Rich Mathematical Tasks John Mason St Patricks Dublin
Rich Mathematical Tasks John Mason St Patrick’s Dublin Feb 2010 1
Outline /What – – – 2 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 ✎ 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? 3
4
Regional /Arrange three coloured regions in order of area Generalise! 5 Dimensions-of-Possible. Variation
Doug French Fractional Parts 6
Triangle Count 7
Reading a Diagram: Seeing As … x 3 + x(1–x) + (1 -x)3 x 2 z + x(1 -x) + (1 -x)2(1 -z) xyz + (1 -x)y + (1 -x)(1 -y)(1 -z) 8 x 2 + (1 -x)2 xz + (1 -x)(1 -z) yz + (1 -x)(1 -z)
Length-Angle Shifts /What 2 D 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 = e 2 x? /What about y = e 3 x? /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 – – – 14 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
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, Routledge. Falmer, London. /Mason, J. 2002, Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing, Chichester 15
- Slides: 15