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Comment by ryeguy_24 | Hacker News Reader

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.

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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.

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.

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.

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?

I probably understand where you are coming from. But, you are may be missing the point?

> The visual explanation of the Markov chains on that web site looks elegant and simple because... it is

I strongly resonate with this statement. However, I disagree with the claim the "You are replacing the active intellectual process". In fact, it is the opposite - it is stimulating.

Let me present an anecdote which will perhaps resonate with many people. Often times, when venturing into a new topic, one tends to read the reference texts. Let us be frank - reference texts are not beginner friendly no matter how much they claim (they are reference for a reason). To get to the meat of the story, a reader has to hold lot of context from past knowledge. This is hard and often towards reaching the actual results, the reader is already exhausted. Instead, such intuitive visualizations provide context to a new reader. The next time the reader goes back to the reference text, this "lookahead" (from the visualization) is valuable. It is easier to internalize that where in the grand scheme of things a particular topic lies.

> 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.

This I believe is nitpicking.

Your intellect can be "activated" in the pursuit of a superficial knowledge of Markov Chains to tell your boss or remember in the future that it might help you with problems. EZ-learning stuff will help you then.

Or your intellect can be "activated" in the sense that further engaging it leads you to deeper, more general knowledge. Is this website doing it? I don't believe so.

"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."

You know, this is actually a good idea. As long as you didn't set any transition matrix elements to zero, the chain will have a stationary distribution that is given by the lead eigenvector of the transition matrix. If you gave each node a score when the chain passed through it, that eigenvector would manifest as the "score" accumulated by each node over a long period of time.

This is one of the cool, deep properties of the chain - and it's not "insight-free" in the same way you describe.

One of my favorite examples of this is the plinko game at the Museum of Math in NYC:

https://www.youtube.com/watch?v=GBu0sfzrGTY

It's a physical manifestation of the central limit theorem, realized by a series of balls that can go either right or left in a random walk downwards. You can even adjust the bias on the left/right choice to supply drift to the random walk. The mass of balls that accumulates in the bottom is a histogram representation of the associated limiting Gaussian curve. Very cool.

I don’t think I agree, but ultimately only feel comfortable speaking for myself.

I’m fairly new to programming so would be interested in hearing if someone else does this too...

On a walk this afternoon (for some reason), I realized I had a visual shape in my head for most design patterns. Some of them look a lot like a knot-tying diagram, for example. I’m not sure I had realized that before.

I also realized the shape representations are about as close to implementation as I would prefer to store design patterns in my mind. Beyond that, they might be limiting or, at worst, strenuous to use as guidelines. They’re surely more useful (to me) than UML or even code.

But the best part is I can tell they’ve been there all along. I wasn’t trying to make them up. I was just mapping out a project while on a walk. I was pretty relaxed at the time and maybe that’s why this came up.

For what they are, I was a little disappointed how long it took to parse the Elements of Reuseable OO Software book when I was just learning to code. And then when I had used the patterns and knew them, I found it a bit sad they were communicated through such rigid forms.

Art actually contends with the same sort of conceptual divisions. There’s the medium which is rigid and cumbersome, the form which is more pure, critical in the abstract but medium-agnostic, and the application, being the interface or the thing out in the world.

So I do agree the ability to transpose these concepts between form and implementation is what matters, but it’s the back and forth that matters, not where one begins.

This disagreement demonstrates why professional teachers are so valuable.

Sometimes the material is important enough that you want to make it as easy to understand as possible. Sometimes the material is a chance to teach your students how to problem solve. Or how to teach themselves new material. Or to help them become mathematically literate (i.e., able to read a book or paper on their own without your help).

A good course/curriculum does both. Some definitions are carefully explained at an intuitive level so that students can struggle with things beyond the defintion. Sometimes the definition is an opportunity to teach students how to go from a formal definition of an intuitive understanding. Or come up with extensions to the definition on their own. Or explore the implications of a definition on their own.

In short, are you teaching the definition of a Markov Chain as key content knowledge, or is this being presented as an exercise in mathematical creativity and mathematical problem solving? There is no "right" answer without more context; a good course will do the former with some topics and the latter with others.

So this is not a topic-level question. It's a course-level or curriculum-level question.

Interestingly, misunderstanding the fundamental tension between "explain well" and "teach people how t learn for themselves" is the source of many public debates on education. E.g., common core Mathematics. Are we teaching students how to add numbers or how to problem solve? Well, ideally both... but the debates always become an either/or.

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.

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.

> ...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.

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.

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.

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.

I think lowering the mental effort to learn something is a really good thing. It leaves room for more mental effort elsewhere.

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.

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.

   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

If you consider the brain like a muscle, then "leaving mental effort for elsewhere" doesn't really make sense. If anything, it's making it more resilient for future work.

It doesn't mean you stop learning with Markov Chains...it means you move on to the next more difficult topic.

Or you wait for the awesome teacher/youtuber to make the next topic easy to understand and grasp withing 20 minuts instead of going on on your own

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