Here are my case studies:
Learning to hear better whether a note is sharp or flat - I am learning to play double bass, and without frets, it is important to be able to hear whether a note is sharp or flat relative to the other notes you are playing. It used to be impossible for me to tell. I had to move the note around and hear if it got worse or better. I have been working for months to improve my ear, doing exercises for a few minutes a day, and I am noticeably (though unfortunately not dramatically) better. I am not "figuring" this out. I am learning this in a totally unconscious way, and when I "know" the note is flat, I just know, I can't tell you how I know.
Similarly, when learning to ride a bike or hit a fast ball, any of the physical skills, there may be some conscious thought going on, but the skill is largely non-verbal and out of our conscious control.
But physical skills are not the only phenomenon where unconscious thought prevails. When solving problems using bar models, suddenly the sequence of steps makes sense and the relationships literally appear in front of you, relationships that were perhaps obscured by the language or mathematical symbols used to pose the problem. When performing fraction operations using visual models, or studying sequences using visual patterns, you can uncover connections and have flashes of insight that seem to come from nowhere. How were those constructed? And how permanent is that new knowledge?
So much of mathematical thinking feels like something you consciously figure out and then you should know it for a lifetime because you understand it. But this other kind of learning ("processing"? - it doesn't seem right to call it thinking) that is more gradual and less predictable, may have a bigger role in learning mathematics than we think. How would that change our teaching?