> 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.
> 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).
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”.