A lot has been written on this subject on the internet by excellent writers and knowledgeable teachers of music, but I felt I had to post something on this subject because it is so important. In fact I would say that if there was one thing I would recommend to any student this would be it. If you want to become better faster - even in a short space of time - practice slowly.
Some weeks ago I was practicing the piano for a concert in which I had to play tunes at a faster tempo than I'm usually able to handle. In the short space of time I had, there wasn't a lot of hope for me to achieve the desired technique, but with several concentrated sessions of slow practice, I was at least able to get to a point where I was relaxed while playing at these faster speeds. I may not having been ripping around all over the piano, but I could play, and I was playing comfortable solos within my limitations. The results were undeniably better than the fumbling, tense improvisations of my previous attempts.
When you are short on time the natural thing to do when you need to improvise on a fast tune would be to crank up the metronome and hope by brute force you'd hit the mark. But does it work? I don't think so. The best you can hope for using this technique are very short term results. Try playing the same solo a few days later and most probably you are back to where you started.
So what happened when I gave those hours to slowly building up the tempo? Starting at a cool medium pace and notching the metronome up every 15 minutes or so. Suddenly everything was different. I was relaxed, I could see the notes in front of me, I was aware, conscious of the movement of my fingers, my brain had time to make musical decisions, my brain had time to cope with those unexpected twists and lines that come when you improvise, I was rhythmically solid, and I was enjoying the moment... I was making good music!
And when I came back several days later and had to play the same solo again cold? Well... it was ok. It took me time to reach where I was, but I know that it was a lot better than if I had done all that practice at speed. I wasn't starting afresh each time I reached the piano. Body and mind knew it was going in the right direction and with a couple more similar sessions I probably would hit that mark.