Why are mathematical standards so low? (And what should they be?)

Another control theorist here. I strongly disagree with your premise of rigor in control. The whole field is ripe with math snobbery and virtually plotted its own funeral to the point of absurdity.

The field's key contribution is in the ideas and the insights it can provide otherwise extremely laborious to obtain. Definitely not in senseless dry exposition of unnecessarily general theorem proof parroting.

I really like mathematics and I spend some time reading obscure math books for kicks.

I was a very bad math student in highschool I don't think it was because I was "ungifted" like some people would describe it.

Towards the end of high school i.e the last year.

I found out that mathematics can be studied like Biology, History (too late for it to make an impact on my final results).

And that the only way of learning mathematics was not just through practising it mindlessly until you grasp it like the teachers said.

I think the real problem with mathematics is that it has really bad comprehensive literature.

Here are the 3 main problems with maths literature. (What I think would have helped me improve my maths skills by alot)

Reason 1.

Implicitly defined notations. There is really no go to comprehensive resource that describes all math notation clearly so jumping into a new topic is difficult.

Reason 2

No clear definitions of algorithms used in respective math topics.

A great example is http://matrixmultiplication.xyz/ that describes the matrix multiplication algorithm. Step by step. Maths education would be better if all important algorithms in different math fields were described as clearly.

Reason 3

There is no comprehensive dictionary for math theorems. A great example would be something like a table of integrals in calculus, but for all fields.

If all these resources are well documented. I really don't see why someone cannot improve their maths skills even via self study up to graduate level.

> Can you imagine a hypothetical education system where students leave secondary school with an advanced-undergraduate level of mathematical maturity?

That would have been my dream.

> Why or why not?

Mathematics is the foundation of science, so that would have been helpful for my (then) future studies.

> What would such a system have to look like?

Not teaching algorithms to resolve problems, teaching from the source of the mathematical principles, I mean the practical issues that caused scientist to develop Math. I suppose that the Russian School of Mathematics[1] teaches Math that way, like in some Soviet books that were mainly practical.

Also, it is needed to have 5 hours per week of Math, to not have a fast pace when teaching kids. Math needs some time to "assimilate", and IMHO, 3 hours/week was not enough.

In this system, once you get out high school you could pursue the proofs of each theorem or a more inner and rigorous approach to learning Math.

[1] https://www.mathschool.com/

It is low probably because the meeting degree they are having already take up most of the time, and may also be because for those better equipped in math they are in research that requires more sophisticated level of math already (ie self selection).

This is extrapolated from my experience in CMB data analysis. People with better math training probably are doing theoretical work via simulation instead. Some, but not all, papers involving CMB data analysis struggles to present their research mathematically, or may perform some statistical analysis in a very naïve way.

>In my experience, most real life systems, organizations, and even psychological patterns can be effectively and practically understood in terms of dynamical systems, feedback loops, optimization, etc., to the ext

So, do the understanding and then apply it and then win! Except, maybe you will find out what thousands of others have, first maths is really big, second it's not big enough to be effectively used as you describe.

This question reminds me of the book A Mathematician's Lament by Paul Lockhart.

He lays out a really good case for why we don't have more mathematicians.

I accept that enhancing fundamental math skills is important, yet I'm skeptical about the feasibility of raising the bar to more advanced levels. Considering a normal distribution, it seems improbable that a significantly larger number of individuals could be educated to this extent.

Look at the CURRENT outcomes produced by mathematical geniuses within the financial sector (and military..). My argument isn't that math's impact is solely detrimental. For me it is about balance and diversity (math is only part of it)

When it comes to systems theory itself and its buddies, back in the day, Cybernetics was more of a big tent idea. But then the math nerds took the wheel, thanks to Shannon's playbook. And that may not always represent the superior path. Again, It is about balance/diversity

In EECS every undergrad does learn a bit of control/systems theory in their sophomore and junior requirements. I loved it but it was mathematically challenging and unfortunately the better background a student already had, the better and the more they got out of the courses. I wish there were a better way, because the result was students eventually got weeded out by an (inhumane, alienating) competitive system, rather than have each student actually learn something well on their own level.

I read about PID controllers and got super excited to use them in a programming context. That was maybe 10 years ago and I still haven't spotted an opportunity.

https://www.mg.edu.rs/en#gsc.tab=0

This would be a horrendous, and perhaps inhuman system. It's already difficult to convince non-math inclined kids (the majority) to study basic algebra; forcing kids to learn a much increased level of abstraction would be borderline abusive.

What do you define as graduate-level maths courses? The scope and depth vary considerably. What I did in the first year of a maths degree may be (is, have seen) considered graduate level in some (some) engineering courses. Could you be more specific?

In my experience, the quantitative aspects of advanced math do not bring you much extra understanding in everyday life situations. The qualitative aspects may be helpful but they can be learned without grokking advanced math.

You are also privileged not to suffer from dyslexia, ADD, ADHD, and so on. Add all the letters up and we see that so many children are affected that we should probably be looking elsewhere for their talents.

The idea that we could actually afford to customize the curriculum for each child is extremely new, but with these AI advances we might finally be able to tap into the great richness of our diversity and celebrate being different.

To have mathematical tools with which to relate to the greater cosmos is a wonderful gift but there are others.

How does a a cashier benefit from systems theory?

There was a system like that. Roll up, roll up, I'll treat you to a story:

It all started in 1957, with the Sputnik. The US was entirely taken by surprise. The belief that the Russians were way ahead instigated an internal crisis, which, in turn, led the US to re-evaluate its national maths curriculum. Thus was born a think tank called the School Mathematics Study Group (SMSG). They developed a radical reform in mathematics education known as 'New Math'.

This was rolled out nationally, to great criticism. The teachers were ill prepared and the parents felt clueless. Look it up, there were 'Peanuts' strips from the period mocking it.

Now, New Maths focused initially on the early years, but then, in the mid '60s came a second round, and one specific initiative was the Secondary School Mathematics Curriculum Improvement Study (SSMCIS), and the guy who heads it happened to be a professor at Columbia University Teachers College. This last tiny fact probably means exactly zero to you, but it is central to why I even know all this stuff.

Anyways...The program's signature goal was to create a unified treatment of mathematics, so that instead of studying the normal curriculum you'd basically study maths the way you're taught it at university: set theory, group theory, axioms proofs and logic, all the way up to calculus. The programme was intended for grades 7 through 12, and was rolled out initially in the NY area and then later in select schools in other affluent cities in the states. It only ever targeted the top 15-20 students in the a class body. That was for sure the right call - this ties to your question, so more on that in a minute.

Eventually two things happened: one, the programme ran out of funding. Two, by the mid-seventies there was a massive backlash against New Maths and the US decided maybe it's okay to just leave it, since the Russians didn't end up winning the space race after all.

I would have known diddly squat about this whole affair were it not for a curious corollary. In 1953 one very specific individual happened to be on a mission in New York. He was a former Russian Jew who studied maths in Canada, served in WWII for the US military and eventually made it to Israel. He was an educator and had somehow caught on to what was happening in Teachers College, and upon his return to Israel, he started devising maths curriculums and translating the original SSMCIS textbooks. This was now dubbed 'The Columbia Programme'.

Fast forward almost 40 years later. In a way I've never managed to uncover, that programme survived, and was still being taught in one of Israel's gifted programmes. I entered my first maths lesson at seventh grade never realizing just how much this would end up influencing the person I'd become. Our textbooks were literally photocopies of the typewritten texts. The teachers has added to it bits of the regular curriculum plus more practice exercises, which the original textbooks lacked, but they left most of it as is.

In the first three years, no one in the class was allowed to drop out and take 'regular' maths. For many, even in a cohort that was already pre-screened for academic achievement, this was a struggle. For sure, once highschool rolled along, anyone who hated it could switch back to the regular national curriculum.

Of the people who stayed, nearly everyone went on to study Maths, Physics or Computer Science to graduate level. This tended to happen in the years above and below as well. Over the years, though, the programme got smaller and smaller. I'm not sure it still exists.

To your question:

You absolutely CAN get highschool students to leave secondary school with advanced-undergraduate level of mathematical maturity. (And, BTW, the Russians are still ahead there...) But you can only do it for a small minority. Not because of elitism, but because most people aren't a good fit for this path.

At the time I intended to write up all of this into a nonfiction essay. But other things took greater priority, and I just left it there. In a way, it's been nice telling this story here.

I failed control-systems in my third year. Was too focused on other more fun subjects. Maybe it is a fun issue?