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> 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 i

> 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 4y ago
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 4y ago
  Can you recommend a book that you think presents,say, calculus, significantly better than the books commonly in use?
  - jacobolus 4y ago
    http://www.science.smith.edu/~callahan/intromine.html 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 https://www.3blue1brown.com/lessons/essence-of-calculus
    - abdullahkhalids 4y ago
      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 4y ago
        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.

jacobolus 4y ago

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 4y ago
Can you recommend a book that you think presents,say, calculus, significantly better than the books commonly in use?
- jacobolus 4y ago
  http://www.science.smith.edu/~callahan/intromine.html 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 https://www.3blue1brown.com/lessons/essence-of-calculus
  - abdullahkhalids 4y ago
    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 4y ago
      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.