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

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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

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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!

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6 Doug French Fractional Parts

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7 Triangle Count

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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)

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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?

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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)?

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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

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

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

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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

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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

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Variation as a Pedagogical Tool in Mathematics

Variation as a Pedagogical Tool in Mathematics

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