I constantly re-iterate "Pitch, Rhythm, Fingering" when students start to learn a new piece: it helps their learning. Pitch and Rhythm alone are not enough.

My understanding of the the science of performance skills, is that you should play the music the same each time - if the fingering is different each time, then the performer never learns the piece in a deep way ("muscle memory") - they could conduct it, because they know how it goes, but they can't deliver it consistently! (I suppose in an ideal situation, one might insist that students also learn dynamics and articulation with their initial study).

But which fingering?

Apparently Claudio Arrau insisted that following the composer's fingering is a "cultural, historical, and perhaps even ethical [question]".(1)

Debussy, in the forward to the Douze Etudes, was of the opposite view. Much more pragmatic:

"One cannot impose a fingering that can suit the different formations of different hands ... One is never better served than by oneself. Let us seek our [own] fingerings!"(2)

I notice that the first book in Thomas Johnston's Read and Play Series - a sight reading course (called Beginner's Tunes) sometimes does not provide fingering at the start of the pieces - that is such a brilliant idea: it forces the student to look ahead and see the whole phrase, so they know which finger to start on.

I tell my students, that unless they have a better idea, they should at least give the supplied fingering the benefit of the doubt, and try it first. If it doesn't suit, then they should change it. On no account do I allow them to change the fingering and not mark it in. I know from experience what happens if you do: in the middle of the concert, you look at the music, and say to yourself "Oh! I'm supposed to put the 4th finger on the C." (which you've never done before!). Then all is lost: you are distracted, the fingering is wrong, you lose your concentration, and the notes.

Be warned!

(1) quoted in The Science and Psychology of Music Performance, Parncutt and McPherson, Oxford University Press, 2002, p297

(2) Debussy, preface to the Douze Etudes, 1916