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Depends what your goal is. I cant imagine anything worse than High School style teaching of mathematics. Its awful. That repetitive calculation, entirely unnece

Depends what your goal is. I cant imagine anything worse than High School style teaching of mathematics. Its awful. That repetitive calculation, entirely unnecessary for almost any real world situation. Almost every time you are doing repetitive calculation what you should have been taught was a second or third conceptualisation of whatever maths you are looking at. So linear algebra I was only ever taught numerically in computer science. WHICH IS CRIMINAL. It has such an obvious and more easily understood geometric interpretation. The same is true of calculus, learn it in physics. Seriously. Do high school physics /and/ maths if you wish to understand calculus. Otherwise you've got someone who probably did a pure maths degree teaching you something that was created by physicists for a reason, that reason has a physical and real world interpretation not to mention a geometric one. Another good example is in chemistry you might learn statistical mechanics, in computer science or electronic engineering you might learn information theory. If those two classes were taught back to back for both groups of people they would actually deeply understand the mathematics behind it. Personally I've always found learning maths in a pure way incredibly challenging. As an adult I think its a lot easier to learn maths through something that has physical or meaningful interpretation. You can then draw on your actual life experience to understand the maths.

elbear 9y ago
Thank you for saying this. I've been learning math from Khan Academy for the last 200 days. I've reached calculus and I feel more and more the need for examples of how to apply the math that I'm learning to truly understand what it's useful for. I was actually thinking of learning how to build video games (or at least parts of it) to properly understand linear algebra. Someone else also suggested physics for calculus.
- Jhsto 9y ago
  > I've been learning math from Khan Academy for the last 200 days. I've reached calculus and I feel more and more the need for examples of how to apply the math that I'm learning to truly understand what it's useful for. Just so that you know this was my impression at a university as well. In fact, calculus courses are translated into "basic mathematics." I think this underlines how much these courses are about learning the tooling for practical applications taught in later classes.
  - Chris2048 9y ago
    Hmm. Which kinds of math? Might be useful to begin a compendium of examples, a problem-to-solution (and vice-versa) phrasebook
- ethbro 9y ago
  Fwiw, I found electromag (typically undergrad physics 2) the best application of calculus. Classical physics of motion (physics 1) used some superficial calculus, but none of the use cases that would let you spread your mathematical wings cropped up in the material we studied.
InitialLastName 9y ago
I disagree about the repetitive calculation being "entirely unnecessary". Background: I (probably like many here) excelled at high school and undergraduate mathematics. I have a graduate degree in a heavily mathematics-focused branch of engineering (digital signal processing) and much of my work involves applying that math at a conceptual level. I'm currently tutoring an adult I'm close to who is approximately at GP's starting level and has a strong anxiety reaction to math. I'm finding that the repetitive calculation aspect is important to being able to developing an intuition for the concepts. The process that's working well for us so far involves alternating between practical word problems (to establish a motivation for learning the material), theoretical/background explanations (to hit the understanding at a high level) and repeated simple problems (to practice the mechanics). I'm finding that, in terms of process, the repeated problem-solving is critical for two reasons. 1) It helps to build a sort of mental "muscle memory", and 2) it helps with developing intuition, since your mind eventually gets bored with the mechanics and starts to notice patterns. Remember that mathematics at that level is all about building blocks. Every concept/problem-solving practice you learn is part of a tower of strategies; if earlier mechanisms aren't almost mindless, it's MUCH harder to build on top of that knowledge.
- zenhack 9y ago
  One of the earliest memories I have of really enjoying math was a unit on modular arithmatic from my 6th grade math class. The teacher introduced the concept with a hands of a clock metaphor, gave us a whole bunch of drill problems, and told us to watch for patterns. We had an easel in the class room where he collected the insights students had gained. Discovering the relationship with remainders by ourselves was incredibly cool.
  - InitialLastName 9y ago
    I had a similar experience when I learned about factoring (not sure what age). I got a little obsessed with finding prime numbers by hand, and eventually figured out that I only had to search for factors up to the square root. I distinctly remember that as the first time I realized how powerful a simple mathematical insight can be to speeding up a process. Now, of course, I use mathematical insight to speed up processes occupationally; I guess that experience stuck with me.
zenhack 9y ago
Having another interest to tie it to (especially if it's something you're more comfortable with) is huge. I have a CS degree, and for me, what bridged the gap into being comfy with "pure" mathematics (and appreciating it outside of any specific application), was my theory of computation class.
wrs 9y ago
My high school calculus and physics teachers synchronized their classes, so shortly after we learned something in calculus, we would use it in physics. Since then I’ve learned this is not as common as it should be! Perhaps it works especially well in that case because calculus was literally invented to do the kind of physics you learn in high school, so it’s a very natural combination. Linear algebra (e.g.) is so general-purpose that the overlap with another course might not be so continuous.
- ams6110 9y ago
  My high school only taught physics without calculus. In 12th grade when I took calculus separately I was sort of annoyed to realize how much easier physics would have been using calculus.
  - ethbro 9y ago
    This was my experience too. I remember doing integrals in high school, but didn't take calculus-based physics until my second year of college. Jesus christ, it explained a lot of things.
timClicks 9y ago
It's always a good idea to anchor new knowledge to pre-existing knowledge. (Perhaps this could be called coral reef learning?) Adult brains, with less plasticity, are able to synthesise new concepts that build on what they already know.

elbear 9y ago

Thank you for saying this. I've been learning math from Khan Academy for the last 200 days. I've reached calculus and I feel more and more the need for examples of how to apply the math that I'm learning to truly understand what it's useful for. I was actually thinking of learning how to build video games (or at least parts of it) to properly understand linear algebra. Someone else also suggested physics for calculus.

Jhsto 9y ago
> I've been learning math from Khan Academy for the last 200 days. I've reached calculus and I feel more and more the need for examples of how to apply the math that I'm learning to truly understand what it's useful for. Just so that you know this was my impression at a university as well. In fact, calculus courses are translated into "basic mathematics." I think this underlines how much these courses are about learning the tooling for practical applications taught in later classes.
- Chris2048 9y ago
  Hmm. Which kinds of math? Might be useful to begin a compendium of examples, a problem-to-solution (and vice-versa) phrasebook
ethbro 9y ago
Fwiw, I found electromag (typically undergrad physics 2) the best application of calculus. Classical physics of motion (physics 1) used some superficial calculus, but none of the use cases that would let you spread your mathematical wings cropped up in the material we studied.

InitialLastName 9y ago

I disagree about the repetitive calculation being "entirely unnecessary". Background: I (probably like many here) excelled at high school and undergraduate mathematics. I have a graduate degree in a heavily mathematics-focused branch of engineering (digital signal processing) and much of my work involves applying that math at a conceptual level. I'm currently tutoring an adult I'm close to who is approximately at GP's starting level and has a strong anxiety reaction to math. I'm finding that the repetitive calculation aspect is important to being able to developing an intuition for the concepts. The process that's working well for us so far involves alternating between practical word problems (to establish a motivation for learning the material), theoretical/background explanations (to hit the understanding at a high level) and repeated simple problems (to practice the mechanics). I'm finding that, in terms of process, the repeated problem-solving is critical for two reasons. 1) It helps to build a sort of mental "muscle memory", and 2) it helps with developing intuition, since your mind eventually gets bored with the mechanics and starts to notice patterns. Remember that mathematics at that level is all about building blocks. Every concept/problem-solving practice you learn is part of a tower of strategies; if earlier mechanisms aren't almost mindless, it's MUCH harder to build on top of that knowledge.

zenhack 9y ago
One of the earliest memories I have of really enjoying math was a unit on modular arithmatic from my 6th grade math class. The teacher introduced the concept with a hands of a clock metaphor, gave us a whole bunch of drill problems, and told us to watch for patterns. We had an easel in the class room where he collected the insights students had gained. Discovering the relationship with remainders by ourselves was incredibly cool.
- InitialLastName 9y ago
  I had a similar experience when I learned about factoring (not sure what age). I got a little obsessed with finding prime numbers by hand, and eventually figured out that I only had to search for factors up to the square root. I distinctly remember that as the first time I realized how powerful a simple mathematical insight can be to speeding up a process. Now, of course, I use mathematical insight to speed up processes occupationally; I guess that experience stuck with me.

zenhack 9y ago

Having another interest to tie it to (especially if it's something you're more comfortable with) is huge. I have a CS degree, and for me, what bridged the gap into being comfy with "pure" mathematics (and appreciating it outside of any specific application), was my theory of computation class.

wrs 9y ago

My high school calculus and physics teachers synchronized their classes, so shortly after we learned something in calculus, we would use it in physics. Since then I’ve learned this is not as common as it should be! Perhaps it works especially well in that case because calculus was literally invented to do the kind of physics you learn in high school, so it’s a very natural combination. Linear algebra (e.g.) is so general-purpose that the overlap with another course might not be so continuous.

ams6110 9y ago
My high school only taught physics without calculus. In 12th grade when I took calculus separately I was sort of annoyed to realize how much easier physics would have been using calculus.
- ethbro 9y ago
  This was my experience too. I remember doing integrals in high school, but didn't take calculus-based physics until my second year of college. Jesus christ, it explained a lot of things.

timClicks 9y ago

It's always a good idea to anchor new knowledge to pre-existing knowledge. (Perhaps this could be called coral reef learning?) Adult brains, with less plasticity, are able to synthesise new concepts that build on what they already know.