What is Fluency?
Efficiency implies that the student does not get bogged down in too many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of sub problems and making use of intermediate results to solve the problem.
Accuracy depends on several aspects of the problem-solving process, among them careful recording, knowledge of number facts and other important number relationships, and double-checking results.
Flexibility requires the knowledge of more than one approach to solving a particular kind of problem, such as two-digit multiplication. Students need to be flexible in order to choose an appropriate strategy for the problem at hand, and also to use one method to solve a problem and another method to double-check the results.
Fluency demands more of students than does memorization of a single procedure
(Russell 2000 http://investigations.terc.edu/library/bookpapers/comp_fluency.cfm)
Caroline Ainsworth – Samples of children’s writing