How I Rewired My Brain to Become Fluent in Math

Maybe to fail the coursework but what about the real world where they won't have to do equations by hand on timed tests anymore?

I got tired of not having an intuitive understanding of particular mathematical concepts. The rote memorization was very limiting for be because I wasn't as easily able to grasp when it might apply to certain concepts.

Pretty sure Kalid is a commenter here, but I have to give him a huge thanks and plug for BetterExplained.com--really helped me grasp some of these higher-level concepts in a more intuitive manner, even if I didn't walk away with a full understanding of the intricacies.

His Cheatsheet is a great starting point if you are interested in a particular topic: http://betterexplained.com/cheatsheet/

It's always the combination of conceptual learning and memorization when it comes to learn math. It's like performing a complex computation job once, and then caching the result for fast retrieval. Memorization is for efficiency: it clears the way for our brains to focus on higher-level thinking. The great Euler "memorized not only the first 100 prime numbers but also all of their squares, their cubes, and their fourth, fifth, and sixth powers. While others are digging through tables or pulling out pencils and paper, Euler could simply recite from memory...". Besides, math is all about making invisible visible, about discovery patterns, or about connecting dots. That means we need to have dots to connect, and need to have patterns to work with. If we don't remember them, what the hell can we use for?

On the other hand, there's no need for rote memory. Just practice by solving interesting problems. There are plenty of opportunities to use math every day. There is also a very effective way of learning: work on slightly harder than usual problems. When learning calculus, I started to work on Demidovich's Problems in Mathematical Analysis, and I thought it was hard. Then, I stumbled upon solution book for college math competitions. Man, that was a huge help. After working through the problems, a lot of concepts became clear to me, and Demidovich's problems became reasonably easy too. It turned out the hard problems were hard because they required me to make non-obvious connections, which nudged me to really understand, from different angles, the concepts that I learned in the classroom.

By the way, when did arithmetic become so hard? It seems kids nowadays are being spoiled by their parents...

It really depends on the goal. From a mathematician's perspective Engineering is still part of the kiddie pool of math. The kind of things some people did in high school mostly though route memorization.

However, for ~98% of the population it's by far the most useful parts of math. Then again you can also say the same thing about just Arithmetic, basic Algebra, Logic and Statistics. So, it's really a question of what your goals are.

There is actually a lot of Math that's been dropped from K-12 education. EX: Understanding logarithms is really fundamental for using a slide rule or understanding floating point arithmetic, but it's not really that useful for most people.

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 at the bottom) and I really, really like her learning strategy.

Her article didn't use the exact phrase "deliberate practice" but I think that captures the essence of what she means by repetition. You need enough conceptual understanding to make sure you're following the path correctly, but then you want to practice -- at the edge of your comfort zone, with feedback, etc. -- to make sure it's really clicking.

It's really easy to fool yourself into thinking "I've got this" when it's untested. In my own case, I realized I didn't "get" imaginary numbers and exponents when I couldn't estimate a^b [a raised to the bth power] in my head with equal fluency for all numbers a and b. I had never really tested every type of number in every type of exponent position (base and power).

For example,

3^4 => this should be a positive real number greater than 1

3^(-4) => this should be a positive real number, very close to 0.

3^i => Hrm.

i^i => Uh oh.

I knew that unless I had fluency with all of these scenarios, I didn't truly "get" exponents or complex numbers. Sure, maybe I had a baby version where I could use them in well-defined ways, but I had a subconscious fear of i appearing as a base and/or exponent. I had to challenge myself and practice thinking through the various permutations before I recognized the gaps. Then I had to deepen my conceptual understanding, and practice again.

(For the previous questions, 3^i should be a complex number on the unit circle, maybe around 50 or 60 degrees, but less than 90, and i^i should be a real number, greater than 0 but less than 1. I can estimate these without calculating them, see http://betterexplained.com/articles/intuitive-understanding-... for more details.)

Again, thanks for the mention!