- 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.
You do not build intuition about things without having some amount of those things in your head which is memorization.
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.
For instance: this one teaches geometric proofs without mentioning the formalism, just letting you "unlock" the tools from a few "axiom" starter tools.
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.
It was only when I got into programming and it taught me math is actually much more than just meaningless letters and numbers that I finally came around to appreciating it. Programming taught me math is the language spoken by the universe, anything in this universe could be explained with math and it was truly a mindblowing paradigm shift.
Math education needs a complete redo from the ground up, it teaches nothing of value and only serves to mass-produce math-haters to the detriment of us all.
My eastern European Calc II professor once complained when the class wasn't immediately grasping Riemann Sums, "This is grade school calculus!"
Ended in shouting and tears. Every. Single. Time.
Does anyone have a recommendation here?
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.
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.
Memorizing is a directed effort to remember something. Remembering something as a side effect of some other task is not memorization.
Plane Geometry track, on the other hand, was incredibly difficult for me to grasp in university.
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.
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.
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.