The language of APOS theory can also be useful in describing this observation. Students will initially exhibit an Action conception. The new formula is perceived as external, as an algorithm to be followed step by step. Repetition and reflection can enable the student to interiorize the Action into a Process, were the Action is now perceived as internal, under control of the student. As a Process the student can imagine applying the action without having to explicitly do it. Processes can be coordinated with other processes and can also be reversed. And so on. You may consider:

Arnon, I., Cotrill, J., Dubinsky, E., Octaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2013). APOS Theory: A framework for research and curriculum development in mathematics education. Springer Verlag: New York.

or an older account such as:

Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D. and Thomas, K.:1996, ‘A framework for research and development in undergraduate mathematics education’, in J. Kaput, E. Dubinsky, & A. H. Schoenfeld (Eds.), Research in collegiate mathematics education II, AMS, Providence, RI , pp. 1-32.

Thank you Michael and Rafael for the very helpful replies. I will try to learn more!

From psychological research, there are three concepts which come to my mind in response to your question:

1. There is a phenomenon which had been described by Luchins and which was called "functional fixedness" or "functional fixation". It basically means that people use and overuse a certain rule which works for them to solve a Problem. Whenever people find an algorithm which they perceive is instrumental in solving a certain task, they employ this algorithm whenever they are confronted with a problem which looks similar to the one that they first solved with this algorithm, even if the new problem could be more easily and more elegantly solved with a different and less complicated method.

2, There is a difference between so-called controlled and automatic processes. Controlled processes demand a high amount of attentional and cognitive capacity and leaves little if any room for extra thinking and adjustments. Problem solvers are so focused on using the (new) rule that they are not able to recognize any alternate routes. Once using a rule or a problem solving strategy has become automatized (= highly practiced to become a routine), less cognitive capacity is needed and the "left-over" capacity can be used for flexible adjustment. This concept goes back to the research of Shiffrin and Schneider (1977).

3. The Novice / Expert paradigm: Novices in a field (e.g., learners who have just mastered to understand and apply a new rule) are less skilled to perceive the distinctive structures in a problem and to develop a creative and adaptive solution. Research has used playing chess to show how novices apply rules in a mechanic manner and need a lot of practice to be able to perceive critical details, to think ahead and see the larger picture, including the consequences of their decision.

You should read Schoenfeld's 1987 paper that describes why naive math students have a hard time selecting appropriate mathematical strategies when solving problems.

http://www.colorado.edu/physics/EducationIssues/T&LPhys/PDFs/Schoenfeld_metacognition.pdf

Great question, John. I'm also curious about this and appreciate the helpful replies! In particular, I'm interested in how we can help students gain the kind of flexibility that more experienced problem solvers have. Is it a matter of giving them more practice with more kinds of problems, of providing explicit instruction, of modeling that flexibility, etc.?

In motor behavior there is Ann Gentile's "GIT" model (1987)- first learners need to "Get the Idea of the Task" so they need to understand the basic parameters, goals, and applications of the movement. Then, depending on if the environment is open (flexible, unpredictable) or closed (fixed, predictable), the learner will either progress from GIT to fixation then diversification (in an open environment) or they will go from GIT to diversification to fixation (in a closed environment). This was originally developed for teaching physical education, but I think it could shed some light on classroom learning, especially when a classroom environment could be manipulated to be more "open" or "closed".

I'm currently interested in naive vs. expert theory in science curricula, especially how students fall back to their naive theories in the presence of cognitive load. If anyone has any suggestions on that, I'd love to read them!