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They are accessible pieces that even the greatest virtuosi find beautiful. In my view, this is how we should be looking at the balance we need to strike in math education.

This is why I love Glenn Gould's recordings of Bach's Two- and Three-Part Inventions. If Glenn Gould can still delight in playing these pieces, then so can I. There are technique and concepts and repertoire that learners need.

Technical proficiency is boring but somewhat mindless.

If you had to listen to anyone (including yourself) only playing Hanon's same 60 exercises day in and day out, you would undoubtedly lose your mind.

This is the Hanon and Czerny mindset shift on which we are focusing this year: elementary things that we consider from an advanced standpoint.

The order of operations becomes more sophisticated.

"Groupings" replace "parentheses" in your thinking about where to start. Sometimes they show up as parentheses, but much of the time they show up wearing a moustache or another costume.

Bach's life as a composer was inextricably bound up in his life as a teacher, and these accessible works are the "rich problems" of piano education.

If you don't understand the basics of scales and harmonies and chord progressions in Western music, you will lack the basic musicianship that is required to make sense of the piano repertoire.

You need to understand how melodic lines or voices can be woven together using harmonies and rhythms and chord progressions to build a piece of music with coherent and reproducible grammar and syntax that can be both encoded and decoded by others.

So this is the basis from which I used Kate Nowak's Speed Dating structure to solidify my Precalculus students' early understanding and integration of transformations of functions on Friday.

As I keep reminding them, the parent functions and their graphs, as well as the transformations of these basic function graphs, are the essential vocabulary development work for calculus.

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