In the method that I use which is the Simply Music Piano playing-based method, teachers constantly reminds students to 'go slow to go fast'. This fantastic article by Graham Fitch titled Enjoying Ultra Slow Practice is a testimony to this motto .
"If you’re serious about playing the piano, there’s no getting away from slow practice. It is a cornerstone of our work from the beginner stages right through to the advanced level, and a practice tool also used by professional pianists and seasoned virtuosos all the time. In this post, I aim to help you not only realise the importance of careful, accurate slow work but also to enjoy it fully! ...
So when do we use this sort of ultra-slow practice?
Use it when learning new pieces, in order for the brain to move faster than the fingers.
Use it for passages requiring fine motor skills that are uneven or that lack control.
Use it for places that sound dull and mechanical, exaggerating the dynamics, hairpins and balances.
Use it to maintain accuracy and finesse in pieces you already know.
Use it to command control of every single note, inflection, dynamic and expression mark, and pedal.
Use it for memory work (if you can play ultra slowly from memory, you know every atom and molecule of the musical structure).
Don’t use it when you are forming the reflexes for fast playing after the initial learning stages. Let’s say you’ve spent some time learning a fast piece slowly and you want to get it up to speed. Now is the time to lay off the slow practice for a while, even though it will feel very comfortable to go back to it each day. Resist this, and make a plan to do little bits fast.