Fluency involves the relationship between the strategy chosen and the knowledge available.
The efficient use of a strategy involves making sensible choices given the numbers and operations involved.
For example, 38 + 59 = ? can be solved by splitting both numbers by place value and recombining the parts.
38 + 59 = ?
30 + 50 = 80 and 8 + 9 = 17
80 + 17 = 97
However, rounding both numbers up and compensating is a much more efficient method.
38 + 59 = ?
40 + 60 = 100
100 – 3 = 97
Strong number sense is developed by trying different strategies and then looking critically at their relative efficiencies.
Flexibility combines a student's disposition and knowledge of computation options.
Students who have the approach of creative mathematicians believe they can generate their own ideas, take risks that often involve speculation about what will work, and have a strong desire to check if the methods work.
Flexibility is encouraged when problems with multiple solution methods are posed and students are prompted to find their own ways to solve these problems.
Establishing classroom norms that value risk-taking and justification are also known to develop fluency and flexibility.
Awareness of alternative mental calculation strategies is promoted when students' strategies are named after them; for example, 'Jerry's doubling and halving method'. Naming of students' strategies gives ownership but also provides a language to share and compare methods.
For further exploration of these ideas read the articles Mentals Moving Along and Numbers + Magic = Answer: Students Explaining: Making the Most of Mental Computation available on the AAMT website.
