Do you have any learners who finish their work quickly and then make a nuisance of themselves disturbing others who have not completed the work? Do you have disaffected or disengaged learners in your classes? Do you have learners who are unmotivated mathematically?

One way to address both of these situations is to challenge learners to make use of their own creativity. Take any procedure or action that they can ‘do’. For example, multiplying whole numbers; expanding expressions of the form (x – a)(x – b); calculating 5% of numbers; finding the arithmetic mean, median and mode of a collection of numbers. Invite them to ‘undo’ those calculations.

• What numbers multiplied together make up a given number?
• What linear expressions multiplied together make up a given quadratic expression?
• Calculate a 5% discount.
• What is the smallest set of numbers that can have a prescribed arithmetic mean, median and mode?

In the process, encourage them to construct examples.

The same idea applies to any exercise or set of exercises: given the answer, make up different questions of the same type that have that same answer. How much freedom of choice do you have? Then make up similar questions that make use of the same solution method. What is the most complicated you can make? The most general? The most ‘interesting’ – whatever that might mean?

With learners who complete their work quickly, establish an implicit or explicit contract with them that they will ‘undo’ the procedure(s) that found the answers in the exercise(s). With learners who are disengaged or unmotivated, use undoing tasks to try to unleash their creativity.

If your students come up with some particularly creative ideas for ‘undoing’ do send them in along with your observations of how this worked with your class.

All the best,

John Mason

John Mason worked for the Open University Mathematics Department, where he designed two of the mathematics summer schools, contributed to numerous courses, and then helped form and run the Centre for Mathematics Education. He has written numerous books and booklets, as well as research articles and book chapters, the best known being ‘Thinking Mathematically’ (1982, 2010).