A proposal for unifying high school mathematics and physics using geometric algebra

[Keith Maynard]

Hello fellow GAers,

I was a bit flummoxed this week reading an abstract that physicists have 'discovered' that quantum physics does not need imaginary numbers. It occurred to me that David Hestenes and the GA community have been shouting that for decades. I decided that we really beed to do something about revolutionising the teaching of mathematics and physics. I attached a PDF document in which I argue that we urgently need to address the mathematical Babel that currently passes for education, We have the mosst elegant system ever, let's introduce it to our high school students so they can become more powerful problem solvers. I would welcome feedback on the idea. I am not an expert just someone who has suffered from the unnecessary complexity of our mathematical systems. Let's create something for our next generation of scientists and mathematicians!
Best regards,

Keith Maynard

[Michael Taylor]

Dear Keith Maynard,

I think what you propose is a worthwhile project.  As I'm busy with a number of projects, I won't join in, though I'll send as attachments some material which might be of interest.

One of these is a pdf which was meant as a short introduction to GA and GC.  The other is a Math Monthly article about the wedge product which was written before I knew anything about GA.  One of the important ideas in the Monthly article was that one could take the fact that a vector could be thought of as a directed line segment which was characterized by thiree properties, namely, (1) magnitude, (2) the line it determined, and (3) its direction within the line, and that those three properties could be generalized to and could characterize simple k-vectors.

I carried out a project similar in spirit to the text you propose when I wrote and published "An Introduction to Geometric Algebra and Geometric Calculus" in 2021.  Only I aimed at math majors at a junior, senior, or beginning graduate level, not at high school students.  Partly I wanted to produce something that was, in a sense, an extension of undergraduate calculus and linear algebra, and, partly, I wanted something that was more mathematically rigorous than the geometric algebra books I saw out there yet was not was not written at a stratospheric level of abstraction.

Good luck with your project.

Regards, Mike Taylor

 The_Wedge_Product_and_Analytic_Geometry.pdf

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[Peter Cameron]

Keith,

In GA the wavefunction is the place to start. 
www.researchgate.net/publication/410517724_The_Lost_Geometric_Language_of_the_Wavefunction

One might imagine including the wavefunction in HS physics classes...

Peter

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[American Citizen]

Peter:

Maxwell's equations suffered damage at the hands of Oliver Heaviside, and physics has suffered even to this very day.

You sense that something is not quite right and your intuition is correct.

GA is the place to start. Electrodynamics needs to be reformed and all elements examined carefully.

Randall

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[Keith Maynard]

Dear Professor Taylor,

Thank you so much for your encouraging words and for taking the time to share these materials with me. Receiving validation from someone who has successfully published a textbook in this domain means a great deal to this project.

I have spent some time diving into the papers you attached, particularly your Math Monthly article on the wedge product. I wanted to share a specific highlight that stood out to me: your framework for generalizing a vector’s properties to higher k-vectors is pedagogically brilliant. Breaking a vector down into its magnitude, the line it determines, and its direction within that line scales beautifully when introducing bivectors to students. It gives them a familiar, rigorous, yet deeply visual anchor to understand how an "area" can possess orientation and handedness.

While your 2021 text masterfully targets the rigor needed for university math majors, our goal is to translate that exact structural integrity into physical, intuitive shortcuts for a 16-year-old student—showing them early on that "coordinate-free" simply means we can solve dynamic systems by tracking how physical objects interact with each other, rather than with an artificial background grid.

I am taking my time to thoroughly study the rest of the Geometric Calculus and integration chapters you sent, and I will be sure to keep you updated as our secondary framework develops. Thank you again for your generosity and support.

Warm regards,

Keith Maynard

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[Keith Maynard]

Dear Dr. Cameron,

Thank you for sharing your fascinating essay, "The Lost Geometric Language of the Wavefunction." Your insights are entirely new to me, but your core philosophy of a "bottom-up" approach aligns perfectly with the goals of this high-school curriculum project.

Like many who encounter standard quantum mechanics, I was always deeply uncomfortable with the abstract, hand-waving characterizations of the wavefunction. Discovering through your work that Geometric Algebra provides an explicit, physical, geometric reality for the wavefunction is absolutely thrilling. It completely vindicates the idea that nature speaks in geometry, not abstract mathematical fictions.

Your bottom-up program mirrors exactly what I have tried to achieve in the classroom. I have previously introduced foundational vector space concepts to 12-year-olds because I wanted them to acquire the authentic mathematical language they would be using for the rest of their careers. By teaching them to use additive and multiplicative identities and inverses to solve equations natively, I watched them grasp concepts that textbooks usually lock behind years of artificial complexity.

If we can teach 12-year-olds the raw axioms of space, we can certainly teach 16-year-olds the unified language of the Clifford geometric product. I am completely open to restructuring our roadmap to provide the absolute best foundation possible for the next generation of scientists. By introducing the geometric product early, we can ensure that when a student eventually encounters the wavefunction, they do not see a ghostly probability cloud, but a real-space geometric transformation.

I am incredibly grateful for your perspective and your essay. It has given this secondary project a profound, foundational destination to aim for.

Warm regards,

Keith Maynard

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[Peter Cameron]

Keith,

Many thanks for your positive upbeat.

Is AAPT collaborating in this project? If so then it has the strongest possible ally. If not, then to start to gain their endorsement seems to me to be the essential next step. Generally the teaching community is more open to new ideas than the mainstream physics community, as their investment in maintaining the status quo is not so deeply entrenched. The 2002 AAPT Oersted Medal award to David Hestenes for "Reforming the Mathematical Language of Physics" exemplifies this.

Best regards,

Peter

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[Dr. Wolfgang Lindner]

Hi,

I have followed your discussion with great interest.

Please, let me tell you that I had written a small text, where I use the CAS Eigenmath

to introduce into the first concepts of GA.

In case you will have a View inside, here is the link:

Best regards,

Wolfgang

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https://LindnerDrWG.github.io

https://independent.academia.edu/LindnerDrWolfgang 

send from my iMac

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[Alex]

Randall,

Do you have any references, links? 

I suspect Maxwell's own lack of understanding regarding the (quaternionic) electromagnetic field—and its translations into measurable physical quantities—is what enabled Heaviside's "simplification". Furthermore, a complete experimental verification was never conducted. And, yes, physics has suffered to this very day but a different interpretation could be tested experimentally.

Alex