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I never understood why we don't do more of this stuff in school, and how calculus instead became the defacto advanced math curriculum in most high-schools. Stud

I never understood why we don't do more of this stuff in school, and how calculus instead became the defacto advanced math curriculum in most high-schools. Students grow up working on their basic algebraic operations, solving equations, etc. Liner Algebra introduces them to the universe that lies just beyond those techniques, has very readily applicable uses, lends itself excellently to simulation/connections to computer science (which is super popular to teach now), etc.

macrolocal 5y ago
I suspect the answer is partly historical; without a computer, calculations in linear algebra are a pain.
Grustaf 5y ago
Both calculus and linear algebra are fundamental for further studies, but I'm not sure linear algebra is more approachable? I personally find geometry and algebra more interesting, but it seems to me that derivatives are more fundamental than matrices. But maybe that's just my bias from being educated like that.
- dls2016 5y ago
  What if I told you the derivative is a linear operator? (matrix meme style)
  - deleted 5y ago
    [deleted]
  - Grustaf 5y ago
    Of course, but there are plenty of linear operators, that property alone doesn't define it.
    - dls2016 5y ago
      I was responding to derivatives being more fundamental than matrices. My rebuttal is that the derivative is but one in a sea of interesting linear operators.
- wbsss4412 5y ago
  This begs the question: more fundamental to what? At a high school level, you have students who are either engaged and will likely learn both subjects eventually, or students who aren’t as engaged and are just looking to take their “last math class”. In my opinion, it makes more sense to offer a choice or at least focus on the curriculum that keeps students are that age to most engaged. As someone who has studied both subjects, I’d go with linear algebra 9 times out of 10, the exception being if someone wanted to also study physics.
  - SavantIdiot 5y ago
    The question seems to be: should integration and differentiation be taught before matrix operations, or after? IMHO, since linear algebra is largely a tool for solving differential equations, I think calculus should be taught first, as the fundamental knowledge.
    - wbsss4412 5y ago
      I’m not really sure where you got the idea that it’s largely a tool for solving differential equations. Certainly that is an application, but that isn’t the only use case.
      - SavantIdiot 5y ago
        That's how it was introduced to me. Which is probably part of the discussion about what should be taught first.
        wbsss4412 5y ago
        I was introduced to the topic on its own, so that colors my perspective as well I suppose. If I were to think about what a high school student would get the most value out of, it would realistically involve a combination of linear algebra and statistics (getting into basic linear modeling/ols). You’d have to hand wave away some of the proofs which require knowledge of calculus, but high school classes aren’t very rigorous anyways.
SamBam 5y ago
I would say that calculus is more relevant to the sciences, is it not? Possibly not Calculus 2, but at least getting an intuitive understanding of rates of change, second derivatives, and area under a curve seems pretty critical for all the sciences.
ivan_ah 5y ago
There is a nice "opinion piece" by Strang that echoes what you're saying here: Too Much Calculus (and not enough LA) https://siags.siam.org/siagla/articles/Strang2001.pdf https://siags.siam.org/siagla/articles/Strang2001.pdf
SavantIdiot 5y ago
I think highschool tries to expose you to as much as possible. I did get a tiny bit of linear algebra at the end of my senior year in calc, but back then PCs pretty much didn't exist. I think today LA is more applicable to programming than integrals, series & expansions, continuity, etc... so maybe times have changed.
edge17 5y ago
My guess is that it's just easier to find Calculus teachers vs Linear Algebra teachers. Maybe it's less obvious in the US but when I have visited schools in poor countries you really get a sense of where knowledge in certain subjects tops out. I find people in the US seem to assume they can magically drum up a supply of linear algebra teachers, or the like. STEM teachers are technical professionals.
chobytes 5y ago
Probably historical. Modern linear algebra is extremely recent as math goes. Definitely much younger than calculus. Also I would add that linear algebra, calculus, and differential equations all go pretty much hand in hand. We could probably stand to teach anyone with an inclination for STEM all of those much sooner.
BeetleB 5y ago
I think it's historical. Linear algebra applications fall into two categories: 1. Theoretical (as in vector spaces, etc). These mostly are useful in advanced courses in engineering/science, and a lot of their applications involve calculus (e.g. Fourier series, function spaces, etc). So calculus needs to be taught first. 2. Computational. These can be subdivided into applications that involve calculus (e.g. differential equations) and everything else (graphics, etc). Many of the latter's applications are relatively recent (last few decades). Whereas calculus was needed in virtually all types of engineering and science. So it made sense to teach calculus. Imagine it's the 1970's. Your in HS. What will you do with all the linear algebra knowledge that won't require calculus? Assume you have no access to computers.

macrolocal 5y ago

I suspect the answer is partly historical; without a computer, calculations in linear algebra are a pain.

Grustaf 5y ago

Both calculus and linear algebra are fundamental for further studies, but I'm not sure linear algebra is more approachable? I personally find geometry and algebra more interesting, but it seems to me that derivatives are more fundamental than matrices. But maybe that's just my bias from being educated like that.

dls2016 5y ago
What if I told you the derivative is a linear operator? (matrix meme style)
- deleted 5y ago
  [deleted]
- Grustaf 5y ago
  Of course, but there are plenty of linear operators, that property alone doesn't define it.
  - dls2016 5y ago
    I was responding to derivatives being more fundamental than matrices. My rebuttal is that the derivative is but one in a sea of interesting linear operators.
wbsss4412 5y ago
This begs the question: more fundamental to what? At a high school level, you have students who are either engaged and will likely learn both subjects eventually, or students who aren’t as engaged and are just looking to take their “last math class”. In my opinion, it makes more sense to offer a choice or at least focus on the curriculum that keeps students are that age to most engaged. As someone who has studied both subjects, I’d go with linear algebra 9 times out of 10, the exception being if someone wanted to also study physics.
- SavantIdiot 5y ago
  The question seems to be: should integration and differentiation be taught before matrix operations, or after? IMHO, since linear algebra is largely a tool for solving differential equations, I think calculus should be taught first, as the fundamental knowledge.
  - wbsss4412 5y ago
    I’m not really sure where you got the idea that it’s largely a tool for solving differential equations. Certainly that is an application, but that isn’t the only use case.
    - SavantIdiot 5y ago
      That's how it was introduced to me. Which is probably part of the discussion about what should be taught first.
      - wbsss4412 5y ago
        I was introduced to the topic on its own, so that colors my perspective as well I suppose. If I were to think about what a high school student would get the most value out of, it would realistically involve a combination of linear algebra and statistics (getting into basic linear modeling/ols). You’d have to hand wave away some of the proofs which require knowledge of calculus, but high school classes aren’t very rigorous anyways.

SamBam 5y ago

I would say that calculus is more relevant to the sciences, is it not? Possibly not Calculus 2, but at least getting an intuitive understanding of rates of change, second derivatives, and area under a curve seems pretty critical for all the sciences.

ivan_ah 5y ago

There is a nice "opinion piece" by Strang that echoes what you're saying here: Too Much Calculus (and not enough LA) https://siags.siam.org/siagla/articles/Strang2001.pdf https://siags.siam.org/siagla/articles/Strang2001.pdf

SavantIdiot 5y ago

I think highschool tries to expose you to as much as possible. I did get a tiny bit of linear algebra at the end of my senior year in calc, but back then PCs pretty much didn't exist. I think today LA is more applicable to programming than integrals, series & expansions, continuity, etc... so maybe times have changed.

edge17 5y ago

My guess is that it's just easier to find Calculus teachers vs Linear Algebra teachers. Maybe it's less obvious in the US but when I have visited schools in poor countries you really get a sense of where knowledge in certain subjects tops out. I find people in the US seem to assume they can magically drum up a supply of linear algebra teachers, or the like. STEM teachers are technical professionals.

chobytes 5y ago

Probably historical. Modern linear algebra is extremely recent as math goes. Definitely much younger than calculus. Also I would add that linear algebra, calculus, and differential equations all go pretty much hand in hand. We could probably stand to teach anyone with an inclination for STEM all of those much sooner.

BeetleB 5y ago

I think it's historical. Linear algebra applications fall into two categories: 1. Theoretical (as in vector spaces, etc). These mostly are useful in advanced courses in engineering/science, and a lot of their applications involve calculus (e.g. Fourier series, function spaces, etc). So calculus needs to be taught first. 2. Computational. These can be subdivided into applications that involve calculus (e.g. differential equations) and everything else (graphics, etc). Many of the latter's applications are relatively recent (last few decades). Whereas calculus was needed in virtually all types of engineering and science. So it made sense to teach calculus. Imagine it's the 1970's. Your in HS. What will you do with all the linear algebra knowledge that won't require calculus? Assume you have no access to computers.