I use two methods in working up a piece. First, yes, I do start with slow practice initially--in a mechanical way--to get the notes, avoid mistakes before committing them whenever possible, working out fingerings in difficult passages, working through accompaniment in the same hand as melody, getting the sense of articulation, rhythm, phrasing, etc. etc. In the next stage I move away from mechanics and toward musicality and musicianship in performance, making many refinements in voicing, dynamics, nuances, etc. If there are fast parts, I work at getting those up to speed, and sometimes find that the slow practice hand motions and/or fingerings that worked famously at first are not really effective at tempo. So I analyze those problems and make modifications that will be more workable for me. If I play at tempo several times and find sloppiness creeping in, then I revert to the discipline of slow practice again until it's rectified. Thus, in that sense there is a going back and forth between slow and fast as may be necessary in the circumstances. I guess everyone's approach is different, but that is how I generally practice.