I agree with tosca1 that so many students would benefit from slow practice! It does take some self-discipline. When preparing a piece, slow practice enables me to pay really close attention to all of the minute details in notation, phrasing, fingering, dynamics, etc, and while working out all the "mechanical" difficulties too. In every piece there is a section or a few measures that pose problems of execution. Thoughtful, precise, and repetitive practice is the cure for those trouble spots.
By the way, I usually do not use pedal during slow practice. That forces me to connect the notes into true legato, listen for balance between the hands, observe matters of touch, voice chords, emphasize inner lines as appropriate, etc. While pedal is the "soul of the piano" as Liszt said, the bad news is that it can also "gloss over" problems. Once glossing over becomes the norm, it is no longer practice--just sloppy playing. Once I am ready to transition to playing closer to tempo and in a far more musical way, I add the pedal (critiquing it constantly with my ear).
With the mechanical difficulties fairly well solved, I can then focus on the important interpretive matters. So what happens if one of those trouble spots rears its head again again? More slow practice, of course. So slow practice does not occur just at the first learning of a piece; rather, it's applied whenever needed while the piece remains in active repertoire.