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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 ea

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/ http://betterexplained.com/cheatsheet/

kalid 12y ago
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 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-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-... for more details.) Again, thanks for the mention!
- shostack 12y ago
  Very happy to give the plug--the quality of your content and your generosity for putting so much out there for free is something I wish more people did. I hope the business side of it is going well for you. Side note--have you considered trying to work with Khan Academy? I feel like Salman's style flows well, but sometimes substitutes learning tricks vs. internalizing the underlying logic. Khan Academy could benefit from a touch of BetterExplained.
  - kalid 12y ago
    Thanks! The business side of things is coming along, I've just finished up a hectic few months (wedding, honeymoon) and am looking to crank on the site again. Long term I want the site to be an ever-growing part of my life. I'm very interested in collaborations, and helping people sprinkle an intuitive approach into whatever teaching they are doing. (I like Bret Victor's approach, where he provides guidance on what programming tools could look like, which might work its way into new programming languages like Swift, etc.). I'm in touch with an internet buddy at Khan Academy and have been meaning to work together, I'll be reaching out soon I think :).
- kev6168 12y ago
  The entries on complex number, exponential functions & e, and many other subjects at betterexplained.com are simply amazing! It helped to clear my decade-long confusions or fears when dealing with them, Thank you for the great work. In my opinion the three essential characteristics of a great teacher are expert level in the subject at hand, great communication skill and enthusiasm for teaching. Possessing two of them already makes a good teacher. You have all three.
  - kalid 12y ago
    That is really humbling and gratifying to hear, thank you.
- wfunction 12y ago
  Is there really an intuition for i^i? I can give you the answer but I have nearly zero intuition for it.

kalid 12y ago

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 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-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-... for more details.) Again, thanks for the mention!

shostack 12y ago
Very happy to give the plug--the quality of your content and your generosity for putting so much out there for free is something I wish more people did. I hope the business side of it is going well for you. Side note--have you considered trying to work with Khan Academy? I feel like Salman's style flows well, but sometimes substitutes learning tricks vs. internalizing the underlying logic. Khan Academy could benefit from a touch of BetterExplained.
- kalid 12y ago
  Thanks! The business side of things is coming along, I've just finished up a hectic few months (wedding, honeymoon) and am looking to crank on the site again. Long term I want the site to be an ever-growing part of my life. I'm very interested in collaborations, and helping people sprinkle an intuitive approach into whatever teaching they are doing. (I like Bret Victor's approach, where he provides guidance on what programming tools could look like, which might work its way into new programming languages like Swift, etc.). I'm in touch with an internet buddy at Khan Academy and have been meaning to work together, I'll be reaching out soon I think :).
kev6168 12y ago
The entries on complex number, exponential functions & e, and many other subjects at betterexplained.com are simply amazing! It helped to clear my decade-long confusions or fears when dealing with them, Thank you for the great work. In my opinion the three essential characteristics of a great teacher are expert level in the subject at hand, great communication skill and enthusiasm for teaching. Possessing two of them already makes a good teacher. You have all three.
- kalid 12y ago
  That is really humbling and gratifying to hear, thank you.
wfunction 12y ago
Is there really an intuition for i^i? I can give you the answer but I have nearly zero intuition for it.