Computational algebraic geometry for high school students | Hacker News Reader

Computational algebraic geometry for high school students

solidangl.es

153 pointsfoxes14 comments

Studying algebraic geometry back in 1980, I nearly dropped out of grad school: Attempting to solve century-old problems with century-old methods stopped making sense to me. Then a friend and I had this vision that algebraic geometry should be computerized. Our computer algebra system "Macaulay" inspired "Cocoa" and "Singular".

"SageMath" is an amazing project. It is best understood as an operating system, that aspires to gradually replace each program it hosts.

Would you credit work you did in Photoshop to your MacOS or Windows operating system? Computational algebraic geometry in SageMath is handled by Singular. The Singular team has dedicated their lives to this project; I cringe whenever I go to a talk and their work is credited to SageMath. This article makes no mention of Singular.

This is quite interesting, and I hope the course material will be made available for others (including myself) to check out.

> None of my students had a course in abstract algebra before. But they didn't need it to get their hands dirty and start playing around.

This really resonates with me, as a deep learning researcher who started out by taking the fast.ai course. The fast.ai course highlighted the same ideas of getting your hands dirty and exploring in a top-down approach. It's great to see how this sort of approach translates to other fields.

Looks like they made their tutorial sheets public [0]. I definitely think abstract algebra can be taught at a more elementary level, this looks like a good data point to confirm that.

[0] https://drive.google.com/drive/folders/11pVN-YZ_ughk-vsDDqow...

I agree -- abstract algebra is elementary and some aspects of it can (and should), be taught in high-school. AA is a domain of maths which people won't even encounter if not studying a STEM field and such crash course could serve as a good outreach/vulgarisation material.

I really liked the materials in the link and think they are suitable for talented high-schoolers. Cox-Little-O'Shea is a fantastic book I studied as an undergrad and learned a lot from it, I wish someone would expose me to it earlier in life.

There's plenty of good math software for algebra which is not very popular (SageMath comes to mind as the most commonly used) and even more code that is simply inside knowledge. The problem is that people in the academia are not being paid for writing software but for publishing articles, even though the community value of a good package outweighs many papers. For example, good luck finding something that will compute non-commutative Groebner bases :)

Macaulay2 will! https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1....

Magma should do it (http://magma.maths.usyd.edu.au/magma/handbook/text/970 claims a Alan Steel's non-commutative generalization of Faugère F4 and noncommutative Buchberger algorithm) and US institutions can get it via the Simmons agreement. Not sure what's good for home users, though.

Curious why you find it so fundamental. In my experience abstract algebras primary benefit was drilling rigorous proofs for group theory etc.

Between physics and comp sci I'd rate abstract algebras contribution to understanding as one of the least important contributors ( although AA does provide a rigorous framework underpinning these fields in one way or another)

When I was in school, I felt it was very important because it taught me how to think of things in terms of mappings and structure-preserving mappings. After learning algebra found learning everything else became much easier.

I'm a CS student, who is super interested in deep learning and ml in general. I've started playing around with pytorch a bit and got some stuff to work, but I don't fully understand the code I wrote.

So instead I started the fastai course and now I have the opposite problem where it feels a little too easy and I fear that that approach would fall flat the moment I want to do something outside of the things fastai can do.

So now I'm thinking about scrapping that too and going to the opposite side and just building something from scratch without any ml framework at all.

Would you say sticking with fastai is good idea, or is it more of way to just pique someone's interest?

If you follow through with the entire fastai course/book, then you will learn how to build an ML framework (the fastai library) from scratch. Yes, it starts out by saying "you can do deep learning with 4 lines of code" but it goes through all the inner workings, and talks about some pretty advanced stuff. Of course, no book can cover everything, but it also functions quite well as a jumping off point for more advanced self-study.

I feel like the entirety of maths could do a bit of more a top down approach in terms of teaching.

At the moment it's a bit like studying only grammar, and spelling without reading a good novel for inspiration.

> In short, we need to stop conflating logical foundation with pedagogical foundation.

YES. I want to put this on a T-shirt and strut around wherever they write high school math curricula.

I’d be willing to bet $20 of reputation that teaching the high-level end results will leave more students with a strong theoretical foundation than starting by teaching the strong foundations. Show the students _why_ the foundation is important, don’t just save the fun bits for a valedictory lap after standardized test week.

I agree, but my experience with the kind of people who care about secondary curricula (people training me when I thought I might graduate in Math Ed, anyway) is that they very much believe that pedagogical foundations should start with something other than logical foundation -- there has been lots of focus on constructivism and manipulatives and geometric intuition and starting with application domains over the last few decades at the tertiary research/training level.

Adoption throughout the primary and secondary system is a different story. There's been some progress, of course, but if you've ever seen people speak about "common core" with contempt, you are seeing some of the primary forms of resistance. There's an entire social psychology and politics around this that has to do with temperaments who fit comfortably into rote execution for the most broadly useful skills and will be allergic to reframing that requires indirect exploration for conceptual foundations -- it messes with sense of status and security on some deep levels. Parents with this profile hate it; then their kids are a bit more likely to hate it too for reasons of nature or nurture. Some teachers will fit this profile too (though probably more rarely). People who influence curriculum are likely to have the politics influence their choices.

Better math pedagogy is certainly not a solved problem even at the cutting edge, but it's more solved than you'd think looking at the overall state and a lot of the reasons are social.

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