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Comment by jakub_g | Hacker News Reader

Never heard about the method before but I've noticed long time ago this is the way that works for me.

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.

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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:

    □²A = J

This is total gibberish to anyone who doesn't already know volumes about Geometric Algebra. It literally states that: Square squared letter A is letter J. What!?

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...

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 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'.

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 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.

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).

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.

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 :)

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.)

> (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.

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).

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...

Same, give me an example, and if I don't get it I'll dig in.

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