What is needed are higher-level abstractions over ML principles that make ML more accessible for people who are interested in programming, but lack the math background to be productive. This doesn't necessarily mean that the lower-level details should be entirely hidden, but that the tools are friendly enough for beginners to easily pick up, and flexible enough for advanced users to tweak to their needs.
There are programmers, though, who are not fluent in the math that makes ML possible, but would be empowered by friendlier tools.
So abstract away, by all means, ensuring the user doesn’t need to know the Biot-Savart law to use a doorbell is the engineer’s job after all, and as semiconductor electronics show, users can include other engineers. (Hell, even the humble resistor is a fairly advanced piece of technology dependent a good chunk of 20th-century physics.) But sooner or later we’re going to have to pay back this “ed debt”. (“Soc debt”?)
[1] See my recent comment at https://news.ycombinator.com/item?id=34389684 for a starting point for a literature dive.
For reference, things known to humanity in 1940 (though not necessarily understood, that was an important difference in my comment) include the Schwarzschild (stationary black hole) and FLRW (homogeneous expanding or contracting universe) solutions to the equations of general relativity (1917 and 1924 respectively), the Darwin term for the effect of special relativity on the spectrum of hydrogen (1928), the severely non-classical exchange interaction responsible for the existence of ferromagnetism (same year), the London equations describing the (non-)penetration of the magnetic field into type-I superconductors (1935), and a way to produce superfluid helium (1937). And that only includes things I was supposed to ( :( ) know at some point. Also Gödel’s incompleteness theorems (1931) and Turing’s proof of undecidability of the halting problem (1936). I’m certainly not proposing to demand high schoolers know all of those.
The only reason I had to go as far as 1940 or so is because linear algebra came relatively late in the history of mathematics, even though it should logically serve as the foundation of multivariable calculus—and was enthusiastically adopted as such once it finally entered the collective mathematical unconscious in the late 19th century. Working out the pedagogy took a couple more decades.
If we were only talking about the “advanced” mathematics of calculus, I’d say Newton’s knowledge as of 1690 would be way overkill. (That would include the convergence speed of power series, a decent theory of ordinary differential equations, the beginnings of the calculus of variations, and even the “Newton polygons”, a theory of formal-series solutions to polynomial equations that properly belongs to algebraic geometry.) At that point J S Bach was five, the city of Philadelphia was a town founded eight years ago, and Peter Romanov (later called the Great) was still twelve years away from founding St Petersburg and thirty years away from proclaiming Russia an empire.
To be honest, I think even if school taught no mathematics beyond basic numeracy, I wouldn’t complain as much. I would still complain, mind you, but only to the extent I do about being able to grow up without knowing what a fugue or the twelve-bar blues is or who Giotto or Niccolò de' Niccoli were. Instead we torture people with years of alleged mathematics bearing no resemblance to the real thing, and thus give them license to think they are not “maths people” (whatever that means) and that whatever maths they did not hear about in school is “advanced” and obscure. At least the average high-school graduate knows he doesn’t have a clue about music history or theory.
My tirade had a point as well, though: post 20th century our collective intuitions have become wildly miscalibrated regarding which things in humanity’s understanding of the world are recent or obscure. It seems that the barrier of “advanced” mostly hasn’t shifted at all since high school became compulsory in most parts of the world, and as decades go by I can see this going from “a goddamn shame” to “existential threat to human culture”. I’m not even sure that hasn’t happened yet—already a high school teacher can rarely, if ever, thoroughly explain a recent development of their choice in their subject.
Derivatives and matrices are high-school math.
(I also think learning linear algebra from the matrices side rather than from the linear transformations side, as was indeed common for mathematicians in the 19th century and for physicists and engineers well into the 20th, is both unnecessarily painful and almost impossible to pull off well. So if a syllabus lists its basics-of-linear-algebra part as “matrices” it’s probably missing the point. Knowing that you do rows times columns is of very little value, the part that makes it worth it is the why.)
Going by dates of introduction to determine if a math topic should be part of general knowledge doesn't make sense. Math was highly advanced by the 20th century, and a lot of it was used to advance the field of physics, which led us to the technological advancements we all enjoy today. None of it needs to be fundamentally understood by the layperson.
You sound well versed in math, but most people just don't need this knowledge in everyday life. Precisely because engineers build tools to make everyday life easier for everyone, regardless of education opportunities. Our society couldn't function any other way, and asking for everyone to have advanced math or engineering knowledge is unrealistic.
On one hand, yes, and that’s part of the point I usually make when I recite how far into the standard university course Newton’s or Gauss’s knowledge can still be found. On the other hand, no, and that’s the other part of the point I want to make when lamenting high-school maths, because the state of the art there is obsolete to a level that’s patently insane. Thank you for making me think about this.
I’ll put it like that. How often do you change your mind? (Or, if you want: how often do you make breaking changes to your APIs?) Obviously it makes to sense to keep a running tally and decide whether to do it or not depending on which side of the target value you’re currently on. Making a performance indicator out of this is a sure-fire way to end up in a completely twisted universe.
On the other hand, if you do it all the time something’s equally obviously gone awry—maybe you just started out being very wrong, and that’s not a problem with the current you, but if after some time there’s still no signs of settling you need to do some soul-searching. Similarly but even more dangerously, if you never do it, it’s of course possible you were simply right to begin with, and that’s happened once or twice in history, but you must seriously consider that you might either be living among fools or avoiding the intrinsic discomfort of the process and thus a fool yourself.
I’d say the recency of a general-education syllabus is an indicator of a similar nature: meaningless in each particular instance, and emphatically not to be used as a target metric, but at the same time as the average goes into the margins it’s increasingly likely it’s failing at its job. And by that measure maths is universally in deep shit. (The natural sciences are as well, just with more variance; I’m not qualified to judge the rest.)
On the top of my list:
- Not teaching students how to learn, and instilling them with the joys of perpetual learning, but forcing them to memorize concepts and regurgitate conclusions made by others, so they can pass an easily gradeable multiple choice test.
- Teaching a _bunch_ of useless concepts for everyday life, including some basic math concepts, let alone the math topics you've mentioned, but not teaching about the basics of finance, law and politics which would be much more relevant for leading a generally successful life.
- High corruption, low teacher wages, bullying, etc.
General education is not as much concerned with actually preparing young adults for adulthood, as it is with getting them through a well-established system designed to squeeze any actual desire to learn from students, while squeezing their parents financially. It's a broken system that won't be fixed by adding advanced math to the syllabus.
We struggled along for 200,000 years without it, so I'd hardly call it an incredible failure that more people don't learn it[1].
[1] The usual disclaimers apply - im not anti-math, and I think more people should learn math, science, etc...
My knowledge embarrassingly stops at Algebra II (USA system).