# 1 Rich Mathematical Tasks John Mason St Patrick’s Dublin Feb 2010.

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1 Rich Mathematical Tasks John Mason St Patrick’s Dublin Feb 2010

2 Outline  What is rich about a task? –The task format? –The task content? –The way of working on the task? –The outer, inner or meta aspects? –Correspondence between: intended, enacted & experienced

3 Seeing As ✎ Raise your hand when you can see something that is 1/3 of something; again differently again differently A ratio of 1 : 2 4/3 of something ✎ What else can you ‘see as’? ✎ What assumptions are you making?

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5 Dimensions-of-Possible- Variation Regional  Arrange the three coloured regions in order of area Generalise!

6 Doug French Fractional Parts

7 Triangle Count

8 Reading a Diagram: Seeing As … x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + x(1-x) + (1-x) 2 (1-z)xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

9 Length-Angle Shifts  What 2D shapes have the property that there is a straight line that cuts them into two pieces each mathematically similar to the original?

10 Tangential  At what point of y=e x does the tangent go through the origin?  What about y = e 2x ?  What about y = e 3x ?  What about y = e λx ?  What about y = μf(λx)?

11 Conjectures  It is the ways of thinking that are rich, not the task itself  Dimensions-of-Possible-Variation & Range-of-Permissible-Change  Specialising in order to re-Generalise  Say What You See (SWYS) & Watch What You Do (WWYD)  Self-Constructed Tasks  Using Natural Powers to –Make sense of mathematics –Make mathematical sense

12 Natural Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Organising & Characterising  Stressing & Ignoring  Distinguishing & Connecting  Assenting & Asserting

13 Mathematical Themes  Invariance in the midst of change  Doing & Undoing  Freedom & Constraint  Extending & Restricting Meaning

14 Reprise  What is rich about a task? –The task format? –The task content? –The way of working on the task? –The outer, inner or meta aspects? –Correspondence between: intended, enacted & experienced

15 Further Reading  Mason, J. & Johnston-Wilder, S. (2006 2 nd edition). Designing and Using Mathematical Tasks. St. Albans: Tarquin.  Prestage, S. & Perks, P. 2001, Adapting and Extending Secondary Mathematics Activities: new tasks for old, Fulton, London.  Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematics Education, RoutledgeFalmer, London.  Mason, J. 2002, Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing, Chichester