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I can recognize some of my experience in what she's saying -- as I've been told several times in math/stats "you don't understand this, you just get used to it.

I can recognize some of my experience in what she's saying -- as I've been told several times in math/stats "you don't understand this, you just get used to it." That's a bit of troubling statement at first, but I think it's sortof a shorthand way of saying that much of the understanding available comes through the process of repeated manipulation and observing outcomes. I've also noticed that a decade or two after going through a math undergrad, the limited material I've retained best was the stuff I learned my freshman year and had to repeatedly re-apply in other coursework. Still, I think she may be overstating the case about the limits of conceptual understanding, though (or perhaps bringing an engineer's perspective to the discussion rather than a mathematician's. ;) Practice is crucial, but when I need to dredge old mostly-forgotten material up out of my brain, the interconnections formed by the mapping of conceptual space we call "proofs" often turn out to be pretty helpful. There's plenty of things I can't remember that I can derive from what I can recall.

shostack 12y ago
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.
ClashTheBunny 12y ago
It's interesting that that famous quote about math is a von Neumann quote. I always use the same quote with language and I think that there is a strong connection between language learning, especially ultra logical and systematized ones like Russian. Still, for the sake of saying the full quote: Young man, in mathematics you don't understand things. You just get used to them.
protonfish 12y ago
If understanding is about "connecting the dots" then you need to learn about all the pertinent dots before that can happen. You can't be expected to have a deep understanding when first introduced to something, you have to trust that it will probably all make sense later.

shostack 12y ago

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.

ClashTheBunny 12y ago

It's interesting that that famous quote about math is a von Neumann quote. I always use the same quote with language and I think that there is a strong connection between language learning, especially ultra logical and systematized ones like Russian. Still, for the sake of saying the full quote: Young man, in mathematics you don't understand things. You just get used to them.

protonfish 12y ago

If understanding is about "connecting the dots" then you need to learn about all the pertinent dots before that can happen. You can't be expected to have a deep understanding when first introduced to something, you have to trust that it will probably all make sense later.