Learn difficult concepts with the ADEPT method (2014)(betterexplained.com)
betterexplained.com
Learn difficult concepts with the ADEPT method (2014)
https://betterexplained.com/articles/adept-method/
49 comments
This is a disease of Mathematics in general.
Actual mathematics teaching, as seen in typical University classes typically introduces a new abstract definition, and then immediately follows that up with a number of "worked examples", either in the lecture itself or a follow-up tutorial class.
However, the reference material will typically only have the abstract definitions, but at least there will be a sequence of them building up the formalities piece by piece. In principle, a gifted mathematician could follow the trail and get to the "aha!" moment without even having to do the homework problems.
Encyclopedias like Wikipedia or Mathworld tend to just mention the most abstract, "end product" of a long chain of mathematics. Terse formulas by themselves are utterly useless to anyone who hasn't gone through the step-by-step learning process and already understands the topic completely.
It's like a historian being given a handful of Egyptian hieroglyphic characters before finding the Rosetta Stone. It's just squiggles and shapes!
An example I came across personally was the simplification of Maxwell's Equations using Geometric Algebra. In GA, it is possible to write the equations of the EM fields incredibly tersely, down to as little as:
The 5 characters in that equation pack in an awful lot: The Geometric Product. Graded vector spaces. The four dimensional spacetime of special relativity. Vector calculus. Natural physical units. Etc...
Pasting that equation into an article can't possibly cover all of that...
Actual mathematics teaching, as seen in typical University classes typically introduces a new abstract definition, and then immediately follows that up with a number of "worked examples", either in the lecture itself or a follow-up tutorial class.
However, the reference material will typically only have the abstract definitions, but at least there will be a sequence of them building up the formalities piece by piece. In principle, a gifted mathematician could follow the trail and get to the "aha!" moment without even having to do the homework problems.
Encyclopedias like Wikipedia or Mathworld tend to just mention the most abstract, "end product" of a long chain of mathematics. Terse formulas by themselves are utterly useless to anyone who hasn't gone through the step-by-step learning process and already understands the topic completely.
It's like a historian being given a handful of Egyptian hieroglyphic characters before finding the Rosetta Stone. It's just squiggles and shapes!
An example I came across personally was the simplification of Maxwell's Equations using Geometric Algebra. In GA, it is possible to write the equations of the EM fields incredibly tersely, down to as little as:
□²A = J
The 5 characters in that equation pack in an awful lot: The Geometric Product. Graded vector spaces. The four dimensional spacetime of special relativity. Vector calculus. Natural physical units. Etc...
Pasting that equation into an article can't possibly cover all of that...
I vaguely recall a presentation from Leslie Lamport on TLA+/formal modelling. I might get some details wrong here, but a member of the audience asked 'why not describe a system by examples' (implying it's easier to see some example interactions to start with), and Leslie's reply was it's not rigorous precise enough (not to mention he was visibly angry at the question). I agree it's not precise enough, but for someone starting, an example is good enough rather than choking on rigor. Sure, rigor does matter, but allow me some toehold first!
Just like 'no taxation without representation', there should be 'no formalization without insight'.
Just like 'no taxation without representation', there should be 'no formalization without insight'.
Out of all resources I have found Wikipedia to be almost uniquely useless for teaching yourself a new concept, its like "explanation by committee". On the other hand if I can understand a Wikipedia article I know I am at least intermediate in a area.
Yep. Highest recommendation for https://betterexplained.com which is at the other pole.
I've also discovered that this type of bottom up approach is faster than the top down approach.
Starting with an example gives a more concrete frame of reference.
I don't think this is just a matter of personal preference either. I believe that the way you learn a new concept is the same way as how you learn to navigate a new physical environment. We don't have a GPS in our brain for mapping, we learn locations by reference to other locations.
Starting with an example gives a more concrete frame of reference.
I don't think this is just a matter of personal preference either. I believe that the way you learn a new concept is the same way as how you learn to navigate a new physical environment. We don't have a GPS in our brain for mapping, we learn locations by reference to other locations.
Same here. I've heard somewhere, in a video about language acquisition, that we have a pattern recognition algorithm in our brains, so all we need is to see examples, and the algorithm does the rest (builds a model by itself).
This is the idea of the "language acquisition device" pioneered by Chomsky in the 60s.
Research from the 60s up to the current day has pretty conclusively shown that language acquisition only occurs through comprehensible input (this is known as the "Input Hypothesis" popularised by Stephen Krashen) and that deliberate learning of grammar and vocabulary has very little to no impact on your language skills. Basically, our brains all learn language in the same manner no matter what language and no matter what age. Children don't learn their native language through reading books on grammar, they acquire it through listening and reading stories which they can comprehend. Adults are exactly the same.
(The theories of the existence of a physical module in the brain or innate universal grammar are not widely accepted science today, but it is inarguable that the above second-order effects have been observed regardless of the mechanism of action.)
Research from the 60s up to the current day has pretty conclusively shown that language acquisition only occurs through comprehensible input (this is known as the "Input Hypothesis" popularised by Stephen Krashen) and that deliberate learning of grammar and vocabulary has very little to no impact on your language skills. Basically, our brains all learn language in the same manner no matter what language and no matter what age. Children don't learn their native language through reading books on grammar, they acquire it through listening and reading stories which they can comprehend. Adults are exactly the same.
(The theories of the existence of a physical module in the brain or innate universal grammar are not widely accepted science today, but it is inarguable that the above second-order effects have been observed regardless of the mechanism of action.)
> (1) language acquisition only occurs through comprehensible input (this is known as the "Input Hypothesis" popularised by Stephen Krashen) and that (2) deliberate learning of grammar and vocabulary has very little to no impact on your language skills
(1) is somewhat true while (2) is utter garbage. In fact, the point of (2) is to reach (1). In his book, Nation (2001) explains very well, with data from numerous studies, that cramming vocabulary is in fact an extremely optimized productive use of time and effort. And given how Nation details the dimension of vocabulary learners, he is above suspicion in respect to being biased towards rote learning.
(1) is somewhat true while (2) is utter garbage. In fact, the point of (2) is to reach (1). In his book, Nation (2001) explains very well, with data from numerous studies, that cramming vocabulary is in fact an extremely optimized productive use of time and effort. And given how Nation details the dimension of vocabulary learners, he is above suspicion in respect to being biased towards rote learning.
Yeah that part of my comment was a bit sloppy (I didn't mention vocabulary originally and then foolishly added it in a later edit). Cramming vocabulary does help make input more comprehensible, but it should be noted that cramming vocabulary doesn't (by itself) cause you to acquire that vocabulary. Graded readers have been shown to also be very effective, despite a lack of vocabulary cramming (meaning that it's not necessary -- which is the point I was unsuccessfully trying to make). I do agree that it is a productive use of time (at least in the early stages of learning a language) and in my own language learning, I am cramming vocabulary.
However, consciously learning grammar definitely does not have significant enough returns to be worth the effort (it can help somewhat in certain cases, but comprehensible input has a much larger effect).
However, consciously learning grammar definitely does not have significant enough returns to be worth the effort (it can help somewhat in certain cases, but comprehensible input has a much larger effect).
Yes, for grammar results are probably more contrasted, I can’t tell for sure since most papers I read are about vocabulary.
Personal experience however tells me that explicit teaching is a good way to know what’s going on, which is very valuable for highly grammatical languages. Again, in a perspective of "bootstrapping". I think what institutional language teaching is missing is that the needs of students, and thus the effectiveness of various methods, change with their proficiency. Like the fact bilingual dictionaries are better for beginners or intermediate learners while monolingual ones are more suited for advanced ones. Sadly, too often teachers (as well as students) are trapped in dogmatic approach where something is either good or bad on the whole spectrum...
Personal experience however tells me that explicit teaching is a good way to know what’s going on, which is very valuable for highly grammatical languages. Again, in a perspective of "bootstrapping". I think what institutional language teaching is missing is that the needs of students, and thus the effectiveness of various methods, change with their proficiency. Like the fact bilingual dictionaries are better for beginners or intermediate learners while monolingual ones are more suited for advanced ones. Sadly, too often teachers (as well as students) are trapped in dogmatic approach where something is either good or bad on the whole spectrum...
[deleted]
I would love to see that video. I have been thinking about this a lot since I don't easily learn new languages. If you do find it please paste it here or pm me :)
That's basically how I learned Japanese grammar. All I needed were 3 to 5 example sentences in an Anki and I could get an intuition for it after that.
Same, give me an example, and if I don't get it I'll dig in.
This is why I like Jeremy Howard’s approach for teaching machine learning. Begin with simple code and examples, then break apart and explain.
https://course.fast.ai/
I've been a full-time technical writer for ~8 years. These seem like solid guidelines to me.
One trick-of-the-trade that I've learned about diagrams is to avoid hardcoding English text into the diagram if you're working on an internationalized doc site. Instead, insert numbers like "(1)" and "(2)" and then provide a numbered list after the diagram explaining each number from the diagram. This makes the content easier to translate. E.g. if you hardcode English into the diagrams, you've got to translate that diagram for every language or (more realistically) your non-English docs are going to have English text hardcoded into their diagrams. Hat tip to David Friedman for teaching me that trick.
I've sometimes had technical people take issue with analogies precisely because they're not completely technically accurate. When that happens I compromise with them by adding an explicit sentence right before or after the analogy along the lines of "This is just an analogy to help you develop an intuition for the topic. It's not 100% technically correct. See X for technical details."
I think the importance and helpfulness of examples is majorly underrated across docs sites.
One trick-of-the-trade that I've learned about diagrams is to avoid hardcoding English text into the diagram if you're working on an internationalized doc site. Instead, insert numbers like "(1)" and "(2)" and then provide a numbered list after the diagram explaining each number from the diagram. This makes the content easier to translate. E.g. if you hardcode English into the diagrams, you've got to translate that diagram for every language or (more realistically) your non-English docs are going to have English text hardcoded into their diagrams. Hat tip to David Friedman for teaching me that trick.
I've sometimes had technical people take issue with analogies precisely because they're not completely technically accurate. When that happens I compromise with them by adding an explicit sentence right before or after the analogy along the lines of "This is just an analogy to help you develop an intuition for the topic. It's not 100% technically correct. See X for technical details."
I think the importance and helpfulness of examples is majorly underrated across docs sites.
It may hark back to the terse crypticness of Unix manpages.
On the other hand, php has a better balance with user contributed examples at the end of the official doc, with exceptional comments and examples eventually becoming part of the official documentation.
On the other hand, php has a better balance with user contributed examples at the end of the official doc, with exceptional comments and examples eventually becoming part of the official documentation.
Books exist. A well-written book usually condenses years and years of knowledge and experience into a few hundred pages.
Honestly the modern trend of having to make all educational content always engaging, with immediate rewards at all times seems wrong to me. Sometimes, its better to just chug through a book and boring formalization for 50 hours until you get to a point of clarity, because getting to the same knowledge via dozens of half-complete blogs and online sources takes MUCH more time in the long run. Books assume you will read the whole thing, so they build up the mental framework "breadth-first", so it seems less useful until you get to a sufficiently high depth. Once you get over the hump though, its great. With quick online learning, they give you a few good "depth-first" dives, but leave the foundational "knowledge/concept tree" underdeveloped.
Books aren't distracting, they aren't internet enabled and make it easy to just sit down and go through page by page. Online stuff is usually a great supplement if you miss some intuition but it hasn't quite replaced books. Everyone is looking for some way to make the fundamental act of paying attention and learning easier.. but paying attention IS difficult. That is why its valuable.
</oldmanrant>
Honestly the modern trend of having to make all educational content always engaging, with immediate rewards at all times seems wrong to me. Sometimes, its better to just chug through a book and boring formalization for 50 hours until you get to a point of clarity, because getting to the same knowledge via dozens of half-complete blogs and online sources takes MUCH more time in the long run. Books assume you will read the whole thing, so they build up the mental framework "breadth-first", so it seems less useful until you get to a sufficiently high depth. Once you get over the hump though, its great. With quick online learning, they give you a few good "depth-first" dives, but leave the foundational "knowledge/concept tree" underdeveloped.
Books aren't distracting, they aren't internet enabled and make it easy to just sit down and go through page by page. Online stuff is usually a great supplement if you miss some intuition but it hasn't quite replaced books. Everyone is looking for some way to make the fundamental act of paying attention and learning easier.. but paying attention IS difficult. That is why its valuable.
</oldmanrant>
No. Books were the most cost-effective medium for transfering knowledge for a long time, but that's in the past. They were never the most efficient - a few hours of one on one chat with domain expert, possibly with a whiteboard (or, failing that, some sand and a stick) at hand, could easily be an equivalent of ten times as much time of reading a book. Now that there's negligible difference in distribution cost between a flat text and a full-blown multimedia presentation, sticking with text is simply a waste of time.
I agree that learning directly from an expert is much more efficient. A multimedia presentation can be just as good as a book or better. Its just that as of today, I haven't seen many websites that match the depth and completeness of books yet.
3blue1brown is awesome, many people are writing linear algebra ebooks, quantum country looks great for quantum computation but as of _now_, none are complete or comprehensive the way a textbook is.
3blue1brown is awesome, many people are writing linear algebra ebooks, quantum country looks great for quantum computation but as of _now_, none are complete or comprehensive the way a textbook is.
> I haven't seen many websites that match the depth and completeness of books yet.
Give it time. We're not even one generation into web as a viable knowledge distribution platform. Authors who wrote books for their entire lives are not going to learn to create good presentations overnight, if at all. That's ok, the following generations, native to interactive, engaging, immersive ways of presenting knowledge, will fill that gap with time.
The potential of the medium, however, is incredible and easily visible today. Better Explained is one of the greatest examples (in my experience). It will only get better from now on. Flat text won't ever disappear, I hope, as there are many subjects where text is the most suitable way of presenting knowledge, like in liberal arts. Most of science, however, will move to interactive presentations, simply because the time/knowledge gained ratio is much better in this case.
Give it time. We're not even one generation into web as a viable knowledge distribution platform. Authors who wrote books for their entire lives are not going to learn to create good presentations overnight, if at all. That's ok, the following generations, native to interactive, engaging, immersive ways of presenting knowledge, will fill that gap with time.
The potential of the medium, however, is incredible and easily visible today. Better Explained is one of the greatest examples (in my experience). It will only get better from now on. Flat text won't ever disappear, I hope, as there are many subjects where text is the most suitable way of presenting knowledge, like in liberal arts. Most of science, however, will move to interactive presentations, simply because the time/knowledge gained ratio is much better in this case.
Agree++.
I really prefer the disconnected mode of a book. The feeling of insight dripping in after being steeped in one thing alone.
I really prefer the disconnected mode of a book. The feeling of insight dripping in after being steeped in one thing alone.
This website's got some great explanations for topics that a lot of people get hung up on. Here's one of their articles on Flux. [1] I was curious how this mental model compares to other ones I had previously encountered and found this article that compares each. [2]
[1] https://betterexplained.com/articles/flux/
[2] https://tomroth.com.au/learning/
[1] https://betterexplained.com/articles/flux/
[2] https://tomroth.com.au/learning/
I've read Tom Roth's article and have to say it's very well written and insightful. Thanks for sharing!
This is actually really good advice. I've tended to try a formal description of something first, in case the other party knows the jargon, but it's been surprising to me how quickly you can shut someone down by throwing words at them they don't understand.
ADEPT method is like using "Uber for X" to explain an idea. Here, I used ADEPT method to explain the ADEPT method.
You used the A part but didn't cover the DEPT part.
That's the idea of analogy, isn't it? It invokes the listener to ask for more information, so that you can explain the DEPT part aka "the pitch deck".
This is the way that I wish more teachers taught. I love finding analogies or visual diagrams for CS concepts to share with my students.
Request: can others share links to books, courses, videos, articles or other resources that employ incredibly thoughtful pedagogy in explaining, rather than diving into the nitty-gritty details immediately? I'd love to start a collection of examples to share with others.
Request: can others share links to books, courses, videos, articles or other resources that employ incredibly thoughtful pedagogy in explaining, rather than diving into the nitty-gritty details immediately? I'd love to start a collection of examples to share with others.
This is a book on Linear Algebra I am working on: https://moallabbad.github.io/Intuitive-Guide-to-Linear-Algeb...
I am going through somewhat lengthy explanations and using interactive graphs in 2D and 3D. My goal is to introduce things gradually and provide different perspectives to increase the chance of at least one clicking.
It’s in no way complete. In fact, it currently only has one chapter. But I am actively working on it. Hope you like it.
I am going through somewhat lengthy explanations and using interactive graphs in 2D and 3D. My goal is to introduce things gradually and provide different perspectives to increase the chance of at least one clicking.
It’s in no way complete. In fact, it currently only has one chapter. But I am actively working on it. Hope you like it.
This looks very cool! Is there a way to subscribe to updates?
In my experience the most difficult for us techies to lay down to somebody outside is the analogy. Is the make or break of entire explanation. And usually I fail 75% of the time right there at the start. Some recover later when I go with examples, but most of them are lost from beginning. Yeah, I know, I'd make a terrible teacher.
I think the key here is the analogy is just a warm up for the actual full detailed explanation. Where I find analogies always fail is in journalistic tech books that just use analogies to explain concepts but never get down to low level details and leave the reader with a vague feeling of understanding something, but if you asked them about any of the implications of the model explained they can't state anything.
I understood nearly zero of all mathematics (beyond the elementary school stuff) before I've found betterexplained and some relevant YouTube videos. I could learn formulae and some typical solutions by heart to substitute values and solve some exam problems but hardly ever understood them. I felt school/college math pure waste of time as I didn't actually understand anything. Things like betterexplained were a blast that opened an entire new beautiful exciting world to me instantly.
Back when I was taught matrix multiplication, I was told matrix is "just a table of numbers", and in order to "multiply" matrix to another, you walk from left to right, and on the other matrix, you go top to bottom. And th... (muffled voices). I thought it was game invented to pracitce more integer multiplication and improve concentation.
Later, during college, I see matrices again! I went, "hang on, I thought... that wasn't a game after all?!".
Decades later I discover 3b1b and Pavel Grinfeld [1] videos. My fist emotion was anger: "This is what you told me 'just a table of numbers'?". What a cruel joke!
[1] https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS1..., https://www.youtube.com/watch?v=I04GHpKc-WQ&list=PLlXfTHzgMR..., https://www.youtube.com/watch?v=odV3oJOpE8s&list=PLlXfTHzgMR...
Later, during college, I see matrices again! I went, "hang on, I thought... that wasn't a game after all?!".
Decades later I discover 3b1b and Pavel Grinfeld [1] videos. My fist emotion was anger: "This is what you told me 'just a table of numbers'?". What a cruel joke!
[1] https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS1..., https://www.youtube.com/watch?v=I04GHpKc-WQ&list=PLlXfTHzgMR..., https://www.youtube.com/watch?v=odV3oJOpE8s&list=PLlXfTHzgMR...
Thank you very much for the links. Please share any other cool educational materials you know.
Negative numbers were hardly distrusted in Indian mathematics - Brahmagupta in the 6th Century just asked called a negative number as "debt". (see for example, [1])
The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact.
[1] https://nrich.maths.org/5961
The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact.
[1] https://nrich.maths.org/5961
> The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact.
Don't sweat it, here's a rough proof. First a proof that a negative times a positive is negative:
Then apply the same idea but with a negative a:
In case you're interested, the proof that the additive element is a multiplicative absorbing element (a×0 = 0, for all a) is also fairly trivial[1].
The above does somewhat implicitly assume that 0 and -0 are equal (or at least that (-a)×0 = (a)×0 = 0, which implies that 0 and -0 are equal).
[1]: https://en.wikipedia.org/wiki/Additive_identity#The_additive...
Don't sweat it, here's a rough proof. First a proof that a negative times a positive is negative:
a*0 = 0 [Assumed but you can prove this too.]
a*(b-b) = 0 [Definition of 0.]
a*b + a*(-b) = 0 [Distributive law.]
Then apply the same idea but with a negative a:
(-a)*0 = 0 [Assumed but you can prove this too.]
(-a)*(b-b) = 0 [Definition of 0]
(-a)*b + (-a)*(-b) = 0 [Distributive law.]
-(a*b) + (-a)*(-b) = 0 [From above.]
(-a)*(-b) + -(a*b) = 0 [Commutative addition to rearrange to X + -X = 0 form.]
In case you're interested, the proof that the additive element is a multiplicative absorbing element (a×0 = 0, for all a) is also fairly trivial[1].
The above does somewhat implicitly assume that 0 and -0 are equal (or at least that (-a)×0 = (a)×0 = 0, which implies that 0 and -0 are equal).
[1]: https://en.wikipedia.org/wiki/Additive_identity#The_additive...
The 'negative' on a number line is just a visualization. If you instead, think of the 'negation' as direction (as in a vector), it becomes easier to understand why multiplying two negative numbers makes the answer positive (You've basically turned around twice).
I actually prefer this visualization rather than the one with a number line with numbers going to the left of 0.
I actually prefer this visualization rather than the one with a number line with numbers going to the left of 0.
Thanks, this was helpful.
To better understand this, and see how complex numbers can be viewed the same way, go through this article, especially up to the first set of slides: https://acko.net/blog/how-to-fold-a-julia-fractal/
Say you have a $100 debt, so your net worth is -$100. If I take that $100 debt away from you, your net worth goes up by $100. -$100×-1 = +$100: a negative times a negative is a positive.
A debt is a negative quantity, and me taking something from you changes your quantity of whatever it was by a negative amount (if I take one apple from you, you gain -1 apples), so a negative times a negative is a positive.
If you have 3 $100 debts. If I take two of those $100 debts away from you, your net worth goes up by -$100×-2 = $200.
A debt is a negative quantity, and me taking something from you changes your quantity of whatever it was by a negative amount (if I take one apple from you, you gain -1 apples), so a negative times a negative is a positive.
If you have 3 $100 debts. If I take two of those $100 debts away from you, your net worth goes up by -$100×-2 = $200.
[deleted]
If curious see also
2016 https://news.ycombinator.com/item?id=12031893
2016 https://news.ycombinator.com/item?id=12031893
[deleted]
This is tangential but does anyone have a "better explanation" for topology?
What are some more sources where I can find things explained in this manner?
[deleted]
When someone starts with definition I get bored. I want to start from example and use case and then I will built a mental formalization on my own. Starting from formalization I just can't be bothered.
You can see the diff e.g. how Wikipedia and Investopedia articles are written.
Each Investopedia article shows an example and repeats some key things multiple times. Whereas trying to learn any economics or maths stuff from Wikipedia is almost impossible. It can only be a reference if you already know the thing and just need to double-check the formula.