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This is a phenomenal example of how to teach math. You can go through theory, formulas and proofs all day long (yes, sometimes rigor is needed), but this type o

This is a phenomenal example of how to teach math. You can go through theory, formulas and proofs all day long (yes, sometimes rigor is needed), but this type of teaching is sticky. It takes a lot more time to display information in the way the author has done, but it reaps massive benefits for those trying to learn. In my mind, math is about concepts and there is no reason why we can't start teaching math like this. Amazing work. The collective intelligence of the world has just gone up. Another amazing example of visual math lectures, https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw.

pseudosudoer 8y ago
3Blue1Brown is an incredible example of how math concepts should be introduced. His videos are a fantastic reminder of why math is a form of art.
- thanatropism 8y ago
  3Blue1Brown is impressive in how it uses visualizations, cute faces, soothing voiceovers, etc. but keeps pulling you down the rabbit hole. So at the end of the video you know that you don't know stuff you either think you knew or didn't know existed. And then you start looking up Wikipedia and just getting lost in the music. This is really really rare. Education as entertainment means that the goal is to get the consumer to feel good about himself, to feel like he learned something. You're getting paid (in money or kudos or whatever) to engage the consumer's narcissism, not his curiosity.
  - florabuzzword 8y ago
    > ...keeps pulling you down the rabbit hole. So at the end of the video you know that you don't know stuff you either think you knew or didn't know existed. And then you start looking up Wikipedia and just getting lost in the music. Education as entertainment means that the goal is to get the consumer to feel good about himself, to feel like he learned something. You're getting paid (in money or kudos or whatever) to engage the consumer's narcissism, not his curiosity. This is a profound observation. I’m not sure it only applies in entertainment. I’ve heard it put other ways, but your version, if applied to education as a whole, might be a pretty big deal. There’s no shortage of evidence, either; from Ivy League campuses to niche interest subcultures. Maybe “narcissism” isn’t the best word, but some sort of similar tension related to confidence seems to be associated with successful educational environments. We commonly attribute it to a one-way causal relationship, but you’re articulating the opposite. Likewise, we are used to acknowledging that ‘not giving up’, or ‘belief in one’s self’ is a necessity, but again you’re making a separate claim. You’re saying, in so many words, the confidence in and of itself defines the will as well as the curiosity. And I can’t argue with it. I want to, but I’m not sure I can.
    - thanatropism 8y ago
      I feel like the key thing about a "growth oriented mindset" or some near-equivalent buzzword is that the normal state of affairs to not know things. This is the foundation (in Socrates) of our civilization. What I've been trying to articulate here and there in the past few weeks that the object of intellectual development should be acquiring structured ways of not knowing. Rigorous mathematics is such a way: in basic analysis class you get red marks for thinking you know things you can't prove. The buddhism of the Heart Sutra is another: the Heart Sutra boils down to "we want to know big truths about emtpiness and suffering to transcend it and achieve Nirvana, but because of emptiness there's no way to know and nothing to know; therefore, recite these magic words because that's what wise men do". (It sounds like a scam, but it's really deep in how the argument is actually made in length). This of course runs counter to beliefs in "ultratechnology" such that we'll soon soon craft our own transcendence. But eh, we shall fight in the shade.
tralarpa 8y ago
I don't know. Is there really a massive benefit? The visual explanation of the Markov chains on that web site looks elegant and simple because... it is, i.e. it's not really a contribution of the visualization, it's because DTMCs are actually easy to understand. In a lecture, the teacher can easily simulate a Markov chain on a black board ("Now we are in this state, now we have to roll a die to decide whether we will go to state X or state Y"). Same for sealf-learners: In a book, you can tell the reader to start at some state X, roll a die, etc. Honestly, I find that the Markov chain website is an example how you should NOT do it. You are replacing the active intellectual process in the head of the student by some trivial animations with scrolling texts and jumping tokens which look nice and dynamic but...what are you learning? Of course, this doesn't mean that visualizations are useless, but they should be carefully designed with some learning goal in mind. Not easy! For example, the visualization could slowly gray out states that are not visited. In that way, the student could get a feeling for concepts like recurrence etc.
- dyarosla 8y ago
  Totally disagree. ‘Same for self-learners: In a book, you can tell the reader to start at some state X, roll a die, etc.’ -> This requires a learner to parse a procedure correctly AND understand how it relates to the topic. This is both doubly harder on the student and doubly likely to force a mistake (either in the procedure, interpretation, or both). Learning with an approach as given in the article approximately halves the work a learner must do, and increases the likelihood that the topic is correctly interpreted. Retention with this method, however, is a whole other matter- one that does correlate with the amount of mental effort required, but how much so is a matter of debate.
- jinfiesto 8y ago
  I both agree and disagree. I've always found an intuitive explanation like the OP to be helpful before diving into a more rigorous study of the material. It helps contextualize the things you learn subsequently.
- ryeguy_24 8y ago
  See my other comment below. If you could learn the same thing at the same level with less mental effort, isn't that a good thing? If the traditional teaching approach for Markov Chains required more intellectual effort, sounds like we've been explaining it poorly, no?
ttoinou 8y ago
I agree. On the other hand, it is lowering mental effort required to study the topic, and that will probably not make every student more clever, maybe more lazy, just waiting for the next youtube video to be told new stuff about maths instead of going research and study on its own.
- ryeguy_24 8y ago
  I think lowering the mental effort to learn something is a really good thing. It leaves room for more mental effort elsewhere.
  - thanatropism 8y ago
    The thing is that 80% of the usefulness of learning is that it's learning practice. This is why you sit for so much crap in K-12 -- you're learning how to learn. This rings even more true for techies who are always chasing a moving target (nonono, object-oriented CORBA RPC, nono, it's REST, it's Kernel SVM, it's Convolutional Networks....). The effort is (almost) the whole points. If you don't keep your learning organs stimulated they will literally wither and shrink.
    - ryeguy_24 8y ago
      I agree with you but I don't think we should make things unnecessarily hard to understand if they aren't so. Do you? There will always be difficult things to learn out there but I don't think we shouldn't artificially make all things difficult if we don't have to.
      - ttoinou 8y ago
        we should make things unnecessarily hard to understand if they aren't so Who said otherwise ? I am just saying that if you as a teacher are concerned about making people/students more clever, you need to think both at vulgarization but also at the meta level where they need to actually make mental effort to incorporate ideas, maths, methodology, calculus, visualization capabilities and in general self-motivation
anonytrary 8y ago
This seems like quite an overreaction. This animation could easily confuse people who don't know as much as you do. The little ball moving back and forth -- what is that? What is doing the moving? What determines the speed of the ball? These are the questions you'll be running into when you show this to middle and high-school students who have no idea what state space is and what matrix operations are. Like most math visualizations, this one seems best for people who are already somewhat familiar with the material. I think it's a cool animation, but I'm almost sure I'm suffering from a hindsight bias. My physics professor used to say that nothing is worse than an analogy taken too literally.

pseudosudoer 8y ago

3Blue1Brown is an incredible example of how math concepts should be introduced. His videos are a fantastic reminder of why math is a form of art.

thanatropism 8y ago
3Blue1Brown is impressive in how it uses visualizations, cute faces, soothing voiceovers, etc. but keeps pulling you down the rabbit hole. So at the end of the video you know that you don't know stuff you either think you knew or didn't know existed. And then you start looking up Wikipedia and just getting lost in the music. This is really really rare. Education as entertainment means that the goal is to get the consumer to feel good about himself, to feel like he learned something. You're getting paid (in money or kudos or whatever) to engage the consumer's narcissism, not his curiosity.
- florabuzzword 8y ago
  > ...keeps pulling you down the rabbit hole. So at the end of the video you know that you don't know stuff you either think you knew or didn't know existed. And then you start looking up Wikipedia and just getting lost in the music. Education as entertainment means that the goal is to get the consumer to feel good about himself, to feel like he learned something. You're getting paid (in money or kudos or whatever) to engage the consumer's narcissism, not his curiosity. This is a profound observation. I’m not sure it only applies in entertainment. I’ve heard it put other ways, but your version, if applied to education as a whole, might be a pretty big deal. There’s no shortage of evidence, either; from Ivy League campuses to niche interest subcultures. Maybe “narcissism” isn’t the best word, but some sort of similar tension related to confidence seems to be associated with successful educational environments. We commonly attribute it to a one-way causal relationship, but you’re articulating the opposite. Likewise, we are used to acknowledging that ‘not giving up’, or ‘belief in one’s self’ is a necessity, but again you’re making a separate claim. You’re saying, in so many words, the confidence in and of itself defines the will as well as the curiosity. And I can’t argue with it. I want to, but I’m not sure I can.
  - thanatropism 8y ago
    I feel like the key thing about a "growth oriented mindset" or some near-equivalent buzzword is that the normal state of affairs to not know things. This is the foundation (in Socrates) of our civilization. What I've been trying to articulate here and there in the past few weeks that the object of intellectual development should be acquiring structured ways of not knowing. Rigorous mathematics is such a way: in basic analysis class you get red marks for thinking you know things you can't prove. The buddhism of the Heart Sutra is another: the Heart Sutra boils down to "we want to know big truths about emtpiness and suffering to transcend it and achieve Nirvana, but because of emptiness there's no way to know and nothing to know; therefore, recite these magic words because that's what wise men do". (It sounds like a scam, but it's really deep in how the argument is actually made in length). This of course runs counter to beliefs in "ultratechnology" such that we'll soon soon craft our own transcendence. But eh, we shall fight in the shade.

tralarpa 8y ago

I don't know. Is there really a massive benefit? The visual explanation of the Markov chains on that web site looks elegant and simple because... it is, i.e. it's not really a contribution of the visualization, it's because DTMCs are actually easy to understand. In a lecture, the teacher can easily simulate a Markov chain on a black board ("Now we are in this state, now we have to roll a die to decide whether we will go to state X or state Y"). Same for sealf-learners: In a book, you can tell the reader to start at some state X, roll a die, etc. Honestly, I find that the Markov chain website is an example how you should NOT do it. You are replacing the active intellectual process in the head of the student by some trivial animations with scrolling texts and jumping tokens which look nice and dynamic but...what are you learning? Of course, this doesn't mean that visualizations are useless, but they should be carefully designed with some learning goal in mind. Not easy! For example, the visualization could slowly gray out states that are not visited. In that way, the student could get a feeling for concepts like recurrence etc.

dyarosla 8y ago
Totally disagree. ‘Same for self-learners: In a book, you can tell the reader to start at some state X, roll a die, etc.’ -> This requires a learner to parse a procedure correctly AND understand how it relates to the topic. This is both doubly harder on the student and doubly likely to force a mistake (either in the procedure, interpretation, or both). Learning with an approach as given in the article approximately halves the work a learner must do, and increases the likelihood that the topic is correctly interpreted. Retention with this method, however, is a whole other matter- one that does correlate with the amount of mental effort required, but how much so is a matter of debate.
jinfiesto 8y ago
I both agree and disagree. I've always found an intuitive explanation like the OP to be helpful before diving into a more rigorous study of the material. It helps contextualize the things you learn subsequently.
ryeguy_24 8y ago
See my other comment below. If you could learn the same thing at the same level with less mental effort, isn't that a good thing? If the traditional teaching approach for Markov Chains required more intellectual effort, sounds like we've been explaining it poorly, no?

ttoinou 8y ago

I agree. On the other hand, it is lowering mental effort required to study the topic, and that will probably not make every student more clever, maybe more lazy, just waiting for the next youtube video to be told new stuff about maths instead of going research and study on its own.

ryeguy_24 8y ago
I think lowering the mental effort to learn something is a really good thing. It leaves room for more mental effort elsewhere.
- thanatropism 8y ago
  The thing is that 80% of the usefulness of learning is that it's learning practice. This is why you sit for so much crap in K-12 -- you're learning how to learn. This rings even more true for techies who are always chasing a moving target (nonono, object-oriented CORBA RPC, nono, it's REST, it's Kernel SVM, it's Convolutional Networks....). The effort is (almost) the whole points. If you don't keep your learning organs stimulated they will literally wither and shrink.
  - ryeguy_24 8y ago
    I agree with you but I don't think we should make things unnecessarily hard to understand if they aren't so. Do you? There will always be difficult things to learn out there but I don't think we shouldn't artificially make all things difficult if we don't have to.
    - ttoinou 8y ago
      we should make things unnecessarily hard to understand if they aren't so Who said otherwise ? I am just saying that if you as a teacher are concerned about making people/students more clever, you need to think both at vulgarization but also at the meta level where they need to actually make mental effort to incorporate ideas, maths, methodology, calculus, visualization capabilities and in general self-motivation

anonytrary 8y ago

This seems like quite an overreaction. This animation could easily confuse people who don't know as much as you do. The little ball moving back and forth -- what is that? What is doing the moving? What determines the speed of the ball? These are the questions you'll be running into when you show this to middle and high-school students who have no idea what state space is and what matrix operations are. Like most math visualizations, this one seems best for people who are already somewhat familiar with the material. I think it's a cool animation, but I'm almost sure I'm suffering from a hindsight bias. My physics professor used to say that nothing is worse than an analogy taken too literally.