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Math Intuition Cheatsheet

betterexplained.com

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Re: Math Intuition Cheatsheet

#51

Post by kalid »

vanderZwan wrote: post #37
kalid wrote: post #8

Earlier quoted context omitted.

(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little ca…

Same thing with logarithms, which were invented to simplify multiplying two huge numbers. Combined with log tables (think paper LUT for humans) that became a simple matter of looking up two log conversions, adding them, then looking up the inverse of the answer. Is that what we learn in high school? Nope. So everyone is left wondering what the hell they are good for (or were, before we had calculators) the first time…

Great tidbit, thanks. I forget if I have a mention that logs were developed before e, but I love it because it's a little mosquito in the ear for the over-rigorous mathematicians who define e up front (as a limit) and then say "the natural logarithm is log base e".

e was only discovered because of logarithms, not the other way around! :)

Re: Math Intuition Cheatsheet

#52

Post by kalid »

mathattack wrote: post #41
kalid wrote: post #8

Earlier quoted context omitted.

(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little ca…

Kalid - First of all, your site kicks ass! Keep doing what you're doing. You probably can't even imagine the 2nd and 3rd order impacts of the benefits you provide. Someone gets one insight from your site, that makes something they learn in school more comprehensible, and then it explains it to someone else, who then can learn more quickly.... Bithive123 - I found that practicing Project Euler has helped my understand…

Thanks so much. My meta goal is to help people gain confidence in what it means to learn (in general, not just math) and ideally encourage people to share their insights with others. No reason for us to all bump our shins on the same coffee table in a dark room, someone turn on the lights on your way out!

Re: Math Intuition Cheatsheet

#53

Post by kalid »

nabla9 wrote: post #40
kalid wrote: post #8

Earlier quoted context omitted.

(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little ca…

I find your emphasis of building intuition very important. Personal anecdote related to Pythagorean theorem/euclidean distance. When in ninth grade, I asked my math teacher to explain why why the the standard deviation is like it is and why the square root and not something else. He could not explain that to me satisfactorily. I spend few hours looking at the formula myself and plotting and drawing different datasets…

Great example, I like that. I started realizing that many instructors don't necessarily have an intuition for the topic they're teaching, and it's often up to the individual student to seek it out. I'm hoping to do more stats concepts down the line, this is a nice analogy for why standard deviation is defined as it is.

Re: Math Intuition Cheatsheet

#54

Post by kalid »

__Joker wrote: post #36
kalid wrote: post #8

Earlier quoted context omitted.

(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little ca…

Unfortunately, most of the mathematical text miss the historical context which they are developed in. I personally enjoy a text in any subject, where the author develops the content with historical perspective. Any learning course should include one textbook which develops the content from an historical perspective.

I agree. A few years after doing articles, I started seeing the power of historical context.

For example, the Fourier Transform was originally _rejected_ as implausible when presented to the mathematicians of the time. They needed a decade of debate to verify its truth, and yet we teach it to students in a week and expect them to internalize it without issue. We need to acknowledge the difficulty/counter-intuitive nature of the idea up front.

Re: Math Intuition Cheatsheet

#55

Post by kalid »

jgrodziski wrote: post #32

For me the best one from Kalid is the explanation of the fourier transform: http://betterexplained.com/articles/an-interactive-guide-to-... This article is simply amazing, moreover Kalid is an humble person and references his inspirations from several people over the web. Thank you very much Kalid for the time you dedicated to share all this wonderful resources.

Thanks so much, and for originally submitting the cheatsheet to HN. (I hope it helps people untangle concepts that confused me for literally a decade.)

I really appreciate the encouragement and hope to keep cranking as long as I can.

Re: Math Intuition Cheatsheet

#56

Post by kalid »

ithinkso wrote: post #26

I like the idea but I personally really enjoy and prefer 'math english'. At first it may seems odd but after a while you get used to it and it's actually easier, you just scan through epsilons, deltas knowing exactly what they mean. Intuition has edge-case scenarios, double-meanings and almost-truths. I don't know why, but I think intuition (pseudocode math as someone had named it) gives a little bit of a 'illusion'…

Great point. I use the word "intuition" but feel it might be overloaded, things with intuitive knowledge, psychics, going with your gut despite the facts, etc.

The sense I mean is closer to "grokking" or "clicking", when symbols and definitions actually mean something. You start thinking with the symbols, not despite them. You actually look forward to checking your understanding with real problems, because when it clicks the problems become easy :).

Re: Math Intuition Cheatsheet

#57

Post by j2kun »

arianvanp wrote: post #33

I started my 1st year in college doing both math and computer science majors. I really love math but I just hate how (especially calculus and linear algebra courses) they just don't work on growing any intuition at all. in contrast, in my CS major we had a linear algebra class (ugh overlapping classes, huh) as well and we used computer graphics as a case study to learn about the basic concepts which was really fun an…

A lot of mathematics people feel this way about programming, too. Both sides expect one to devote the time to build intuition to grok their subject. I think it's easier to learn programming coming from the mathematics side, because you're going from nobody holding your hand to a compiler giving you error messages. You can't interactively test whether your intuition in mathematics is sound (you can, but you have to play both the role of the compiler and the tester).

Re: Math Intuition Cheatsheet

#58

Post by kalid »

RobinL wrote: post #27

This September, I'm starting tutoring A level maths to underprivileged kids for a charity after helping them with GCSEs last year. This looks like it will be really helpful. Thanks!

That's awesome! I hope you can make use of the site. One goal for the site is to provide "ammo" to teachers who are looking for other ways to present difficult concepts.

Re: Math Intuition Cheatsheet

#59

Post by kalid »

JadeNB wrote: post #43
kalid wrote: post #19

Earlier quoted context omitted.

Thank you Jacob, I really appreciate it. I try to imagine that I'm writing to a younger version of myself, who hasn't seen the material yet. It helps me remove potential anxiety about not knowing everything, or trying to impress -- there's no reason to deceive yourself about what you do or don't know.

Since we've mentioned the article, I can't help asking about a particular tit-bit that caught my attention --what do primes have to do with quantum mechanics?

Definitely not an expert, but I found a few good articles on this (just updated the article):

http://seedmagazine.com/content/article/prime_numbers_get_hi... Nice summary: http://carbonatoms.wordpress.com/2009/03/13/prime-numbers-ar...

Primes appear to be zeroes of the Reimann Zeta Function. Nature loves minimizing energy, so there might be some reason the Zeta Function models how atoms behave. Then, primes are the most "stable orbits" or have a similarly useful property.

Super high level, but I think it's interesting.

Re: Math Intuition Cheatsheet

#60

Post by lfischer »

richardjdare wrote: post #35
chaoxu wrote: post #29

Earlier quoted context omitted.

> Embarrassingly, years after a poorly-timed calculus course left me thinking I had to be able to prove the central limit theorem in order to use calculus Why would you need to prove central limit theorem in order to use calculus? What kind of course teach it that way? It would sound like something to prove for a honors calculus class that prepares students to become mathematicians. Anyways, about math education. Mat…

>>Why would you need to prove central limit theorem in order to use calculus? What kind of course teach it that way? It would sound like something to prove for a honors calculus class that prepares students to become mathematicians. Yes, but not knowing calculus, how would he know that? Or any of those things that you have pointed out? They are certainly not taught at school (where we are unhelpfully taught an entire…

What subjects are you comparing with? I don't know whether the situation is that much better for physics, for instance. I think part of the problem is that mathematics books are in general either targeted at experts or at professors who need to assign a textbook. It looks like publishers believe the market for self-taught mathematicians is too small. In contrast, programmers are a pool of people who are willing to spend money on books to teach themselves, because they hope it will have a direct career benefit. That said, mathematics curricula must vary a lot from country to country. I was taught the fundamental of calculus in my penultimate year of high school!
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