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Rich Mathematical Tasks

Rich Mathematical Tasks. John Mason St Patrick’s Dublin Feb 2010. 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. Seeing As.

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Rich Mathematical Tasks

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  1. Rich Mathematical Tasks John Mason St Patrick’sDublin 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 A ratio of 1 : 2 4/3 of something • What else can you ‘see as’? • What assumptions are you making?

  4. Regional • Arrange the three coloured regions in order of area Generalise! Dimensions-of-Possible-Variation

  5. Doug French Fractional Parts

  6. Triangle Count

  7. Reading a Diagram: Seeing As … x2 + (1-x)2 x3 + x(1–x) + (1-x)3 x2z + 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)

  8. 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?

  9. Tangential • At what point of y=ex does the tangent go through the origin? • What about y = e2x? • What about y = e3x? • What about y = eλx? • What about y = μf(λx)?

  10. 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

  11. Natural Powers • Imagining & Expressing • Specialising & Generalising • Conjecturing & Convincing • Organising & Characterising • Stressing & Ignoring • Distinguishing & Connecting • Assenting & Asserting

  12. Mathematical Themes • Invariance in the midst of change • Doing & Undoing • Freedom & Constraint • Extending & Restricting Meaning

  13. 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

  14. Further Reading • Mason, J. & Johnston-Wilder, S. (2006 2nd 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

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