What was Godowsky's method of teaching? As everybody knows, he was reported to be "a wizard of technique" (ein Hexenmeister der Technik, as he was dubbed unanimously by the German and indeed the world press). For this reason numerous young pianists from all over the world flocked to him, mainly in the hope of getting his recipe for attaining "virtuoso technique". Alas for them! Godowsky hardly ever said a word about technique in the sense in which these youngsters understood it; all his comments during a lesson were aimed exclusively at music, at correcting musical defects in a performance, at achieving maximum logic, accurate hearing, clarity, plasticity, through a scrupulous observance and a broad interpretation of the written score.
In his class, he valued above all the real musician and approached with obvious irony those pianists whose fingers were fast and agile while their brains were slow and dull (and there were several such in my time). He would immediately lose all interest in a pupil whose hearing was inaccurate, who memorized wrong notes or showed bad taste.
Heinrich Neuhaus, The Art of Piano Playing (1958), Ch. 1. The Artistic Image of a Musical Composition