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My two cents. We do a disservice to students in the US by treating Algebra/Calc as advanced math. Typically, American schools judge student math aptitude prior

by lumost 4y ago

My two cents. We do a disservice to students in the US by treating Algebra/Calc as advanced math. Typically, American schools judge student math aptitude prior to algebra based on rote memorization of multiplication tables. Potentially with the added benefit of more rote memorization problems if one is in the honors curriculum. Aptitude for this type of problem does not directly translate into aptitude for higher math. There is nothing particularly advanced about Algebra or Calculus. At most, it's a question of how much time one is willing and able to put into the subject. The fact that most American students do not exit high school with a functional knowledge of single variable calculus is purely an artifact of our curriculum priorities.

ocbyc 4y ago
YouTube is probably the best source of education delivered by passionate teachers at this point.
- daniel_rh 4y ago
  I've been looking for a really engaging solid intro to number theory course on youtube, including rings, Abelian groups, etc for early teens. Does anyone have a recommendation here?
polio 4y ago
I was pretty annoyed when I got to university and figured out that math could be about proofs rather than calculation. If I had been raised with that idea of math, I would have loved it so much sooner and would probably be pretty good at it by now.
- clnq 4y ago
  I was also annoyed for the opposite reason. I had assumed until then that math was more about application of what’s been proven rather than revisiting the proofs. I think the majority of students feel what I felt; that math got a lot less applied and a lot more esoteric in the academia.
  - noobermin 4y ago
    Some people out there do both. As a physicist, I mainly use applications of what's been proven.
- mlyle 4y ago
  IMO, good math ed is balanced: - Enrichment, visual, and speedy stuff to build intuition - Proofy stuff to build rigor and reasoning ability - Rote stuff to build arithmetic and symbolic manipulation skills Ideally, it would be balanced this way from kindergarten to mid-undergrad. In practice, most students rotate through these.
  - noobermin 4y ago
    Yeah, people who say the rote stuff isn't part of math don't get it, or they have a simplistic view of math that isn't as bad as the ones who don't know of the proofy side of math and think it is only rote and bland calculation. It's all of it and you need the rote stuff to build intuition too in addition to the things like visual "proofs" and graphs. You do not build intuition about things without having some amount of those things in your head which is memorization.
- danielvaughn 4y ago
  Same. I feel robbed of something that could have radically changed my life. I’ve always suspected that I could have had an aptitude for math, but was too young to be able to articulate it. I just knew that something was wrong with how it was being taught. It’s a bit late for me now at 38, or at least it feels that way. I never took college math so HS-level algebra and geometry is my current level of understanding and sometimes it really hampers my programming ability.
  - anonymouskimmer 4y ago
    If you find it too tedious to self-study on your own (there are books for this purpose), then you can consider taking a trig or calc course at a local community college. Just think of it as a temporary hobby.
    - daniel_rh 4y ago
      Or go for one of these gamified math tutorials. For instance: this one teaches geometric proofs without mentioning the formalism, just letting you "unlock" the tools from a few "axiom" starter tools. https://www.euclidea.xyz/ https://www.euclidea.xyz/
      - sanctified384 4y ago
        Wtf this is awesome! This should be better publicized. Thank you!!
      - dxbydt 4y ago
        This is very nicely done. I run a training group and I will use this for my students. Thank you for posting this resource.
  - pknomad 4y ago
    I feel for you. I stopped after discrete math and analysis-lite class but I wish I took more algorithm and higher level math classes to be better at my craft. It's also hard to take classes when you have competing priorities when you're 30+. I don't know your current situation but I feel like you can totally do it since you have renewed determination for learning.
- sacrosancty 4y ago
  That's more your parents' job since a career or hobby as a mathematician is extremely niche. School math is really a foundation to enable you do to all sorts of other things, as well as being directly useful. You can go a long way in engineering with only high-school calculus and engineer is a hugely popular profession and trade. Proofs are almost completely useless there. An engineer can usually "prove" something satisfactorily by testing it in a couple of cases and guessing that it generalizes to all real-world-scale numbers. That takes a kind of intuition about how typical equations typically behave which you get from doing all those tedious exercises at school.
- dboreham 4y ago
  Unfortunately there are not sufficient good mathematics teachers around to impart this level of understanding to students in school. Not that the teachers we have are trying to be bad, just realistically if you have a mathematics competency and good critical thinking skills, you are able to earn 2-3X as much not being a teacher, so most people do that.
- fsckboy 4y ago
  you do get proofs in HS geometry
mullingitover 4y ago
> We do a disservice to students in the US by treating Algebra/Calc as advanced math. My eastern European Calc II professor once complained when the class wasn't immediately grasping Riemann Sums, "This is grade school calculus!"
- akiselev 4y ago
  Fond memories of my parents trying to teach me the pythagorean theorem and algebraic manipulation in second or third grade. Ended in shouting and tears. Every. Single. Time.
  - tanseydavid 4y ago
    You made your parents cry?!? </sarc>
- owlbite 4y ago
  We certainly covered it about age 15/16 in the UK.
icelancer 4y ago
I agree with that. Calculus is made out to be some boogeyman when in reality is not that unapproachable. Plane Geometry track, on the other hand, was incredibly difficult for me to grasp in university.
dheera 4y ago
Yes, this. I knew how to do derivatives in 8th grade, came to the US in 10th grade and took AP Calculus because I was more than ready for it. I took differential equations at a community college in 11th grade, multivariable calculus in 12th. Some school admins went nuts about letting a student take calculus in 10th grade because they didn't think it was doable. But I tested out of everything else. I was the first person in the history of the school to do this, only because I came from internationally having completed a trig course in 9th grade. In the US they wouldn't have allowed that in the first place, but here I was, having already done it, because I was given the opportunity outside the US. I never saw myself having any special learning abilities, I was just moving onto the next thing to learn. I'm not any sort of prodigy, I just hated team sports and liked science and math, and spent lots of my extracurricular time learning about it.* I only moved faster in math because I put so many hours into studying it outside of school, not because I had any innate talent for it. It also made it more fun, because by learning ahead of class, I was learning for myself with much less pressure than learning for some stupid exam. This "no child left behind" stuff is pure nonsense in the way they are implementing it. *You see, I was previously growing up in Asia, where STEM is what the "cool kids" do.
jhanschoo 4y ago
I disagree that Calculus is not hard. Integration and differentiation on certain elementary functions, sure. Once you go beyond you can't mechanistically manipulate symbols anymore. The foundation of limits, too, properly treated requires quite a bit of foundational and proof machinery as well. So now you need to talk about how first-order logic is related to mathematics.
- noobermin 4y ago
  You can build intuition without giving a full on analysis class. Also, algebra can be tedious (just look at any spinor calculation in quantum field theory). That's in fact one of the problems, focus on tedious integrals rather than the substance which you can glean as a teenager.
nradov 4y ago
A college math professor who taught calculus told me that on the first day of class he gave students a quiz on basic arithmetic. The questions were easy enough that most students could score 100%. But he found that the speed with which students completed the quiz was a reliable predictor of their final course grade. Rote memorization of basic concepts does have some value in reducing the mental load and making it easier to focus on more complex concepts. As a practical matter though, I think that most people working in STEM fields — including engineers — would benefit more from learning statistics instead of calculus.
- Swizec 4y ago
  > But he found that the speed with which students completed the quiz was a reliable predictor of their final course grade Funnily all my college math professors and their TAs were notoriously bad at arithmetic. They regularly joked that arithmetic isn’t real math so not a big deal. My physics professor was an arithmetic magician though. The way he could take a ridiculously complex equation covering the whole blackboard and guesstimate the solution in his head was a marvel to behold.
  - toast0 4y ago
    You can have either calculus or arithmetic in your head. But, it sounds like it's good to have arithmetic in your head before you replace it with calculus.
- lmm 4y ago
  > But he found that the speed with which students completed the quiz was a reliable predictor of their final course grade. Rote memorization of basic concepts does have some value in reducing the mental load and making it easier to focus on more complex concepts. Maybe. Or maybe it's useless but skill at it is a good measure of how conscientious those students were in high school (e.g. the extent to which they bothered to do their homework). Or any number of other possible explanations.
- WalterBright 4y ago
  I never memorized anything in math. After working enough problem sets, you just know it. Frankly, rote memorization of math is a worthless endeavor.
pknomad 4y ago
I agree with you on this point, especially the part about rote memorization. I feel like when you get exposed to calculus for the first time, you're almost expected to accept the definitions for what they are (central limit theorem, etc) without really understanding where it's coming from. 3Blue1Brown's videos, fwiw, seem to do better at explaining the core concepts. On a tangent, I also feel like college math curriculum kinda don't make sense sometimes because you're hit with more computational heavy classes like calc and stats courses and then they break everything down and rebuild you with proof-based and analysis course at the beginning of your junior year.

by lumost 4y ago

ocbyc 4y ago

YouTube is probably the best source of education delivered by passionate teachers at this point.

daniel_rh 4y ago
I've been looking for a really engaging solid intro to number theory course on youtube, including rings, Abelian groups, etc for early teens. Does anyone have a recommendation here?

polio 4y ago

I was pretty annoyed when I got to university and figured out that math could be about proofs rather than calculation. If I had been raised with that idea of math, I would have loved it so much sooner and would probably be pretty good at it by now.

clnq 4y ago
I was also annoyed for the opposite reason. I had assumed until then that math was more about application of what’s been proven rather than revisiting the proofs. I think the majority of students feel what I felt; that math got a lot less applied and a lot more esoteric in the academia.
- noobermin 4y ago
  Some people out there do both. As a physicist, I mainly use applications of what's been proven.
mlyle 4y ago
IMO, good math ed is balanced: - Enrichment, visual, and speedy stuff to build intuition - Proofy stuff to build rigor and reasoning ability - Rote stuff to build arithmetic and symbolic manipulation skills Ideally, it would be balanced this way from kindergarten to mid-undergrad. In practice, most students rotate through these.
- noobermin 4y ago
  Yeah, people who say the rote stuff isn't part of math don't get it, or they have a simplistic view of math that isn't as bad as the ones who don't know of the proofy side of math and think it is only rote and bland calculation. It's all of it and you need the rote stuff to build intuition too in addition to the things like visual "proofs" and graphs. You do not build intuition about things without having some amount of those things in your head which is memorization.
danielvaughn 4y ago
Same. I feel robbed of something that could have radically changed my life. I’ve always suspected that I could have had an aptitude for math, but was too young to be able to articulate it. I just knew that something was wrong with how it was being taught. It’s a bit late for me now at 38, or at least it feels that way. I never took college math so HS-level algebra and geometry is my current level of understanding and sometimes it really hampers my programming ability.
- anonymouskimmer 4y ago
  If you find it too tedious to self-study on your own (there are books for this purpose), then you can consider taking a trig or calc course at a local community college. Just think of it as a temporary hobby.
  - daniel_rh 4y ago
    Or go for one of these gamified math tutorials. For instance: this one teaches geometric proofs without mentioning the formalism, just letting you "unlock" the tools from a few "axiom" starter tools. https://www.euclidea.xyz/ https://www.euclidea.xyz/
    - sanctified384 4y ago
      Wtf this is awesome! This should be better publicized. Thank you!!
    - dxbydt 4y ago
      This is very nicely done. I run a training group and I will use this for my students. Thank you for posting this resource.
- pknomad 4y ago
  I feel for you. I stopped after discrete math and analysis-lite class but I wish I took more algorithm and higher level math classes to be better at my craft. It's also hard to take classes when you have competing priorities when you're 30+. I don't know your current situation but I feel like you can totally do it since you have renewed determination for learning.
sacrosancty 4y ago
That's more your parents' job since a career or hobby as a mathematician is extremely niche. School math is really a foundation to enable you do to all sorts of other things, as well as being directly useful. You can go a long way in engineering with only high-school calculus and engineer is a hugely popular profession and trade. Proofs are almost completely useless there. An engineer can usually "prove" something satisfactorily by testing it in a couple of cases and guessing that it generalizes to all real-world-scale numbers. That takes a kind of intuition about how typical equations typically behave which you get from doing all those tedious exercises at school.
dboreham 4y ago
Unfortunately there are not sufficient good mathematics teachers around to impart this level of understanding to students in school. Not that the teachers we have are trying to be bad, just realistically if you have a mathematics competency and good critical thinking skills, you are able to earn 2-3X as much not being a teacher, so most people do that.
fsckboy 4y ago
you do get proofs in HS geometry

mullingitover 4y ago

> We do a disservice to students in the US by treating Algebra/Calc as advanced math. My eastern European Calc II professor once complained when the class wasn't immediately grasping Riemann Sums, "This is grade school calculus!"

akiselev 4y ago
Fond memories of my parents trying to teach me the pythagorean theorem and algebraic manipulation in second or third grade. Ended in shouting and tears. Every. Single. Time.
- tanseydavid 4y ago
  You made your parents cry?!? </sarc>
owlbite 4y ago
We certainly covered it about age 15/16 in the UK.

icelancer 4y ago

I agree with that. Calculus is made out to be some boogeyman when in reality is not that unapproachable. Plane Geometry track, on the other hand, was incredibly difficult for me to grasp in university.

dheera 4y ago

Yes, this. I knew how to do derivatives in 8th grade, came to the US in 10th grade and took AP Calculus because I was more than ready for it. I took differential equations at a community college in 11th grade, multivariable calculus in 12th. Some school admins went nuts about letting a student take calculus in 10th grade because they didn't think it was doable. But I tested out of everything else. I was the first person in the history of the school to do this, only because I came from internationally having completed a trig course in 9th grade. In the US they wouldn't have allowed that in the first place, but here I was, having already done it, because I was given the opportunity outside the US. I never saw myself having any special learning abilities, I was just moving onto the next thing to learn. I'm not any sort of prodigy, I just hated team sports and liked science and math, and spent lots of my extracurricular time learning about it.* I only moved faster in math because I put so many hours into studying it outside of school, not because I had any innate talent for it. It also made it more fun, because by learning ahead of class, I was learning for myself with much less pressure than learning for some stupid exam. This "no child left behind" stuff is pure nonsense in the way they are implementing it. *You see, I was previously growing up in Asia, where STEM is what the "cool kids" do.

jhanschoo 4y ago

I disagree that Calculus is not hard. Integration and differentiation on certain elementary functions, sure. Once you go beyond you can't mechanistically manipulate symbols anymore. The foundation of limits, too, properly treated requires quite a bit of foundational and proof machinery as well. So now you need to talk about how first-order logic is related to mathematics.

noobermin 4y ago
You can build intuition without giving a full on analysis class. Also, algebra can be tedious (just look at any spinor calculation in quantum field theory). That's in fact one of the problems, focus on tedious integrals rather than the substance which you can glean as a teenager.

nradov 4y ago

A college math professor who taught calculus told me that on the first day of class he gave students a quiz on basic arithmetic. The questions were easy enough that most students could score 100%. But he found that the speed with which students completed the quiz was a reliable predictor of their final course grade. Rote memorization of basic concepts does have some value in reducing the mental load and making it easier to focus on more complex concepts. As a practical matter though, I think that most people working in STEM fields — including engineers — would benefit more from learning statistics instead of calculus.

Swizec 4y ago
> But he found that the speed with which students completed the quiz was a reliable predictor of their final course grade Funnily all my college math professors and their TAs were notoriously bad at arithmetic. They regularly joked that arithmetic isn’t real math so not a big deal. My physics professor was an arithmetic magician though. The way he could take a ridiculously complex equation covering the whole blackboard and guesstimate the solution in his head was a marvel to behold.
- toast0 4y ago
  You can have either calculus or arithmetic in your head. But, it sounds like it's good to have arithmetic in your head before you replace it with calculus.
lmm 4y ago
> But he found that the speed with which students completed the quiz was a reliable predictor of their final course grade. Rote memorization of basic concepts does have some value in reducing the mental load and making it easier to focus on more complex concepts. Maybe. Or maybe it's useless but skill at it is a good measure of how conscientious those students were in high school (e.g. the extent to which they bothered to do their homework). Or any number of other possible explanations.
WalterBright 4y ago
I never memorized anything in math. After working enough problem sets, you just know it. Frankly, rote memorization of math is a worthless endeavor.

pknomad 4y ago

I agree with you on this point, especially the part about rote memorization. I feel like when you get exposed to calculus for the first time, you're almost expected to accept the definitions for what they are (central limit theorem, etc) without really understanding where it's coming from. 3Blue1Brown's videos, fwiw, seem to do better at explaining the core concepts. On a tangent, I also feel like college math curriculum kinda don't make sense sometimes because you're hit with more computational heavy classes like calc and stats courses and then they break everything down and rebuild you with proof-based and analysis course at the beginning of your junior year.