How I Rewired My Brain to Become Fluent in Math • Hacker News Board

Re: How I Rewired My Brain to Become Fluent in Math

#131

Post by kalid » Fri, Oct 03, 2014, 5:51 PM UTC

wfunction wrote: post #130

kalid wrote: post #119

Earlier quoted context omitted.

Yep, I think so. My philosophy, at least, is that any idea can eventually be intuitive, no matter how difficult at first, if we find the right analogies. I see any exponent like a^b as starting at 1.0, intending to apply a rate of change of (a), but modifying that rate by (b). For example, 3^2 is an initial rate of change of 3x, which is then applied for 2 units of time [leading to 9]. So, 1.0 would turn into 9.0. Mo…

Your intuition is nice for figuring out the directions, but not so nice for explaining the numbers. By which I mean, I can understand why it's real, but I can't explain why i^i = e^(-pi/2)... I would expect it to be a nicer number instead (like 1/2, or 1/e, or something like that). I can't really explain why both pi and e end up int he formula.

Oh yeah, after getting the direction, figuring out the numbers is the next step :).

Having i as a base means "we plan on rotating 90 degrees" which actually means pi/2 radians.

e^rt models growth rate of r, for time of t. so e^(i · pi/2) creates a 90 degree turn (we intend on rotating, i, and do this enough to get a full 90-degree turn, pi/2).

This is all a fancy way of saying:

90 degree turn = i = e^(i · pi/2)

Now, with i^i, we're planning on modifying that growth rate (e^(i · pi/2)) that we just figured out! We're going to twist the "rate" from i (90 degrees) to i · i (180 degrees):

i^i = e^(i · i · pi/2) = e^(-pi/2)

which is the real number less than 1 (about 0.2).

I'm summarizing on the fly, but the full explanation for i^i and i^i^i are here: http://betterexplained.com/articles/intuitive-understanding-...

Re: How I Rewired My Brain to Become Fluent in Math

#132

Post by dropit_sphere » Fri, Oct 03, 2014, 5:51 PM UTC

jxm262 wrote: post #121

icantthinkofone wrote: post #112

I remember struggling with Calculus in high school. I sat in the library next to the smartest guy in the class and explained to him that I couldn't figure out why a certain formula was to be used to solve a problem. It went something like: Steve: "You have to use that formula to solve this." Me: "I understand that but why do we use that formula?" Steve: "Cause that's the only formula that works." Me: "OK but how did…

I've constantly struggled with something along these lines too. I've always had a difficult time in Chemistry because i wanted to know the why's of things as opposed to just memorizing things. Even in programming, my intro Java class everyone learned the basic public static void main(String[] args){} and we were expected to take it for granted. The first Hello World assignment took most students a few minutes to comp…

I would remember that knowledge is not fungible, while prestige and money are. Do enough things for prestige and money, and you'll have time to do things for knowledge.

Re: How I Rewired My Brain to Become Fluent in Math

#133

Post by akavi » Fri, Oct 03, 2014, 5:53 PM UTC

xsace wrote: post #115

ivansavz wrote: post #78

My observation from years of experience teaching calculus and mechanics is that every student has a different learning style. In particular, there are "theory people" who need to understand the reasoning behind the concepts first and "practice people" who best understand concepts by looking at worked examples. There may be other subdivisions, but these are the main types. I had two cases when my students showed no si…

不: not 闻: hear 若: good 见: see 之: have 知: know looks easy :)

'若' is closer to 'as' or 'equivalent to'

Re: How I Rewired My Brain to Become Fluent in Math

#134

Post by B-Con » Fri, Oct 03, 2014, 6:07 PM UTC

> memorized the equation so I could carry it around with me in my head and play with it. If m and a were big numbers, what did that do to f when I pushed it through the equation? If f was big and a was small, what did that do to m? How did the units match on each side? Playing with the equation was like conjugating a verb. I was beginning to intuit that the sparse outlines of the equation were like a metaphorical poem, with all sorts of beautiful symbolic representations embedded within it.

I cannot emphasize how useful this approach is. What's more, I am very surprised by how many students lack the basic ability to sit and "play around" with math concepts.

I used to tutor algebra, trig, and pre-calc. I was surprised by how few students had the ability to look at something and break it into pieces. If you gave them piece A which they knew, piece B which they knew, and put them together, the result was something new the student couldn't understand and they'd sit and wait for an explanation.

This was sad because this is largely what math learning is. Given pieces you do understand, you put together bigger pieces and then build an understanding of those bigger pieces.

I have always advocated something similar to what what the author said here:

> What I had done in learning Russian was to emphasize not just understanding of the language, but fluency. Fluency of something whole like a language requires a kind of familiarity that only repeated and varied interaction with the parts can develop.

I love this point. I am borderline convinced it it is near impossible to learn math concepts more than one or two steps beyond the point that you are fluent in. Beyond that, it's just memorization, guessing, and poor heuristics that just get you to skate past the test and do little or nothing for understanding or retention.

The bigger picture here is that the author has learned how to learn. It amazes me how significant a divide there is between people who know how to learn and people who don't. Forget IQ or test scores, I think knowing how to learn is the biggest indicator of how far someone will go in life. So perhaps most of all I love the fact that they began this journey relatively late (most students going through a similar math path would have done in their late teens or very early 20s what the author was doing in their late 20s).

Re: How I Rewired My Brain to Become Fluent in Math

#135

Post by Zelphyr » Fri, Oct 03, 2014, 6:08 PM UTC

icantthinkofone wrote: post #112

I remember struggling with Calculus in high school. I sat in the library next to the smartest guy in the class and explained to him that I couldn't figure out why a certain formula was to be used to solve a problem. It went something like: Steve: "You have to use that formula to solve this." Me: "I understand that but why do we use that formula?" Steve: "Cause that's the only formula that works." Me: "OK but how did…

I feel like this touches on the Montessori Method quite a bit. Most schools assume the student is an empty vessel and that the knowledge needs to be poured in. Montessori takes the approach that humans are naturally curious and that we in fact LOVE to learn so it encourages understanding through exploration.

I would say the rote memorization that the author of this article lauds is actually unnecessary. When one is learning through exploration of a subject and subjects you in fact are exposed to the same concept many times (like with memorization) but you have the added benefit of association with other concepts.

In other words; learning doesn't have to be boring to be effective!

Re: How I Rewired My Brain to Become Fluent in Math

#136

Post by larssorenson » Fri, Oct 03, 2014, 6:15 PM UTC

Reading this reminds me of how much I really want to be better at math, and especially the quick fire recall and understanding. I'm taking courses with a lot of math that I've learned before, and encountering similar struggles this time around as last time. Maybe I'll try a new approach, one such as this, to insure my success.

Re: How I Rewired My Brain to Become Fluent in Math

#137

Post by wfunction » Fri, Oct 03, 2014, 6:27 PM UTC

kalid wrote: post #131

wfunction wrote: post #130

Earlier quoted context omitted.

Your intuition is nice for figuring out the directions, but not so nice for explaining the numbers. By which I mean, I can understand why it's real, but I can't explain why i^i = e^(-pi/2)... I would expect it to be a nicer number instead (like 1/2, or 1/e, or something like that). I can't really explain why both pi and e end up int he formula.

Oh yeah, after getting the direction, figuring out the numbers is the next step :). Having i as a base means "we plan on rotating 90 degrees" which actually means pi/2 radians. e^rt models growth rate of r, for time of t. so e^(i · pi/2) creates a 90 degree turn (we intend on rotating, i, and do this enough to get a full 90-degree turn, pi/2). This is all a fancy way of saying: 90 degree turn = i = e^(i · pi/2) Now,…

Two things:

1. I said earlier that I already know how to find the numbers mathematically. What I don't have any intuition for is why the numbers are correct (such as why both e and pi should be in the answer).

2. You can't just distribute the exponents without justifying that it's valid. (I don't know the appropriate conditions for doing this myself either, but I know they exist.) For example (e^(2 i pi))^(1/2) is not e^(i pi), it's e^0. In other words, you need to first reduce it mod 2pi. Now if you're dealing with complex numbers I have no idea what the conditions should be.

Re: How I Rewired My Brain to Become Fluent in Math

#138

Post by aluhut » Fri, Oct 03, 2014, 6:41 PM UTC

throwaway6822 wrote: post #101

ivansavz wrote: post #76

Earlier quoted context omitted.

Since we are on the topic of math textbooks, I will suggest the No bullshit guide to math and physics which is a math textbook written specifically for adult learners. See http://minireference.com/ for more info. I'm the author

I've been looking for something just like this. Do you plan to release an ePub or mobi-format book? I'd prefer to read it on my ereader, and PDFs don't reflow on smaller screens.

Just get Calibre and convert your pdf to epub.

Re: How I Rewired My Brain to Become Fluent in Math

#139

Post by RankingMember » Fri, Oct 03, 2014, 6:56 PM UTC

spion wrote: post #126

jxm262 wrote: post #121

Earlier quoted context omitted.

I've constantly struggled with something along these lines too. I've always had a difficult time in Chemistry because i wanted to know the why's of things as opposed to just memorizing things. Even in programming, my intro Java class everyone learned the basic public static void main(String[] args){} and we were expected to take it for granted. The first Hello World assignment took most students a few minutes to comp…

YMMW, but I use "I can always fill in that gap later if I have the time" and move on. Later, if I do indeed have the time or I find out that I'm struggling because of missing details, I'll get back to it. Its like making a temporary mess in a project while trying to understand the problem you are solving and making a promise that you will refactor it later, once you get to something that works.

This is where OCD really kicks my ass. I want to understand everything that I'm typing, and it's a major hurdle to have to accept that, no, I just can't have all the answers.

Re: How I Rewired My Brain to Become Fluent in Math

#140

Post by afarrell » Fri, Oct 03, 2014, 6:58 PM UTC

jxm262 wrote: post #121

icantthinkofone wrote: post #112

I remember struggling with Calculus in high school. I sat in the library next to the smartest guy in the class and explained to him that I couldn't figure out why a certain formula was to be used to solve a problem. It went something like: Steve: "You have to use that formula to solve this." Me: "I understand that but why do we use that formula?" Steve: "Cause that's the only formula that works." Me: "OK but how did…

I've constantly struggled with something along these lines too. I've always had a difficult time in Chemistry because i wanted to know the why's of things as opposed to just memorizing things. Even in programming, my intro Java class everyone learned the basic public static void main(String[] args){} and we were expected to take it for granted. The first Hello World assignment took most students a few minutes to comp…

What I do is keep a notepad next to me. When I start something, I first write what my goal is. This guides me as to how deep I am willing to go. If my goal is just to learn things, I start out with a tutorial and write notes about what I don't understand as things that I can follow-up on later.

It helps that I've spent some time explicitly trying to learn things on multiple levels. If you like doing that, look at MIT's class 6.004 and it will give you a deeper understanding of computer architecture.