Music is made up of three parts: pitch, tone, and rhythm.
The Overall Goal
My goal in teaching music is to teach all three of these parts in a synthetic and comprehensive manner. I leave no stone unturned, and my students learn to approach music in a way as to not merely play but to actually understand. Indeed, depending on the student, I usually load the first few lessons relatively heavily with theory, as they can quickly get to the point of understanding scales and chords and therefore can help find the very chords that most teachers would spoon-feed them.
My goal in teaching music is to enable the student to learn on his/her own. To this end, I avoid merely teaching them chords. It would be impossible to teach them every kind of chord that they may want to play in any case, once you include inversions, extensions, and just plain old variable voicing into the mix. But I teach students how to understand scales, how to use those to create chords, how to manipulate those chords, and then how to combine all of those to create different genres and sounds.
However, I do not confuse comprehensive with boring. Quite the opposite, in fact – I intentionally aim to make lessons fun! We laugh, we make jokes, we make fun of our mistakes. Piano is such a limitless enterprise, when teaching chords and not only sheet music, that students usually become hooked on their own!