Sadly, mathematics classes are like that as well. Instructors start throwing equations on the board, expecting us to somehow connect it all together. The best math textbook (Theory of Algebra) I ever read had little sections about the person who revealed a particular subject, why they were studying it, and how the subject is used.

Do you remember what the book was called, or the writers.

Sadly many professors fall victim to the Curse of Knowledge. It doesn't help they need to follow a tight schedule and intuition isn't something you can develop in a single lecture. I suppose self-study and repetition is the most likely solution.

>The best math textbook (Theory of Algebra) I ever read had little sections about the person who revealed a particular subject, why they were studying it, and how the subject is used.

I've found the best type of books provide motivation for concepts, how they have evolved, etc.

Take Computer Science for example, many of its concepts were area of research for decades but from a student's perspective it seems these concepts were always here instead of being constantly refined until the states they're now in.

Exactly, understanding the intentions and history behind a concept is key to achieving comprehension. I've had to teach myself nearly all of the more advanced mathematical concepts I know, and I'm finally starting to reach a point where I feel I have a nice approach toward achieving an understanding:

- Learn the definition - Learn the motivations and history - Peruse a few examples - Try to map the above to a brief synopsis that explains the concept in intuitive terms that relate to your own life. Rely on pictures.

Finally, I find it helpful sometimes to try and "deduce" identities from "first principles". e.g. assume I didn't know that n^0 = 1. How might I reach that conclusion? If I have an understanding of what exponentiation means I should be able to come up with a few different propositions (these could even be relatively informal) that make such a conclusion make sense.

My childhood was rife with mathematics teachers that focused more on rote memorization of identities instead of careful explanation of definitions and development of "intuition". There's pretty much no better way to ensure you'll produce students that dislike and suck at math for the rest of their lives than proceeding by mind-numbing rote memorization.

I think that's a great approach. If possible, you can add another step: Teaching it to someone else.

>There's pretty much no better way to ensure you'll produce students that dislike and suck at math for the rest of their lives than proceeding by mind-numbing rote memorization.

I couldn't agree more.

I recently learned what solfege actually is from a simple ChatGPT conversation after hearing about it for many years.

Any good resources that do it the right way?

I've been using Punkademic and I'm very happy with it.

Looks like I have to pay before I can see what "Tools you will need to learn Music Theory quickly and efficiently".

https://www.punkademic.com/course/music-theory-comprehensive...

That's how a paid course works?

Yeah, but I'm not going to start a course if I don't know what tools and equipment I need for it, upfront.

They have a free trial. And a YouTube. And the courses are republished on tons of platforms, each of which has some sort of preview.

Michael New on Youtube.

This site is nice https://www.musictheory.net/lessons

Music theory is not inherently complex, but the notation adds incidental complexity -- it's sort of like the QWERTY keyboard; once you're stuck on a suboptimal way of doing things it's hard to move off.

Oh my god. Music and photography are my two pet peeves. They have their own twisted vocabulary for everything that surely exists for historical reasons, but might as well be purpose-built to obstruct polymaths from connecting their concepts to other concepts in other fields.

My single largest goal when I'm teaching, is to find an appropriate analogy to a concept that my student already grasps, and bridge it to the new concept. Some fields make this super easy -- MechE and electronics, for instance, if you understand one, you're well on your way to understanding the other -- and some make it super hard.

While I agree with your goal of finding analogies when teaching new concepts, I think your first paragraph is a pretty unsympathetic take on music theory. (I don't know anything about photography.)

Music theory is descriptive, not prescriptive. Is it any surprise that something we had to invent a whole new symbolic notation for is difficult to connect to concepts we can describe in natural language?

If you want to go really deep into the philosophy of music, to understand it from "first principles" so to speak, I highly recommend Leonard Bernstein's lecture series "The Unanswered Question" - https://en.wikipedia.org/wiki/The_Unanswered_Question_(lectu... (available on YouTube).

To paraphrase Bernstein's comment from his book "The Joy of Music": we need to stop comparing Beethoven to grassy fields and mountain streams. A "major seventh" or a "plagal cadance" are themselves the description of those qualities. By all means use music as metaphor, but when you're trying to describe it you're going to be using terms specific to its domain.

I can't find the link. But I saw a minidocumentary about a person who trained his ears to listen to harsh noises. Once he attuned his ears, he could write music in those new musical notes.