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Honestly, every time that a discussion on mathematics comes up in HN I feel the same way: people seem to think that mathematicians are some kind of elitist peop

Honestly, every time that a discussion on mathematics comes up in HN I feel the same way: people seem to think that mathematicians are some kind of elitist people that don't know how to explain concepts, and that mathematics would be like coding (note: projecting everything into the tech world is not the best way to understand the world) if it weren't because mathematicians don't explain things well. But the reality is way different: * We know that teaching mathematics is difficult. But, contrary to what most people think, there is no magic pill to solve it. Motivation and intuition are important, yes, but so are definitions and formality. Also, even when giving them to the students, they might not be useful: sometimes each person needs a different motivation of a concept, sometimes the intuition relies on more advanced concepts... Furthermore, things don't usually click the first time you are exposed to them, but only after you have worked with them for some time. Don't think that just because you are given a motivation and intuition behind the concepts everything is just going to become easy to understand. Even motivations and intuitions need work to be understood. * The mathematics people study in high school and first-year engineering courses is very different from the mathematics that mathematicians study. Funnily enough, most mathematicians I know don't like the approach of mathematics as a tool for other disciplines, mainly because it focuses on things that are hardly interesting (here's the 123th trick for how to solve this weird integral that we are teaching you only because you will need it to solve some engineering equation) and avoids proofs that, despite being hard, most of the time helps understanding the concepts. * Related to the last points, an important number of math books are written not for laypersons but for mathematicians. They go straight to the point, avoid long motivations/intuitions that are not specially useful and assume familiarity with a lot of concepts. Sometimes they are written as reference books for people that already know the ideas. That, of course, makes them hardly approachable by non-mathematicians, but there are already books that do that. Just because a book or a way to teach math is not of your liking does not mean it needs to be changed. * Not all mathematical concepts need to have a direct application. Sometimes, things are studied just because they seem interesting or beautiful. Searching for motivation needlessly in these cases tends to be a waste of time. It's like reading a novel or solving a puzzle: you don't need to know "why", just enjoy it. If you're lucky applications can appear later, but there's nothing wrong if they don't.

marcosdumay 9y ago
> Motivation and intuition are important, yes, but so are definitions and formality. Yet, many (maybe even most) mathematics professors and books insist on only teaching the later two... Every time a similar discussion comes up in HN, there's a post like yours, pointing that the thing is too difficult to break into small problems like the one pointed. Yes, the thing is all very complex. Yet, the pointed problem is still a problem, and people are still not solving it.
- gjulianm 9y ago
  Honestly, most mathematics professors I've had have attempted at giving motivation or intuition about things. Not always useful, not always correct and sometimes dangerously misleading, but the attempts are there. Regarding the posts like mine, they come up because the majority is the other view on the matter: that there must be some way to hack this that somehow is going to be found in the HN comments, and that professors/book writers don't care. And the reality is different: most mathematicians care about intuition and motivation. It's their basic tool of work. But transmitting those ideas and transmitting them effectively is not easy. In fact, there's this funny phenomenon that I've seen in myself and it seems to apply to a lot of people: we forget the intuition we are given for a problem if it doesn't click right away. I also refuse completely the premise that "people are still not solving it". In this post there are people linking to resources on books/pages with new ways to teach mathematics. People are solving it and each time getting better, and however it's never going to be straightforward to teach mathematics (just like it's never going to be straightforward to teach anything, mathematics is only seen as special because it has a lot more presence in education that other subjects).
mathperson 9y ago
amen. the comments in this post are entirely frustrating.
Chris2048 9y ago
But how do you know this? I've come across articles on certain subjects that are far more illuminating than usual - these shouldn't exist if mathematics is just irredeemably difficult to learn. The only conclusion I can draw is mathematicians care more about advancing mathematics, than concepts in teaching; How many lecturers are actively involved in psychology of learning, or real testing of their students comprehension? A lot of books are perfectly happy to bullet point list the properties of some concept, like feeding data to a machine, then try a combination of explanations and diagrams to describe how these properties fit together, or an intuition of their emergent properties. It's funny, I've often gotten a lot more help from offhanded comments of characterisations of a concept, than from their formalities. It seems some lecturers are scared of making such observations, in case they mislead students, so they stick to the "safety" of irrefutable formality. set theory might be sufficiently abstract that it won't strictly mislead, but it won't enlighten either. { as an aside, it was years after my first undergraduate class that I finally realised that the integral notation represented a summation, and that it was literally the continuous form of the discrete "sigma" summation notation, I was surprised I hadn't realised it before! being able to relate the two gave me a stronger basis to reason about them in some ways - for example, I now thing of the variable of differentiation (dx, etc) like a 'counter' variable (the n=.. atop the sigma in summations), which helped when trying when reasoning about integrals etc with more than one (obv. if you try to derive a limit as some variable goes to inf., there is ambiguity if there are two unrelated variables), plus most continuous forms are based on discrete concepts taken to some limit. in fact, it even gives a little intuition about why an integral might be followed by "* dx", in such a way that the 'd' variable can be treated as an independent unit (taken to the other side of an equation, for example) - it is bound to the integral in some sense, but also represents a quantity in itself that can be shared amongst integrals } I also get the feeling that mathematics is too "pen and paper", and ripe for an overhaul wrt anything that can be easily done using a math program. I feel the human part of maths should be high level concepts, not the actual calculation (although some understanding of what a calculation does is important) - hence, it is more important I understand what an integral represents, than know how to calculate it. > an important number of math books are written not for laypersons but for mathematicians But some math is useful outside of mathematical study, so 'maths for engineers' books are sadly too uncommon. Consider a software engineer trying to understand cryptography better, reading about elliptical functions.. > Not all mathematical concepts need to have a direct application All but a few can at least be characterised somehow, even by mapping to similarly abstract social constructs.

marcosdumay 9y ago

> Motivation and intuition are important, yes, but so are definitions and formality. Yet, many (maybe even most) mathematics professors and books insist on only teaching the later two... Every time a similar discussion comes up in HN, there's a post like yours, pointing that the thing is too difficult to break into small problems like the one pointed. Yes, the thing is all very complex. Yet, the pointed problem is still a problem, and people are still not solving it.

gjulianm 9y ago
Honestly, most mathematics professors I've had have attempted at giving motivation or intuition about things. Not always useful, not always correct and sometimes dangerously misleading, but the attempts are there. Regarding the posts like mine, they come up because the majority is the other view on the matter: that there must be some way to hack this that somehow is going to be found in the HN comments, and that professors/book writers don't care. And the reality is different: most mathematicians care about intuition and motivation. It's their basic tool of work. But transmitting those ideas and transmitting them effectively is not easy. In fact, there's this funny phenomenon that I've seen in myself and it seems to apply to a lot of people: we forget the intuition we are given for a problem if it doesn't click right away. I also refuse completely the premise that "people are still not solving it". In this post there are people linking to resources on books/pages with new ways to teach mathematics. People are solving it and each time getting better, and however it's never going to be straightforward to teach mathematics (just like it's never going to be straightforward to teach anything, mathematics is only seen as special because it has a lot more presence in education that other subjects).

mathperson 9y ago

amen. the comments in this post are entirely frustrating.

Chris2048 9y ago

But how do you know this? I've come across articles on certain subjects that are far more illuminating than usual - these shouldn't exist if mathematics is just irredeemably difficult to learn. The only conclusion I can draw is mathematicians care more about advancing mathematics, than concepts in teaching; How many lecturers are actively involved in psychology of learning, or real testing of their students comprehension? A lot of books are perfectly happy to bullet point list the properties of some concept, like feeding data to a machine, then try a combination of explanations and diagrams to describe how these properties fit together, or an intuition of their emergent properties. It's funny, I've often gotten a lot more help from offhanded comments of characterisations of a concept, than from their formalities. It seems some lecturers are scared of making such observations, in case they mislead students, so they stick to the "safety" of irrefutable formality. set theory might be sufficiently abstract that it won't strictly mislead, but it won't enlighten either. { as an aside, it was years after my first undergraduate class that I finally realised that the integral notation represented a summation, and that it was literally the continuous form of the discrete "sigma" summation notation, I was surprised I hadn't realised it before! being able to relate the two gave me a stronger basis to reason about them in some ways - for example, I now thing of the variable of differentiation (dx, etc) like a 'counter' variable (the n=.. atop the sigma in summations), which helped when trying when reasoning about integrals etc with more than one (obv. if you try to derive a limit as some variable goes to inf., there is ambiguity if there are two unrelated variables), plus most continuous forms are based on discrete concepts taken to some limit. in fact, it even gives a little intuition about why an integral might be followed by "* dx", in such a way that the 'd' variable can be treated as an independent unit (taken to the other side of an equation, for example) - it is bound to the integral in some sense, but also represents a quantity in itself that can be shared amongst integrals } I also get the feeling that mathematics is too "pen and paper", and ripe for an overhaul wrt anything that can be easily done using a math program. I feel the human part of maths should be high level concepts, not the actual calculation (although some understanding of what a calculation does is important) - hence, it is more important I understand what an integral represents, than know how to calculate it. > an important number of math books are written not for laypersons but for mathematicians But some math is useful outside of mathematical study, so 'maths for engineers' books are sadly too uncommon. Consider a software engineer trying to understand cryptography better, reading about elliptical functions.. > Not all mathematical concepts need to have a direct application All but a few can at least be characterised somehow, even by mapping to similarly abstract social constructs.