The following are some of the common definitions for part
a piece or portion of a whole
an integral constituent of something ⇒ dancing is part of what we teach
an amount less than the whole; bit ⇒ they only recovered part of the money
Both part and a part can be used to refer to a discrete countable thing
We teach several skills. Dancing is ...
Actually, the person who mentioned countability was close to the mark.
It is a question of whether we can predict the size or number of parts.
If we can predict it, then we use a qualifier; otherwise, no quantifier.
Note that articles (a, an, the) are quantifiers; also words such as "some", "any", "no", etc.
There's probably a better term than "...