One the central ideas of the RME (realistic mathematics education) is that models go through three stages of development.
1) beginning in a realizable context, a model is first the way in which learners represent the context. So when I gave a group of students a problem about walking the streets of Manhattan, they used a model of a number line to represent that context.
2) Then I used the open number line as a way to represent their thinking during number strings. So, if a learner told me that she solved 206 – 199 by starting from 199 and adding up to 200, then 206, I might model the problem like so:
3) Finally, models become tools for thinking. I know this is happening when learners tell me that they used “the giant number line in my head.” Actual quote from a seventh grade student.
This is my absolute favorite article about the development of models: