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

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

#111

Post by maxerickson » Fri, Oct 03, 2014, 2:07 PM UTC

clarry wrote: post #108

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

It's an interesting looking book, and I'm somewhat inclined to buy it. Bookmarked! I agree that you don't need to read thousands of pages to learn calculus. However, I don't want to stop at calculus. Basically, what I'd really love to have is a "mother of all maths textbook" -- a thick and heavy tome that compacts information from all the other thick and expensive books (which I'm never going to read) and different f…

I hope you get a well informed answer. There are famous (so to speak) examples of such things:

http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_math%C3%A...

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

#112

Post by icantthinkofone » Fri, Oct 03, 2014, 2:09 PM UTC

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 they come up with that one formula?"

Steve: "Cause it just works!!"

He didn't know. That's when I first realized a lot of his knowledge was just memorization while I needed to know the why. It wasn't till my first year in college that I finally found a math instructor who always shows us the reasons why that I was able to take off and grasp math by connecting the dots.

So, no, I do not think memorization is the key but, like most zen things, understanding is.

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

#113

Post by ilitirit » Fri, Oct 03, 2014, 2:10 PM UTC

chestervonwinch wrote: post #96

ilitirit wrote: post #91

He highlights one of the problems I had with mathematics when I reached University. I was in the top 2% of students in the first few months. I considered it a trivial subject. "Here's a class of problems, here's how you solve them." Easy peasy. Then we started integrating and differentiating forms of problems we weren't familiar with. All of a sudden, everything we were taught didn't seem to apply any more. Our lectu…

>> I switched to Applied Math and aced it. What sort of integration and differentiation were you doing originally that is not included in an applied math curriculum?

My Applied Math curriculum (roughly):

- Logic of Compound/Quantitative statements.

- Elementary Number Theory and Proof Methods

- Sequences and Mathematical Induction

- Graphs/Tree/Set theory

- Algorithms

- Functions and Application

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

#114

Post by lordnacho » Fri, Oct 03, 2014, 2:15 PM UTC

I don't understand why there seems to be a languages/math dichotomy. I remember in school teachers used to bucket people in one or the other.

But from what I can tell, and also from the article, they have a lot of common ground. Learning either requires understanding of structure, but also a fair bit of memorization. Learning conjugations by heart is not all the different to learning trig relations. You also need both on the fly when you're using your language/calculus.

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

#115

Post by xsace » Fri, Oct 03, 2014, 2:57 PM UTC

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 :)

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

#116

Post by MilnerRoute » Fri, Oct 03, 2014, 3:22 PM UTC

Dylan16807 wrote: post #42

MilnerRoute wrote: post #25

Earlier quoted context omitted.

This is very true. A corporate trainer once told me there's specific kinds of learners. They've actually identified each type, and which specific methodologies you need to reach each one. The best teachers find a way to synthesize it all into an effective single presentation. (This may also be why tutors are so helpful -- they can customize their material to the audience...)

Although there is ample evidence that individuals express preferences for how they prefer to receive information, few studies have found any validity in using learning styles in education.[2] Critics say there is no evidence that identifying an individual student's learning style produces better outcomes. There is evidence of empirical and pedagogical problems related to the use of learning tasks to "correspond to di…

Heh. Okay, that's good to know. These comments prompted me to look up "Learning Styles" on Wikipedia, which confirms that scientific studies have not confirmed the validity of the "different learning styles" theory.

http://en.wikipedia.org/wiki/Learning_styles#Criticism

That probably says something about corporate trainers. I still remember the HR department at one company where I worked that insisted on giving the Myers-Briggs personality test to every employee. So maybe this also says something about junk science and the way it lingers on in our workplaces...

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

#117

Post by Taek » Fri, Oct 03, 2014, 3:32 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…

The point of this article is that they need equal emphasis. If you don't know why you're using a certain formula, you're missing half the picture, but you're still going to be able to solve problems faster than someone who has to redo the derivation every time they see a problem.

In professional life (as a programmer), it's very valuable to be able to look at a problem and think 'hash table!' before even fully understanding what sort of problem I'm approaching. I can get things done a lot faster because I've practiced my algorithms and data structures and I have strong intuitions for where each is useful.

At the same time, having a well explained, fundamental understanding of how everything works is critical to innovation. If you're staring at a problem that nobody has ever been able to sovle before, it means that the general basket of tools isn't going to help, because if it would somebody would have already solved it.

The point of this article is that both styles of learning are critical, but America has largely thrown one of them out the door.

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

#118

Post by mafafu » Fri, Oct 03, 2014, 3:34 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…

You have a good point, but I think the author was making the case that a conceptual understanding AND memorization/repetition are required to make the learning stick. I don't see a case for not explaining concepts, but a caution to not go too far the other way. And I saw repetition as being more important than memorization. That by repeating the application of your understanding of some concept, as being the key to making it stick.

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

#119

Post by kalid » Fri, Oct 03, 2014, 3:45 PM UTC

wfunction wrote: post #90

kalid wrote: post #59

Earlier quoted context omitted.

Wow, thanks for the plug and the kind words! Glad to hear the site is helping -- most of the time I miss the intricacies too, and end up adding things years later [after a re-read] or in response to a comment from the article. As a more general reply to the article, I was fortunate enough to work with Prof. Oakley on her Coursera Class ( https://class.coursera.org/learning-001/lecture , I'm the last guest interview a…

Is there really an intuition for i^i? I can give you the answer but I have nearly zero intuition for it.

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.

More here: http://betterexplained.com/articles/understanding-exponents-...

So what's i^i?

Well, our rate of change is "i", which means we plan on starting at 1.0 and having our rate of change be a rotation: 1.0 * i = 90 degrees on the unit circle.

If we are using i^i, that means we are rotating our rate of change. That is, we intended to rotate around the unit circle at 90 degrees, but now I'm going to turn the "rocket boost" which is applying that rotation by another 90 degrees (the i as the exponent). This means the rocket is facing 180 degrees (backwards) and our growth is going to be an exponential decay. We'll be on the real number line, but shrinking.

How about (i^i)^i? That's another 90-degree twist on the rocket, so it's pointing 270 (downward) and we'll rotate around the circle clockwise. It's probably a negative imaginary number, and (i^i)^i actually equals -i.

This is a quick brain dump, and not very clear without diagrams, check out the articles above if you'd like!

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

#120

Post by clueless123 » Fri, Oct 03, 2014, 3:49 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…

Right on man.. When you think you know something, you earned your Bachelors. When you realize you dont Know a thing, you get your Masters. When you realize no one else knows either you get a PHD!