Doing And Undoing
If you can do something, can you undo it? That's when you often have to be creative!
Whenever you find that you can do some calculation or solve some kind of problem, it enriches your appreciation of what you have learned if you ask yourself whether you can go backwards: whether given the answer you could re-construct the question, or even all possible questions of that type with that answer.
Doing & Undoing is not only a ubiquitous and powerful mathematical theme, but a rich source of pedagogic choices.
Main Section
If adding 3 to something gives you 8, what was the something? This is the undoing of ‘adding 3’. If adding 3 to something and then multiplying by 2 gives you 16, what was the something? Thus is the undoing of ‘adding 3 then multiplying by 2’. It proceeds by undoing the operations one by one but in reverse order, and this is an awareness that most learners who can add and multiply appreciate without having to be told. All it takes is becoming explicitly aware of what they implicitly appreciate. These ideas can be the beginning of a sequence of doings and undoings leading to solving linear equations in one unknown.
Given two numbers you can write down their sum and their product (doing). Given the sum and product, can you write down the numbers(undoing)? This is the first of a huge collection of number theory problems first set down (as far as is known) by Diophantus in about 250CE. It has appeared in most collections of books which straddle arithmetic and algebra ever since! (see Ball 2007 for a modern ICT version).
You can write down the equation of a polynomial with given roots (doing) but finding the roots of a polynomial varies from straightforward (linear) to challenging (cubic, quartic) to impossible (in general for quintics and above).
Undoing something you can do is a theme which reappears all through mathematics. It is often the source of interesting and challenging problems. Usually there are several or many possible answers to an undoing, which brings in an element of creativity. Usually the ‘doing’ is algorithmic or technique based, whereas the undoing involves choices and sometimes a deeply insightful experience of the domain in order to find answers.
Reflecting a point in the x-axis is easy to do algebraically: you change the sign of the second coordinate. Reflecting a point in other lines through the origin (for the moment) is not so easy. If you could somehow rotate the line to be the ixi axis, then do your reflection, then rotate back, you could solve the problem. This idea of ‘shifting something awkward to somewhere convenient, operating, then shifting back, makes integral use of doing (shifting one way) and undoing (shifting back), and is one of the most powerful tools in mathematics (Gardinaer 1992, Melzak 1983).
Probes & Prompts
What ‘actions’ do you carry out (in mathematics) which could be thought of as a doing. What might the undoing be?
Taking Action
Take any typical task that you might give to learners, and ask yourself what the ‘undoing’ might be. Try to characterise all possible answers to the undoing question.
Case Studies
Research Sources
Ball, B. (2007). Knowing The Answers. Mathematics teaching 200 p17-18.
Melzak, Z. (1983). Bypasses: a simple approach to complexity. New York: Wiley.
Gardiner, A. (1992). Recurring Themes in School Mathematics: Part 1 direct and inverse operations. Mathematics in School, 21(5) p5-7.
Categories
Constructs, Themes