In the previous example, the numbers and their distance from each other provide a big clue about an efficient means of solving the problem. Efficient strategies are ones that can easily be applied to given types of prob lems after an examination of the numbers involved. They are not invented anew each time a problem is presented. Instead, fluency means developing robust strategies that work efficiently for different kinds of problems. It also involves having a sense of the appropriate ballpark for the solution, keeping track of how one is solving the problem, and double-checking results.
Flexibility requires the knowledge of more than one approach to solving a class of problems.
For second graders it is essential to know that subtraction problems can be solved either by starting with the smallest number and adding up or by starting with the biggest number and subtracting. Finally, a student who is mathematically flexible can always check her accuracy by solving the problem a different way, recognizing that there are many strategies for checking accuracy, just as there are different ways of solving the problem. Fluency can only be achieved if all three ingredients are taught and learned in concert. Focusing on fluency, rather than other forms of balance, can provide a strong sense of direction for educators. To the classroom teacher, it means that a mathematical idea isn t covered, it is worked on over time until children have achieved efficiency, accuracy, and flexibility.