Flexible (or facile) stage
The flexible or facile counting stage is characterised by using number properties combined with number facts. Typically, flexible strategies can be described as using ‘this’ to work out ‘that’. A student may determine that 7 + 6 is 13 because double 6 is 12 (or double 7 is 14) and 7 + 6 is 1 more (or 1 less than double 7). However, recalling doubles alone is not sufficient to indicate that a student is able to use numbers flexibly. Flexible strategies make use of the properties of numbers and do not employ counting by ones. For example, compensation is a flexible arithmetical strategy.
Flexible use of numbers
Compensation is based on the idea of making adjustments to numbers so as to maintain a common difference in subtraction and the same total in addition. That is, the difference between two numbers stays the same as long as we add or subtract the same quantity from both numbers.
Compensation involving addition is different from compensation involving subtraction. Although it is possible to describe seven plus three as the same as eight plus two (7 + 3 = 8 + 2, adding one to the first and subtracting one from the second) as compensation, in practice this can also be thought of as an example of progressive accumulation. Compensation with addition involves redistributing the parts of the numbers in a way that is similar to balancing equations. While 38 + 17 = 40 + 15 can be thought of as compensation, this can also be described as resting on a multiple of ten using the jump strategy. Clearly, knowing combinations that make ten is part of the knowledge needed to carrying out the process of resting on tens.
Compensation involving subtraction requires adjusting both numbers by the same amount to keep the difference the same. For example, 34 – 28 can be calculated by adding two to each number to make 36 – 30. Compensation involving subtraction is the basis of the “equal addends” method of teaching the subtraction algorithm, sometimes described as “add ten, add ten”.