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Did bad high-school mathematics education conditioned us all to hate everything that's related to mathematics? Label a concept "this is just engineering practi

Did bad high-school mathematics education conditioned us all to hate everything that's related to mathematics? Label a concept "this is just engineering practice" or "this is just physics. this equation describes how universe works" and people will happily accept it as is. But label it "mathematics" and explanations are in order because maths is hard to understand... I don't see any posts where authors concern themselves with metaphors for Maxwell's equations (or anything else in physics; maybe I'm hanging out in the wrong circles though). What is a process of understanding a mathematical equation? Stare at it until you grok it? Use it to solve problems to get a feel of what it means? Write proofs about it so you can state some properties with certainty? I don't know what the best/structured process is, but the core is "take this equation and do something with it". Do the same with monads! I think people having problems understanding monads are those that defer writing any code until they feel "they get it". Instead of "now I get it, monads are burritos, time to write some code" one should write some code in order to understand monads

ch 11y ago
I think it has much to do with the abstractness of mathematics (especially category theory). Some just have a hard time anchoring the ideas to something concrete, and they are in need of that anchoring to make the ideas stick. With engineering and physics equations there is always something physical to relate them to (at least somewhere in the universe(s)) -- not to say there aren't harder equations to grasp in those fields too, but again that is typically trending into the more abstract (less macro) areas of the field. The trouble with math is sometimes you just have to work through equations enough until you form some intuition about how they work. Some folks just want to skip that step and others Oblidge them by assisting their mental model with metaphors which seem to never be universal enough -- if it were it would just be math.
- GreaterFool 11y ago
  Just thought about something that I think is analogous to explaining monads. I remember reading few articles about "core math" in the States and it had example problems related to addition, subtraction, multiplication. The complaint was that those problems where incomprehensible gibberish and kids and adults alike had really hard time figuring them out. From what I recall the core of the issue was that the question authors had some specific mental model of how addition and multiplication works and the questions were structured in a way that required figuring out what that model was (otherwise it was very hard to understand what was being asked). Why does 2 + 3 = 5? Well, because it does. I don't even know what my mental model of addition is. I just do it. I stared at it long enough as a kid and I developed neural circuits that compute the result. Instead if I asked: I have two stars in a box. How many squares do I have to put in the box to have 5 shapes? | * * | + | ??? | = | * * # # # | Which is how I recall those weird maths questions were structured. Now you a have a (terrible IMHO) model of addition. That's what I think those monad tutorials do for you.
  - TuringTest 11y ago
    In order to first learning how to add, examples with apples and bananas are helpful. Learning happens by relating some new concepts to something you already know; that applies also to monads. Those monad metaphors provide the same benefit. Even if some intuitions are wrong and you still need to learn the match to get the whole picture, at least you get a sense of (partial) understanding and purpose during the whole learning process thanks to the metaphor, instead of feeling lost the whole time.
nazka 11y ago
> Did bad high-school mathematics education conditioned us all to hate everything that's related to mathematics? Label a concept "this is just engineering practice" or "this is just physics. this equation describes how universe works" and people will happily accept it as is. I won't say it better than in this post![1] It's really too bad that people in general are struggling with the different problems related to Mathematics, or because they don't have a good environment to learn, or other things like that. The thing is when someone starts to learn Mathematics, the beginning is really tough. You have a new language to learn, you have to do everything the "hard way", you have to learn the "handshakes", and because your mind is your only compiler/error check. But the longer you train yourself the better you get, until you have enough basic knowledge to read and learn anything. At the end it is so grateful to overcome that, and so useful because Mathematics is such a powerful tool. [1] http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-guide-to-mathematics-for-programmers/ http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-...
jerf 11y ago
"Did bad high-school mathematics education conditioned us all to hate everything that's related to mathematics?" Yes. https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... (A frequent HN link, but, well, for good reason.)
- platz 11y ago
  While I think Lockhart correctly points out there is something wrong with math education, I disagree with the solution he proposes to fix it. Read page 16+17 and tell me who you agree with more, SIMPLICIO or SALVIATI
  - Chinjut 11y ago
    I'm Team Salviati.
    - platz 11y ago
      I assume you also disapprove of zed shaw's "Learn X the Hard Way" books (which, btw, have many adherents and praises), which he explains partly here? https://news.ycombinator.com/item?id=7207283 https://news.ycombinator.com/item?id=7207283 The irony is that even in painting, there is a fair amount of "rote" or repetition needed before ones "creativity" can be explored. This may be somewhat tangential/distracting, but I find the whole notion of 'creativity' problematic - many artists also think so (“Amateurs look for inspiration; the rest of us just get up and go to work.” ― Chuck Close).
      - jerf 11y ago
        I tend to think that a certain amount of rote arithmetic is required, and, well, if the student doesn't like it, I'm not sure what to do about that. Reading and spelling are the same way... if you want to be a fluent reader, you need to put the time in. What I learned in elementary school is OK. (There my objections would center more around the cohort system being terrible, but that's another objection. Also, I have to qualify that because the education system considers screwing with elementary math education one of their primary mandates.) But math education flings itself off the rails when it starts doing symbolic manipulation. (And if anything, every time they screw with it they're flinging themselves off the rails harder and sooner.)
Lambdanaut 11y ago
I've written code with monads plenty of times. I've even made my own. I'm still not sure I know what a monad is in the mathematical sense of the word. I don't have any idea how it interacts with other mathematical constructs or fits into Category Theory. Using Monads in a programming context teaches you how to use monads in a programming context.
- AnimalMuppet 11y ago
  Which, if I'm trying to program, is good enough. I'm not trying to write a paper in category theory...
mindslight 11y ago
I loved school math classes, but a decade later I do have a hard time understanding monads and category theory. Compared to programming (even functional), "math" is a completely different way of thinking. Abstract properties are stated, from which some arbitrary (partial) structure is implied. Whereas with programming, the structure of "doing" is always concrete, regardless of paradigm (algorithmic complexity is everpresent, even when you're ignoring it). Compared to abstract math, school classes are more akin to programming in that you're mostly following algorithms and applying patterns with a slight intuition, even when you get into algebra and calculus. My biggest hurdle to understanding is the use of completely different terminology for concepts that are the same, at least intuition wise. My thought process is based on intuitions first, rather than manipulating symbols in the abstract (which seems to be more common? or maybe it just seems this way?). So when confronted with terms like 'conjunction' and 'disjunction', it just throws me off that they're 1. the "same" as AND/OR, 2. the nuance of difference is rarely stated. So the way I cope is by directly thinking AND/OR, while being painfully aware that there is some distinction I'm not aware of. This leads to a lot of reading that's completely disconnected from anything until I find the gem that illustrates the actual difference. I'd learned the workings of practical Haskell monads some time ago, but what made monads/category theory finally click is reading Moggi's "Notions of computation and monads". Going back to the "source" let me see the specific motivation and actually understand how Haskell objects differ from perfect ones from category theory. Apparently it's just non-termination, but I had to do a lot of searching to find where that was actually stated, rather than assumed and vaguely referenced. It feels like to someone who thinks "more mathematically" this is just a small detail, since they deal with each type of structure in isolation. But until I can relate them, I'm just out in the weeds. Of course now that I understand this it seems like quite a simple concept. But I had to do an awful lot of work to get to the point where my thought gamut was "expanded" to be able to include this one concept. So I don't really know what my exact point is. But to connect to something you said, it seems like experience from writing code and desire to understand abstract concepts ("monads are burritos") come from two different places, and the latter isn't necessarily served by increasingly outlandish metaphors, but by making the details accessibly explicit.
imh 11y ago
>I don't see any posts where authors concern themselves with metaphors for Maxwell's equations (or anything else in physics; maybe I'm hanging out in the wrong circles though). This is quantum mechanics through and through. While the equations are agreed on, interpretations vary and you get weird varied metaphors all over the place.
superuser2 11y ago
High-school mathematics education taught us to hand-execute algorithms to manipulate symbols into equivalent forms. The rules of arithmetic, then the rules of algebra, properties of trigonometry, more algebra, then the obscure edge-case tricks of integration and differentiation. That is 100% of "math" to any high school graduate. Nothing remotely related to reasoning about monads. There is no reason at all that a straight-A high school math student should be expected to be even slightly good at writing proofs.

ch 11y ago

I think it has much to do with the abstractness of mathematics (especially category theory). Some just have a hard time anchoring the ideas to something concrete, and they are in need of that anchoring to make the ideas stick. With engineering and physics equations there is always something physical to relate them to (at least somewhere in the universe(s)) -- not to say there aren't harder equations to grasp in those fields too, but again that is typically trending into the more abstract (less macro) areas of the field. The trouble with math is sometimes you just have to work through equations enough until you form some intuition about how they work. Some folks just want to skip that step and others Oblidge them by assisting their mental model with metaphors which seem to never be universal enough -- if it were it would just be math.

GreaterFool 11y ago
Just thought about something that I think is analogous to explaining monads. I remember reading few articles about "core math" in the States and it had example problems related to addition, subtraction, multiplication. The complaint was that those problems where incomprehensible gibberish and kids and adults alike had really hard time figuring them out. From what I recall the core of the issue was that the question authors had some specific mental model of how addition and multiplication works and the questions were structured in a way that required figuring out what that model was (otherwise it was very hard to understand what was being asked). Why does 2 + 3 = 5? Well, because it does. I don't even know what my mental model of addition is. I just do it. I stared at it long enough as a kid and I developed neural circuits that compute the result. Instead if I asked: I have two stars in a box. How many squares do I have to put in the box to have 5 shapes? | * * | + | ??? | = | * * # # # | Which is how I recall those weird maths questions were structured. Now you a have a (terrible IMHO) model of addition. That's what I think those monad tutorials do for you.
- TuringTest 11y ago
  In order to first learning how to add, examples with apples and bananas are helpful. Learning happens by relating some new concepts to something you already know; that applies also to monads. Those monad metaphors provide the same benefit. Even if some intuitions are wrong and you still need to learn the match to get the whole picture, at least you get a sense of (partial) understanding and purpose during the whole learning process thanks to the metaphor, instead of feeling lost the whole time.

nazka 11y ago

> Did bad high-school mathematics education conditioned us all to hate everything that's related to mathematics? Label a concept "this is just engineering practice" or "this is just physics. this equation describes how universe works" and people will happily accept it as is. I won't say it better than in this post![1] It's really too bad that people in general are struggling with the different problems related to Mathematics, or because they don't have a good environment to learn, or other things like that. The thing is when someone starts to learn Mathematics, the beginning is really tough. You have a new language to learn, you have to do everything the "hard way", you have to learn the "handshakes", and because your mind is your only compiler/error check. But the longer you train yourself the better you get, until you have enough basic knowledge to read and learn anything. At the end it is so grateful to overcome that, and so useful because Mathematics is such a powerful tool. [1] http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-guide-to-mathematics-for-programmers/ http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-...

jerf 11y ago

"Did bad high-school mathematics education conditioned us all to hate everything that's related to mathematics?" Yes. https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... (A frequent HN link, but, well, for good reason.)

platz 11y ago
While I think Lockhart correctly points out there is something wrong with math education, I disagree with the solution he proposes to fix it. Read page 16+17 and tell me who you agree with more, SIMPLICIO or SALVIATI
- Chinjut 11y ago
  I'm Team Salviati.
  - platz 11y ago
    I assume you also disapprove of zed shaw's "Learn X the Hard Way" books (which, btw, have many adherents and praises), which he explains partly here? https://news.ycombinator.com/item?id=7207283 https://news.ycombinator.com/item?id=7207283 The irony is that even in painting, there is a fair amount of "rote" or repetition needed before ones "creativity" can be explored. This may be somewhat tangential/distracting, but I find the whole notion of 'creativity' problematic - many artists also think so (“Amateurs look for inspiration; the rest of us just get up and go to work.” ― Chuck Close).
    - jerf 11y ago
      I tend to think that a certain amount of rote arithmetic is required, and, well, if the student doesn't like it, I'm not sure what to do about that. Reading and spelling are the same way... if you want to be a fluent reader, you need to put the time in. What I learned in elementary school is OK. (There my objections would center more around the cohort system being terrible, but that's another objection. Also, I have to qualify that because the education system considers screwing with elementary math education one of their primary mandates.) But math education flings itself off the rails when it starts doing symbolic manipulation. (And if anything, every time they screw with it they're flinging themselves off the rails harder and sooner.)

Lambdanaut 11y ago

I've written code with monads plenty of times. I've even made my own. I'm still not sure I know what a monad is in the mathematical sense of the word. I don't have any idea how it interacts with other mathematical constructs or fits into Category Theory. Using Monads in a programming context teaches you how to use monads in a programming context.

AnimalMuppet 11y ago
Which, if I'm trying to program, is good enough. I'm not trying to write a paper in category theory...

mindslight 11y ago

I loved school math classes, but a decade later I do have a hard time understanding monads and category theory. Compared to programming (even functional), "math" is a completely different way of thinking. Abstract properties are stated, from which some arbitrary (partial) structure is implied. Whereas with programming, the structure of "doing" is always concrete, regardless of paradigm (algorithmic complexity is everpresent, even when you're ignoring it). Compared to abstract math, school classes are more akin to programming in that you're mostly following algorithms and applying patterns with a slight intuition, even when you get into algebra and calculus. My biggest hurdle to understanding is the use of completely different terminology for concepts that are the same, at least intuition wise. My thought process is based on intuitions first, rather than manipulating symbols in the abstract (which seems to be more common? or maybe it just seems this way?). So when confronted with terms like 'conjunction' and 'disjunction', it just throws me off that they're 1. the "same" as AND/OR, 2. the nuance of difference is rarely stated. So the way I cope is by directly thinking AND/OR, while being painfully aware that there is some distinction I'm not aware of. This leads to a lot of reading that's completely disconnected from anything until I find the gem that illustrates the actual difference. I'd learned the workings of practical Haskell monads some time ago, but what made monads/category theory finally click is reading Moggi's "Notions of computation and monads". Going back to the "source" let me see the specific motivation and actually understand how Haskell objects differ from perfect ones from category theory. Apparently it's just non-termination, but I had to do a lot of searching to find where that was actually stated, rather than assumed and vaguely referenced. It feels like to someone who thinks "more mathematically" this is just a small detail, since they deal with each type of structure in isolation. But until I can relate them, I'm just out in the weeds. Of course now that I understand this it seems like quite a simple concept. But I had to do an awful lot of work to get to the point where my thought gamut was "expanded" to be able to include this one concept. So I don't really know what my exact point is. But to connect to something you said, it seems like experience from writing code and desire to understand abstract concepts ("monads are burritos") come from two different places, and the latter isn't necessarily served by increasingly outlandish metaphors, but by making the details accessibly explicit.

imh 11y ago

>I don't see any posts where authors concern themselves with metaphors for Maxwell's equations (or anything else in physics; maybe I'm hanging out in the wrong circles though). This is quantum mechanics through and through. While the equations are agreed on, interpretations vary and you get weird varied metaphors all over the place.

superuser2 11y ago

High-school mathematics education taught us to hand-execute algorithms to manipulate symbols into equivalent forms. The rules of arithmetic, then the rules of algebra, properties of trigonometry, more algebra, then the obscure edge-case tricks of integration and differentiation. That is 100% of "math" to any high school graduate. Nothing remotely related to reasoning about monads. There is no reason at all that a straight-A high school math student should be expected to be even slightly good at writing proofs.