It’s more like our current public transit system: it gets some people where they’re going, somewhat on time, but it’s generally pretty crummy and full of historical inequity.
And no, this is rarely the most intuitive or contextual way to learn math. Another analogy - a library doesn't sort their books by which ones were best reads or most influential, but by topic and author. Similarly, math curriculums are organized by a hierarchy of which theorems can prove the next theorem with no explanation of which ones are important. Organization doesn't always provide intuition.
This is not always feasible or effective. Sometimes it's just better to start by doing some simple reasoning about things in isolation, and build the proper connection and context afterwards.
This is absolutely not true. If anything, math education has a tendency to keep losing intuition over time as it's refactored for modern approaches and notation.
Once you get to math grad school you can find more material that uses approaches and notations that are only about 50 years old.
The most significant “recent” change to be found from the 20th century is the “Bourbaki-zation” of mathematics, especially sources intended for expert readers: cutting out pictures, intuition, and leading examples in favor of an extremely spare and formal style that alienates many newcomers and chases them out of the field. And I guess at the high school level, there’s the domination of pocket calculators (displacing slide rules) which came about in the 1970s–80s.
There is massive, massive room for improvement across the board.
If you read works by e.g. Euler, other than being in Latin they still seem pretty much modern (we did tighten up some of the details in the century or two afterward), because much less has changed in the way we approach those subjects than you would expect. By contrast, if you read Newton or his contemporaries/predecessors, the style is often completely different and almost unrecognizable/illegible to modern audiences, building on the millennia old tradition of The Elements and Conics.
For another serious transformation, look to the way computing is taught, which has changed quite dramatically in the past 50 years. Nothing remotely like that is happening in up-through-undergraduate mathematics.
Just look what you can do with high-production-value video animations: https://www.3blue1brown.com/lessons/essence-of-calculus
It’s hard to simply say this is “better”: it depends what skills and content you are trying to teach. The more computing-heavy version arguably does a lot better job quickly preparing students to engage with scientific research literature (because differential equations are a fundamental part of the language of science). But it might make it harder for students to e.g. dive into a traditional electrodynamics course intended for future physicists, full of gnarly integrals to solve.
Most of the people proposing even more significant departures (in content or style) aren’t writing introductory undergrad textbooks.
However, different approaches don't just teach different "skills and content" as you say, but entire paradigms of thinking. There is mathematical thinking and there is computational thinking (and other types as well), and any course helps you step up the ladders of these paradigms by different amounts.
My experience teaching undergrad math/physics/cs for several years is that computational thinking is in the short term time and effort cheap, and this causes a fixed point in how students think. If you give them the concept of say differential equations, and teach them some computational methods and some mathematical methods to solve these equations, they will always lean towards just using the computational methods. This seems all fine and dandy, except when you go to more advanced mathematical abstractions, and in the previous step the students had not mastered the mathematical way of thinking, they are lost. They simply don't have the mathematical capacity to grasp the higher abstractions. And no amount of 3B1B fixes it - this lack of long term investment into an important thinking paradigm.
The point is that we treat the differential of a real valued function as a function/vector/matrix for historical reasons. The simpler perspective that always works is that the differential of a function is the best linear approximation of the function at a given point. But for historical reasons most math textbooks restrict themselves to "first order functions" and avoid, e.g., functions returning functions.
This also leads to ridiculous notational problems when dealing with higher order functions, like integration and all kinds of integral transforms.
Granted the academic profession has historical inequality but what about the math itself displays that?
Statistics is even worse. A mix of old tricks developed to avoid computations when these were expensive. See [2].
[1] A Radical Approach to Real Analysis https://www.davidbressoud.org/aratra/
[2] The Introductory Statistics Course: A Ptolemaic Curriculum? https://escholarship.org/uc/item/6hb3k0nz
However, if you want to get really serious about that, you'll need that zero plus an infinitessimal be equal to zero. This is impossible in classical logic due to the excluded middle (which forces each number to be either equal to zero or non-zero).
For a more algebraic treatment, and its important applications to automatic differentiation, I'd suggest starting with the relevant wikipedia articles:
Computing science is when notation came into its own. Younger mathematicians have taken those lessons to heart, but as the old saying goes, progress comes one funeral at a time.
Being forced to mechanically parse and interpret a syntax has a way of really bringing out any ambiguity.
It's not to save paper or because of the product. You don't know the solution to the problem you are working on from the beginning and most of the time is spent writing and writing and writing in a scratchpad trying to solve what you need. Anything longer than a single glyph for variables would be too tedious so everyone evolved to use single letters. And then the papers are written with the same convention since it's natural. You have variable names though with the use of subscripts with the added benefits that it can be (and is) used to elegantly group relevant variables together giving you some sort of abstraction
I once wrote a comment about it here on HN - language in maths is not a programming language used to tell a computer how to go from A to B, but a natural language used to talk about maths between peers. Every natural language have idioms, inconsistences and other quirks. Polish will not change for you so it's easier for you to learn it, it will change in the way that let's polish people communicate better with each other which also include a lot of historical and cultural happenstances. Same with maths
There are attempts like Esperanto and other artificial languages like that and I think any attempts at 'codification' of maths to use some programming language has the same chance of success of wide adoption
Aren't existing programming languages already types of codified artificial math dialects which have seen wide adoption
Programming languages are more for humans than for computers. Otherwise we’d be writing our programs in 1s and 0s, and extending our editors in Emacs Binary and VSCode BinaryScript.
Right, we're on the same page, I just think this is a bad thing and you evidently think it's a good thing. I'm well aware many mathematicians don't, because it's how they were trained and unlearning is the hardest kind of learning. The ambiguity[1] of natural language is observably ill-suited for formal reasoning, and the experience of computing science has shown this conclusively.
Do bear in mind that the pioneers in our field were virtually all trained mathematicians. They were well aware of the historic faults of the field because having to make programs actually work forced them to be.
The legacy fuzzy pencil and paper approach of traditional mathematics is going to end up being to proper formal mathematics just as what's now called philosophy is to formal logic.
[1] Let's not confuse ambiguity with generality.
This is absolutely beautifully said user23. I as a programmer often struggle with understanding notations used in some papers.
Programmers tend to have this lack of fluency with written math that they completely miss: the concise names are not to save paper or make writing easier or anything like that. They're because they make the structure of expressions easier to visually identify and parse. The shapes of expressions are an incredibly important feature of the language and often contain implicit structural analogies. You need to be able to see those analogies to correctly read mathematics, and long variable names would obscure that part of the language.
I suppose it's similar to having enough fluency in a natural language to mechanically translate the words of a poem, but you can't properly read things like the metre, so you've unknowingly missed half of what the author originally wrote and lost it all in translation.
Generally you'd use upright text in square brackets to denote longer variable names, the notation is often seen in applied fields. But this quickly becomes clunky with longer expressions.
Students who have a lot of practice/experience by the time they get to be teenagers (often via extra-curricular help and support) are much better prepared than those without that practice. Which is of course not a problem per se, you see the same in any field and it’s great if kids want to learn ahead of their peers. But then the content, curricular design, and pedagogy of mathematics courses leave students with the impression that those differences in preparation are due to innate differences in aptitude (“I suck at math”; “she’s just a math person”; ...), toss less well prepared students into the deep end to sink without enough support, and ultimately chase a huge number of people away who might otherwise find the subject beautiful and interesting, and could meaningfully contribute.
Until then it will not be wise to break what works (even for a minority of students).
What I mean is that there are incentives for the people involved in those systems which are extremely difficult to reform, and as long as the current incentives prevail it is all but impossible for anyone to refactor things like basic mathematical notions and notations.
Switching and retraining costs are high, gaps in inter-operability are expensive, and there is almost nobody who will achieve any career advancement through promoting changes to the high school and early undergraduate curriculum.
Mathematicians are generally most interested in pushing on the shiny boundaries of the field rather than trying to clean up the centuries-old material for novices. Teachers have their hands full enough with their students to do much new research in pedagogy. Practitioners in industry have their own problems to solve.
I don’t think this is actually the most effective way to train analog electrical engineers, or the most effective possible set of conceptual/notational tools for practical electrical engineering.