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Math is presented in a way that's supposed to be organized, compact, and categorical. If we taught math the same way math was proven and discovered, it would be

Math is presented in a way that's supposed to be organized, compact, and categorical. If we taught math the same way math was proven and discovered, it would be so slow and inefficient that we would still be covering linear algebra in post grad. As an analogy: The 1,000th person to climb Mt. Everest takes a well defined path that has already been mapped out as the most efficient path to the top. If every single person had to go through the treachery of finding the dead ends, cliffs, crevices, and death traps that the first few climbers endured, it would be a journey only a few could accomplish. Most people (computer scientists, engineers, chemists, physicists) using math only need to reach the top and see the view from the peak. The few climbers that are really dedicated to climbing (ie, the math researchers who reach the frontier of math) will naturally learn about the rest of the jagged, unmapped landscape as they climb harder and unconquered mountains.

jacobolus 4y ago
The current pedagogy and curriculum is nowhere close to the “most efficient path” (nor is it the most intuitive, best organized, easiest to extend, ...). 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. 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.
- nyokodo 4y ago
  > full of historical inequity. Granted the academic profession has historical inequality but what about the math itself displays that?
  - le-mark 4y ago
    My mind goes to various, unfortunate notational conventions.
    - xnyan 4y ago
      Is there something you could say about these unfortunate notations or an article you you could point me to so that I can understand what they are?
      - nextos 4y ago
        For example, calculus uses notation and terminology that predates the modern limit-based field Weierstraß and others built. It's really confusing [1]. 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/ https://www.davidbressoud.org/aratra/ [2] The Introductory Statistics Course: A Ptolemaic Curriculum? https://escholarship.org/uc/item/6hb3k0nz https://escholarship.org/uc/item/6hb3k0nz
        aeneasmackenzie 4y ago
        Limits are not an inherent part of calculus. You can do all calculus relevant for the physical world just fine with nilpotent infinitesimals if you but give up excluded middle.
        jqgatsby 4y ago
        I've heard of this constructivist approach to calculus, but hadn't made the connection with nilpotents. that's really interesting, could you explain why nilpotenxy and forgoing the law of the excluded middle relate to each other?
        enriquto 4y ago
        You can use nilpotents with classical logic and the excluded middle. This is called dual numbers and it's already a good model for "calculus without limits". They are like complex numbers, but instead of x^2=-1 you set x^2=0. 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).
  - jacobolus 4y ago
    The biggest problem is that it is very unfriendly to uninitiated newcomers and makes insufficient effort to draw people in. You end up with a culture that is unfortunately insular and has trouble engaging with even engineers and scientists, much less the general public. It’s also not very friendly to people who approach problems in different ways: symbol pushing has been elevated and anyone who has difficulty with symbol pushing (for whatever reason) ends up at least partly excluded. 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.
    - red75prime 4y ago
      Well, we can't know that until we find (or won't find) the more effective way of teaching (or a way to do math without "symbol pushing" for that matter). Until then it will not be wise to break what works (even for a minority of students).
      - jacobolus 4y ago
        Current incentives are set up to make even the most trivial attempts to run against the mainstream definitions and notations extremely difficult.
        simonh 4y ago
        I don’t think it’s fair to say they are set up to do that. They weren’t conceived with that purpose. It’s just a fact of life that once we’ve invested a huge amount of effort in one set of conventions it’s very costly to change those conventions.
- WoahNoun 4y ago
  > 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
- robot_no_419 4y ago
  I never said it was the most intuitive, just the most streamlined and most efficient. I'm having a hard time understanding why you think it's not the best organized. Most math courses are taught with a pretty straight forward approach: start with the axioms and definitions, prove the easy and auxiliary theorems that are easily derived from the axioms, prove the fundamental theorems that make the subject useful. In other words, the shortest path from the axioms to the important theorems. I don't see a way to make it more organized or compact, but I'm open to hear what you think is a more condensed or organized way to teach math. 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.
  - trasz 4y ago
    What you describe is probably the least efficient way to learn math - or anything else for that matter - because you’re trying to learn things you can’t, at that point and for a while afterwards, reason about. They don’t have connection to anything else. Compare that with the opposite - learning stuff you can already mentally associate with parts of known reality.
    - zozbot234 4y ago
      > learning stuff you can already mentally associate with parts of known reality. 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.
- scythe 4y ago
  I would counter that the current pedagogy -- at least high school through early undergrad -- is the most efficient path, or close to it, for teaching students to become electrical engineers in the analog era. Historically, that was the most math-heavy profession that had a lot of jobs (not just professors/researchers). We just haven't updated it in a long time.
  - jacobolus 4y ago
    That’s probably not too far off the mark... with the proviso that we are fixing the notation, terminology, and problem solving methods for electrical engineering to what was historically used in the 1950s, and not allowing any more radical “refactoring” of those ideas or methods. 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.
joe_the_user 4y ago
I think that's one useful way to view it. Math is infinite "mountain" and the higher you get, the rougher the summaries. One thing I should mention. I have been reading The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age by Jan von Plato. One thing I notice is that a lot of mathematicians activity is not proving more and more complex theories but rather, producing a framework that takes a series of complex results and shows them to be much simpler within the framework. And that's just to say, the organization of math isn't just a matter of simplification for the layman, it's part of the progress of math itself.
sidkshatriya 4y ago
Good points. This is essentially compression of knowledge at play. To make the next discovery it makes sense to get compressed information about the previous discoveries. Historians are often more interested in the various routes attempted to achieve scientific discovery -- which failed, which succeeded etc. Scientists are interested in climbing to the next peak (of knowledge) with just sufficient knowledge of how we came to the current location. It always helps to know a bit of history. You might encounter problems while climbing to the next peak and knowing a bit of history might give you some additional tools to solve problems you may encounter. However, you must be judicious. Learn too much about the past and you won't have much time to create the future. Also if you learn too little about the past you may not be well equipped to deal with upcoming challenges. It is a balance.
victor106 4y ago
That’s a great analogy. Thank you.

jacobolus 4y ago

The current pedagogy and curriculum is nowhere close to the “most efficient path” (nor is it the most intuitive, best organized, easiest to extend, ...). 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. 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.

nyokodo 4y ago
> full of historical inequity. Granted the academic profession has historical inequality but what about the math itself displays that?
- le-mark 4y ago
  My mind goes to various, unfortunate notational conventions.
  - xnyan 4y ago
    Is there something you could say about these unfortunate notations or an article you you could point me to so that I can understand what they are?
    - nextos 4y ago
      For example, calculus uses notation and terminology that predates the modern limit-based field Weierstraß and others built. It's really confusing [1]. 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/ https://www.davidbressoud.org/aratra/ [2] The Introductory Statistics Course: A Ptolemaic Curriculum? https://escholarship.org/uc/item/6hb3k0nz https://escholarship.org/uc/item/6hb3k0nz
      - aeneasmackenzie 4y ago
        Limits are not an inherent part of calculus. You can do all calculus relevant for the physical world just fine with nilpotent infinitesimals if you but give up excluded middle.
        jqgatsby 4y ago
        I've heard of this constructivist approach to calculus, but hadn't made the connection with nilpotents. that's really interesting, could you explain why nilpotenxy and forgoing the law of the excluded middle relate to each other?
        enriquto 4y ago
        You can use nilpotents with classical logic and the excluded middle. This is called dual numbers and it's already a good model for "calculus without limits". They are like complex numbers, but instead of x^2=-1 you set x^2=0. 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).
- jacobolus 4y ago
  The biggest problem is that it is very unfriendly to uninitiated newcomers and makes insufficient effort to draw people in. You end up with a culture that is unfortunately insular and has trouble engaging with even engineers and scientists, much less the general public. It’s also not very friendly to people who approach problems in different ways: symbol pushing has been elevated and anyone who has difficulty with symbol pushing (for whatever reason) ends up at least partly excluded. 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.
  - red75prime 4y ago
    Well, we can't know that until we find (or won't find) the more effective way of teaching (or a way to do math without "symbol pushing" for that matter). Until then it will not be wise to break what works (even for a minority of students).
    - jacobolus 4y ago
      Current incentives are set up to make even the most trivial attempts to run against the mainstream definitions and notations extremely difficult.
      - simonh 4y ago
        I don’t think it’s fair to say they are set up to do that. They weren’t conceived with that purpose. It’s just a fact of life that once we’ve invested a huge amount of effort in one set of conventions it’s very costly to change those conventions.
WoahNoun 4y ago
> 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
robot_no_419 4y ago
I never said it was the most intuitive, just the most streamlined and most efficient. I'm having a hard time understanding why you think it's not the best organized. Most math courses are taught with a pretty straight forward approach: start with the axioms and definitions, prove the easy and auxiliary theorems that are easily derived from the axioms, prove the fundamental theorems that make the subject useful. In other words, the shortest path from the axioms to the important theorems. I don't see a way to make it more organized or compact, but I'm open to hear what you think is a more condensed or organized way to teach math. 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.
- trasz 4y ago
  What you describe is probably the least efficient way to learn math - or anything else for that matter - because you’re trying to learn things you can’t, at that point and for a while afterwards, reason about. They don’t have connection to anything else. Compare that with the opposite - learning stuff you can already mentally associate with parts of known reality.
  - zozbot234 4y ago
    > learning stuff you can already mentally associate with parts of known reality. 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.
scythe 4y ago
I would counter that the current pedagogy -- at least high school through early undergrad -- is the most efficient path, or close to it, for teaching students to become electrical engineers in the analog era. Historically, that was the most math-heavy profession that had a lot of jobs (not just professors/researchers). We just haven't updated it in a long time.
- jacobolus 4y ago
  That’s probably not too far off the mark... with the proviso that we are fixing the notation, terminology, and problem solving methods for electrical engineering to what was historically used in the 1950s, and not allowing any more radical “refactoring” of those ideas or methods. 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.

joe_the_user 4y ago

I think that's one useful way to view it. Math is infinite "mountain" and the higher you get, the rougher the summaries. One thing I should mention. I have been reading The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age by Jan von Plato. One thing I notice is that a lot of mathematicians activity is not proving more and more complex theories but rather, producing a framework that takes a series of complex results and shows them to be much simpler within the framework. And that's just to say, the organization of math isn't just a matter of simplification for the layman, it's part of the progress of math itself.

sidkshatriya 4y ago

Good points. This is essentially compression of knowledge at play. To make the next discovery it makes sense to get compressed information about the previous discoveries. Historians are often more interested in the various routes attempted to achieve scientific discovery -- which failed, which succeeded etc. Scientists are interested in climbing to the next peak (of knowledge) with just sufficient knowledge of how we came to the current location. It always helps to know a bit of history. You might encounter problems while climbing to the next peak and knowing a bit of history might give you some additional tools to solve problems you may encounter. However, you must be judicious. Learn too much about the past and you won't have much time to create the future. Also if you learn too little about the past you may not be well equipped to deal with upcoming challenges. It is a balance.

victor106 4y ago

That’s a great analogy. Thank you.