The coming revolution in physics education

[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.