The trouble is that too many teachers, mostly piano teachers, put eye before E(ar). Notation becomes, in that case, merely a set of motor instructions: Put this finger here, that finger there, etc.
Rather than deal with actual sound the teacher proceeds, usually, to supply the sound that is supposed to be heard, usually an imaginary sound implied through over-stimulation of the [teacher's] theoretical nerve. To give the example cited at the Music Perception and Cognition conference, the Bach Minuet in G, the famous one that every beginner plays.
There is a chord in the first phrase, but only one, the opening sound, a full G major triad in the left hand. After that the left hand moves in single tones against the right hand melody. But at the conference a well-meaning graduate student was analyzing the harmonies of the minuet. What harmonies? And who besides me was objecting to this falsification of the score?
Ever since this demonstration I have been re-working the difference between implied and explicit harmony in my own ear. Two tones do not a triad make. The suggestion of a triad is only a suggestion. Imagining it as a full-blown triad distorts entirely the composer's intention, which must have been to imply rather than specify a triad, especially if the composer is Bach (pick one!) or Mozart.
Think about this simple fact: Every major and minor triad is a combination of one major third and one minor third, the difference between them being only which one is on the bottom. A minor third floating around is literally doing just that unless explicated by the missing major third that will pronounce its modality.
Implied modality is the art. What do we accomplish by presuming to fill in the missing third?
