Slow practice at the piano
I've had teachers advocate slow practice, in fact, most of my piano teachers advocate slow practicing.
But I've known a few people who advise against it: our technique works differently at slower and faster tempos. Even fingerings can work at one tempo, but not at another. One of my former teachers, Paul Barnes, used the analogy that you can't water-ski under-speed! The alternative-because these teachers don't expect fast tempos upon sight reading-is to practice small sections of a new piece at performance tempo.
I'm sympathetic to this idea. Ultimately, I often have to catch myself, and many students, in getting 'trapped' at a slow tempo. Many of my piano practice strategies are geared towards practicing in small sections, and building towards performance tempo as quickly as possible.
But I do value slow practicing for pianists, in the end. But I always try to frame it this way: Practice slow in order to think fast.
If our fingers and our brains are slow, we aren't pushing towards the goal of perfecting a piece for performance. We're not ingraining it in our minds. We aren't doing slow practice, we're doing slothful practice.
If your fingers are moving slow, but your brain is moving fast though, you're still making progress. Your brain needs the chance to make connections between new notes, to chunk information into efficient modes of memory. Practice piano so that your eyes and fingers are working at a pace that allows your brain to do that crucial work.
"Modern performers seem to regard their performances as texts rather than acts, and to prepare for them with the same goal as present-day textual editors: to clear away accretions. Not that this is not a laudable and necessary step; but what is an ultimate step for an editor should be only a first step for a performer, as the very temporal relationship between the functions of editing and performing already suggests." -Richard Taruskin, Text and Act