1. Make up problems, and solve them. (Calculus: what is the volume of a sphere? Not using ‘the’ formula—integrate it from top to bottom. (Getting to the general formula eventually requires plugging in variables or functions rather than numbers.) There’s also a^b=b^a.)
2. Find different ‘languages’ and convert between them. (It may seem obvious how to convert between addition and multiplication: 5+5+5 = 3x5 = 3+3+3+3+3. But what is 1+2+3+...+100?)
3. Generalizing (The easiest explanations are ‘make functions/algorithms’. 3+5=5+3=8 replaced by f(3, 5) = 8, perhaps because f(3, 5) = f(2, 6) = f(1, 7) = f(0, 8) = 8. Less trivially: Everything is a function. Derivative operator? Turns functions into (usually other) functions. (Looking into what functions it doesn’t change/are their own derivatives might be useful for reasons similar to why knowing about 1 is good for understanding/working with multiplication.))
4. What makes sense to you? Some people find it easier to understand derivatives by analogy to the (related) discrete case: If one of your character’s stats in a video game automatically go up by some amount when your character levels up, then the amount a stat goes up between level 11 and 12 is like the derivative of that stat with regards to level.* Aside from the particular mathematical connections between these two things, the advantage is that something has been put in other terms: discrete rather than continuous.
5. Just practicing might be more important than note taking. (If I want to get good at trig functions, I don’t have amazing notes on trig functions—I have a set of starting equations (“identities”) which I can take and combine and fiddle with, and then I will be better with handling trig functions.) I guess I’m saying “you don’t have to work with “first” principles, but working from a starting point gets you somewhere.”
Note that 4 ties into 2.
*There’s a post here on LW about that, though it didn’t dive into why that was a great connection.
It might depend on the area of math.
Things I have found useful:
1. Make up problems, and solve them. (Calculus: what is the volume of a sphere? Not using ‘the’ formula—integrate it from top to bottom. (Getting to the general formula eventually requires plugging in variables or functions rather than numbers.) There’s also a^b=b^a.)
2. Find different ‘languages’ and convert between them. (It may seem obvious how to convert between addition and multiplication: 5+5+5 = 3x5 = 3+3+3+3+3. But what is 1+2+3+...+100?)
3. Generalizing (The easiest explanations are ‘make functions/algorithms’. 3+5=5+3=8 replaced by f(3, 5) = 8, perhaps because f(3, 5) = f(2, 6) = f(1, 7) = f(0, 8) = 8. Less trivially: Everything is a function. Derivative operator? Turns functions into (usually other) functions. (Looking into what functions it doesn’t change/are their own derivatives might be useful for reasons similar to why knowing about 1 is good for understanding/working with multiplication.))
4. What makes sense to you? Some people find it easier to understand derivatives by analogy to the (related) discrete case: If one of your character’s stats in a video game automatically go up by some amount when your character levels up, then the amount a stat goes up between level 11 and 12 is like the derivative of that stat with regards to level.* Aside from the particular mathematical connections between these two things, the advantage is that something has been put in other terms: discrete rather than continuous.
5. Just practicing might be more important than note taking. (If I want to get good at trig functions, I don’t have amazing notes on trig functions—I have a set of starting equations (“identities”) which I can take and combine and fiddle with, and then I will be better with handling trig functions.) I guess I’m saying “you don’t have to work with “first” principles, but working from a starting point gets you somewhere.”
Note that 4 ties into 2.
*There’s a post here on LW about that, though it didn’t dive into why that was a great connection.