1 Getting Children to Make Mathematical Use of their Natural Powers The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking John Mason ‘Powers’ Norfolk Mathematics Conference Norwich Nov

2 Conjectures  Everything said here today is a conjecture … to be tested in your experience  The best way to sensitise yourself to learners … … is to experience parallel phenomena yourself  So, what you get from this session is what you notice happening inside you!

3 Tasks  Tasks promote Activity;  Activity involves Actions;  Actions generate Experience; –but one thing we don’t learn from experience is that we don’t often learn from experience alone  It is not the task that is rich … – but whether it is used richly

4 Memory  Rhythms  Counting

5 Glimpsed  Say What You Saw  Sketch what you think you saw  Compare with what others drew  How did you go about it? Draw What You Saw Say What You See

6 More or Less grids MoreSame Less More Same Less Perimeter Area With as little change as possible from the original!

7 Circle Round a Square  Imagine a Square  Now imagine a circle in the same plane as the square, so that the two are touching at a single point  Now imagine the circle rolling around the outside of the square, always staying in touch  Pay attention to the centre of the circle as it rolls  What is the path the centre takes, and how long is it?

8 Numberline Movements  Imagine you are standing on a number line somewhere facing the positive direction. (Make a note of where you are!)  Go forward three steps;  Now go backwards 5 steps  Now turn through 180°  Go backwards 3 steps  Go forwards 1 step  You should be back where you started but facing in the negative direction.

9 ThOANs  Think of a number between 0 and 10  Add six  Multiply by the number you first thought of  Add 4  Subtract twice the number you first thought of  Take the square root (positive!)  subtract the number you first thought of  You (and everybody else) are left with 2!

10 Ride & Tie  Imagine that you and a friend have a single horse (bicycle) and that you both want to get to a town some distance away.  In common with folks in the 17 th century, one of you sets off on the horse while the other walks. At some point the first dismounts, ties the horse and walks on. When you get to the horse you mount and ride on past your friend. Then you too tie the horse and walk on…  Supposing you both ride faster than you walk but at different speeds, how do you decide when and where to tie the horse so that you both arrive at your destination at the same time?

11 Ride & Tie  Imagine, then draw a diagram!   Does the diagram make sense (meet the constraints)? Seeking Relationships

12 Two Journeys  Which journey over the same distance at two different speeds takes longer: –One in which both halves of the distance are done at the specified speeds –One in which both halves of the time taken are done at the specified speeds distance time

13 Named Ratios  Now take a named ratio (eg density) and recast this task in that language  Which mass made up of two densities has the larger volume: –One in which both halves of the mass have the fixed densities –One in which both halves of the volume have the same densities?

14 Elastic Multiplication  Make a mark about 1 cm from each end of your elastic … this is your thumbnail mark  Make a mark half way between your thumbnail marks  Make marks one-third and two-thirds of the way between your thumbnail marks

15 Counter Scaling  Someone has placed 5 counters side-by-side in a line  Someone else has made a similar line with 5 counters but with one counter-width space between counters.  By what factor has the length of the original line been scaled?  How many counters would be needed so that the scale factor was 15/8? “Fence-post Reasoning” Generalise!

16 Outer & Inner Tasks  Outer Task –What author imagines –What teacher intends –What students construe –What students actually do  Inner Task –What powers might be used? –What themes might be encountered? –What connections might be made? –What reasoning might be called upon? –What personal dispositions might be challenged?

17 Powers  Every child that gets to school has already displayed the power to – imagine & express – specialise & generalise – conjecture & convince – organise and categorise  The question is … – are they being prompted to use and develop those powers? –or are those powers being usurped by text, worksheets and ethos?  In each lesson, does every child in the class have an opportunity to use (and develop) one or more powers?

18 Problem Solving Skills

19 Reflection  Tasks promote activity; activity involves actions; actions generate experience; –but one thing we don’t learn from experience is that we don’t often learn from experience alone  It is not the task that is rich –but the way the task is used  Teachers can guide and direct learner attention  What are teachers attending to? –Powers –Themes –Heuristics –The nature of their own attention

20 Mathematical Powers Those with Mathematical Powers are Super Heroes!

21 Captain CC I have the power of convincing I can prove what I think to others I have the power of conjecture I can say what I think will happen

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26 Follow Up  open.ac.uk  mcs.open.ac.uk/jhm3  Presentations  Thinking Mathematically (Pearson)  Questions & Prompts (ATM)  Learning & Doing Mathematics (Tarquin)  Developing Thinking in Algebra (Sage)