Title is a bit confusing, but I'm trying to figure out what's something I should know by heart/know in my head vs. something I don't really need to know but should be able to understand if I refreshed on it.

Something like the circle equation or trig properties - I feel like that's just memorization and not something that people memorize, right? But on the other hand, not knowing how to integrate or differentiate would be a pretty big setback?

I was looking at how to factor polynomials and realized I had forgotten how to do so, but a quick refresher course on Khan and YouTube allowed me to regain the ability.

Or the powers problem here - https://www.reddit.com/r/learnmath/comments/9vap0t/help_with_11_yearolds_maths_homework_needed_yeah/

I also don't know how to do powers, but upon reading the comments it came back to me that it was just adding/subtracting if they have the same base.

I'd feel silly for not knowing how to differentiate/integrate - should I feel silly for not knowing the circle equation/trig/factoring polynomials/powers off the top of my head?

The problem is a lot of this stuff I have never used in real life and was always a purely academic pursuit. After graduating 3 years ago, I've forgotten a lot of specifics.

## hh26

If you learn the methods behind things well enough, not only can you rederive the formulas yourself when needed, but for me at least it helps prevent me from forgetting them in the first place.

The circle equation around the origin, x

^{2}+ y^{2}= r^{2}is just the pythagorean theorem with different labels. If you understand why that's true, really internalized it, then it's impossible to forget it unless you forget the pythagorean theorem. The general circle equation, (x-h)^{2}+ (y-k)^{2}= r^{2}, is just the same graph with a horizontal shift h and vertical shift k. If you know the basic circle equation and you know how horizontal and vertical shifts work, and have internalized those methods, then this becomes obvious and you don't need to memorize it separately, you can just start from x^{2}+ y^{2}= r^{2}and modify it appropriately.Same sort of stuff with factoring polynomials. It just follows from the distributive rule. If this expanded polynomial were secretely (x+a)(x+b), what would a and b have to be to make that true? You can rederive the basic rules by asking the right questions and then answering them.

Maybe I'm biased because I'm in grad school for math and keep doing this stuff regularly so I haven't had a chance to forget. I do occasionally forget lots of specific details about subjects I haven't touched in several years, but I retain enough information about the subject to A) recognize when there is a formula to solve a certain type of situation, so even if I don't remember what the formula is I can google it to find out, and B) remember enough information that when I try to relearn these subjects I can do it five times faster than when I learned it originally

And that's probably good enough.

## we_are_privacy

So it's better to seek to understand the underlying reason and explanation behind why something is done rather than memorizing each component separately.

I have a hard time seeing the interconnectedness of the different aspects of math though, and it's kinda always been this way. Every time I've been taught math, it seems it's always about how to do things, rather than why things are done the way they care. Would you happen to know of any resource that focuses on the reason/logic behind math rather than purely how to do things? Thanks.

## hh26

I don't know any specific places. I think to some degree every book and teacher has some level of justification for why they do what they do, it's just a matter of how much time they spend and how much they emphasize the reasons versus the practical applications.

I think one of the main distinctions between between people who find math easy and people who find math hard is the ability/desire to figure out the reasons for formulas and methods by thinking about it as soon as the formula or method is introduced.

The teacher explains what sine is, I mentally record "okay, sine is the height of an angle". The teacher explains what cosine is, I mentally record "okay, cosine is the length of an angle". The teacher says sin

^{2}+ cos^{2}= 1, I'm like "why would that be true? okay, sine tends to get larger whenever cosine gets smaller, so it sounds reasonable. But they get larger/smaller at different speeds, so it wouldn't need to be constant... oh if I draw a triangle then these are just the sides of the triangle and this is actually just the pythagorean theorem." And then I blink and realize I haven't been paying attention to the teacher for the past 3 minutes and he's about halfway through the explanation for why this formula is true because of the pythagorean theorem.So, um, try to do that, I guess? I think most times the teacher will explain the reasons, so maybe just payig more attention to those explanations will help, but if you can, try to make as many connections and explanations and analogies on your own as you can. If someone tells you a formula is true, then clearly something is forcing that to be true, something would break if it wasn't true, find out what that something is. Usually it's something fairly simple based on the definitions of things, like "sine is y, cosine is x, so the pythagorean theorem forces x

^{2}+ y^{2}= 1"## bluesam3

Teaching wise: A degree at COWI or an equivalent university elsewhere. Later years of degrees everywhere else. Postgraduate study basically anywhere.

Online: Chen's Napkin is a good place to start.

For your specific things: the circle equation has been covered.

I've never bothered to keep those dreary equations they make you memorise at school in my memory. I've also needed one... I think once, in the course of a degree, masters, and PhD. Most of them can be derived quickly from one of a handful, as I recall, but you're vanishingly unlikely to ever need to do so after the course in which they are taught. Your mileage may vary if you do something like engineering in the future, but that's what computers are for.

It's literally just addition and multiplication, and distributivity (apart from being impossible in general). Not much to memorise.

For integer powers, it's literally just repeated multiplication, and all properties follow immediately from that. For non-integer powers, it's just the obvious generalisation (and the unique generalisation that keeps all of those properties).

## chahud

What’s the highest level of math you’ve taken?

## we_are_privacy

Calculus