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One thing I will always appreciate about Feynman as a teacher is that he never shies away from telling you why things work the way they do. I've spoken with a l

One thing I will always appreciate about Feynman as a teacher is that he never shies away from telling you why things work the way they do. I've spoken with a lot of people who "hate math" and it's almost always because they never had a teacher that went into the whys and wherefores behind equations, tables, graphs, etc. Here we have a plainspoken yet rigorous explanation of what algebraic operations actually mean (addition is iteratively adding 1 b times; multiplication is iteratively adding a b times; exponentiation is iteratively multiplying by a b times, etc.). I wish more students would have the chance to build up such intuition, for such intuition is the key to keeping oneself interested in math and not finding that one "hates" it.

acpetrov 7y ago
I stopped enjoying math for about this reason. My university tried to teach with proofs, which to me is about as far from an intuitive understanding as possible Looking for recommendations on lectures / course that goes over linear algebra in this way
- jihadjihad 7y ago
  It's honestly harder than it should be to find well-written primers on linear algebra with a geometric focus, which I find to be the most intuitive way to "grasp" the subject and the motivations for it. I like the video lectures on MIT's website a lot for a variety of topics, and the one on linear algebra is pretty solid IMHO: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra... I also like the posts on this blog as well: https://dhruvp.netlify.com/2018/12/31/matrices/ https://dhruvp.netlify.com/2018/12/31/matrices/
- Ntrails 7y ago
  > I stopped enjoying math for about this reason. My university tried to teach with proofs, which to me is about as far from an intuitive understanding as possible Interesting, because until I had the proofs for eg differentiation, or heck even the quadratic formula - they were meaningless rote learnings. The proof is the intuition imo
  - wnewman 7y ago
    Both proof and explanation seem to be important. Two experiences with explanation made a particular impression on me. First I found it hard to get much out of a Real Analysis course when I was a grad student. Only partly because of lack of explanations, admittedly. Probably even more of a problem was another kind of cultural mismatch between physical scientist me and the math prof that taught the course. My interest was mostly that I was actually doing path integral Monte Carlo calculations (for my Chemistry thesis) and wanted to make sure I understood the fundamentals. The prof, like many (most?) mathematicians seemed to be more interested in investigating ingeniously weird boundary cases. So the course didn't seem to teach me much about the gotchas that might come up in actual statistics or numerical analysis, and instead more about the ingenuity of mathematicians in constructing absurdly farfetched abuses of e.g. the axiom of choice. But besides that cultural mismatch, lack of an explanatory framework sure didn't help. Thus, I was very happy decades later when I ran across Terence Tao's book on measure theory (available as a free manuscript online), which had a lot of the same kind of material with quite a good framework of motivation and explanation wrapped around its proofs. I also like Vapnik's _The Nature of Statistical Learning Theory_ which as I understand it is highly parallel to a much longer proof-heavy version of most of the same material. I much preferred this book to the approach in my undergraduate course in statistics. Again, the difference wasn't only lack of explanations (also, e.g., not enough enough grounding in proof or plausibly provable propositions, and too narrowly based in a frequentist worldview assumed without any explicit justification), but the lack of explanatory framework sure didn't help, so later I welcomed Vapnik's explanations of his approach. I have never been motivated to read the proof-heavy parallel book by Vapnik, but I do find it reassuring that it exists in case I ever work with problems that were weird enough to start raising red flags about breaking the comfortable assumptions in my informal understanding of the statistics.
  - aschismatic 7y ago
    Exactly! Proofs are why I love mathematics so much. There's nothing quite like the great "ah-ha!" moment when you find that one leap of logic that topples the rest of the dominoes in a proof. Of course, I understand that some people are better at teaching proofs by involving students in the discovery. Maybe the parent commenter was taught proofs in that same rote manner. I've had professors that have done that, and it's like someone sucked all the fun and learning out of the subject. From the preface of Computation: Finite and Infinite Machines by Marvin L Minsky: > The reader is therefore enjoined not to turn too easily to the solutions; not unless a needed idea has not come for a day or so. Every such concession has a price—loss of the experience obtained by solving a new kind of problem. Besides, even if reading the solutions were enough to acquire the ability to solve such problems (which it is not), one rarely finds a set of ideas which are at once so elegant and so accessible to workers who have not had to climb over a large set of mathematical prerequisites. Hence it is an unusually good field for practice in training oneself to formalize ideas and evaluate and compare different formalization techniques.
- thomasahle 7y ago
  You might enjoy "Linear Algebra Done Right" by Sheldon Jay Axler. The main thesis is that proving everything using determinants hides the intuition for what's really going on.
- ziddoap 7y ago
  This guy[1] doesn't necessarily hit all of the meat & potatoes, but covers some of the more interesting topics and does so in an amazing visual way. I'd highly recommend his videos (especially the playlist covering linear algebra: "Essence of Linear Algebra") as a supplement to more traditional mediums. 3Blue1Brown[1]: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
  - mettamage 7y ago
    Sorry for being pedantic, your comment is amazing for mentioning him. However, I'm a fanboy of 3B1B and I just have to say this. It's 3Blue1Brown, not the other way around. He named it after his eye color which apparently has 3 parts blue and 1 part brown.
    - ziddoap 7y ago
      Haha, thanks for pointing that out. I've watched literally every video of Grant, and am familiar with the source of the name - but for some reason I always mix it up. In my head I just say "3B1B", which I think is where I confuse myself.
- davinic 7y ago
  I learned a ton from the linear algebra section of the Stanford Machine Learning course on Coursera: https://www.coursera.org/learn/machine-learning https://www.coursera.org/learn/machine-learning
djmips 7y ago
It is thought that multiplication is better explained to mean scaling a by a factor of b. This is better illustrated by a diagram. I believe that 3Blue1Brown covers this graphically in several videos. For one, it better unifies division with multiplication as an inverse scaling.
- EvanAnderson 7y ago
  It took me awhile to grok why thinking about multiplication as scaling made sense, but given how much it comes up in discussion I thought it was pretty important to understand. I think having been taught that multiplication was repeated addition as a kid made it difficult for me to think about multiplication as scaling. Once the "light came on", however, it was very intuitive. It's definitely better to think about repeated addition as being a "trick" that just happens to work like multiplication on natural numbers, versus thinking that multiplication IS repeated addition. Edit: I've heard a similar statement made re: exponentiation-- that it's not "just repeated multiplication". I'd love to have the similar flash of understanding re: exponentiation that I had when I learned to think about multiplication as scaling. Is the "exponentiation is not repeated multiplication" statement true, and if so, does anybody have a recommendation on something I can read about it? I would dearly love to be more in touch with math than I am. I stopped at intro college algebra and never felt overly comfortable with trig. From talking with friends who have deeper math backgrounds I can tell that calculus would add a richness to my experience that I lack, if only because I can see a richness that my basic understanding of algebra brings when I talk to people for whom algebra isn't intuitive. I hope I can get it together enough to spend some time learning more about math before I end up old(er) and grey(er).
curiousgal 7y ago
I'm sadly now convinced that that approach only works for "simple" concepts. For instance I am yet to find a single book, course or teacher that can explain measure theory in a way that cultivates intuition.
- sorokod 7y ago
  They are simple because they are close to intuition one has already. Mathematical intuition is not something one is born with, it takes time and effort to develop.

acpetrov 7y ago

I stopped enjoying math for about this reason. My university tried to teach with proofs, which to me is about as far from an intuitive understanding as possible Looking for recommendations on lectures / course that goes over linear algebra in this way

jihadjihad 7y ago
It's honestly harder than it should be to find well-written primers on linear algebra with a geometric focus, which I find to be the most intuitive way to "grasp" the subject and the motivations for it. I like the video lectures on MIT's website a lot for a variety of topics, and the one on linear algebra is pretty solid IMHO: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra... I also like the posts on this blog as well: https://dhruvp.netlify.com/2018/12/31/matrices/ https://dhruvp.netlify.com/2018/12/31/matrices/
Ntrails 7y ago
> I stopped enjoying math for about this reason. My university tried to teach with proofs, which to me is about as far from an intuitive understanding as possible Interesting, because until I had the proofs for eg differentiation, or heck even the quadratic formula - they were meaningless rote learnings. The proof is the intuition imo
- wnewman 7y ago
  Both proof and explanation seem to be important. Two experiences with explanation made a particular impression on me. First I found it hard to get much out of a Real Analysis course when I was a grad student. Only partly because of lack of explanations, admittedly. Probably even more of a problem was another kind of cultural mismatch between physical scientist me and the math prof that taught the course. My interest was mostly that I was actually doing path integral Monte Carlo calculations (for my Chemistry thesis) and wanted to make sure I understood the fundamentals. The prof, like many (most?) mathematicians seemed to be more interested in investigating ingeniously weird boundary cases. So the course didn't seem to teach me much about the gotchas that might come up in actual statistics or numerical analysis, and instead more about the ingenuity of mathematicians in constructing absurdly farfetched abuses of e.g. the axiom of choice. But besides that cultural mismatch, lack of an explanatory framework sure didn't help. Thus, I was very happy decades later when I ran across Terence Tao's book on measure theory (available as a free manuscript online), which had a lot of the same kind of material with quite a good framework of motivation and explanation wrapped around its proofs. I also like Vapnik's _The Nature of Statistical Learning Theory_ which as I understand it is highly parallel to a much longer proof-heavy version of most of the same material. I much preferred this book to the approach in my undergraduate course in statistics. Again, the difference wasn't only lack of explanations (also, e.g., not enough enough grounding in proof or plausibly provable propositions, and too narrowly based in a frequentist worldview assumed without any explicit justification), but the lack of explanatory framework sure didn't help, so later I welcomed Vapnik's explanations of his approach. I have never been motivated to read the proof-heavy parallel book by Vapnik, but I do find it reassuring that it exists in case I ever work with problems that were weird enough to start raising red flags about breaking the comfortable assumptions in my informal understanding of the statistics.
- aschismatic 7y ago
  Exactly! Proofs are why I love mathematics so much. There's nothing quite like the great "ah-ha!" moment when you find that one leap of logic that topples the rest of the dominoes in a proof. Of course, I understand that some people are better at teaching proofs by involving students in the discovery. Maybe the parent commenter was taught proofs in that same rote manner. I've had professors that have done that, and it's like someone sucked all the fun and learning out of the subject. From the preface of Computation: Finite and Infinite Machines by Marvin L Minsky: > The reader is therefore enjoined not to turn too easily to the solutions; not unless a needed idea has not come for a day or so. Every such concession has a price—loss of the experience obtained by solving a new kind of problem. Besides, even if reading the solutions were enough to acquire the ability to solve such problems (which it is not), one rarely finds a set of ideas which are at once so elegant and so accessible to workers who have not had to climb over a large set of mathematical prerequisites. Hence it is an unusually good field for practice in training oneself to formalize ideas and evaluate and compare different formalization techniques.
thomasahle 7y ago
You might enjoy "Linear Algebra Done Right" by Sheldon Jay Axler. The main thesis is that proving everything using determinants hides the intuition for what's really going on.
ziddoap 7y ago
This guy[1] doesn't necessarily hit all of the meat & potatoes, but covers some of the more interesting topics and does so in an amazing visual way. I'd highly recommend his videos (especially the playlist covering linear algebra: "Essence of Linear Algebra") as a supplement to more traditional mediums. 3Blue1Brown[1]: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
- mettamage 7y ago
  Sorry for being pedantic, your comment is amazing for mentioning him. However, I'm a fanboy of 3B1B and I just have to say this. It's 3Blue1Brown, not the other way around. He named it after his eye color which apparently has 3 parts blue and 1 part brown.
  - ziddoap 7y ago
    Haha, thanks for pointing that out. I've watched literally every video of Grant, and am familiar with the source of the name - but for some reason I always mix it up. In my head I just say "3B1B", which I think is where I confuse myself.
davinic 7y ago
I learned a ton from the linear algebra section of the Stanford Machine Learning course on Coursera: https://www.coursera.org/learn/machine-learning https://www.coursera.org/learn/machine-learning

djmips 7y ago

It is thought that multiplication is better explained to mean scaling a by a factor of b. This is better illustrated by a diagram. I believe that 3Blue1Brown covers this graphically in several videos. For one, it better unifies division with multiplication as an inverse scaling.

EvanAnderson 7y ago
It took me awhile to grok why thinking about multiplication as scaling made sense, but given how much it comes up in discussion I thought it was pretty important to understand. I think having been taught that multiplication was repeated addition as a kid made it difficult for me to think about multiplication as scaling. Once the "light came on", however, it was very intuitive. It's definitely better to think about repeated addition as being a "trick" that just happens to work like multiplication on natural numbers, versus thinking that multiplication IS repeated addition. Edit: I've heard a similar statement made re: exponentiation-- that it's not "just repeated multiplication". I'd love to have the similar flash of understanding re: exponentiation that I had when I learned to think about multiplication as scaling. Is the "exponentiation is not repeated multiplication" statement true, and if so, does anybody have a recommendation on something I can read about it? I would dearly love to be more in touch with math than I am. I stopped at intro college algebra and never felt overly comfortable with trig. From talking with friends who have deeper math backgrounds I can tell that calculus would add a richness to my experience that I lack, if only because I can see a richness that my basic understanding of algebra brings when I talk to people for whom algebra isn't intuitive. I hope I can get it together enough to spend some time learning more about math before I end up old(er) and grey(er).

curiousgal 7y ago

I'm sadly now convinced that that approach only works for "simple" concepts. For instance I am yet to find a single book, course or teacher that can explain measure theory in a way that cultivates intuition.

sorokod 7y ago
They are simple because they are close to intuition one has already. Mathematical intuition is not something one is born with, it takes time and effort to develop.