Before I had a guitar instructor, I would learn a new piece or exercise by breaking it into chunks. I would learn each chunk individually and then merge it with prior chunks until I could play the entire piece.
I still do this, but I’m also doing something else that’s at least as effective: I practice each new piece at a tempo slow-enough to be able to play it in the correct time from the get-go. At first, this tempo is very slow (like 30 beats per minute). This is harder to implement than it seems, because our tendency is to hurry up and learn the piece yesterday.
I started using this practice strategy after my guitar instructor told me that the best guitarists he’s known have all started out by practicing very slowly. He even suggested that the more slowly and correctly you’re willing to initially practice, the higher your eventual zenith of skill.
By implementing this strategy, I’ve witnessed a seeming paradox: practicing very slowly helps me to progress more quickly. Because I’ve always been learning piece X in the correct time, I no longer suffer from a major problem I had before: awkward pauses “baked” into the song because I was faster in some sections than in others, thereby practicing these differences into the song itself even though they didn’t belong.
This strategy of slowing down to speed up is useful in other domains, too:
Running: you want to start out slowly, both during each run (to warm up), and also when you’re just starting out as a runner (too many people make the mistake of doing too much too soon, thereby getting injured and not being able to run for weeks or months).
Strength training: lifting weights slowly (and lowering them even more slowly) may help you gain strength more quickly than otherwise, at least in the initial stages of a strength-training program. It also reduces the risk of injury.
Learning to draw: you start out very slowly, creating contour drawings, respecting negative space, assessing value, and carefully gauging perspective at every step of the way.
Mathematics/hard sciences: you don’t rush over complex material; instead, you work through it carefully, taking more time to explore ideas you haven’t fully understood.
There is a time to speed up, but that time arrives only after you’ve worked slowly enough to gain significant skill. For example, after you’ve mastered a song at a slower tempo, you can speed it up. After you’ve gained some fitness, you can consider incorporating interval training in your runs. After you’ve learned to draw with as little internal processing (stereotyping) as possible, you can start to incorporate gesture drawing into your practice sessions to increase your speed of drawing. After you’ve learned, say, a section in your physics text and have solved some problems, you can consider timing yourself. After you’ve learned a language, you can start to read more quickly to improve your overall comprehension (because questions that arise are often resolved by info you encounter later in the reading).