Asking Questions in order to promote Mathematical Reasoning

1. The Open University Maths Dept University of Oxford Dept of Education Asking Questionsin order to promoteMathematical Reasoning John Mason Hampshire Nov 2009

2. Elastic Mathematics (1/3) • With marks towards you, fingernails at the end marks, stretch the elastic. What do you notice? • Stretch and release the elastic so that the green mark stays still (in terms of left-right movement; you can’t stop it thickening and thinning!). • Hold one hand still and stretch the elastic. What is the same and what is different? • What is the same, and what is proportionate?

3. Elastic Moments (2/3) • Now stretch the elastic so that the red mark stays still. What are you doing with your hands? • Using the marked line on the handout, align the end marks with marks; then stretch the band until the green mark reaches the original end mark. What is the scale factor? • What is the ratio of the new (stretched) green-red segment to the original? • Start again and stretch the band so that the green mark aligns with where the red mark was when at rest. Where is the red mark now, in relation to the original marks when at rest?

4. Scaling a Drawing • Place a point P to the left of and level with the roof of the house. • Now for each point Q on the corners of the house and windows, go from P beyond Q by half the distance PQ;add in the lines • Now place a point R to the left of and below your new house. Mark points one third of the way from points on the house to R • What do you notice? Why must this be the case?

5. Revealing Shapes The coloured shapes on the right are somewhere in the white cells on the left. By clicking on a colour and then on a cell, you can test your conjectures as to which shape is where

6. 6 7 2 1 5 9 8 3 4 Sum( ) – Sum( ) = 0 Magic Square Reasoning What other configurationslike thisgive one sumequal to another? 2 Try to describethem in words 2 Any colour-symmetric arrangement?

7. Sum( ) – Sum( ) = 0 More Magic Square Reasoning

8. Questions Used (1/2) • Say What You See (SWYS) • What do you predict? What do you expect? • What stays the same? • What can you fix (hold constant)? • Can you (stretch it) so that …? • What actions are available on what objects? • What does it feel like in your body to … ? • Holding … fixed, perform action on … • By how much is the elastic scaled to perform that action? • What have you found? • Those who have not spoken much say what you think has been said about the situation by others or what you see

9. Questions used (2/2) • What’s going on when … (action performed)? • When I’ve done this action, where does (x) end up? • Where is (x) in relation to (y) now? • positions; gaps between positions • What can be said about the relation between (x) and (y) before and after the action? • Note the build up to this relational question • Make a record so that someone else can reconstruct • The relationship • What happens when action is performed • Do-Talk-Record • Each both arises from and informs the others • Catch yourself reasoning