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> Figuring them is easy from experience but the math behind that is way harder; anything beyond simple equations is like alien language to me ... It took me a

> Figuring them is easy from experience but the math behind that is way harder; anything beyond simple equations is like alien language to me ... It took me a long (and often frustrating) time to figure out that the perception of difficulty is largely because of how implicit most everything is in mathematics. It's not like programming where eventually there is a compiler with a definite structure that's going to make your program behave in some exact specific way. Even worse, is the way we're typically taught mathematics, it's always kinda starting in the middle (as opposed to the beginning). We aren't taught about what it is our how it works, so it's really hard to situate particular mathematical concepts we're taught into a larger coherent framework. The way I interpret mathematical works changed so much (and for the better), once I realized that any significant bit of math you're using is the result of something a human was trying to accomplish. In the same way you can point to a piece of code and ask it's developer what the motivation was for some abstraction or algorithm, there are always reasons behind the development of mathematical systems. Once I understood that, rather than mathematical systems being these disconnected things that materialized out of nowhere, I started seeing commonalities and considering the reasoning behind them, and seeing more of the network that relates them. It became a lot more enjoyable. I'd recommend checking out some history of mathematics if you'd like to get a better sense of that (my personal intro recommendation: "Men of Mathematics" by E.T. Bell or maybe "Mathematics and the Imagination," which isn't history so much, but serves a similar goal). If you're looking for something on algorithms specifically the book recommended by the author ("The Algorithm Design Manual") is a pretty friendly read and largely non-mathematical, while still containing all the useful bits (unless you're looking for a reference work on computational complexities or something).

finaliteration 8y ago
> We aren't taught about what it is our (sic) how it works My biggest frustration learning math when I was younger was not knowing -why- I was doing something a certain way or how it worked. It was incredibly discouraging because it didn’t just “click” like it seemed to with my peers.
- westoncb 8y ago
  I was the same way. I think the way we typically teach math (in the U.S. at least) to young people is particularly hard for those who insist on understanding rather than memorizing things. I was the same as how you describe yourself, and I always just assumed the kids who were best in the class were doing what I was trying to do also, but were much better at it. And then you see later how fragile/limited one's understanding is when taking the memorization approach—but as a kid you don't have many options.
- joshvm 8y ago
  There is a certain danger in trying to learn maths with this mindset though. A lot of topics seem intuitively easy, but the proofs only come somewhere near the end of a degree level course. It's a very healthy mindset to have, but it's a blessing and a curse. Some people are just very good at just abstracting away the details and getting on with things; what seems like "clicking" is not always the same as intuition. Probably it's good to operate in both modes. It's a really really useful skill to know when you need to care about the internals and when you can safely ignore them.
  - finaliteration 8y ago
    I’ve definitely learned that now after 10 years of development experience. But as a curious kid who didn’t just want to memorize and regurgitate it was quite a challenge. I’ve debated restarting mathematics all the way from basic algebra and geometry to see if I’d do any better these days.
    - ficklepickle 8y ago
      I noticed my high school math(s) teachers were very bad at explaining why and what we were doing. Arbitrary example: One teacher kept saying "f(x)" but couldn't explain what a function is. He just said "it's anything", then "don't worry about it". If he had even said "a function is like a machine that takes number(s) as input, changes them with a formula and outputs the new number(s)", I think it would have helped us grok. I think he understood math(s) so well that he couldn't relate to someone who didn't know what a function was.
diffeomorphism 8y ago
> because of how implicit most everything is in mathematics. Which is a completely wrong impression due to how it is taught in school and to undergrads in the US. Speaking as a mathematician, since one is reasoning about abstract objects everything necessarily is pedantically explicit (otherwise proofs could not possibly work). Hence, in US graduate courses or EU undergrad courses you start from scratch and rigorously define every symbol you ever write and justify every step you take (e.g. given a field, why is "1" distinct from "0"? What is a derivative? Prove that it is actually well-defined, exists for such functions etc.). What many people confuse for being implicit is the heavy polymorphism and terseness in mathematical notation. For instance, one often identifies a function, its graph or its image depending on type makes sense in context. Here, rigorous courses spent much time to prove that any possible ambiguity in notation is actually no ambiguity at all; one is allowed to use short-hand notation because all interpretations are equivalent.

finaliteration 8y ago

> We aren't taught about what it is our (sic) how it works My biggest frustration learning math when I was younger was not knowing -why- I was doing something a certain way or how it worked. It was incredibly discouraging because it didn’t just “click” like it seemed to with my peers.

westoncb 8y ago
I was the same way. I think the way we typically teach math (in the U.S. at least) to young people is particularly hard for those who insist on understanding rather than memorizing things. I was the same as how you describe yourself, and I always just assumed the kids who were best in the class were doing what I was trying to do also, but were much better at it. And then you see later how fragile/limited one's understanding is when taking the memorization approach—but as a kid you don't have many options.
joshvm 8y ago
There is a certain danger in trying to learn maths with this mindset though. A lot of topics seem intuitively easy, but the proofs only come somewhere near the end of a degree level course. It's a very healthy mindset to have, but it's a blessing and a curse. Some people are just very good at just abstracting away the details and getting on with things; what seems like "clicking" is not always the same as intuition. Probably it's good to operate in both modes. It's a really really useful skill to know when you need to care about the internals and when you can safely ignore them.
- finaliteration 8y ago
  I’ve definitely learned that now after 10 years of development experience. But as a curious kid who didn’t just want to memorize and regurgitate it was quite a challenge. I’ve debated restarting mathematics all the way from basic algebra and geometry to see if I’d do any better these days.
  - ficklepickle 8y ago
    I noticed my high school math(s) teachers were very bad at explaining why and what we were doing. Arbitrary example: One teacher kept saying "f(x)" but couldn't explain what a function is. He just said "it's anything", then "don't worry about it". If he had even said "a function is like a machine that takes number(s) as input, changes them with a formula and outputs the new number(s)", I think it would have helped us grok. I think he understood math(s) so well that he couldn't relate to someone who didn't know what a function was.

diffeomorphism 8y ago

> because of how implicit most everything is in mathematics. Which is a completely wrong impression due to how it is taught in school and to undergrads in the US. Speaking as a mathematician, since one is reasoning about abstract objects everything necessarily is pedantically explicit (otherwise proofs could not possibly work). Hence, in US graduate courses or EU undergrad courses you start from scratch and rigorously define every symbol you ever write and justify every step you take (e.g. given a field, why is "1" distinct from "0"? What is a derivative? Prove that it is actually well-defined, exists for such functions etc.). What many people confuse for being implicit is the heavy polymorphism and terseness in mathematical notation. For instance, one often identifies a function, its graph or its image depending on type makes sense in context. Here, rigorous courses spent much time to prove that any possible ambiguity in notation is actually no ambiguity at all; one is allowed to use short-hand notation because all interpretations are equivalent.