USF will study central force motion in introductory physics and in classical mechanics, spending a few weeks on the subject. The goal of studying central force motion is to be able to determine trajectories of orbits and rockets, and yet USF students do not even plot orbits (unless they take the elective senior computational physics course) much less rocket trajectories . So, the graphs in the my course, e.g. the Juno trajectory, would be, let's say, spectacular at USF. Even more, when Newton solved the problem of plotting orbits from his laws of gravity and motion, this marked the beginning of modern math and science. This is THE problem in the history of science, unsolved for thousands of years. It's nice to see a solution.

The DEs for small electrical circuits, as in the course, are solvable analytically for sine wave inputs, and the physics dept. (I think, could be EE dept) electrical circuits class at USF, text Hambley, does, in one section in one chapter, use the MATLAB ODE solver to analyze circuits for steady state sinusoidal inputs. Whereas, in my course the circuits are analyzed for sine, step, and impulse inputs. Yep, fantastic once again.

Rigid-body motion is given short shrift in the US physics dept, the classical dynamics text, Thornton, has analyses of pendulums and spinning tops, which are somewhat spectacular themselves, but these objects are not of great interest to most of us. It's not in the paper but the first and simplest rigid-body analysis in my course is for a tumbling rod. Completely outside the scope of the physics program at USF, and really neat, hence, fantastic. Ditto the launch to orbit. Even more so the electro-mechanical analysis of the VEX robot described in the paper.

Here's how the paper puts it

"Differential equations have extraordinary analytic and explanatory power, as we’ve seen in this short paper. But, it takes computational calculus to unleash this power. Introducing students early to modeling with differential equations and analyzing a wide range of physical systems using computational calculus, from the Juno space probe now orbiting Jupiter to the VEX robot on the classroom floor, will put computers, differential equations, and computational calculus at the center of technical education from the beginning. In the future each will play an increasingly important role in many of the courses in the physics curriculum."

This sense of mastery of the analysis of a wide range of real physical systems is completely missing from the typical university physics program today, because it requires analyzing mostly unsolvable DEs and that requires computational math which is given minimal coverage at USF.

 

[Vanadium 50]

for(;;)  {
   printf("Valid criticism.\n");
   printf("You're wrong.  This is the most stupendously stupendous idea ever!\n");
}

 

[Dr. Courtney]

I never said the examples were random. But the accomplishment isn’t the specific problems solved, it is using a numerical method to solve an ODE with no analytical solution. So, again, the accomplishment isn’t super stupendous amazing wonderful fantastic even for college, it is solid sophomore level work.

If we're talking about physics, my view is that there need to be enough problems of interest solved to demonstrate the generality of the approach to useful physics. The double pendulum problem is interesting, but in isolation, it is of little practical value. A small subset of more interesting problems would be: projectile motion with drag, rocket motion with thrust and drag, multi-body satellite, and relativistic rocket ship.

Is it extraordinary? Well, it is certainly not ordinary, as most sophomore physics courses do not have it. I was pleased that my son's physics professor included some of this in his sophomore level course. Of course, most of the students in the class couldn't handle the programming, and as soon as the prof saw my son could handle the programming, he was snatched up for his research group. The course had the opportunity, but there were so few points attached to the numerical integration projects that most students could botch the projects (learn nothing from them) and pass the course.

 

[Will Flannery]

Of course, most of the students in the class couldn't handle the programming, and as soon as the prof saw my son could handle the programming, he was snatched up for his research group.

Trying to attach programming to an existing physics course is probably not a good idea, a better idea is to have an introductory course in ... Scientific Programming :)

 

[Dale]

Let's just say fantastic, even at the college level.

Overselling again. It is “good at the sophomore level”, anything more than that is an exaggeration.

A (physics) student at USF

Maybe you just need to look at better schools. I have no idea how USF is ranked. I was at TAMU which is a good engineering school, but not at the MIT or Stanford level.

So maybe “fantastic, even at the USF college level” or “good at the TAMU sophomore level”.

 

[Dale]

If we're talking about physics, my view is that there need to be enough problems of interest solved to demonstrate the generality of the approach to useful physics. The double pendulum problem is interesting, but in isolation, it is of little practical value.

In my case it was a differential equations class. So the entire class was considered generally useful techniques regardless of the practical value of specific problems.

Well, it is certainly not ordinary, as most sophomore physics courses do not have it. I was pleased that my son's physics professor included some of this in his sophomore level course.

Well, your son’s physics course had it, and my DE class had it. And Will’s school doesn’t do it at all. Other schools may reserve it for a dedicated numerical methods course. So I am not sure that “most” is right.

 

[Dale]

for(;;)  {
   printf("Valid criticism.\n");
   printf("You're wrong.  This is the most stupendously stupendous idea ever!\n");
}

I agree. This loop is boring now. I won’t bother responding to another iteration of non responsive overselling, I will just close the thread.

@Will Flannery the continuation of this thread is up to you. Either respond to some of the substantive points in a factual manner and we can have a useful conversation or repeat your non-responsive hype and it ends. Your choice.

 

[Dr. Courtney]

Well, your son’s physics course had it, and my DE class had it. And Will’s school doesn’t do it at all. Other schools may reserve it for a dedicated numerical methods course. So I am not sure that “most” is right.

At my son's school, only the courses for physics majors have it. The physics courses for engineers and other science majors do not have it. So most of the sophomore physics students are missing it. (My son is a tutor for all the physics courses and an assistant in mechanics course for engineers, so he's familiar with all the content.) None of the physics courses at institutions where I've attended or taught have had numerical integration of diff eqs, including LSU (Baton Rouge), MIT, a community college in Ohio, Western Carolina University, and the United States Air Force Academy. It also was not part of the Physics curriculum when my wife taught at West Point. Definitely not common or ordinary.

If it is reserved for a numerical methods course, many physics and engineering majors are going to miss it. Lots of physics and engineering degrees do not require a numerical methods course.

 

[Dale]

There's nothing like it in the USF physics DE class http://ewald.cas.usf.edu/teaching/2015F/3113/syllabus.pdf

There is a Modelling and Analysis of Engineering Systems course at USF that does this stuff. Either this course or a DE course is required for engineers. So this material is standard sophomore level at USF also.

http://www.rc.usf.edu/~kaw/download/today/EGN3433.pdf

None of the physics courses at institutions where I've attended or taught have had numerical integration of diff eqs, including LSU (Baton Rouge), MIT, a community college in Ohio, Western Carolina University, and the United States Air Force Academy.

LSU has ME 2543 which is required for mechanical engineering sophomores

MIT has 18.03 Differential Equations or 2.087 Engineering Mathematics: Linear Algebra and ODEs one of which is required and both of which cover numerical ODEs. I can't tell what year these are scheduled.

Western Carolina University has Math 320 which is required for EE majors in the sophomore year.

The Air Force Academy has Aero Engr 351, 352, 442, and 457 all are required for Aeronautical Engineers and use numerical methods starting in their 2nd class year.

I cannot confirm that none of the physics courses teach numerical methods, but it appears to be standard fare for engineering curricula at all of the institutions mentioned in this thread, typically at the sophomore level.

Edit: apparently MIT’s 18.03 is required for physics majors too. So it is part of the physics curriculum.

 

[bob012345]

Classical physics is difficult because it is based on differential equations, and the differential equations of interest are usually unsolvable. The student must invest a lot of time in learning difficult math, and still can only analyze very simple systems.

I disagree with your original premise. Basic physics courses don't need to over complicate things by focusing on things like the fact that the acceleration due to gravity varies minutely and requires numerical integration. That obscures the physics concepts . Besides, one could argue your formula for acceleration is also an approximation. Unless you stop at the level of String Theory or whatever, it's an approximation so what's the point of complicating it?

 

[Will Flannery]

The 'revolution' in the OP is to teach numerical methods for computing solutions to differential equations, first ODE and then PDEs, very early in the physics curriculum and then to use those methods in subsequent courses in the physics (and engineering) programs.

One important question is 'To what extent is it already being done?'

I am familiar with the (nearby to me) USF program, and in the physics curriculum numeric methods appear briefly in EGN 3373, text Hambley, where the MATLAB ODE solver appears in a 5 pg. section on electrical circuit AC analysis, then again in the senior year when there is an elective course in Computational Physics, PHY 4151C, text Giordano, Computational Physics.

I wanted to check UC Berkeley, my alma mater, and I found that there is PHY 77 which is a freshman/sophmore course in Computational Methods but I couldn't find a synopsis so I emailed the instructor. The text is Newman's Computational Physics (!) and the course is being sold more as a lead into the calculus sequence rather than the physics program. It is not a prerequisite to any courses in the physics department, and as far as I can tell, that's the extent of numeric methods in physics at Berkeley. The synopsis shows that 2 days are spent on Ch. 8 in the text, ODEs, and 0 days on Ch. 9, PDEs.

I replied to the instructor:
I agree that computational methods for DEs should precede the math classes, I remember from my undergraduate days that the math classes were not only abstract, they were unmotivated as the study of DEs was necessarily preceded by differential and integral calculus and on top of that I had no idea of the significance of DEs in analyzing physical phenomena. And the kicker of course is that most DEs are unsolvable anyhow. I've examined 5 physics texts starting with Goldstein and each avoids the issue that Newton's model for central force motion is unsolvable for r as a function of t, while never mentioning that fact.

Ideally, IMO, programming and numeric methods would be taught early and used extensively in the remainder of the physics/technical curriculum. That is not what is happening at USF or UCB. I think these schools are representative, but a survey of more schools would be interesting.

And, the survey should be extended to include engineering programs, where numeric methods are more likely to appear especially in upper division courses. A brief look at the USF ME department core courses https://www.usf.edu/engineering/me/documents/core-classes-availability.pdf shows that there are several courses where computational methods are used, but it's not clear which methods, so a closer look is required, which I'll do this week :).

 

[bob012345]

The 'revolution' in the OP is to teach numerical methods for computing solutions to differential equations, first ODE and then PDEs, very early in the physics curriculum and then to use those methods in subsequent courses in the physics (and engineering) programs.

One important question is 'To what extent is it already being done?'

I am familiar with the (nearby to me) USF program, and in the physics curriculum numeric methods appear briefly in EGN 3373, text Hambley, where the MATLAB ODE solver appears in a 5 pg. section on electrical circuit AC analysis, then again in the senior year when there is an elective course in Computational Physics, PHY 4151C, text Giordano, Computational Physics.

I wanted to check UC Berkeley, my alma mater, and I found that there is PHY 77 which is a freshman/sophmore course in Computational Methods but I couldn't find a synopsis so I emailed the instructor. The text is Newman's Computational Physics (!) and the course is being sold more as a lead into the calculus sequence rather than the physics program. It is not a prerequisite to any courses in the physics department, and as far as I can tell, that's the extent of numeric methods in physics at Berkeley. The synopsis shows that 2 days are spent on Ch. 8 in the text, ODEs, and 0 days on Ch. 9, PDEs.

I replied to the instructor:
I agree that computational methods for DEs should precede the math classes, I remember from my undergraduate days that the math classes were not only abstract, they were unmotivated as the study of DEs was necessarily preceded by differential and integral calculus and on top of that I had no idea of the significance of DEs in analyzing physical phenomena. And the kicker of course is that most DEs are unsolvable anyhow. I've examined 5 physics texts starting with Goldstein and each avoids the issue that Newton's model for central force motion is unsolvable for r as a function of t, while never mentioning that fact.

Ideally, IMO, programming and numeric methods would be taught early and used extensively in the remainder of the physics/technical curriculum. That is not what is happening at USF or UCB. I think these schools are representative, but a survey of more schools would be interesting.

And, the survey should be extended to include engineering programs, where numeric methods are more likely to appear especially in upper division courses. A brief look at the USF ME department core courses https://www.usf.edu/engineering/me/documents/core-classes-availability.pdf shows that there are several courses where computational methods are used, but it's not clear which methods, so a closer look is required, which I'll do this week :).

That effectively converts most subjects to computations and programming as more than tools but as a paradigm. Stephen Wolfram views the world that way but I'm not sure it's the best way. It de-emphasizes the mathematical form and relationships and treats the world as algorithms. It can also lead to the false idea that merely doing computations and simulations is as good as or a replacement for doing real experiments.

 

[Dale]

The 'revolution' in the OP is to teach numerical methods for computing solutions to differential equations, first ODE and then PDEs, very early in the physics curriculum and then to use those methods in subsequent courses in the physics (and engineering) programs.

One important question is 'To what extent is it already being done?'
...

I appreciate this substantive and moderate response!

My impression from the searches I did above is that it is common in the engineering curricula typically at the sophomore level and usually as part of the required math courses.

I think that in the engineering curriculum subsequent engineering courses rely on this knowledge, but not the basic physics courses which are typically in the freshman year.

That effectively converts most subjects to computations and programming as more than tools but as a paradigm. Stephen Wolfram views the world that way but I'm not sure it's the best way.

I wonder if any studies have been done to determine if a computational paradigm leads to better or worse understanding of conceptual questions.

 

[atyy]

Do you introduce calculus before Euler's method?

I agree there is no coming revolution. Numerical integration is already part of some high school mathematics syllabi: http://studywell.com/maths/pure-maths/numerical-methods/

 

[Dr. Courtney]

Do you introduce calculus before Euler's method?

I agree there is no coming revolution. Numerical integration is already part of some high school mathematics syllabi: http://studywell.com/maths/pure-maths/numerical-methods/

I do not. No need to. Just the idea that shortening the time steps with the kinematic equations accounts for the fact that accelerations are changing.

If students understand the kinematic equations (for constant acceleration) and the single idea of making the time steps short to account for changes, there is no need to know calculus.

 

[atyy]

I do not. No need to. Just the idea that shortening the time steps with the kinematic equations accounts for the fact that accelerations are changing.

If students understand the kinematic equations (for constant acceleration) and the single idea of making the time steps short to account for changes, there is no need to know calculus.

I guess the system I was in had physics in 2 separate places. We had physics without calculus, and just constant acceleration in the mechanics part of the course (but it was a 2 year course, and covered circuits, electromagnetism, old quantum physics). We did do circular motion, and I can't remember how that was approached without calculus. In the mathematics part of the course, we had calculus and followed by numerical integration, and mechanics again (this was also a 2 year course, covering many other things like vectors in 3D).

 

[Will Flannery]

Do you introduce calculus before Euler's method?

I agree there is no coming revolution. Numerical integration is already part of some high school mathematics syllabi: http://studywell.com/maths/pure-maths/numerical-methods/

The paper describes a high school course that can be taught independently of the math program beyond high school algebra and geometry. From #55 - computational methods for (computing solutions to) ODEs should precede the calculus classes, I remember from my undergraduate days that the calculus classes were not only abstract, they were unmotivated as the study of DEs was necessarily preceded by differential and integral calculus and on top of that I had no idea of the significance of DEs in analyzing physical phenomena ...

Anecdotally, I've asked two high school (top private schools) math teachers why they were teaching calculus, why is it useful? One answer "I don't know", second answer "It's like weight-lifting, even though you don't do it outside the gym it makes you strong for other activities". I swear I am not kidding !

The revolution has already happened outside the university. Each of the three topics in the paper, central force motion, electric circuit analysis, and rigid-body motion, are analyzed in the real world using computers start to finish, for one reason - it is the only way complex systems can be analyzed.

And, why is that? It is because the process models are written as differential equations, and the differential equations cannot be solved analytically but they can be analyzed using computational calculus. This is the way it is done. But they are teaching physics in the U just as they have for the last 50 years. The revolution is inevitable, it's just a question of when.

 

[symbolipoint]

Will Flannery, in #61,
Maybe the revolution is pushing students to be brilliant and fashionable, before they learned enough of the fundamentals.

 

[Dale]

If students understand the kinematic equations (for constant acceleration) and the single idea of making the time steps short to account for changes, there is no need to know calculus.

In fact, it motivates calculus later on. Calculus is what you get when you make the steps infinitesimal. You can’t do that by the numerical approach (infinite memory and computation time), but for some problems you can do it analytically using calculus.

 

[Dale]

Anecdotally, I've asked two high school (top private schools) math teachers why they were teaching calculus, why is it useful? One answer "I don't know", second answer "It's like weight-lifting, even though you don't do it outside the gym it makes you strong for other activities". I swear I am not kidding !

Given your tone in this thread I have no doubt that you asked such a question and received such responses. It is extraordinarily easy to provoke such answers by a suitable choice of tone or wording.

 

[atyy]

The paper describes a high school course that can be taught independently of the math program beyond high school algebra and geometry. From #55 - computational methods for (computing solutions to) ODEs should precede the calculus classes, I remember from my undergraduate days that the calculus classes were not only abstract, they were unmotivated as the study of DEs was necessarily preceded by differential and integral calculus and on top of that I had no idea of the significance of DEs in analyzing physical phenomena ...

Anecdotally, I've asked two high school (top private schools) math teachers why they were teaching calculus, why is it useful? One answer "I don't know", second answer "It's like weight-lifting, even though you don't do it outside the gym it makes you strong for other activities". I swear I am not kidding !

The revolution has already happened outside the university. Each of the three topics in the paper, central force motion, electric circuit analysis, and rigid-body motion, are analyzed in the real world using computers start to finish, for one reason - it is the only way complex systems can be analyzed.

And, why is that? It is because the process models are written as differential equations, and the differential equations cannot be solved analytically but they can be analyzed using computational calculus. This is the way it is done. But they are teaching physics in the U just as they have for the last 50 years. The revolution is inevitable, it's just a question of when.

Mechanics and Euler's method are already part of some high school mathematics syllabi. In the AQA syllabus, Euler's method is referred to as a "step-by-step method based on the linear approximations"
https://www.aqa.org.uk/subjects/mathematics/as-and-a-level/mathematics-6360/subject-content/further-pure-1

Here is the A-level Further Mathematics syllabus in Singapore. It includes Euler's method, as well as mechanics and electrical circuits.
https://www.seab.gov.sg/content/syllabus/alevel/2017Syllabus/9649_2017.pdf

Here is a YouTube video by Jack Brown explaining Euler's method. His channel has lots of videos for people studying high school mathematics.


I think mechanics and Euler's method were part of my high school mathematics syllabus. I'm sure we did mechanics, but I'm not entirely certain about Euler's method. Currently Euler's method seems to be only found in the more advanced "Further Mathematics" syllabus, which I did not do. I only did the more basic "Mathematics" syllabus. However, I'm certain I learned proof by induction in high school under "Mathematics", whereas proof by induction now seems to be only in the "Further Mathematics" syllabus. Nonetheless, these syllabi show that it is not uncommon for mechanics and Euler's method to be taught in high school mathematics.

The Further Mathematics A-level (ie. knowledge of Euler's method) are stated to be useful preparation to study eg. physics at Cambridge University, and mechanical engineering at Imperial College.
https://www.natsci.tripos.cam.ac.uk/subject-information/part1a/phy
https://www.imperial.ac.uk/study/ug...chanical-engineering-meng/#entry-requirements

 

[kith]

I also don't think that there's a revolution to come but I do think that there are interesting decisions to be made with regard to teaching dynamics.

Teaching students how to solve constant acceleration problems and than emphasizing that every problem can be solved by approximating the acceleration as constant during short time steps certainly appeals to me. If I get the OP right, he isn't concerned that much with whether the Euler method is introduced or not, but more with the emphasis one puts on simple numerical methods and their power.

Personally, I have encountered numerics quite late in the university physics curriculum and it wasn't presented as conceptual but more as a tool which we need to resort to if analytic solutions fail. Feynman on the other hand introduces numerical solutions to the harmonic oscillator and planetary motions in the same lecture in which he introduces Newton's second law. And he promotes their importance: Euler's method is introduced in subsection 9-5 which is titled "Meaning of the dynamical equations".

 

[Dale]

If I get the OP right, he isn't concerned that much with whether the Euler method is introduced or not, but more with the emphasis one puts on simple numerical methods and their power.

I don’t know. He was quite insistent that it must be the Euler method and nothing else. His reasoning seemed pretty flimsy to me, but the OP was quite fixated on that specific method.

 

[kith]

I don’t know. He was quite insistent that it must be the Euler method and nothing else. His reasoning seemed pretty flimsy to me, but the OP was quite fixated on that specific method.

Ok, he thinks that the introduction of the Euler method is necessary. But he doesn't think that it is sufficient. Pointing out that it is already widely taught at some point in the curriculum isn't enough to convince him that his approach isn't revolutionary if his main concern is how the method is used in the teaching.

 

[atyy]

The paper describes a high school course that can be taught independently of the math program beyond high school algebra and geometry. From #55 - computational methods for (computing solutions to) ODEs should precede the calculus classes, I remember from my undergraduate days that the calculus classes were not only abstract, they were unmotivated as the study of DEs was necessarily preceded by differential and integral calculus and on top of that I had no idea of the significance of DEs in analyzing physical phenomena ...

Is this really before calculus? At the bottom of page 1, you say:

"Newton’s model for a falling object consists of state variables for position, r, and velocity, v, and the rate equation
r' = v(t)
v' = -Gm/r^2"

How is that to be understood without calculus - aren't r' and v' derivatives?

 

[Will Flannery]

It's important to understand the power of computational calculus, i.e. methods of computing solutions to differential equations, and its potential effect on the entire technical curriculum. I had two tables in the OP to demonstrate this, but I didn't upload them, so they didn't appear. The first one is

The point is that all these results are obtained with essentially the same math as used to compute the trajectory of a falling apple. The physical laws are simple and intuitively clear, the model derivations are one or two lines of simple algebra, and Euler's method does the rest.

Looking more closely at the results:

Central force motion - this is the Apollo trajectory; the method of analysis, i.e. simulation, is the state of the art. This is how it is done in the real world. The real life sims are 3-d and more accurate and much more detailed, but simulation using computational calculus is the state of the art method for analyzing central force motion.

Electric circuit analysis - Again, this is the state of the art method for analyzing electric circuits. In the real world SPICE (simulation program with integrated circuit emphasis) is an industry standard simulator that automates the procedure for the EE, who enters a circuit description and the program does the rest.

Rigid-body motion - here we had to pull back a little, from 3-d to 2-d. Analysis of 3-d rigid-body motion requires Euler angles/quaternions and the moment of inertia tensor. But, 3-d simulation using computational calculus is the state of the art method for analyzing rigid body motion.

If this course is taught to good high school science students, it will begin the transformation of the technical curriculum so that modeling with differential equations, and using computational calculus to analyze the models, are central features from the start.

The situation is even more dramatic when it comes to branches of physics based on partial differential equations. The table below is for a follow up university course that uses the finite difference method (FDM), which is Euler's method extended to PDEs, to analyze partial differential equation models. I'll save the discussion of the table for later if anyone is interested - hint: heat is easy, waves are easy, the primary difficulty, beginning with stress and strain, is ... ?