Going to go on a tangent here in case other musicians are reading. I found Heinrich Neuhaus' book  (he that taught Richter and Gilels) particularly inspiring. On what "breathes life into music":
> Music is a tonal process and being a process and not an instant, or an arrested state, it takes place in time. Hence a simple logical conclusion: these two elements, tone and time, are fundamental [...] they are decisive and determine all the rest. [...]
> it is [...] frequently - and not without reason - compared to the pulse of a living organism [...]
> It is very difficult to speak of rhythmic harmony although it is extremely easy to feel it. It is irresistible. When it is achieved in a performance, it is felt by literally everyone. When I listen to Richter, very often my hand begins spontaneously to conduct. The rhythmic element in his playing is so strong, the rhythm so logical, so organized, strict and free and is so much the result of his total conception of the work he is performing that it is impossible to resist the temptation to take part in it by gesture, although such participation is somewhat ridiculous and reminds one of Faust's Du glaubst du schieben und du wirst geschoben (You think you push but you are being pushed). Strictness, coordination, discipline, harmony, sureness and mastery, this is the real freedom! With a performer such as Richter, two or three departures from strict rhythm are more effective, more expressive, more meaningful than hundreds of "rhythmic liberties" in a pianist in whom this feeling of harmony, this total concept is absent.
Two examples:  Celibidache conducting Brahms, which is terrifying like lying paralysed in the path of a very slow steamroller, and  Richter playing his favourite Schubert sonata, an interpretation that Glenn Gould, having seen it live, credits as putting him "in a trance" and making him enjoy Schubert for the first time .
> Since music is a tonal art, the most important task, the primary duty of any performer is to work on tone. [...]
> I never cease repeating what Anton Rubinstein said about the piano: "You think it is one instrument? It is a hundred instruments!" Carl Czerny [...] established that it is possible to render on the piano one hundred dynamic gradations encompassed between limits which I shall term "not yet tone" and "no longer tone". How curious that two such entirely different personalities as Rubinstein and Czerny should have arrived at the same figure of 100.
> How could we explain why a good pianist plays so well on a bad piano and a bad pianist plays badly on a good piano; why a good conductor with a bad orchestra can create an incomparably greater impression than a bad conductor with a good orchestra? [...] There is an incident in the life of Liszt which applies to this whole question of the role of tone in a piano composition. When Liszt heard for the first time Henselt, who had an extraordinary "velvet" tone, he said: Ah, j'aurais pu aussi me donner ces pattes de velours! (I, too, could have given myself these velvet paws!) For Liszt, with his immense horizon as composer-performer, the "velvet touch" was merely a detail in his technical arsenal, whereas for Henselt it was the main purpose. I write all this in order to stress once more that tone (together with rhythm) is the first and most important among other means of which a pianist should be possessed, but that it is a means, and not the purpose.
On both, as an addendum:
> I wrote the whole of this section mainly because I wanted to show how inextricably tone is bound up with rhythm and how a mistake in tone quality can result in a mistake in rhythm. Again and again: everything is part of the same entity.
 https://www.amazon.com/Art-Piano-Playing-Heinrich-Neuhaus/dp... - even if you are not a pianist, or a classical musician, it is worth a read for the deep insights - from experience and thinking - of someone who worked with some of the greatest in the 20th century.
Any competent music teacher (I write as the husband of a piano teacher) can perform music with musical expression, and thus show students examples of the beauty of music. But the very best music teachers are also intimately familiar with all the isolated subskills that build into understanding a piece of music, and controlling the performer's muscle movements, and responding to the audience in a live setting to build a coherent, musically expressive performance. My wife teaches skills such as "music mapping,"
proper hand position,
and how to tie those and many other skills together
as part of a comprehensive process of teaching making music.
The basic problem with mathematics education at the elementary school level in the United States (see my previous reply to this thread
generally commenting on the submitted article) is that elementary school mathematics teachers can do NONE of the comparable things with mathematics that a good music teacher can do with music. They cannot isolate and focus on useful techniques, they cannot put on an example performance of solving an interesting, challenging problem, and they cannot make connections between their (poor) understanding of the problems found in elementary mathematics and their students' (often better, but different) understanding of the same problems. In the United States, people mostly seek music instruction for young learners from the private enterprise system. My wife gains most of her new clients, who are crammed into her busy teaching schedule as previous clients graduate from secondary school and go off to university, from friends of current clients who are happy with her work. By contrast, an elementary school teacher in a typical government-run school in the United States teaches on a take-it-or-leave-it basis, with attendance being compulsory in default of a government-approved alternative, and funding guaranteed to the school, and thus employment guaranteed to the teacher, whether the learners learn mathematics or not. Systemic change is necessary to get mathematics taught as music is taught to elementary-age pupils in the United States.
For an eye-opening look at how elementary mathematics teachers could be prepared, and how that would help learners, see Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma.
That book is a very enjoyable--but rather shocking--read, full of information about how to teach mathematics for "Profound Understanding of Fundamental Mathematics."
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