I wish I would have been introduced to Geometric Algebra or calculus of forms or whatever it is called during my physics studies. We learned all the conventional things you need for classical mechanics and electromagnetism, like div and curl and BAC-CAB. But there were a couple of things that we were not tought well, which caused problems later. One thing is that at first, a vector was just an N-tuple. But in physics, something is only a vector or a tensor if it behaves under a very specific way during transformations. (The infamous "a tensor is an object that transforms like a tensor"...)

The other thing is that I hit a wall when trying to read theory papers, because nobody ever explained what a wedge operator or a two-form and so on is. I was able to mechanically follow calculations, but never got a real good intuition.

We learned math together with the mathematicians in a very axiomatic way. Without a geometric intuition, I had a hard time understanding why you need dual vectors like one-forms or covariant vectors. They seem just like a convoluted way to write the scalar product.

If you want an "intuitive" approach, you might check out this new book: https://www.amazon.com/Visual-Differential-Geometry-Forms-Ma....

Disclaimer: I haven't read it (but I have heard good things).

I have read (part of) it, and it is excellent. I highly recommend it (alongside Needham’s VCA book).

But the lessons learned from there are largely orthogonal to the lessons you will learn from working with GA for a while.

To the grandparent, I would also recommend the very pretty book http://matrixeditions.com/5thUnifiedApproach.html

> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation

> the stuff that's already taught is better

These two statements seem contradictory.

> But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free)

The author made no claims about tensor notation or differential forms; perhaps those might displace all vector-like notation (GA or otherwise) in 100 years, in which case the author's claim can be weakened to "in 100 years, GA will be the dominant form of vector notation".

As for "the stuff that's already taught is better", notice that such stuff includes:

- Complex numbers

- Pseudovectors

- Cross-products

- Matrix algebra

- Dirac notation

You seem to agree that GA is "more powerful" than the Gibbs-style vector algebra normally taught. The article is arguing that GA is also a simpler and more consistent approach, which I agree with (complex numbers are certainly simpler on their own, but are a little redundant if we're using GA for the rest).

From my own experience in formal education (UK high school and undergraduate physics), I never encountered tensors or differential forms. I've since learned a little about tensors (for general relativity), but that's been due to my own curiosity; I've learned a little about GA for the same reason. I never used quaternions (hence why I left them out of the above list), although I'm aware of them and that they're used e.g. in computer graphics. I used vectors, pseudovectors, and matrices a lot, and I'm certain those topics would have been easier to learn and comprehend if they'd used GA instead.

spekcular is correct. One could draw the following graph very roughly capturing how these concepts developed:

Grassman’s exterior algebra -> Differential forms

Cartan’s Exterior Derivatives -> Differential forms

Quaternions -> Clifford Algebra

exterior algebra -> Clifford Algebra -> geometric algebra

Differential forms are a particular kind of tensor and tensors can be defined in terms of multilinear maps. As spekcular says, the standard curriculum covers differential forms, tensors and vectors. This entails becoming familiar with multivectors, the wedge product and multilinear algebra, making geometric algebra a relatively small delta to pick up.

On the other hand, the standard course will also prepare you for mathematical topics like lie derivatives, differential geometry and de Rham cohomology.

Other than physics, the standard approach equips you with the mathematical machinery underlying many topics in machine learning and statistics like Hamiltonian monte carlo, automatic differentiation, information geometry and geometric deep learning.

The central advantage of geometric algebra over the standard approach isn't that it's better or more general, it's that pedagogical material for it is generally leagues and magnitudes better than those for the standard course.

> As spekcular says, the standard curriculum covers differential forms, tensors and vectors... making geometric algebra a relatively small delta to pick up.

Could you be a bit more specific about which "standard curriculum"/"standard approach" you're talking about?

For example, in my formal education (high school; masters with physics major, comp. sci. minor; 4 years of a comp. sci. PhD (abandoned)), I did not encounter differential forms, tensors, multivectors, the wedge product or multilinear algebra (or quaternions, lie derivatives, differential geometry, (co)homology, etc.).

Maybe you're talking about a "standard approach" for a pure mathematics curriculum, or perhaps physics/math grad school?

All I can say is that high school and undergraduate physics (in the UK, circa the late naughties) (a) does not standardise on those topics, (b) is filled with tricky operations which are easy to mix up or perform the wrong way around (e.g. cross products, matrix multiplication, pseudovectors), and (c) many of those annoyances would simplify-away under GA.

It's a cliche that physicists (certainly when teaching) cherry-pick the parts of mathematics they find useful. All of those concepts would certainly be useful in a physics course, but would perhaps be too much to fit in; yet there's certainly enough scope to cherry-pick GA (since we can drop Gibbs-style vector algebra[0] to make room). Perhaps something else, like differential forms, might be even better; I honestly don't know (maybe I'll do some reading about it).

[0] By "Gibbs-style" I mean the 'cross product and dot product ought to be enough for anyone' approach that permeated my undergraduate learning.

By standard approach I mean the typical material covered for someone studying vector calculus properly. This will be stuff like differential forms and the basics of tensors, manifolds and multilinear maps at the undergrad level. Differential geometry and cohomology are examples of courses which build on them.

I agree with you that pseudovectors, cross products and vector calculus are a terribly adhoc way to teach this stuff but a course covering linear algebra with differential forms elegantly unifies, corrects and generalizes them. Standard is also in contrast to the geometric algebra/calculus alternate path.

> for someone studying vector calculus properly

If you can’t invert vectors, you aren’t studying vector calculus properly. ;-)

Differential forms are a half-baked formalism.

Unfortunately I don’t know of any great undergraduate level geometric calculus textbooks. Ideally there would be something like Hubbard & Hubbard’s book (http://matrixeditions.com/5thUnifiedApproach.html) written using GA as a formalism.

Hestenes & Sobczyk’s book (http://geocalc.clas.asu.edu/html/CA_to_GC.html) is a hard slog, and not appropriate for an undergraduate audience.

> Differential forms are a half-baked formalism.

I can't emphasize enough how wrong this is. It's the standard formalism in research-level physics and math for good reasons.

From my view, it goes both ways: geometric algebra/calculus is a more transparent version of the standard approach and the translation back to it is also a relatively small delta to pick up.

Either way of going about what is in essence the same material entails becoming familiar with multivectors, the wedge product, and multilinear algebra, whether you do it through geometric algebra or the standard approach.

That makes sense but my argument is since further material (some examples which I listed) assumes and builds upon the standard approach, you'll likely be better off taking that path.

?? You'll likely be better off seeing different perspectives of the same thing.

> I never encountered tensors or differential forms.

Your UK undergraduate physics must have been a bit different to mine. About a third of my physics course was taught by the maths dept, and tensors/algebras were very much a part of that.

I recall, after freshers week, the dean getting everyone together. He said two things:

- Hopefully you all had a great fresher's week, now it's down to business, and

- Make sure you have fun at college.

He also had a projection on the overhead saying "If you can't blind them with science, baffle them with bullshit". I'm reasonably certain the second statement above was the latter, because...

He then casually mentioned a "maths refresher" 2 week course that all freshers had to take before "the real stuff" started. That "maths refresher" was the entire Further Maths 'A' level syllabus. In two weeks. Those of us who had done Further Maths at school were fine. Those that hadn't were shell-shocked.

> That "maths refresher" was the entire Further Maths 'A' level syllabus. In two weeks. Those of us who had done Further Maths at school were fine. Those that hadn't were shell-shocked.

Heh, that reminds me of my first physics course in an under-graduate computing degree (in Romania). The curriculum was so well designed overall that this Physics course needed linear algebra concepts that would be taught halfway-through the semester in Algebra, integration along a surface and similar that would be taught at a similar in the Calculus course, and some Statistics I don't remember that would be only be taught in the second semester.

The prof's solution? He taught a 3-hour course covering all of the above, and considered that good enough for all future courses. This particular Physics course later went on to cover analytical mechanics (generalized coordinates, Lagrangians, Hamiltonians), electricity, general relativity, statistical thermodynamics, and quantum mechanics, all in a single semester.

Needless to say, 99% were happy they passed and couldn't tell you a single thing about any of these subjects a few minutes after the final exam.

> That "maths refresher" was the entire Further Maths 'A' level syllabus. In two weeks. Those of us who had done Further Maths at school were fine. Those that hadn't were shell-shocked.

Heh, I recall managing to coast for a short time thanks to having done AS Further Maths.

The Further Maths syllabus was quite modular, and the modules our teachers picked had some discrete math (sorting algorithms, Dijkstra's algorithm, bridges of königsberg, etc. which was useful for comp. sci.), and some which complemented the regular maths course (complex numbers and more calculus, which was certainly useful for physics).

Our Further Maths was a lot less modular. Preparation for it started in the 2nd year (so 12/13 years old), when we were streamed for maths - if you were in set-1, you studied to take 'O' level (showing my age here) in 4th year (so a year earlier than most) on an accelerated schedule.

That meant you could take AO (a halfway house between O and A) when everyone else was taking their normal O levels. The thing is that the extra stuff in AO was all Pure Maths, and formed a fair amount of the easier "P1" maths syllabus for the normal A level maths exam, which had P1 and Me1 (Maths with mechanics 1, basically statics).

Because you'd done that work already prior to the A level years, you could take "A level maths" after only 1 year (which looked really good on UCCA applications :), so you've now done an exam consisting of the two 'P1' and 'Me1' papers in the first year of your A levels.

Which meant that in the second year of your 'A' levels, you could do 'Pure Maths' (P1, P2) and 'Further Maths' (Me1, Me2) for a total of 3 maths A levels.

On top of that, you had your other two subjects (mine were Physics and Chemistry), and because it was the JMB board, everyone got to do "General Studies".

Getting all of them gave you 6 A levels, even though some of the work was duplicated in the maths arena (over different years of course :)

S levels were a bonus on top - there was no fudging for those, though, you just took what you thought would be useful to study. They gave me maths and physics because I'd said I was going to do physics at college... :)

> it is deficient in various ways when compared to [...] differential forms (e.g. if you want to work basis-free)

There is nothing basis-dependent in Geometric Algebra. This presentation started from a basis, but then again so do many presentations of differential forms, leading to 2-forms like dx \wedge dy and so on.

The actual difference is that Geometric Algebra requires a choice of inner product (actually, you can get away with any bilinear form), while differential forms do not. However, some of the important operations on differential forms in physics do require an inner product (e.g. the hodge star operator and the codifferential), so you end up back on equal footing with GA again.

I worded that in an unclear way. My main beef is that GA is just way clunkier than differential forms, which are clearly the "right" approach if you want to approach the subject from a theoretical perspective. I see no advantage over the usual treatment, and many disadvantages.

I'm a skeptic too, but I might not be the intended user for the GA formalism. Please explain your reasons.

My skepticism of the supposedly superior pedagogy of Geometric Algebra is the following:

- 3D vector algebra with the cross product operation and the dot product operation is fairly easy and intuitive. Its replacement by GA might not be so easy. So maybe GA should be introduced after the vector formalism.

- An arbitrary element of a Geometric Algebra might not have a geometric meaning. For instance, some elements of a GA are vectors, while some are scalars, but there are also these exotic mixed quantities which are scalars plus vectors. This is pretty hard for me to understand intuitively.

- An arbitrary matrix has a geometric meaning. It's essentially just a linear transformation. By contrast, I don't feel that an arbitrary element of a geometric algebra has a geometric meaning.

- Consider those elements of a Geometric Algebra which represent rotations -- they are called rotors. Observe that if "z" is a rotor then the element "-z" is also a rotor which stands for the same rotation as "z". So there is more to a rotor than whatever rotation it describes. This seems very unintuitive and advanced. (I know that this behaviour has applications for the study of spin-1/2 particles in quantum physics).

I also have trouble understanding where the rule for multiplying two elements of a geometric algebra comes from. It's an operation, introduced from seemingly nowhere, which happens to have some applications in some areas. But I'm not comfortable with a multiplication rule being introduced out of nowhere without being derived out of something. The claim that it has a consistent geometric meaning from which it can be derived is never justified. My criticisms are therefore largely pedagogical.

I think your reasons are good and essentially what I would give. I also think differential forms are a much theoretically cleaner way to express the same concepts.

Further, teaching differential forms prepares my students to engage with the (vast majority of the) existing math and physics literature. Teaching geometric algebra doesn't.

The practical reasons boil down to: I have to teach the standard stuff because otherwise they can't read the literature. Having done that, what's the marginal benefit of teaching GA? Not a lot.

On the other hand, maybe GA can help with differential forms. Differential forms involves exterior algebra, and I feel like some aspects of exterior algebra are elaborated upon in an insightful way by GA. For instance, the grade-2 elements of an Exterior Algebra can be understood as angular velocities in many circumstances. In GA, this is captured by the exponential map that sends grade-2 elements to rotors. I don't know if this can be helpful for teaching purposes.

> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation. But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free). [I'm oversimplifying a bit, but a full discussion is too long for a comment here.]

I have no opinion about the claims, but I loved the article, as I quickly saw the gains from this algebra for my very basic needs.

After checking out your 2 suggested alternatives, I'm not so convinced they are easier to understand.

>Geometric algebra is, as the article points out, a more powerful version of the usual vector notation

That's not just a gross oversimplification, this is also flat out wrong if what you meant was that it only has vectors. It has more general objects called multivectors through pretty much the same process you get one, two, etc. forms from the wedge product.

In fact, both GA and differential forms build from the exterior algebra, and you can go from the former to the latter through geometric calculus (one key difference e.g. would be the method of reciprocal bases to compute inner products with non-orthonormal bases, rather than explicitly working out a basis and then its dual). So I'm confused about your remark regarding its alleged deficiency vs. differential forms if you pretty much reconstruct it within the GA/GC system (especially regarding working basis-free).

With regards to tensor notation in terms of calculations, if you mean all that index gymnastics, well GC still openly provides that way of computing things out from what you're used to.

What I like about geometric algebra/geometric calculus is precisely the way in which it's nothing new: it's putting everything people use in one system by clarifying the connections between these seemingly disparate systems. Even lie groups/lie algebras can be constructed rather efficiently in the algebra.

Another appealing feature of GA is its ability to make pretty transparent an old theorem from Cartan and Dieudonne that says you can view geometric transformations like rotations, and even translations (in projective geometry) as compositions of reflections.

There's other appealing features like this in terms of classifying and relating different geometries together that harken back to the Erlangen program, but my point is even in terms of concrete calculations, it's not quite right to say it's just a "more powerful version of the usual vector notation" as it includes more general objects than vectors, and still includes a lot of very similar ways of doing calculations (almost a kind of "backwards-compatibility?") you're used to with tensor index calculations, just with the added bonus of making the transition to the tensors used from vector calculus seamless, alongside other added relation to other systems made more transparent.

To put it in concrete terms, where does GA really fit into the story of undergraduate physics (or mathematics)?

Suppose I want to teach first-semester mechanics. I can get through this fine with the usual vector notation. Vectors and dot products are intuitive when taught well (the latter just being projections), and while cross products are a little hairy, they don't play a major role in the course. There's no time for GA, and it would confuse more than illuminate in any case.

Next, I want to teach E&M. Here, I'd probably lead with the usual vector calculus notation (because even if it's ugly, it's standard and students should know it), and then follow with an explanation in terms of differential forms. [I assume this is a more theoretical, or honors, class; I might stick with vector calculus if it's more computational.] So now students know differential forms, they can do everything in a coordinate-free way and on manifolds, and they can access a significant amount of standard physics and mathematics literature.

Having proceeded in this way, what does introducing GA do except suck up a lot of class time? To me, it seems clunky and without any distinctive advantages.

Another question to think about: if this notation system is so good, why don't working mathematicians or physicists actually use it? For example, people thought Feynman diagrams were strange at first, but they proved their value and consequently caught on.

Again, my argument is that this is not some revolutionary esoteric knowledge, it's well-understood stuff that people don't teach for good reasons.

> Suppose I want to teach first-semester mechanics

If you need to teach undergraduate mechanics, I highly recommend you at least read some of Hestenes’ New Foundations for Classical Mechanics http://geocalc.clas.asu.edu/html/NFCM.html

> without any distinctive advantages

The most basic distinctive advantage is that you can invert vectors (which is incredibly useful!!) without needing to pretend that vectors are matrices, complex numbers, or some other kind of object.

GA takes most of the advantages of complex numbers vs. R² for representing plane geometry, but extends them to arbitrary dimension, and extends them further (when using complex numbers for plane geometry you end up representing vector–vector products via the obscure z̄w product involving complex conjugation, and it is easy to get confused about the difference between a vector vs. a scalar+bivector).

But there are a wide variety of other powerful (and geometrically interpretable) algebraic identities which can be applied to vectors, blades, and multivectors, ranging from awkward to impossible to express using the language of differential forms, Gibbs-style vectors, etc. Physicists often end up resorting to tedious coordinate-by-coordinate calculations for stuff that would end up being an easy vector expression in GA. Learning these identities and how to apply them takes years and a lot of practice solving problems using GA.

My own experience for the first few years of knowing that GA existed but not being too fluent with it was that I would work some problem (mostly 2–3 dimensional geometry problems) out in coordinates, spending like 2 pages of scratch paper for the opaque intermediate calculations, with high chance for mistakes, then eventually find that most of the ugly bits along the way canceled and yielded a nice result. Then I would think a bit more about the problem, skim through a list of GA identities, and find I could have shortened that 2 pages of work to 3 lines, each of which had an obvious geometric interpretation.

Can you give an example of a problem that might appear in an undergraduate physics or math course, whose solution is lengthy and tedious by "usual methods" but dramatically simplified by the use of GA?

I have seen examples proposed before and been distinctly unimpressed. Any serious simplifications in solutions are usually due to some notation-agnostic insight.

The nicest written-up example I know is https://www.shapeoperator.com/2016/12/12/sunset-geometry/

* * *

As a relatively recent personal example I spent a few months (in bits and pieces) working out a bunch of metrical spherical geometry for myself without reference to past work, with points represented as displacement vectors to stereographically projected points at https://observablehq.com/@jrus/planisphere with the eventual goal of implementing computational geometry / cartography code using that as a canonical representation, which I think is superior to representations used currently in practical software.

The same spherical relationships are comparable (some things slightly slightly easier, some slightly trickier) to represent as displacement vectors in an embedded sphere. But there again relationships are clearer to express in GA terms.

Most of the material there is stated without proof (maybe eventually full proofs should be included), but several of the identities there I worked out very tediously with pages of scratch work in coordinates, then realized afterward the same results could be arrived at with only a few lines of GA.

Only a bit of the material is truly novel (after doing the work myself, I hunted around for sources and found some of the same formulas worked out previously using classical spherical trigonometry 200+ years ago), and e.g. some very similar material where the stereographically projected points are represented as complex numbers can be found at http://fer3.com/arc/img/110279.applications%20of%20complex%2...

In theory most of the rest could be also worked out using complex numbers or matrices, but (a) some ideas end up awkward and unidiomatic there so you would never think to do it, so that many identities that are slightly obscure in GA are almost unheard of written in other formalisms, (b) the algebraic manipulation is at least 2–3x more cumbersome.

That is a strange and unconvincing article. In outline, it goes:

1) Look at this slick solution using geometric algebra.

2) Look at how ugly the trigonometric solution is. GA is so great!

3) And, by the way, one can mechanically translate the GA solution into the usual vector notation.

Point 3 is even a bit understated: GA concepts are really only used for a few lines under "Solving for the Earth's radius." Once you hit the equation for epilson^2, it's just standard algebra and trig.

Anyway, the relevant comparison is not between trig and the GA solution; it's between GA and the usual vector language. It's really the same method in different notation. You take the cross product and separate into parallel and perpendicular components, and then you reach the epsilon^2 equation, and it's the same from there.

Also, I think the author is far too hard on the trigonometric solution. The vector solution is somewhat clever, and it for any given problem that gets placed in front of me, it's not obvious a slick solution exists. On the other hand, the philosophy of trigonometry is that given a completely determined problem about triangles, you can just trig-bash mechanically to get an answer (and here you can even before starting that small-angle approximations will make life easier, so trig is even more attractive). It's really not that bad here. Especially, the comment about it being tricky because one must find a "non-trivial relationship between the four angles" is puzzling. Anyone who's spent time with geometry problems like this knows that the first step is to angle-chase and write in all the value and relations, from which this falls out immediately (and again, totally mechanically). Then you just turn the algebra crank and win.

[I don't have time to think about the spherical geometry stuff. Sorry!]

>It's really the same method in different notation. You take the cross product and separate into parallel and perpendicular components, and then you reach the epsilon^2 equation, and it's the same from there.

Bivectors and cross-product aren't just different notation for the same thing if that's what you meant. They're distinct (but very much so related) mathematical structures. For one thing, one's associative while cross products break associativity.

As far as GA sharing a lot with the more comment vector algebra/calc methods. Personally, I'm happy that GA has an attitude of "if it's not broke, don't fix it". It also means there's really not a lot of time lost in the transition due to the compatibility. Hell it's even backwards compatible in the sense that you can still easily retrieve your axial vectors the cross product gave you if you so wish (which cleared up instantly what the exterior algebra folks were doing with their hodge star business when I decided I wanted to explore that perspective later on).

> ... by clarifying the connections between these seemingly disparate systems.

This is the big reason I like it. I remember learning bits and pieces of linear algebra whose rules seemed so entirely random (vectors vs pseudo-vectors, curl, quaternions, some spin calculations I can't even remember anymore, etc), that turned out to have more unified geometric interpretations once I learned GA. It definitely slowed me down as a student who hated 'just memorize it' pedagogy.

> People know about it and have decided not to teach it

Generally teachers don’t (can’t) individually decide this. Decisions about what to teach have incredible historical inertia, and are largely decided based on what the teacher learned when they themself went to school decades ago, what everyone else is teaching, what materials are easily available, what notations are used in past literature, etc. Substantial transitions in the teaching of existing material take generations.

In 2020 our basic math/science curriculum and pedagogy in high schools and universities has all been pretty well statically fixed for 50+ years (many parts are unchanged in 200+ years), except in computer science where some of the basic ideas are newer than that, and in graduate-level courses that get closer to the cutting edge.

* * *

The place where geometric algebra has seen most rapid adoption is in computer programming, where code actually has to work, and a more effective formalism makes correct code easier to write and reason about, saving a ton of time and effort even for basic examples.

Even in physics, where a better formalism leads to improved physical intuition and deeper conceptual understanding, a transition is an uphill struggle, because symbolic fluency with geometric algebra takes years of practice.

(Some) pure mathematicians on their high horses scoff at anything that doesn’t advance their own obscure abstract research, which is unconcerned with conceptual obstacles faced by undergraduate students, scientists, or engineers. They can hand-wave a better formalism away with “this is isomorphic to X and Y other structures, so there’s no value in it”.

> Substantial transitions in the teaching of existing material take generations.

> In 2020 our basic math/science curriculum and pedagogy in high schools and universities has all been pretty well statically fixed for 50+ years (many parts are unchanged in 200+ years), except in computer science where some of the basic ideas are newer than that, and in graduate-level courses that get closer to the cutting edge.

This is wildly incorrect. Even in the past ~20 years we've seen a sea change in our understanding of science pedagogy. Look up the work of Carl Wieman on active learning, or https://www.pnas.org/content/111/23/8410. Inclusive classroom practices are another thing that's come into fashion in the last ~10 years. The curriculum has also evolved; the most obvious thing to point to is the new emphasis on connections to data science in math/stats courses.

If you're someone who doesn't stay up-to-date on pedagogy, then yes, it takes your retirement to bring about a change. But a lot of people, especially those teaching at small liberal acts colleges, have continually evolving teaching practices. There are entire conferences where people get together to talk about college teaching.

> Even in physics, where a better formalism leads to improved physical intuition and deeper conceptual understanding, a transition is an uphill struggle, because symbolic fluency with geometric algebra takes years of practice.

Do you really believe this? To anyone to recognizes that it's the standard stuff in (a clunkier) disguise, it shouldn't take years.

Appealing to these two frictions does not offer a convincing theory of why GA has not been adopted despite being around for, what, 50+ years? The fact that it's worse than existing notation does.

Have you ever spent a few months trying to solve a wide variety problems using GA as a formalism, or tried teaching it to e.g. undergraduates? If not, you are speculating beyond your experience.

I've taught math to plenty of undergraduates – enough to know what does and doesn't play well – and I made an honest attempt to find problems where GA might have some advantage. It is the fact that I sunk several hours of my life into this with no reward that explains why I'm a little salty in this thread, and motivated to warn other people away.

I'd suggest that your comment that math and science pedagogy have been static for the last 50+ years reveals that you are the one speculating beyond their experience.

There has been continuous research into alternative pedagogy, but the typical undergraduate intro math/science course looks pretty much unchanged in both pedagogy and curriculum. My undergraduate math and physics courses circa 2005 were only slightly different than similar courses from 1960 (the main differences were things like an online discussion board in some courses, some courses with power point slides instead of a chalkboard, videotaped lectures in some courses, use of computers to type up papers instead of typewriters/handwriting), and the typical course is still not that much different 15 years later.

One of my hobbies is skimming old math textbooks; Lacroix’s textbook from about 1800 is not essentially different in structure or content than a typical 2020 intro calculus textbook for undergraduates or high school students, or almost any book in between. Way less radical or era-appropriate than something like http://www.math.smith.edu/~callahan/intromine.html

If you hunt you can find teachers trying new ideas (and you could also find teachers trying non-mainstream pedagogy 20, 40, or 60 years ago), but it takes generations for ideas to turn over.

If you are interested in better introductory physics pedagogy in particular, David Hestenes (in other work, the chief promoter of GA for decades) is a real pioneer and a huge influence on e.g. Eric Mazur. http://geocalc.clas.asu.edu/html/Modeling.html https://mazur.harvard.edu/files/mazur/files/rep_557.pdf

> the typical course is still not that much different 15 years later.

For context, I checked your profile to see where you did your undergraduate degree. I am familiar with the way calculus is currently taught at that university, and it looks quite similar to the "radical [...] era-appropriate" textbook that you linked (at least based on a quick read of a few chapters). Those courses are also taught in a quasi-active learning style (though nothing as extreme as a flipped classroom, etc.). Your observations may have been accurate 15 years ago, but that's thankfully no longer case. There's also pressure from the department/admin to make these changes in upper-level courses. See e.g. https://people.math.harvard.edu/~community/inclusive-classro... or materials from https://bokcenter.harvard.edu/active-learning.

I’m glad to hear that. I never interacted with the intro calculus course there. My impression is that most intro calculus courses around the US today still use some book like Stewart, Larson, or Thomas, and still teach in traditional lecture style.

In poking around I am also glad to see they switched from Griffiths’s to Townsend’s book for intro QM. Much more conceptually clear with less focus on mindless computation. (Disclaimer: I went to high school with Townsend’s daughter.)

I wonder if anything similar can be done for the undergrad electrodynamics course, which was more or less an experiment of “how many gnarly multiple integrals can you grind before burning out?”

> I wonder if anything similar can be done for the undergrad electrodynamics course, which was more or less an experiment of “how many gnarly multiple integrals can you grind before burning out?”

The classical field theory course was one of my favorites at the master level. Classical EM is beautiful in the sense that by sprinkling some math magic you can basically calculate everything from a few basic laws. Everything sort of fits together in a coherent tight package. TBH, the class did have a fearsome reputation for being math-heavy, and many of my class mates struggled (which was weird, because I was never super-strong in math compared to many of them).

If deficiency to tensors and differential forms is a reason not to be taught, then why do we learn about vector notation?

Oh right, because it's a natural language for talking about geometry and mechanics, by far the most common and important type of reasoning that the average student will need to do. And geometric algebra is demonstrably superior for that domain. So your comment is pointless.

This may be true in some places, but my undergraduate physics education spent a lot of time on standard Gibbs-style vector calculus. Taylor Classical Mechanics and Griffiths Electrodynamics especially depend on them. Maybe there is a case to be made that first years should start with differential forms, but until that happens I think geometric algebra could be a big improvement.

Maybe those are better for calculations but are they better for learning?

Most people need calculations to help them learn...