The coming revolution in physics education

[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 ... ?

 

[Will Flannery]

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?

r'(t) = v(t) is an equation for the velocity of the object at time t, the name of the function is r', so, it's just the name of a function. When you take your calculus course you'll find out that the velocity of a function r(t) is usually denoted r'(t) and it's called a derivative.

Ditto for v'(t) = -Gm/r(t)^2

That is, a process model is a set of variables, var1, var2, ... varn, and an equation for the rate of change, i.e. velocity, of each. So I need a name for the function that is the rate of change of var1, (and var2, etc.) and I called it var1' just as is done in calculus. I could have called it rate_of_change_var1 or any other name.

Try this - read thru the paper attached in the OP, up to 'end of first lecture'. You should be able to understand every step. If not, it's on me - post a question.

 

[Tom Hammer]

I taught high school physics for 15 years with some success. The difficulties were not in solving problems once the underlying physics was understood but in undoing the years of “false physics” that most students arrived having internalized. In the 90s there was a movement to measure the basic physics beliefs of incoming students and then to do a second measurement after taking basic physics. Most students in traditional courses never improved their basic understanding, so in response a course was developed with much success at Dicinson College and elsewhere. Unfortunately, thought the course had a high rate of success in improving basic understanding, it required a lot of technology to implement and never caught on widely.

I hope that any proposed “revolution” is designed with some testing and use of the scientific method instead of simply the belief by some that “this seems more logical and easier”.

 

[Dale]

I hope that any proposed “revolution” is designed with some testing and use of the scientific method instead of simply the belief by some that “this seems more logical and easier”.

I agree wholeheartedly with this. I had asked a similar question earlier. I don’t know if there is solid evidence that the use of numerical methods improves conceptual understanding.

https://www.physicsforums.com/threa...-physics-education.954664/page-3#post-6055004

@Will Flannery is clearly of the opinion that it does, and @bob012345 is clearly of the opinion that it is detrimental. I would like to see scientific evidence of its efficacy one way or the other.

 

[atyy]

r'(t) = v(t) is an equation for the velocity of the object at time t, the name of the function is r', so, it's just the name of a function. When you take your calculus course you'll find out that the velocity of a function r(t) is usually denoted r'(t) and it's called a derivative.

Ditto for v'(t) = -Gm/r(t)^2

That is, a process model is a set of variables, var1, var2, ... varn, and an equation for the rate of change, i.e. velocity, of each. So I need a name for the function that is the rate of change of var1, (and var2, etc.) and I called it var1' just as is done in calculus. I could have called it rate_of_change_var1 or any other name.

Try this - read thru the paper attached in the OP, up to 'end of first lecture'. You should be able to understand every step. If not, it's on me - post a question.

If v' is just the name of a function, then it has no relation to v?

 

[Will Flannery]

*

If v' is just the name of a function, then it has no relation to v?

I'm using this convention, which is standard: v' is the name of the function that returns the velocity of v, but it's just a convention. I could have used a different one: v# is the name of the function that computes the velocity of v (maybe I should have).

 

[atyy]

*

I'm using this convention, which is standard: v' is the name of the function that returns the velocity of v, but it's just a convention. I could have used a different one: v# is the name of the function that computes the velocity of v (maybe I should have).

So what is "velocity"?

 

[Dr. Courtney]

I taught high school physics for 15 years with some success. The difficulties were not in solving problems once the underlying physics was understood but in undoing the years of “false physics” that most students arrived having internalized.

My experience has been much different. Pretty good Physical Science courses early in high school have done a good job with my students undoing the "false physics." The challenges I've had were more on the math side: very poor algebra skills, no trig to speak of, complete inability to solve word problems involving multiple steps, etc.

I hope that any proposed “revolution” is designed with some testing and use of the scientific method instead of simply the belief by some that “this seems more logical and easier”.

My view is that the focus of the lab side of the course should be the scientific method. How many high school classes spend 25-40% of the class time on real labs (that test hypotheses, require data analysis, and written discussions)? How many high school classes complete 15-20 real physics experiments?

It is very hard to teach and learn the scientific method in a meaningful way when the lab portion of the courses consistently get short shrift.

 

[Zeynel]

https://mail.aol.com/webmail/getPar...ope=STANDARD&saveAs=PDE+chart+small.jpg[/IMG]

* what Newton did: see http://farside.ph.utexas.edu/teaching/336k/lectures/node32.html

Sorry, a nitpick. This language does not reflect reality. Newton did none of the things shown in the link. Newton did not work with equations but with proportions. Newtonian dynamics as presented in the link was developed years after Newton's death. Newton did not even know about "Newton's constant G". Why is this important? I think it is important to conform to historical developments.

 

[Will Flannery]

Sorry, a nitpick. This language does not reflect reality. Newton did none of the things shown in the link. Newton did not work with equations but with proportions. Newtonian dynamics as presented in the link was developed years after Newton's death. Newton did not even know about "Newton's constant G". Why is this important? I think it is important to conform to historical developments.

You're right - although I wouldn't worry about the gravitational constant which just adjusts for the units of measurement - the reference should have read something like ...
* There is no analytically defined r(t) such that r'' = G∙m2 / (r∙r). Newton derived Kepler’s laws, from which Newton (and Kepler) derived numerical procedures to approximate r(t). For a modern treatment see ... link.

 

[grandbeauch]

The answer to that question seems obviously to be the black boxes. It will take slightly less time to teach the inputs and outputs of the black box than to teach Euler’s method, and it will take far less time to use it.

When they are done they will have the same understanding of physics in a shorter amount of time and they will know how to use a computational tool that they can use professionally for the rest of their life even to solve problems where Euler’s method flat out fails.

 

[Dr. Courtney]

https://aapt.scitation.org/doi/10.1119/1.5055324

Nice article in this month's TPT on using Euler's Method (in cognito) in a spreadsheet to solve for motion with a varying force.

My classroom approach is very similar and accessible to high school students who don't even know what Calculus really is yet.

 

[Will Flannery]

Since Newton, the basic paradigm for the analysis of physical systems in classical physics has been:
1. State the physical laws governing the system
2. Derive a differential equation model of the system
3. Solve or otherwise analyze the model from step 2.

The basic paradigm is not taught in high school or the university because differential equation models of physical systems are difficult or impossible to solve, and require years of study of calculus. And even then the differential equations remain impossible to solve. Therefore the educational system deemphasizes differential equations for as long as possible. At USF (U of South Florida) a physics major takes a course in DEs in his/her junior year!

The consequence of ignoring the basic paradigm is that university students typically spend two years studying what I'm calling pre-physics, that is, physics without differential equations. At the nearby USF a physics major takes only two physics courses in their first two years, the text for both courses is 'University Physics' by Young, Freedman, it is one of the world's most popular physics texts according to wiki. It covers mechanics, central force motion, electric circuit analysis, rigid body dynamics, heat transfer, wave phenomena, stress and strain in materials, fluid dynamics, and electrodynamics, all without differential equations !

The remedy is to teach the paradigm, along with computational calculus in the form of Euler's method for ODEs and the finite difference method for PDEs, at the very beginning of physics/STEM education, in a course dedicated to that purpose and called Scientific Programming. I've written a paper that describes the course titled 'A New Curriculum for Classical Physics', that fills in many of the details, you can see it here ...
http://www.berkeleyscience.com/ANewCurriculum.pdf

 

[mpresic3]

The basic paradigm is not taught in high school or the university because differential equation models of physical systems are difficult or impossible to solve, and require years of study of calculus. And even then the differential equations remain impossible to solve. Therefore the educational system deemphasizes differential equations for as long as possible.

This is NOT the reason the "basic paradigm" , as you call it, is not taught in high-school and university. One important reason simple harmonic motion for a pendulum is taught with simplifying assumptions, is to show how much mileage can be obtained from solving a (elementary and straightforward) second order differential equation. Solving the same problem with elliptic functions, or a numerical method will lead to a better solution, but at the expense of time and effort. The time and effort may be manageable for students in physics, but what about the pre-med, or social science student, does he or she need the differential equations?
The simple pendulum is only one example. I am not a professional educator but my time as a recitation instructor, and time in post graduate education (probably around 10-15 years in total), suggests that physics educators should stress how approximations are useful in getting to the heart of physics problems, and how (as another example), the central force problem for the path of a body under the inverse-square law can be (cleverly) solved using conservation laws, changes of variable, and other techniques.

This is not to say that we should dismiss your idea of including scientific programming, and numerical solutions to differential and partial differential equations. I make my living doing just this. A course (or maybe even two) should probably added to the physics curricula, but the curricula is pretty tight these days with quantum mechanics, electricity and magnetism, laboratory, statistical physics and so on.

I just take issue with trying to justify the effort to introduce this course by creating a new "paradigm". I also do not buy into the idea that the educational system deemphasizes differential equations for as long as possible. Lately, there have been other posters to this forum, to deemphasize calculus in high-schools
I think it would be better to call to mind that companies that hire and graduate schools for research are interested in solving "practical problems" not "textbook" problems, and a course in numerical methods is important to these ends

 

[hutchphd]

I would like to be a little harsher in my criticism of your plan.
The introduction of numerical solutions and canned programs as the initial exposure to physics education is a terrible idea. It promotes what I like to call the Oracle Approach to physics which I consider anathema:
Student uses computer program to show it takes a ball 1s to fall 16 ft.
Ask student how far it will fall in 2s. Student says "Let me plug it into the program"
Anathema.Excuse me but "paradigm" is one of my trigger words
'

 

[Will Flannery]

I realized that I had a hard sell on my hands, and I thought my first post was a winner ! But ... clearly my initial post was unconvincing, so I've added an illustrative example that will hopefully improve it, in the last paragraph below. (Note: part of my reason for posting is to develop a concise intriguing, if not convincing, argument.)

Since Newton, the basic paradigm for the analysis of physical systems in classical physics has been:
1. State the physical laws governing the system
2. Derive a differential equation model of the system
3. Solve or otherwise analyze the model from step 2.

The basic paradigm is not taught in high school or the university because differential equation models of physical systems are difficult or impossible to solve, and require years of study of calculus. And even then the differential equations remain impossible to solve. Therefore the educational system deemphasizes differential equations for as long as possible. At USF (U of South Florida) a physics major takes a course in DEs in his/her junior year!

The consequence of ignoring the basic paradigm is that university students typically spend two years studying what I'm calling pre-physics, that is, physics without differential equations. At the nearby USF a physics major takes only two physics courses in their first two years, the text for both courses is 'University Physics' by Young, Freedman, it is one of the world's most popular physics texts according to wiki. It covers mechanics, central force motion, electric circuit analysis, rigid body dynamics, heat transfer, wave phenomena, stress and strain in materials, fluid dynamics, and electrodynamics, all without differential equations !

The remedy is to teach the paradigm, along with computational calculus in the form of Euler's method for ODEs and the finite difference method for PDEs, at the very beginning of physics/STEM education, in a course dedicated to that purpose and called Scientific Programming. I've written a paper that describes the course titled 'A New Curriculum for Classical Physics', that fills in many of the details, you can see it here ...
http://www.berkeleyscience.com/ANewCurriculum.pdf

To illustrate using planetary motion: in the USF text 'University Physics', and in high school physics, the section on central force motion begins with Newton's law of gravity and Newton's 2nd law of motion. So, the first step of the paradigm is accomplished. It is a trivial matter to derive the differential equation model for a falling object, A = GM/RR, and as I recall that's done in high school physics. However, 'University Physics' doesn't derive this equation, instead it introduces, not derives, Kepler's Laws, and uses them for the section on planetary motion. So, for planetary motion the student studies pre-Newtonian physics. The alternative is to use Euler's method to compute solutions to the differential equation as is shown in the paper. Euler's method is simple, intuitive, and can be taught to high school science students and used to compute 1-D trajectories of falling objects in a single one-hour lecture. Orbits can be calculated in a second lecture. In terms of analyzing physical systems, we've replaced two years of study of analytic calculus that is ineffective for analyzing complex systems with a one-hour lecture on computational calculus that is not only effective in analyzing real complex systems, it is the only way that real complex systems can be analyzed and it has revolutionized science and engineering outside the university ! The paper demonstrates how computational calculus is applied in most areas of classical physics.

 

[Will Flannery]

Note: I published a paper on the subject of the thread, i.e. using computers in physics education, in 'The Physics Teacher' for 10/19, you can see it here ... http://www.berkeleyscience.com/TheComingRevolution.pdf.

 

[pbuk]

I think this is a terrible idea. Plugging equations into black box doesn't give any understanding and creates a reliance on that black box to solve problems. And Euler's method is a terrible black box anyway.

Then that isn’t a physics course. So sell it as what it is: a numerical methods course. It is not a revolution in teaching physics. One of the problems I think that you are having is that you are mislabeling the course and people rightly object.

If it is a numerical methods course then that is even worse - presenting Euler's method as some kind of universal solution generator without an understanding of its limitations and how they arise from Taylor's theorem (with also an understanding of round-off error) is about as useful as teaching multiplication by repeated addition.

 

[Vanadium 50]

Spamming the forum with links to your paper is not an effective way to get your ideas across.

If your ideas have merit, once is enough. If they don't, even a thousand won't help.

 

[atyy]

The consequence of ignoring the basic paradigm is that university students typically spend two years studying what I'm calling pre-physics, that is, physics without differential equations. At the nearby USF a physics major takes only two physics courses in their first two years, the text for both courses is 'University Physics' by Young, Freedman, it is one of the world's most popular physics texts according to wiki. It covers mechanics, central force motion, electric circuit analysis, rigid body dynamics, heat transfer, wave phenomena, stress and strain in materials, fluid dynamics, and electrodynamics, all without differential equations !

Is this true? https://www.amazon.sg/University-Physics-Modern-Hugh-Young/dp/0135159555 says it's for courses in calculus-based physics.

That University Physics by Young and Freedman is for calculus-based physics is also mentioned at https://web.mit.edu/physics/prospective/undergrad/transfer.html
"The Physics Validation Exams are three-hour, closed book exams covering Classical Mechanics (8.01) or Classical Electromagnetism (8.02) at a level of calculus-based introductory physics texts for science and engineering students such as: University Physics by Young and Freedman; Physics by Halliday, Resnick and Krane; Physics for Scientists and Engineers by Serway; Physics for Scientist and Engineers by Fishbane, Gasiorowicz, and Thornton. The exams will be similar to the final exams given in 8.01 and 8.02, with problems based on a selection of the topics listed below. Neither calculators nor formula sheets may be used during Validation Exams."

 

[kith]

Is this true?

I don't know about the US but it is far from true in Europe. Physics students at university not only encounter calculus right from the start but usually take a course in real analysis in the first semester.

 

[Will Flannery]

Is this true? https://www.amazon.sg/University-Physics-Modern-Hugh-Young/dp/0135159555 says it's for courses in calculus-based physics.

The text for PHY 2048-2049 is 'calculus based', calculus is a prerequisite for the courses at USF. Yet, the text is almost completely differential equation free ... you can download a free pdf, for example at https://www.academia.edu/41736532/University_Physics_With_Modern_Physics_14th_Edition_by_Hugh_D_Young_Roger_A_Freedman

Checking the curriculum guide for the University of South Florida (USF) [1], a physics major takes one calculus course, MAC 2311 – Calculus I, and no physics courses in the freshman year and two math courses, MAC 2312 – Calculus II and MAC 23113 – Calculus III, and two courses, PHY 2048 – General Physics I and PHY 2049 – General Physics II, in the sophomore year. Differential equations are not covered in the three math courses [3]. Differential equations are covered in the third-year course PHZ 3133 – Mathematical Method for Physics.
Note: I've got links for all this.

The first mention of differential equations in Young and Freedman (use the search function on the pdf) is on page 276, they are mentioned in passing. The fourth mention, on page 415 in the section on planetary motion, reads

These results can be derived by a straightforward application of Newton’s laws and the law of gravitation, together with a lot more differential equations than we’re ready for.

Thus planetary motion is covered without the DE model, i.e. A=GM/RR, electric circuit analysis without the DE models of capacitor and inductor, heat transfer is covered without Fourier's law, fluid mechanics without the Navier-Stokes equations, electrodynamics without Maxwell's equations in differential equation form, etc.

 

[hutchphd]

So USF would not be my first choice for a physics major.
This is not evidence for your thesis however.

 

[Vanadium 50]

If your ideas have merit, once is enough. If they don't, even a thousand won't help.

This was not a suggestion you work your way up past a thousand.

 

[Dale]

university students typically spend two years studying what I'm calling pre-physics, that is, physics without differential equations

This may be the case at USF, but I dispute your claim that it is “typical”. Do you have a scientific reference supporting your claim?

I gave several counter-examples earlier in the thread, so I am highly skeptical about the correctness of your claim here.

 

[Will Flannery]

This may be the case at USF, but I dispute your claim that it is “typical”. Do you have a scientific reference supporting your claim?

I gave several counter-examples earlier in the thread, so I am highly skeptical about the correctness of your claim here.

I base it on the following: the Young and Freedman text is one of the world's most popular introductory physics texts according to wiki.

I've just checked the Resnick and Halliday text used at UF (a free pdf download is available) which is similar but does use differential equations in the section on electric circuit analysis.

Also the OpenStax text is similar (a free download is available).

All of the texts cover planetary motion without Newton using Kepler's laws, heat transfer without Fourier's law, fluid mechanics without the Navier-Stokes equations, and electrodynamics without Maxwell's equations in differential equation form.

Also from a post above ... https://web.mit.edu/physics/prospective/undergrad/transfer.html ... the way I read it MIT has allows a student to transfer credit for their introductory physics courses from another college by taking a test ...

The Physics Validation Exams are three-hour, closed book exams covering Classical Mechanics (8.01) or Classical Electromagnetism (8.02) at a level of calculus-based introductory physics texts for science and engineering students such as: University Physics by Young and Freedman; Physics by Halliday, Resnick and Krane; ...

So, I don't claim the avoidance of DEs is universal (the UC Berkeley text does give an analytic derivation of Kepler's laws I think, but that's atypical for an introductory course, the junior level course in classical mechanics at USF doesn't, for example), but I think it is typical.

 

[Dale]

None of which constitutes adequate support for your claim on this forum.

 

[gmax137]

just as a single meaningless data point, when I was in school the intro classes for physics majors was mechanics by Kleppner & Kolenkow followed by EM by Purcell. if they offered a "Halliday..." intro physics course, it was for the pre-meds, not physics majors.

maybe this wasn't all to the good - the physics majors all knew our path in high school or even earlier. Maybe the root of the diversity problem often discussed.

 

[Will Flannery]

just as a single meaningless data point, when I was in school the intro classes for physics majors was mechanics by Kleppner & Kolenkow followed by EM by Purcell. if they offered a "Halliday..." intro physics course, it was for the pre-meds, not physics majors.

maybe this wasn't all to the good - the physics majors all knew our path in high school or even earlier. Maybe the root of the diversity problem often discussed.

Kleppner is the UC Berkeley text (I downloaded a free pdf), and ... I'm glad I didn't take that class, and I was math major. The chapter on central force motion is way tough ... yet I note that it does not solve Kepler's problem, i.e. position as a function of time ... instead we have ...

Equation (10.11) formally gives us r as a function of t, although the integral may have to be done numerically in some cases.

where 10.11 is a nasty looking integral from r0 to r involving expressions for total mechanical energy and effective potential energy.

 

[hutchphd]

There is a considerable difference between showing analytically that certain functions solve differential equations of motion and developing the skillset to solve ab initio those equations. For the most part that latter skillset is learned in the junior and senior level courses. But one certainly should not delay providing the differential equations and analytic solutions.
When I arrived at Cornell 50 yrs ago I literally did not know what an integral sign was and took "noncalculus" freshman physics from Sears and Zemansky (no Young). Even in that context I was exposed to the details of the harmonic oscillator first semester much to my edification. The solutions to uniform constant accelerated motion were presented and conjoined with initial conditions to solve the differential equations. By the beginning of sophomore year I was much the better for it. Incidentally the calculus based course was taught from Halliday and Resnick.
If remedial work is required by some students then by all means we need to be creative in supplying it efficaciously. But to abjure the teaching of analytic methods central to the subject is ridiculous, and the notion that existing programs postpone these subjects as you describe is absolutely not true for any institution that I know.

 

[Will Flannery]

Here is what happens - (as I recall) - in high school you derive the equation A = GM/RR, but that is deemed to difficult to solve so it is simplified to A = -9.8 and it's easy even in high school to solve this heuristically, so that is done, and you calculate the trajectories of falling and thrown objects near the surface of the earth.

Here's what they don't tell you. You will never see the equation A = GM/RR solved, it does not have a closed form solution and I'll challenge anyone to find an analytic series solution in any text. I don't know of one. (I do know of a Texas A&M website where Kepler's laws are derived from the DE, and a solution is given for Kepler's problem (I think.), i.e. computing position as a function of time from Kepler's equation)

Let's see what Kleppner does ... and I've seen this elsewhere ... he alludes to a solution, noted in my previous post, but doesn't provide it. Instead he writes ...

Often we are interested in the path of the particle, which means knowingr as a function of θ rather than as a function of time.

and that problem is solved.

However, I worked for years on space projects, Star Wars !, the first use of laser gyros in space, Space Station Freedom stability control, and the Mars Observer. and in every project position as a function of time was basic, and position as a function of angle was never a consideration. That is, incidentally, when I discovered that computational calculus is how problems are solved. Not in the U. It was a revelation to me.

The reality is that the two-body problem doesn't have a closed form solution, and the three body problem, e.g. rocket trajectories between to Earth and moon, are completely intractable analytically.

They never tell you this at the university. So ...

There is a considerable difference between showing analytically that certain functions solve differential equations of motion and developing the skillset to solve ab initio those equations. For the most part that latter skillset is learned in the junior and senior level courses.

No, the differential equation models of real systems are almost always completely intractable. The skillset to solve them doesn't exist.

But to abjure the teaching of analytic methods central to the subject is ridiculous, and the notion that existing programs postpone these subjects as you describe is absolutely not true for any institution that I know.

I'm not suggesting to delay teaching analytic calculus, and I'm not saying that current programs do. I'm saying they delay teaching differential equations. What I am suggesting is teaching modeling physical processes with differential equations at the very beginning, in and of themselves they are not difficult at all, they don't even involve calculus. It's when you try to solve them analytically that things get difficult, and I would delay that just as it is done now.

However, computational calculus is trivially easy, it computes solutions to all differential equations, even analytically unsolvable ones, no problem, and it can be taught to high school students in a single one hour lecture and immediately be used to analyze any number of real complex physical systems, e.g. the Apollo trajectory, electric circuits, etc. This will completely transform physics and STEM education.

And computational calculus is not a 'black box', in its simplest form, Euler's method, it is a trivially easy formula, distance = velocity * time, or generally change = (rate of change)* time. That's it ! It is intuitively obvious and transparent, it completely demysitifies differential equations.