Learn difficult concepts with the ADEPT method (2014)
betterexplained.com
betterexplained.com
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.
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) 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.
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).
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...
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.
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...
Just like 'no taxation without representation', there should be 'no formalization without insight'.
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.
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.
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>
I really prefer the disconnected mode of a book. The feeling of insight dripping in after being steeped in one thing alone.
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.
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.
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.
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.
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...
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.
I actually prefer this visualization rather than the one with a number line with numbers going to the left of 0.
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...
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.