Themes Which Pervade mathematics
What connects mathematical topics together to make them belong to 'mathematics'? Mathematics often comes to learners as a smorgasbord of disparate topics, concepts and techniques with little rhyme or reason.
What makes mathematics a domain of enquiry is the persistent use of core themes. Apparently disparate topics can often be seen as examples of the use of one or more mathematical themes.
Main Section
Doing & Undoing
Many mathematical topics, not to say creatively challenging problems arise by turning a straightforward action (a doing) into an undoing, by starting with the answer, the result, and asking what other similar situations or starting conditions would get the same result. This is a form of characterising: trying to express properties which characterise a collection of objects.
Invariance in the midst of change
When something is detected that remains (relatively) stable or unchanged, despite other things changing, then you have an invariance in the midst of change.
The sum of the angles of a Euclidean triangle is invariant (180°) no matter how the triangle itself varies.
The sum of an even number of odd numbers is always even: evenness is an invariant property of the sum, no matter how the numbers themselves vary (as long as they are integers!).
It is really useful when a property is being discussed, to ask yourself what can change, and in what ways? (dimensions of possible variation).
Freedom & Constraint
Every task and exercise is an example of freedom and constraint.
For example, solve the equation 2x + 3 > 5. Start with a general or unknown number, x. It can be anything: get a sense of the freedom it has. Now consider 2x + 3. It is a shift along the number line from x. There is a constraint that 2x + 3 has to be greater than 5. So adding 3 makes it greater than 5, so 2x must be greater than 2 and so on.
For example, find a number which leaves a remainder of 1 when divided by 2, 3 and 4. Start with complete freedom: any number (but it must be an integer!). Now constrain it to leave a remainder of 1 on dividing by 2. It must be odd; of the form 2t – 1 for some t. But it also has to have a remainder of 1 on dividing by 3. So it must be of the form 3s + 1. So subtracting 1 from it makes it divisible by 2 and by 3. And so it goes. Introducing constraints one at a time, getting a sense of the freedom available after each constraint.
Fill in the boxes to make a true statement: 12 =
Start with a general number or other object of the kind sought by the task. What is the freedom it enjoys (what are its dimensions of possible variation)? Now place one of the constraints on it.
Extending & Restricting
Probes & Prompts
In what way do or could these themes play a role in the topic you are currently teaching?
Taking Action
Take one of the themes and consider how the tasks for the next topic you are going to teach could be modified or augmented so as to exploit, manifest or drawn upon that theme?
Use this as a topic for a team meeting.
Case Studies
Research Sources
Gardiner, A. 1992, Recurring Themes in School Mathematics: Part 1 direct and inverse operations Mathematics in School, 21(5) p5-7.
Gardiner, A. 1993a, Recurring Themes in School Mathematics: Part 2 Reasons and Reasoning, Mathematics in School, 23(1) p20-21.
Gardiner, A. 1993b, Recurring Themes in School Mathematics: Part 3 generalised arithmetic Mathematics in School, 22(2) p20-21.
Gardiner, A. 1993c, Recurring Themes in School Mathematics, Part 4 infinity, Mathematics in School, 22(4) p19-21.
Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematics Education, RoutledgeFalmer, London.
Mason, J. & Johnston-Wilder, S. (2006 2nd edition). Designing and Using Mathematical Tasks, Open University, Milton Keynes.
Categories
Themes