1 Learning Mathematics as a domain for Creativity John Mason Tower Hamlets June 2008

2 Outline  Some tasks (paration)  Some reflection (post-paration)  Some pre-paration

3 Another & Another  Write down a pair of numbers whose product is 12  and another pair

4 Another & Another  Write down a pair of numbers whose product is 13  and another pair  and a pair that you think no-one else in the room will write down  and a pair that perhaps no human being has ever written down

5 Example Spaces  The examples that come to mind when you hear a word or see symbols  Dimensions of possible variation  Ranges of permissible change

6 Difference  Write down two fractions that differ by 3/4  and another pair  and a pair that make it as obscure as possible

7 Constrained Number  Write down a decimal number between 2 and 3  and which does not use the digit 5  and which does use the digit 7  and which is as close to 5/2 as possible

8 Constrained Quadrilateral  Draw a quadrilateral  with a pair of equal edges  and with a pair of perpendicular edges  and with a pair of parallel edges  How many different ones can you find?

9 Perpendicularity  Draw a quadrilateral which has both pairs of opposite sides perpendicular  Trouble? –Try just one pair of opposite sides perpendicular

10 Sentenced 37 + – 37 = 49 Make up your own like this 3 ÷ 4 = 15 ÷ Make up your own like this What is the ‘like this’ of your example?

11 Distribution  Write down five numbers whose arithmetic mean is 5 –What are the dimensions of possible variation: how much freedom?  and whose median is 6 –how much freedom now?  and whose mode is 7 –how much freedom now?

12 Trig Construction  Draw an angle whose tangent is 3/4  Draw an angle whose tangent is 2/3  Draw an angle which is the sum of your two angles. What is its tangent?  Can you describe how to do this ‘in general’? Use this to write down a formula for the tangent of the sum of two angles whose tangents are known rationals On squared paper:

13 Iteration  Write down a number between 0 and 1  add 1 and divide by 2  repeat this over and over …  What happens in the long run?

14 Iteration  Write down a positive number  Take its square root  Keep taking the square root of the result … what happens?  Did you try a number between 0 and 1?

15 Mathematics & Creativity  Creativity is a type of energy  It is experienced briefly  It can be used productively or thrown away  Every opportunity to make a significant choice is an opportunity for creative energy to flow  It also promotes engagement and interest  For example –Constructing an object subject to constraints –Constructing an example on which to look for or try out a conjecture –Constructing a counter-example to someone’s assertion