The coming revolution in physics education

[physicsponderer]

I suppose if I had pressed and dug I could have uncovered at least one misconception with each. Since eminent physicists like to make wagers on physics and since at least one side of the wager must have a misconception, I don't think that having one misconception is a substantive criticism. I am sure I have many more than one.

I guess I have a fascination with physics puzzles and misconceptions of all sorts, regardless of subjects. Perhaps I overvalue them. Physics graduates do understand the physical world well, compared to graduates in other subjects, but I just think they should understand it better than they do.

 

[Dale]

I guess I have a fascination with physics puzzles and misconceptions of all sorts, regardless of subjects. Perhaps I overvalue them. Physics graduates do understand the physical world well, compared to graduates in other subjects, but I just think they should understand it better than they do.

OK, but do you honestly believe that a mere change in curriculum would change that?

 

[physicsponderer]

OK, but do you honestly believe that a mere change in curriculum would change that?

I think changing the curriculum would have some effect, but I have no idea how much. Maybe if to get onto a degree course in the first place students had to show exceptional ability at physics puzzles, that might help more.

 

[Dale]

I think changing the curriculum would have some effect, but I have no idea how much.

Really? I suspect that you would just come up with different tricky puzzles and make the same complaint. I.e. I do not think that the complaint indicates a real deficiency, but an unrealistic expectation. There will always be some set of tricky puzzles that would trick a good number of graduates.

From my experience, physics graduates are well equipped to work at my company where I was a hiring manager for about 15 years. I didn't spend a lot of time deliberately finding puzzles to trick them, but I am sure that such puzzles could have been found. But again, that they should be puzzle experts or untrickable is an unrealistic expectation in my opinion, and wholly unnecessary for my real-world needs.

 

[Dr. Courtney]

I would have disagreed. You don't get to redefine English words. Physics does not only mean a good complete physics degree course.

Of course, but both the overall context of this thread and the specific context of the part of my post you failed to quote make it clear that most of the discussion here is talking about complete physics courses. One does not have a "coming revolution in physics education" without discussing complete courses.

Recall that I wrote:

Not at all. What if I had said, "Without reading, one cannot really teach law" or "Without reading, one cannot really teach history"?

At the high school and college levels, one is not really teaching law or history with the appropriate level of rigor if one does not require the students to _READ_.

So sure, there are some elementary physical principles that can be taught to students without requiring them to do math. But with the exception of physics courses with "Conceptual" in the name, one is not being honest about the rigor if one is teaching high school or college physics without requiring students to do the math.

What would you think of a law school that did not require their law students to read? This is how I regard physics teachers who do not require their students to solve quantitative problems.

You don't get to redefine physics courses (by removing the math) without getting your mathless course descriptions approved by the appropriate bodies and accrediting agencies. And that is the sleight of hand being attempted in many high school and college physics courses these days - they are telling the accrediting agencies and downstream stakeholders (courses, employers, etc.) that their physics courses are still based heavily in quantitative problem solving, yet students who pass these courses are barely able to solve the kinds of problems the course descriptions lead the downstream stakeholders to believe the students can solve.

The most important question I think teaching candidates need to ask is, "Are you going to fire me if I am unwilling to pass students who cannot do the math required in my courses?" Many teaching candidates are afraid to ask this question, and many teachers go ahead and gift grades to keep their jobs. This is just as fraudulent as history and law professors passing students who cannot read.

 

[abcd112358]

The revolution is to introduce differential equations, computational calculus, and computers into the curriculum at the start and to use them to analyze physical systems in all classes in classical physics, specifically mechanics, electric circuit analysis, dynamics, heat transfer, wave phenomena, stress and strain in materials, fluid dynamics, and electrodynamics.

In (a part of) germany there was a change in the curriculum about 15 years ago. With this change there was an introduction to a (simple) numerical method where studets of 10th grade use Excel to analyze position, speed and velocity of a pendulum or a falling object including air resistance. Although your proposal seems to be more sophisticated, the basic ideas sound related. There are two (personal und subjective) observations I made:
1. Nearly all teachers I asked about said part of the curriculum told me they were less than impressed by the effect these classes had on their study progress regarding more complex mechanical problems. The only teachers that told me their students actually learned something meaningful were those teachers that made the students calculate the first few steps of the numerical method by hand (and with the use of a simple calculator).
2. After said 15 years there is again a change in the curriculum. The numerical method was dropped. My guess is that it did not have the expected/hoped effects. Else I would assume the decision makers would have extended the application of this (and maybe additional) numerical method(s) to even more parts of the physics curriculum.

My conclusion would be that the introduction of a numerical method to the physics curiculum did not revolutionize anything.

My personal experience (without learning any numerical method at school) is that during my time at university no one I met had real problems teaching themselves the use of Matlab and the likes. Same goes for the application of numerical methods.

 

[Will Flannery]

Look at how well the availability of circuit simulation has improved the analytic capabilities of analog electrical engineers! Its a revolution! Paradigm shift! Everybody knows it!
...

Up to this point your post is exactly right. You've made my case. Let's have a look ...wiki - SPICE

Unlike board-level designs composed of discrete parts, it is not practical to breadboard integrated circuits before manufacture. Further, the high costs of photolithographic masks and other manufacturing prerequisites make it essential to design the circuit to be as close to perfect as possible before the integrated circuit is first built. Simulating the circuit with SPICE is the industry-standard way to verify circuit operation at the transistor level before committing to manufacturing an integrated circuit.
...
The birth of SPICE was named an IEEE Milestone in 2011; the entry mentions that SPICE "evolved to become the worldwide standard integrated circuit simulator".[13] Nagel was awarded the 2019 IEEE Donald O. Pederson Award in Solid-State Circuits for the development of SPICE.[14]

 

[Dr.D]

I did, and recommend using prepackaged ODE solvers instead of hand coding Euler’s method. All the way back in post 2

Looks like I missed that! Mea culpa, mea ...

But it seems like a lot of others did also. There is simply no point in learning to apply an unstable method. Agreed that Euler helps to explain the ideas behind numerical solution of ODEs, but it should never be used if you want valid results.

 

[hutchphd]

Up to this point your post is exactly right. You've made my case. Let's have a look ...wiki - SPICE

Was there any discussion of pedagogy in this article?
Revolution?
paradigms (shifted or otherwise)?
Design method?
Integrated circuits are expensive up front. SPICE is a fabulous tool and I use it myself to test that a circuit performs as expected. Perhaps my paradigm shifted when I wasn't looking...geez I didn't even notice. Again give me a break.

 

[Will Flannery]

Was there any discussion of pedagogy in this article?

? There was no discussion of pedagogy. The article demonstrates that the computer and computational calculus have revolutionized the analysis of physical systems outside of the university. A complete paradigm shift. It is inevitable that they will revolutionize physics/STEM education as well.

 

[atyy]

? There was no discussion of pedagogy. The article demonstrates that the computer and computational calculus have revolutionized the analysis of physical systems outside of the university. A complete paradigm shift. It is inevitable that they will revolutionize physics/STEM education as well.

How can there be a coming revolution if numerical methods have long been part of undergraduate physics and engineering curricula?
https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/
https://faculty.math.illinois.edu/~laugesen/285/syll.html
https://faculty.math.illinois.edu/~laugesen/286/blog.html

 

[atyy]

https://sites.lsa.umich.edu/michigan-math-in-action/2017/03/06/the-effectiveness-of-change/
The Effect(iveness) of Change
By Gavin LaRose

Article about numerical methods and other stuff in the differential equations course at Michigan. Euler's method has been part of the course since at least the mid-90s.

 

[jasonRF]

I'll try a different tack. You have apparently condensed my 'revolution' down to taking a numerical methods course early, but that's only part of it.

Here is the rest of the paragraph that didn’t all make it into the part you quotes. Here I have italicized the part you are bringing up

A freshman course is certainly better than nothing, but if a student we are hiring is to take such a course I would much prefer an upper-division version than a freshman version. That way they would learn more sophisticated techniques, have deeper understanding of assumptions and limitations, etc. For example, they would know not to use Euler's method to design something that will cost my project time and $$$ if it is wrong ! I suspect an upper-division course would be more useful for those students going to physics graduate school as well. The pedagogical benefits would have to be significant to prefer the freshman version. This likely means that the syllabi of the subsequent physics courses would need to change. I wonder what topics you propose to eliminate from each course to make room for this new numerical work? Or do you think it can be added without removing anything at all? I doubt it...

 

[jasonRF]

#2 - the central force motion examples in the paper dramatically demonstrate the enormous benefits of teaching computational methods. Newton's solution to the Kepler problem represents the beginning of modern math and science, and it is almost unsolvable analytically, you have to use the computer. And yet, I have not found one traditional university physics text, upper or lower division, that gives a solution.

And white Kepler's problem is almost unsolvable analytically, the three-body problem, e.g. a rocket trajectory from the Earth to the moon, is completely intractable analytically.

What is true for central force motion is true for every branch of classical physics, that is, after the physical laws are stated and the system model derived, the student is faced with unsolvable or nearly unsolvable differential equations.

The paper includes examples from electric circuit analysis and 2-D rigid body dynamics that illustrate how these systems are analyzed outside the classroom. My new paper includes examples for heat transfer, wave phenomena, stress and strain in materials, fluid dynamics, and electrodynamics.

The bottom line is computational calculus is the only way real systems can be analyzed.

The fact that most problems have no exact analytical solution should not be news to anyone with a technical degree. I would hope everyone agrees that a good physics or engineering education should include at least some numerical methods. The issue in the thread is whether your proposed "numerical methods before we have learned anything about calculus or thermodynamics or waves or electrodynamics...", followed by an overhaul of most courses in the curriculum, is preferred to some other approach.

Here I will expand on the sentences I italicized in my prior post:

Part of the concern for me is that it isn't clear to me how much change you would make to upper-division courses in mechanics, electrodynamics, thermal physics, etc. The word "revolution" usually implies a lot of change, much more than one week of lecture and one numerical project in place of one of the current weekly homework assignments. Also, would subsequent physics courses teach more advanced numerical techniques or simply apply those learned in the freshman class? The more they do, the more decisions need to be made about what to eliminate from the current syllabi. As a concrete example, if you assume a current electromagnetics sequence teaches everything in Griffith's book, what sections would you eliminate in order to make room for the new numerical work? How many weeks would that allow you to spend on your new content?

A more conservative, incremental approach to better incorporating numerical methods would be to make no changes to the standard theory and experimental courses, but ensure that all graduates learn some numerical methods along the way. Many departments already do this to some degree. Those that don't may simply need to strongly recommended students use one of their electives for that purpose. The debates here would be whether the brief introductions provided in may differential equations classes are adequate, or if a dedicated numerical methods course should be required.

If students are to take a dedicated numerical course we could argue about the level. I believe Dale mentioned he had a sophomore-level option, which is late enough that it could cover similar topics as your course but at a higher level and with more sophisticated methods. I also had options starting sophomore year, but took a senior-level course which freely used material from the junior-level prerequisites (electrodynamics, Fourier analysis, ...), so could include things like spectral methods, assume we knew electrodynamics when we developed models for and simulated electromagnetic waves in nonlinear media, etc. By the way, the three-body problem ( spacecraft -moon-earth) was the first project.

My opinion is that courses taken after the students have at least learned calculus and intro physics would be more useful post-graduation than your pre-calculus version. So the pedagogical benefits need to be significant to prefer your course over a later course. I'm also skeptical that it makes sense to reduce the analytical content of the current curriculum. Physics graduates need strong analytical skills and they take many hours to develop - many more than numerical skills do. Indeed, I think it is much easier to learn to use numerical techniques on the job than it is to gain analytical skills.

jason

 

[Jarvis323]

On PF it is expected that all posts be consistent with the scientific literature. It is common to ask for references here, and such requests should always be honored, even if you think the point is obvious. If one cannot provide such a reference then it is expected that one will retract the unsupported claims. This is a key part of the PF culture that keeps our quality high compared to other science forums.

To play devils advocate, has your opinion on the topic been oversold? Have you provided references supporting the belief that black boxes are good teaching tools?

 

[Jarvis323]

How can there be a coming revolution if numerical methods have long been part of undergraduate physics and engineering curricula?
https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/
https://faculty.math.illinois.edu/~laugesen/285/syll.html
https://faculty.math.illinois.edu/~laugesen/286/blog.html

I think that a lot of people in this thread are missing the point of the OP's idea.

It's not like the idea is to teach them numerical methods for the sake of gaining a practical skill that they take with them along their education and career.

The idea is to give them a simple intuition about what differential equations are and what we do with them. The other side is that a simple hands on approach might introduce them to the subject in a way that is less scary, less abstract, and more fun. The possibility is that for some students this could inspire and motivate them to want to and not be afraid to get into physics, because they can wrap their heads around it to some extent to begin with. So the point is that it is an early course, rather than a later one. And the measure of success is more about the potential students subsequent confidence and interest.

Whether subsequent courses in numerical methods are redundant or will replace what was learned, is only of concern if the students end up deciding they want to be physicists.

 

[Dale]

To play devils advocate, has your opinion on the topic been oversold? Have you provided references supporting the belief that black boxes are good teaching tools?

Please quote any claims that I made which you would like to see supported. I am happy to provide references for any factual claims I made (or retract/modify the claim).

The issue wasn’t @physicsponderer’s opinion, he is entitled to his opinion (as am I). It was the “facts” that he asserted in support of his opinion. He has now corrected the fact claim with no need to change his opinion.

 

[Jarvis323]

Please quote any claims that I made which you would like to see supported. I am happy to provide references for any factual claims I made (or retract/modify the claim).

The issue wasn’t @physicsponderer’s opinion, he is entitled to his opinion (as am I). It was the “facts” that he asserted in support of his opinion.

I don't see the line. It looks like the "fact" he asserted (something like, physics students seem to lack intuition about the math they're using) is no less an opinion than your opinion that they don't seem to.

As an example, post 150 from hutchphd, in bold, "the easiest way..." is just as easily interpreted as a statement of fact. And you even threw in a like.

I'm just playjng devils advocate.

 

[Dale]

I don't see the line.

Ok, let me know when you find it.

As an example, post 150 from hutchphd, in bold, "the easiest way..." is just as easily interpreted as a statement of fact. And you even threw in a like.

If you object to that claim then, by all means, ask him to provide a reference.

 

[Jarvis323]

Ok, let me know when you find it.

If you object to that claim then, by all means, ask him to provide a reference.

It just seems that the authority a mentor has should not be leveraged to further their own opinion, it should be applied fairly to maintain quality and civility. So I think it is equally your responsibility to demand hutchphd provides a reference as it is to demand physicsponderer does if the line is crossed. I guess optimally, that line should be clearly drawn by mentors and acted on logically and consistently, independent of one's own leanings. Of course that never happens in the real world, where we're all human, and both interesting and purely objective discussions aren't easy to have.

 

[hutchphd]

Anyone who finds any statement of mine to be blatantly false or not supportable is free to question it. The full statement from me in this case was "at some point (in the process of learning all physics) the easiest way is to bite the bullet and learn the mathematics" . This is clearly an opinion but to me an obvious one. Similar to "if you go out in water over your head far enough you will drown".
If anyone can provide cogent reason why this is not obvious I will try to provide further justification. (And someone playing devil's advocate is not sufficient cogent reason. } A request from a Mentor would be sufficient on its face.
As would my use of the term "paradigm shift" in any context..

 

[atyy]

I think that a lot of people in this thread are missing the point of the OP's idea.

It's not like the idea is to teach them numerical methods for the sake of gaining a practical skill that they take with them along their education and career.

The idea is to give them a simple intuition about what differential equations are and what we do with them. The other side is that a simple hands on approach might introduce them to the subject in a way that is less scary, less abstract, and more fun. The possibility is that for some students this could inspire and motivate them to want to and not be afraid to get into physics, because they can wrap their heads around it to some extent to begin with. So the point is that it is an early course, rather than a later one. And the measure of success is more about the potential students subsequent confidence and interest.

Whether subsequent courses in numerical methods are redundant or will replace what was learned, is only of concern if the students end up deciding they want to be physicists.

George Jones and Dr Courtney have both taught Euler's method in high school. See post #3 and post #8. I agree that's a good idea. And you can see it's pretty standard in the introductory differential equations course that many physics and engineering majors take at university (it's Chapter 2 of Edwards and Penney, one of the standard texts; Boyce and DiPrima, another standard text, have it later, but mention early in the text that elements of the chapter can be taught early). If Euler's method is already commonplace in university and at least sometimes taught in high school (as George Jones and Dr Courtney relate), why is the OP's proposal a "coming revolution"?

 

[Dale]

It just seems that the authority a mentor has should not be leveraged to further their own opinion

I didn’t. Absolutely anyone can request references. Frankly, it is offensive that you would say that. Nothing I did in that exchange was leveraging my authority as a mentor.

So I think it is equally your responsibility to demand hutchphd provides a reference as it is to demand physicsponderer does if the line is crossed.

I neither accept your charge that I abused my authority nor your assertion that therefore I need to demand references from everyone.

If YOU want a reference from someone then YOU can ask for it precisely because asking for references is not a mentor function in the first place. It is something everyone can and should do as they see fit.

Also, asking for references is not always something done for claims that you dispute. I have also asked for references for ideas that I found interesting and wanted to learn about more.

I find your accusation here quite offensive and completely unfounded.

 

[Jarvis323]

I didn’t. Absolutely anyone can request references. Frankly, it is offensive that you would say that. Nothing I did in that exchange was leveraging my authority as a mentor.

I neither accept your charge that I abused my authority nor your assertion that therefore I need to demand references from everyone.

If YOU want a reference from someone then YOU can ask for it precisely because asking for references is not a mentor function in the first place. It is something everyone can and should do as they see fit.

Also, asking for references is not always something done for claims that you dispute. I have also asked for references for ideas that I found interesting and wanted to learn about more.

I find your accusation here quite offensive and completely unfounded.

I think you're exaggerating what I said. We were going down the path, of what is considered a statement of fact, vs a statement of opinion. My argument was just that it's sometimes not clear. As a devils advocate, I was encouraging a deeper inspection of what is the line, so that as a mentor, you could more fairly apply your authority without risk that you let your own opinion bias how you apply your authority.

The bit about what I think a mentors responsibility is, was rather what I think the ideal is in terms of resolving these issues, not a full on indictment of you. I'm not saying you're infallible.

This is a human level thing that we all ought to think about. It's something to strive for. I apologize if I caused offense.

 

[Jarvis323]

A correct course design must be ruthlessly narrow in only teaching that which only that course will teach.

Do you have a supporting reference for this?

 

[Dale]

I was encouraging a deeper inspection of what is the line, so that as a mentor, you could more fairly apply your authority

Nothing in that exchange had anything to do with applying my authority!

If I had been unfairly applying my authority I would not have asked for references. I would simply have deleted his post and thread banned him.

 

[hutchphd]

This is a human level thing that we all ought to think about. It's something to strive for. I apologize if I caused offense.

But it is already something that has been well and thoroughly thought about.
Please do not miss the other point here. If someone had requested that I provide elucidation and I refused, then @Dale should use his judgment and authority regarding my acquiescence. Conversely that doesn't preclude him from making a request on his own initiative or not. He is a participant in the forum as much as anyone else and certainly should have the prerogative of such requests on his own. His authority to demand factual references is no different from yours or mine.

 

[Jarvis323]

Nothing in that exchange had anything to do with applying my authority!

If I had been unfairly applying my authority I would not have asked for references. I would simply have deleted his post and thread banned him.

Then I retract my criticism.

 

[Dale]

Do you have a supporting reference for this?

My papers on instructional design are at the office, so I don't have the reference where I originally learned it at hand. But making a course narrowly focused in the context of an overall curriculum is not an idea unique to that reference. Here is another: "Department faculty need to agree on a set of emphases for each course in order to function as a building block within the overall curriculum. The course may focus on only one or two of the curriculum goals, but those few goals must guide the learning outcomes that the course pursues"

Idea Based Learning by E. Hansen, 2011, p 30

One comment, in my current role I teach short intense classes to adults. They last from 1 to 3 weeks, 8 hours per day, so the need for a narrow focus is extreme. These courses are always designed in the context of other education. But even in a longer course there is typically more that a teacher would like to teach than there is time allotted in class, so choices must be made and there needs to be a sound basis of what to include and what to exclude. That relies heavily on understanding the course in the context of the overall curriculum and trusting your fellow-teachers to teach the rest of the curriculum.

 

[Auto-Didact]

To get back on topic:

I think that a lot of people in this thread are missing the point of the OP's idea.

It's not like the idea is to teach them numerical methods for the sake of gaining a practical skill that they take with them along their education and career.

The idea is to give them a simple intuition about what differential equations are and what we do with them. The other side is that a simple hands on approach might introduce them to the subject in a way that is less scary, less abstract, and more fun. The possibility is that for some students this could inspire and motivate them to want to and not be afraid to get into physics, because they can wrap their heads around it to some extent to begin with. So the point is that it is an early course, rather than a later one. And the measure of success is more about the potential students subsequent confidence and interest.

Whether subsequent courses in numerical methods are redundant or will replace what was learned, is only of concern if the students end up deciding they want to be physicists.

Quoted for relevance. One cannot start early enough with exposing children to mathematical ideas; there is a joke often made about teaching a course in analysis in kindergarten but I cannot find the image. The key point to be made is that one cannot underestimate the value of having a true understanding of something compared to just being able to recall lifeless facts; mathematics is one of the best ways to spark such an understanding. This feeling of understanding is empowering for individuals and naturaly spreads across all domains of an individuals life: this is the true goal of any education.

George Jones and Dr Courtney have both taught Euler's method in high school. See post #3 and post #8. I agree that's a good idea. And you can see it's pretty standard in the introductory differential equations course that many physics and engineering majors take at university (it's Chapter 2 of Edwards and Penney, one of the standard texts; Boyce and DiPrima, another standard text, have it later, but mention early in the text that elements of the chapter can be taught early). If Euler's method is already commonplace in university and at least sometimes taught in high school (as George Jones and Dr Courtney relate), why is the OP's proposal a "coming revolution"?

The difference is that teaching this at university is already after having done a postselection, i.e. only the STEM students - those who are able to navigate the current education system successfully - will learn this, while teaching in high school all students learn this. The issue of what teaching method is best for which specific aims at a certain age range is essentially a classic epidemiologic problem of how to evaluate a new intervention versus a control intervention within the context of high school education; i.e. the problem is best decided by using a double blind placebo controlled randomized controlled trial, or a less ideal variant if the DBPCRCT is not realistic.

Having mentored hundreds of students, from high school level up to masters level, across dozens of fields, there is one and only one conclusion I have come to: mathematics and physics education in high school is typically quite abysmal. The paradox is that this is not necessarily because the textbooks or teachers are bad, but because both the textbooks as well as the teachers almost universally reliably fail to engage with all the students, i.e. this is to a large extent a marketing problem.

Take note that I am not considering the students who are already interested in mathematics or physics from a young age, but all students. For the students already interested in mathematics and physics, typically about 10% of any given classroom, the current system works quite well; for the other 90% things aren't usually so optimistic. The numbers may not be completely accurate, but we are literally talking about almost the entire population, apart from the few that end up in STEM jobs.

For most, dare I say for all, of the students for which the current system doesn’t work, they do not usually comprehend the impact not grasping elementary mathematics will have on their life; this is after having factored out those who do grasp this through their teachers, mentors or parents upplaying it and making clear that it will be necessary down the line in their careers, obviously if they want to choose a STEM career but actually pretty much independent of which career they end up choosing.

In very expensive private schools we somehow see this problem far less, i.e. somehow they have figured out a way how to engage all students, not just the ones who go onto do a STEM degree. How have they done this? Easy: by spending enormous amounts of money to let hugely influential charismatic public intellectuals - i.e. guest teachers such as Leonard Susskind, Hannah Fry or Brian Cox - come and convince their kids that STEM is awesome.

This educational strategy seems to work with these students, in multiple ways, namely they understand why science is important, they understand how it benefits them and society, they understand what mathematical literacy can do for them, they even understand the beauty of mathematics. This is revolutionary especially for those not going into STEM in that they typically end up having or wanting a strong mathematics background.

A digression: when I was in university, there were two girls majoring in psychology and law which I came across during my undergrad physics classes. Both of them came from two such private schools and both of them were taking complete course in calculus with the one a course in differential equations and the other mathematical methods for physicists. Coming from medicine, and therefore being an intruder in physics, I was extremely interested in their reasons for taking these courses; their answer was simple, like was mine: they understood mathematical beauty and wanted to expose the mathematical beauty within thrir own disciplines.

Can you imagine what an edge a psychologist - or literally any social scientist or scholar within the humanities for that matter - can have within their own discipline if they can create mathematical models at the same level of sophistication as a physicist, instead of relying purely on less than stellar undergrad statistics course to do quantitative research? Based on this experience, while still a student myself, I started mentoring students in all areas, teaching them not to be afraid of mathematics, but I digress.

To come back to the main question of how to engage high school students in mathematics, without necessarily spending millions of dollars on celebrity scientists to personally come dazzle the kids, it is almost obviously that a change to the curriculum is necessary. There have been large educational experiments in the past who have tried this and there are as I illustrated many smaller experiments which try this, but nothing as of yet in a truly integrative manner.

Garnishing an understanding for the average student - be it by demystifying something as relatively simple as differential equations from physics using something as simple as Euler's method - is a step in the right direction. Suffice to say the advocation of teaching black boxes is failing to see the forest for the trees; this is similar to arguing that all that is needed for one to be able to produce literature such as the works of Shakespeare is an understanding of English grammar.

 

[Dale]

Suffice to say the advocation of teaching black boxes is failing to see the forest for the trees; this is similar to arguing that all that is needed for one to be able to produce literature such as the works of Shakespeare is an understanding of English grammar.

I think you misunderstand the suggested role of black boxes in this context, and as a result you have this exactly backwards. The idea is not "teaching black boxes". The idea is to teach physics, that is the forest. My recommendation is to use the black boxes so that you can focus on the physics and not on the details of the numerical methods.

Euler's method is not physics, and it isn't even a good numerical method. It is a tree, and not even a particularly good tree at that. Any time spent teaching physics students how to program Euler's method is wasted time. That time is doubly wasted because we are not teaching physics and we are not teaching a good numerical method.

That would be failing to see the forest for the trees. We would have lost the opportunity to teach good physics and squandered it to teach a bad numerical method.

 

[jasonRF]

I wasn't asking for neutral words. I guess I would like you to expand on 'throw the computer at' because I don't know what you mean. The minuscule amount of knowledge I have about coding has led me to believe that the computer is a tool that needs to be used with great care and insight, or otherwise you almost always get unexpected results.

"Throwing the computer" at a problem is a phrase I sometimes use (but probably shouldn't, at least in a post online) when I've exhausted whatever low-hanging analytical work I can do, and have no choice but to use numerical methods. I don't use the term for a simple numerical integral or to evaluate an analytical expressions.

For example, at work right now I am working on some electromagnetics problems. I spent a fair amount of time deriving analytical approximations for some "toy" versions of the problems and plotted the results to gain physical insight and intuition. Now I am at the point where I need to get real numbers for a real design, so I am using expensive commercial software to run large numerical simulations. I would call this "throwing the computer" at the problem.

Again, I should not have used the phrase in my post, since it could easily be interpreted as derogatory. In reality I highly value numerical simulations, but I usually gain more physical insight from simpler analytical or semi-analytical approximations. Plus, comparing the simulation to the simpler approximation often yields insights into the effects of features that are not captured by the "toy" models.

jason

 

[atyy]

To come back to the main question of how to engage high school students in mathematics, without necessarily spending millions of dollars on celebrity scientists to personally come dazzle the kids, it is almost obviously that a change to the curriculum is necessary. There have been large educational experiments in the past who have tried this and there are as I illustrated many smaller experiments which try this, but nothing as of yet in a truly integrative manner.

Garnishing an understanding for the average student - be it by demystifying something as relatively simple as differential equations from physics using something as simple as Euler's method - is a step in the right direction. Suffice to say the advocation of teaching black boxes is failing to see the forest for the trees; this is similar to arguing that all that is needed for one to be able to produce literature such as the works of Shakespeare is an understanding of English grammar.

Euler's method is part of the A-level Further Mathematics syllabus. This is not required, but recommended as one of the subjects for entry to electrical engineering at Imperial College and to physics at Oxford. It's not quite what you are thinking, as it still refers to the better students, but Euler's method is already routinely taught to many high school students.

https://www.seab.gov.sg/docs/default-source/national-examinations/syllabus/alevel/2020syllabus/9649_y20_sy.pdf
https://pmt.physicsandmathstutor.com/download/Maths/A-level/FP3/Worksheets-Notes/AQA FP3 Textbook.PDF
https://www.imperial.ac.uk/electrical-engineering/study/undergraduate/entry-requirements/
https://www.ox.ac.uk/admissions/undergraduate/courses-listing/physics

In the US, AP Calculus BC also requires Euler's method. It seems that about 14% of US high school students take calculus, about 7% of them take an AP Calculus test, and about 2% of them take a Calculus BC course, with about 1% taking the Calculus BC exam. You can also find Euler's method taught in some AB courses.
https://www.maa.org/external_archive/columns/launchings/launchings_06_09.html
https://fiveable.me/ap-calc/unit-7/...ler-s-method/study-guide/XZF01jg29LPjZaV7jKjE
https://cty.jhu.edu/online/courses/advanced_placement/ap_calculus_ab.html

 

[Jarvis323]

My papers on instructional design are at the office, so I don't have the reference where I originally learned it at hand. But making a course narrowly focused in the context of an overall curriculum is not an idea unique to that reference. Here is another: "Department faculty need to agree on a set of emphases for each course in order to function as a building block within the overall curriculum. The course may focus on only one or two of the curriculum goals, but those few goals must guide the learning outcomes that the course pursues"

Idea Based Learning by E. Hansen, 2011, p 30

One comment, in my current role I teach short intense classes to adults. They last from 1 to 3 weeks, 8 hours per day, so the need for a narrow focus is extreme. These courses are always designed in the context of other education. But even in a longer course there is typically more that a teacher would like to teach than there is time allotted in class, so choices must be made and there needs to be a sound basis of what to include and what to exclude. That relies heavily on understanding the course in the context of the overall curriculum and trusting your fellow-teachers to teach the rest of the curriculum.

This makes sense, but don't see it as so clear cut that any overlap between courses is "incorrect". I don't know what correct should mean in this context. There is a difficult trade-off. Like you've pointed out, time spent teaching one thing takes away from the amount of time that can be spent teaching something else. Teachers have the difficulty that they have to make sure they get through all of the material, and the curriculum has to be completely covered through all of the courses. But a lot of students get left in the dust when everything is flying by so fast, and they're already lost to begin with.

On the one hand, you have all of the information that needs to be shown to the students at some point. On the other hand, students benefit from also exercising their minds, and learning how to think in general. This opportunity will be squandered if the student's experience is reduced to memorization, and if they are distracted, get lost, or lose interest. If the OP's idea has merit, it implies that there might be conflict between having 100% disjoint courses and achieving these other goals.

I suspect that a lot of people on PF were those exceptional students who didn't get left in the dust, so it might be hard for a lot of you to relate.

I got an A in all of my science and math courses in college. But a whole lot of the material that was covered along the way has exited my mind. I feel like all that I am left with is the things I actually understood deeply, had fun with, or applied in some meaningful way. I guess I am one of those students who excelled, even though my intuition/understanding of what I was doing was left in the dust to an uncomfortable degree. The sheer volume of mathematics that one must master to be a physicist has somewhat scared me away. Maybe weeding me out wasn't a bad thing, because I went into computer science instead.

In my education as a computer scientist, I've ran into quite a bit of redundancy in the courses I've taken. For example, discrete mathematics, theory of computation, and combinatorics. However, I found that the overlap rather solidified my understanding in a valuable way. In each of these courses, it's not memorization of facts that are so valuable, it's learning how to think abstractly, and how to find and write proofs. I also liked these subjects a lot because I was easily able to understand them from the bottom up, and a really solid understanding of the fundamentals could go a long ways. In physics, maybe that's not the case? Or maybe I just never found that beginning thread in physics that I could latch onto like I did in computer science.

I like the OP's idea because I suspect that such an approach would have worked better for me. Maybe using black boxes would have worked also. But I figure that some approach to better capture the attention of some students like me is warranted. Maybe the exact idea the OP has will work, maybe not. But it sounds promising to me. I'm not sure it is the perfect solution, or that it will help all students. I suspect it would be in large part dependent on the teacher and how they engage the students as individuals.

And I will admit that I am one of those people that looked at differential equations, and just saw a bunch of symbols, then learned how to manipulate them in abstract ways, into different forms, and so forth, while having little intuition about what the point was, or what they meant. It seems like it was only down the line, after a lot of abstract manipulation of equations, and memorization of a lot of rules and procedures, that I ever did anything meaningful with them.

I think that people learn differently, and one approach cannot be optimal for every person, and I have myself as evidence of that. For me, I probably would have been better off starting with analysis (in some limited and simple enough introductory form), before calculus, and taking foundations of mathematics before geometry and algebra. Maybe the revolution in education will be to figure out how to teach different people with different approaches. Maybe the OP's idea could be an approach that works better for some people.

 

[Jarvis323]

Any time spent teaching physics students how to program Euler's method is wasted time. That time is doubly wasted because we are not teaching physics and we are not teaching a good numerical method.

Isn't there value in understanding what makes a good numerical method? Wouldn't seeing those flaws and limitations in action be valuable? How will they know the appropriateness of one black box model over another, if they don't understand them at all besides their inputs? Should there be a rule book or diagram they memorize that tells them which one to use and why?

The way I see it, experience running into problems and limitations with methods, is valuable experience that teaches you generalizable skills and pragmatism. A lot of the courses I've taking in computer science begin with the simplest solution. Then we break it. Then we analyze why it broke, and we find a better solution. Then we break that solution, and find a new solution, and so forth. Finally, we might learn theories about the problem domain that say something about our limitations and trade offs. I generally liked going through this type of thing.