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:

Van Den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. *Educational Studies in Mathematics*, *54*(1), 9–35. doi:10.1023/B:EDUC.0000005212.03219.dc

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