My favorite math!
After two years of studying math independently through Khan Academy, I had my learning technique on lock: notetaking style, practice process, spaced repetition cards, conceptual toolkit... everything came together for Calculus. I struggled with the AP test—fast computation is not my strong suit—but ended the year with a really solid understanding of calc, and the skills to practice it.
I had a lot of fun with this one. Calculus is so much less about computation and so much more about elegant algebraic & geometric manipulation. It's figuring out how to approach the impossible by substituting it with the possible, and then proving that they are ultimately the same thing. That's really cool! Calculus was my favorite subject in the first semester, and it's only moved down the list because I started enjoying the others, once I developed some better methods for learning.
I had a lot of fun with this one. Calculus is so much less about computation and so much more about elegant algebraic & geometric manipulation. It's figuring out how to approach the impossible by substituting it with the possible, and then proving that they are ultimately the same thing. That's really cool! Calculus was my favorite subject in the first semester, and it's only moved down the list because I started enjoying the others, once I developed some better methods for learning.
Explainers & creative work
What e actually is
When I learned that the derivative of eˣ is eˣ, my first question was why.
That's actually not a real question. But I didn't realize that yet. When I figured it out (with help from math YouTube channel 3blue1brown), I wrote a little explainer to solidify it for myself. Explainer document |
Trying to calculate an integral without knowing how
Several years ago, I studied the concepts behind integrals and derivatives—what they mean, how they're used, what they represent—without any real idea of how to calculate them.
Once I had a pretty decent understanding of derivatives and limits, I though I might be able to anticipate the AP curriculum and figure out how to prove an integral myself, through trapezoid approximation, as trapezoid slices got thinner and thinner. It didn't work out exactly as I was hoping, but talking it through with Steven did help me see what kind of intuitions for infinity I was lacking. Homebrewed integrals Q1 |
Derivatives of matrices
On a walk through the school one day, it occurred to me that if you can take the derivative of a function as the rate at which it changes a number, then you could theoretically take the derivative of a matrix as the rate at which it changes a vector. There are a couple of ways to look at this, and it's still an area of open curiosity for me.
Matrix derivatives explainer Thinking notes |
Single neuron-neural network
Throughout my high school career, every time I tried to get into the math of neural networks, I'd run up against a knowledge barrier. First it was just programming experience, then linear algebra, then single-variable calculus. Now I still don't get everything… but I finally had enough to build the world's crappiest machine learning algorithm that still used the fundamental principles of neural networks—from scratch. And I plan to keep building it out this summer!
Planning notes Source code |
Studying
Notes
So what is my learning approach? Well, it usually consists of:
Q1: limits, angle addition identities, approximating derivatives, basic algebraic derivatives |
Q2: chain rule, trig derivatives, e & exponentials, derivatives of logarithms, implicit differentiation, L'Hôpital's rule, the Mean Value Theorem
Q3: extrema, inflection points, optimization, Riemann sums, basic definite and indefinite integrals
Q4: u-substitution, differential equations, integrals between curves, solids of rotation, deriving some volume equations
Q3: extrema, inflection points, optimization, Riemann sums, basic definite and indefinite integrals
Q4: u-substitution, differential equations, integrals between curves, solids of rotation, deriving some volume equations
Practice
I've been using spaced repetition to build formulae, concepts, etc. into my long-term memory for over a year now. But I couldn't figure out how to properly used spaced repetition for practice problems. If you just paste a practice problem on a flashcard and put the answer on the back, you're not going to memorize how to solve the problem; you're going to memorize the answer. You want to be encountering novel applications of the same skills... but how do you put that on a static flashcard?
For a while, I thought about making some sort of elaborate Anki add-on that would allow me to dynamically generate problems by including random number syntax and configurable templates... before I realized that I could just copy and paste the link to a Khan Academy practice test (which has random questions) on the front of the flashcard, click the link, and mark the card as "remembered" if I got the test right.
Anki's configured for recall flashcards that take a couple seconds, not 10 minutes, so there's been a lot of trial and error getting the settings right. At the moment, I have a decent equilibrium figured out, but I'm still working on how to stay fresh moving forward throughout my education, without accumulating a massive set of practice problems I have to do all the time.
For a while, I thought about making some sort of elaborate Anki add-on that would allow me to dynamically generate problems by including random number syntax and configurable templates... before I realized that I could just copy and paste the link to a Khan Academy practice test (which has random questions) on the front of the flashcard, click the link, and mark the card as "remembered" if I got the test right.
Anki's configured for recall flashcards that take a couple seconds, not 10 minutes, so there's been a lot of trial and error getting the settings right. At the moment, I have a decent equilibrium figured out, but I'm still working on how to stay fresh moving forward throughout my education, without accumulating a massive set of practice problems I have to do all the time.
End of the year: cards I've learned. Orange cards are for "still learning," light green I know pretty well, dark green are known (but all of them get practice). Each card represents one quiz or unit test on the Khan Academy AP Calculus course.
I didn't get as far with Khan Academy unit mastery as I normally would—because I moved so quickly in Q3, only did new tests as they came up in Anki, and set my Anki new card limit to 1/day max, there were some areas (like differential equations) I just didn't get to practicing much. I still feel confident in my understanding, but I'm a little weaker on fast computation than I'd like to be. More room to grow over the summer and in the years to come!
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