What number is three more than twice as much as 7?

This looks like a fully determined traditional type of word-problem. But think of it another way.

Think of a number … any number … all numbers … et a sense of the freedom available to you.

Now think of a number that is twice as much as 7. How much freedom do you have now?

Now think of a number which is three more than 14 … how much freedom of choice have you now?

The story of seven (see Banwell, Tahta & Saunders) provides an even sharper example. Compare the tasks 3 + 4 = ? and 7 = ? + ?. In the first there is no freedom, only constraint.  Children are encouraged to see arithmetic as the getting of answers, preferably as quickly as possible, and answers are either right or wrong.  The second is an invitation to creativity, to making choices, and to becoming aware of the freedom available.  For learners familiar only with whole numbers, there is some freedom in the choice of one of the ?’s, namely numbers from 0 to 7.  For learners familiar with fractions or decimals, there are infinite choices for the first ?.  For learners familiar with negative numbers there are further choices for the first number.  (This is an example of exploring the range-of-permissible-change.)  But once the first number is chosen, the freedom suddenly disappears.

Every mathematical task can be re-interpreted as a construction task, and as an exploration of the freedom available under imposed constraints (how many solutions are there?, what is the most general solution)?).  When an unfamiliar task is encountered, it often helps to remove some of the constraints and see what freedom of choice there is then available, because by becoming aware of the class of objects that satisfy the reduced constraints, you an sometimes see how to then impose further constraints. [see also dimensions-of-possible-variation, another and another].