[WoahNoun]

> It’s just an arbitrary history-dependent path people happened to come up with, mostly centuries ago, and “refactoring” any of it is almost impossible.

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.

[jacobolus]

There are few (if any) important differences between algebra textbooks from 400 years ago, trigonometry textbooks from 300 years ago, and calculus textbooks from 200 years ago vs. their current counterparts. The way we teach vector calculus is more than a century old. Introductory statistics courses still often haven’t caught up with the existence of computers. Undergraduate level math textbooks from 60–90 years ago are still among the most popular course sources across most subjects, including abstract algebra, analysis, etc. Hot “new” material comes from the 19th–early 20th century. The curriculum (at least say 8th grade through undergrad level) is calcified and dead, like a bleached coral.

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.

[solveit]

Can you recommend a book that you think presents,say, calculus, significantly better than the books commonly in use?

[jacobolus]

http://www.science.smith.edu/~callahan/intromine.html is one idea (and read that page/book for a critique of the kind of typical ~200 year old textbook/course we still use today), but this could be a lot better with a bigger budget and more support.

Just look what you can do with high-production-value video animations: https://www.3blue1brown.com/lessons/essence-of-calculus

[abdullahkhalids]

This is a fairly well written book. But I don't see anything qualitatively better about it than the best standard math textbooks. What am I missing?

[jacobolus]

It largely dropped the "memorize this set of symbol-pushing rules then apply them to a long list of exercises" version of differential/integral calculus found in typical introductory textbooks, in favor of a "make up a model for a situation, then program a simulation into a computer and see what happens" approach. That makes for a radically different experience for students.

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.

[abdullahkhalids]

Using simple computer simulations to teach introductory math courses is definitely a change that has been slowly happening over the past couple of decades.

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.

[Qem]

"Elementary Calculus: An Infinitesimal Approach", by Jerome Keisler. Learning calculus is made harder than necessary by the legacy of clumsy epsilon and delta formalism. This formalism is not the intuitive approach Newton and Leibniz used to develop Calculus, based on infinitesimals, that was shunned later because it took time until Abraham Robinson made it rigorous in the 1960s. The author made the entire book available for free online: https://people.math.wisc.edu/~keisler/calc.html See also: https://en.wikipedia.org/wiki/Nonstandard_analysis

[fmap]

Open any book on differential geometry and compare the treatment of differentiation with the needlessly index heavy treatment in any undergraduate calculus textbook.

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.