I've been teaching a beginner's guitar class at the office with another coworker. Teaching to adults is interesting because as a kid I was satisfied without necessarily knowing the underlying foundational concepts of music (something that held back my development for a while) but adult students are quick to ask questions and dive deeper. "What do you mean play a minor chord?" is easy to explain away to a child and get the to accept an intuitive definition (to be explained more thoroughly later) but totally throws off adults who are used to having acquired deep knowledge in some other field.

As with music, math, programming, one does need a deep foundation to be good but also one needs to develop a feel by repeated effort so that your mind begins to recognize patterns.

As a kid I refused to learn music because no-one could give me a good explanation of the difference between the black piano keys and the white keys. The explanations were always circular: white notes are special because they are in C-Major; C-Major is special because it contains all the white notes.

(The correct answer is that people's hands aren't wide enough to deal with all twelve notes on the same level, and that particular subset of the notes contains a lot of consonant intervals and barely any dissonant ones.)

I suspect your correct answer is wrong.

Organ keyboards originally had only the long keys (the white keys) because they only had the natural notes of the scale. The accidentals were added later as the short keys, and were added as short keys so you could still play as though they weren't there.

Piano keys are the same because they copied the keyboard from the harpsichord, which copied it from the organ.

The reason the long keys are in C is because a piano is naturally tuned to C. Tighten all the strings a bit to bring it up to D, and all the long keys will be in the key of D.

"Tighten all the strings a bit to bring it up to D, and all the long keys will be in the key of D"

... briefly, before the piano implodes.

But what do you mean by "natural notes of the scale"? Why is it those notes that are in C and not the others?

By the "natural notes" I just mean the notes that would make up an octave in that key, like "Do, Re, Mi, Fa, Sol, La, Ti, Do" in whatever key you like. On a piano, the long (white) keys are tuned to C for that. If you start on a D and just play the white notes up, it won't sound right.

If you're asking "why is it tuned to C", it's probably because C is the key that historically was assigned the letters without any sharps or flats. There's no physical reason for this, it's just a convention (as opposed to, say, octaves, where each octave is half the frequency of the previous).

If you're asking why C got the letters without sharps or flats, I believe you have the Romans to blame for that. They did start with A for their scale without sharps or flats, but it was what we would now call "A minor". Which leaves C as the major scale.

it's defined somewhat arbitrarily but those 7 notes are the notes of the C Major scale. If you just pay them in sequence, that is the scale. Western music is based around 12 tones, but each scale only has 7 of those tones. A piano with black keys allows you to play in any key/scale (based around semitones) since all 12 tones are present but only 7 constitute the major scale.

>As a kid I refused to learn music because no-one could give me a good explanation of the difference between the black piano keys and the white keys.

Doesn't that seem really stupid to you now or do you still support that stance? Refusing to learn the entire field of music because of one irrelevant detail is pretty appalling.

It makes you distrust the people teaching you - if they're going to answer your legitimate question with circular logic, it shows that they're missing some info and not willing to admit it. So they're either lying or wrong about this thing, and who know what else they're lying or wrong about?

Similar logic puts some people off of Christianity - x is in the Bible, and we know the Bible is true because the Bible says it is true, therefore x is also true. Hmm. Someone who tries to sell you that logic is also either lying or wrong, now you can't trust either that person or the Bible.

Imagine people had refused to use the calculus because no one could really answer what an infinitesimal value really was. Sometimes you just need to make progress.

My understanding is that it was rejected by many mathematicians for precisely that reason, but that many others took up the challenge to answer the question rigorously, resulting in some major leaps in theory. On the other hand, it's good that many others were able to overlook the issue of the infinitesimals and put the techniques to lots of practical use. The lesson I take from this is that both first principles questioning and suspension of disbelief can be valuable tools for learning and discovery.

I don't think the GP (Kluny) or GGP (OscarCunningham) disagree with you on that point. If I understand them correctly, it's fine to have unresolved mysteries in your understanding, so long as you are honest about it and explicitly label it[1]. In this case, "hey, we can't quite put infinitesimals on a solid grounding (or Dirac Delta, or whatever), but it doesn't break anything else, so let's see if we can figure it out some other time."

There's a huge difference between "it's not worth our time to resolve this now" and "non-rigorous thought is okay, reality doesn't have to make sense, just do what the high-status people say, they're smarter".

[1] There's a practice in some groups to explicitly say "it works by magic" to avoid false understandings and to flag unresolved issues.

"If I understand them correctly, it's fine to have unresolved mysteries in your understanding, so long as you are honest about it and explicitly label it[1]."

Yes, exactly.

I think it's more of a rejection of the instruction, not the concept being instructed.

I had a similar experience with music instruction, everyone taught to the instrument and to the book of music, never any concepts of music theory even to the point of "What's with the sharps and flats over here by the squiggle? Why not put them where I can see them easier?" "Yeah, I don't know, it's kind of a pain, I just write them back in with pencil." "How do you write a new song?" "Oh, it's really hard, you have to be really smart like Beethoven. Don't worry about that, we have plenty of stuff to practice in the book."

Later, in self educating, I discovered key signatures, playing to those keys (and even improvising!), and that I don't have to think about sharps and flats except for accidentals and it makes so much goddamn sense, but no adult I had access to in childhood ever knew or at least none could explain something which was so fundamental, something that actually made music fun and allowed me to create a positive-feedback loop required for consistent practice.

Had I stood firmer on the grandparent's question, rejected instruction sooner (instead of just through confused burn-out), and begun to self educate sooner, I could have avoided wasting a lot of time.

A good teacher can get all of this for you. I knew all of that stuff because my mother was a violin teacher and she new musical theory in and out. Some of her first lessons are in theory before you get to sight reading so you don't have any questions like that. You had shitty teachers it sounds like and you should never reject formal education just because you had 1 bad experience.

It's funny, I "learned" piano for years as a child, but somehow missed a lot of the theory (e.g. the circle of fifths). A year ago, I got a deal on guitar lessons from a student who was looking for work for the summer. It was a crash course, pretty much taught the basic cords (G, C, D, Em, Am, etc) but we didn't have enough time to go into much theory there either. But it was enough of a crash course to start looking at the guitar and figuring out how it works, relative to my base knowledge on piano.

My partner is quite musical though (formally trained), and she's help fill in a lot of the gaps. I'm at the point now where I can fiddle around with different chord progressions and get good-sounding melodies to come out on my own. I get inversions and how they work on the guitar, etc. It's quite cool and way more interesting to me than just learning songs from a book.

If you would have rejected instruction, you would not likely be playing any music at all.

Grit is hard to come by for kids. You never find yourself interested in 'the whole field of music'. You find yourself drawn to the immediate. Seeing a piano in front of you, you might want to learn about it. You might get turned off the exact same way.

Unless your parents are directing you, childhood is a random walk through the universe of creative possibility.

Kids aren't known for being rational.

Kids haven't learned what yak shaving is yet.

(Kalid from betterexplained here)

Thanks for the comment, totally agree with moderation. My goal is to ask "Is the structure I'm providing helpful right now?".

There's an implicit assumption that "more structure is better" (especially in technical subjects that can be structured) because it seems official and defensible. It reminds me of many Wikipedia articles -- they grow in length (why not?) and become more precise (why not?) and you're left with a description inscrutable to a beginner.

Calculus, for example, gets my goat because we spend weeks on limits, continuity, L'Hospital's Rule, etc. so students can "properly" understand derivatives. (Never mind that Newton and Leibniz never used them and seemed to do ok.) Showing a circle non-rigorously transform into a triangle to measure its area (Archimedes) gets closer to the meaning of Calculus than a page of epsilons, deltas and tangent lines. One you have the idea, you put in the rigorous skeleton by showing how the formal rules describe the same result.

Appreciate the comment!

I also agree that you need a mix for optimal learning. I noticed myself that when trying to really grok something, most of the work happens after I both know the formal description and the intuitive idea behind it - it's figuring out how exactly the two fit together that leads to actual understanding.

Exactly. Terry Tao calls this having a "post-rigorous" understanding (https://terrytao.wordpress.com/career-advice/there’s-more-to...). We can't think the formal rules are sufficient. (And in the case of adjectives, it seems explicit knowledge of the formal rules aren't even necessary for fluency.)

You probably like this http://cr.yp.to/papers/calculus.pdf It develops calculus in just 12 pages without being hand-wavy.

Thanks!

This is everything I hate about language learning through pure immersion (Duolingo, Rosetta Stone, most language schools). For example, I struggled for an absurdly long time with genders in German when I started because I assumed it was like French (male, female, plural). Turns out there's 4 genders, and often the female article is the same as a the plural. Once English sentence could have solved that.

I think that learning the rules is an absolute necessity, even if your goal is to develop your intuition well enough that you don't need them. (Edit: I think this implies a little more than I meant to say. Being exposed to the rules is a necessity, but you don't necessarily have to go out and memorize them from beginning to end or anything.) They're important scaffolding. Even native speakers of a language need to be taught the formal rules if they're to speak well.

I'm going to make a lame attempt to coin a term and call this the Arch Fallacy, which is when you look at your goal and assume anything not present in it is unnecessary to achieve it. A completed arch contains no supports within it, but you'll have a tough time building one without any.

> Even native speakers of a language need to be taught the formal rules if they're to speak well.

Native speakers are taught some rules, but many are rarely taught unless a student has learning difficulties. The adjective ordering that OP mentions is a solid example. Others include aspiration of stops; voicing assimilation in words like "bets" and "beds"; the difference between "putting up" and "putting out"; why it's (sometimes) okay to say "Hey fucker" to your best friend, but not to your teacher; and the deletion of nominative arguments in subjunctive phrases.

Second language learners often (maybe always) find more rules useful, for several reasons. However, those who learn the language in highly rule-focused manner often have difficulty actually communicating in the same way that Computer Science professors are sometimes quite knowledgeable about CS theory, but unable to produce good code. Rules are good at telling you what not to do, but bad at helping you be creative.

Yeah, native speakers won't typically be exposed to the same rules, and there aren't a lot of rules taught, but they are there to an extent.

I wouldn't say that one should learn in a rule-focused manner, but neither should one learn completely by example. A mix is best, even when your goal is to have the rules fall away in the end.

The new eastern span of the Bay Bridge is a similar example. They had to build a complete temporary bridge of iron girders to support the new roadway before the tower and cables were in place. (The bridge design didn't allow the tower and cables to be built first and then the roadway suspended like a conventional suspension bridge.)

This page has a good video of the combined structure and the process of shifting the roadway load from the temporary bridge to the cables:

http://baybridgeinfo.org/projects/sas

The background image on the page is also nice, showing all three bridges together:

http://baybridgeinfo.org/sites/default/files/images/backgrou...

"contains no supports" (was "contains to supports")

Funny thing is on first reading, I didn't notice that at all and read it the way you meant. I only saw the typo on a second reading. Interesting autocorrect the brain does.

In any case I like your Arch Fallacy! I think I will use it.

> In any case I like your Arch Fallacy! I think I will use it.

I'll use it too, it's great and I don't recall any term for that particular error in thinking. I've already started spreading it (the name, not the error) among friends, hoping it will catch on.

Thanks for finding the typo. Glad you like the idea.

There are three genders in German. And four cases. Multiply them together, throw in plurals, add in uses of the indefinite article and the definitive article, and, well, it is far too much for a single sentence to explain.

But in principle I agree with you. We need a bit of book learning to go with the "just start speaking" approach.

Hence the 3-dimensional chart for determining adjective ending -- it depends on gender, case, and the word preceding it (i.e., definite article, indefinite, or nothing). Although I found a chart on this page that "reduces" it to a four-branch flow chart:

http://www.nthuleen.com/teach/grammar/adjektivendungenexpl.h...

After having lived 4+ years in Germany and struggling to learn German, I would say that the only way you can learn truly by pure immersion is from your parents. In any other case you need at least some theoretical basis because no one else is willing to spend hours trying to communicate with someone who's basically babbling.

I'm using Duolingo now and after I'm finished with it I want to get some formal classes to (hopefully) get to a B1 level. I wouldn't call Duolingo immersion learning though -- they explicitly provide grammatical rules (unless you're using the mobile app which lacks some features) and it's very difficult and frustrating to try to do it with your intuition alone.

I also liked the explanation of cases and adjective endings from this site: http://yourdailygerman.com/learn-german-online-course/

In the language tutorials I wrote, I decided to write a whole section for people who didn't want to learn grammar. (http://langintro.com/kintro/grammar/hategram.htm) My example was Russian adjectives, where a 3 by 5 chart reduces to a 5-entry table plus two spelling rules.

My best math classes have been where the professor took the intuitive approach but prefaced it with a rigorous definitions so I always had a safety net to fall back on.