I was at South Dakota State University last week and I asked some future math teachers to define the word "abstract" in a sentence. All of them defined it as an adjective, not a verb. They were more aware of "abstract" as something you are, not something you do.
- A thought or idea that cannot be made tangible or concrete.
- Abstract is something that is different, non mainstream, and requires higher level thinking.
- Anything that is out of the ordinary or requires creative thought.
- A concept or idea that is not easily or not able to be put into concrete or physical terms.
- Beyond the logical ways of thinking about problems and ideas.
- Not concrete. Imaginary. Out of the box thinking.
John Mason, in a great piece called "Mathematical Abstraction as the Result of a Delicate Shift of Attention":
When the shift occurs, it is hardly noticeable and, to a mathematician, it seems the most natural and obvious movement imaginable. Consequently it fails to attract the expert's attention. When the shift does not occur, it blocks progress and makes the student feel out of touch and excluded, a mere observer in a peculiar ritual.
If they don't understand their own power, how will their students?
BTW: Also great. Frorer, et al:
… we rarely find [abstraction] explicitly discussed let alone defined. You can pick up a book entitled Abstract Algebra and not find a real discussion of abstraction as a process, or of abstractions as objects …