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

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Rich Mathematical TasksRich Mathematical Tasks

John MasonJohn Mason

St Patrick’sSt Patrick’sDublinDublin

Feb 2010Feb 2010

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OutlineOutline

What is rich about a task?What is rich about a task?– The task format?The task format?– The task content?The task content?– The way of working on the task?The way of working on the task?– The outer, inner or meta aspects?The outer, inner or meta aspects?– Correspondence between:Correspondence between:

intended, enacted & experiencedintended, enacted & experienced

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Seeing AsSeeing As

✎ Raise your hand when you can see Raise your hand when you can see something that issomething that is1/3 of something; 1/3 of something;

again differentlyagain differently

A ratio of 1 : 2A ratio of 1 : 2

4/3 of something4/3 of something

✎ What else can you ‘see as’?What else can you ‘see as’?✎ What assumptions are you making?What assumptions are you making?

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5Dimensions-of-Possible-Variation

RegionalRegional

Arrange the three coloured Arrange the three coloured regions in order of arearegions in order of area

Generalise!

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Doug French Fractional PartsDoug French Fractional Parts

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Triangle CountTriangle Count

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Reading a Diagram: Seeing As …Reading a Diagram: Seeing As …

a

a

x3 + x(1–x) + (1-x)3

x2 + (1-x)2

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)

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Length-Angle ShiftsLength-Angle Shifts

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

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TangentialTangential

At what point of y=eAt what point of y=exx does the does the tangent go through the origin?tangent go through the origin?

What about y = eWhat about y = e2x2x?? What about y = eWhat about y = e3x3x?? What about y = eWhat about y = eλxλx?? What about y = μf(λx)?What about y = μf(λx)?

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ConjecturesConjectures

It is the ways of thinking that are It is the ways of thinking that are rich, not the task itselfrich, not the task itself

Dimensions-of-Possible-Variation &Dimensions-of-Possible-Variation &Range-of-Permissible-ChangeRange-of-Permissible-Change

Specialising in order to re-Specialising in order to re-GeneraliseGeneralise

Say What You See (SWYS) Say What You See (SWYS) & Watch What You Do (WWYD)& Watch What You Do (WWYD)

Self-Constructed TasksSelf-Constructed Tasks Using Natural Powers toUsing Natural Powers to

– Make sense of mathematicsMake sense of mathematics– Make mathematical senseMake mathematical sense

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Natural PowersNatural Powers

Imagining & ExpressingImagining & Expressing Specialising & GeneralisingSpecialising & Generalising Conjecturing & ConvincingConjecturing & Convincing Organising & CharacterisingOrganising & Characterising Stressing & IgnoringStressing & Ignoring Distinguishing & ConnectingDistinguishing & Connecting Assenting & AssertingAssenting & Asserting

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Mathematical ThemesMathematical Themes

Invariance in the midst of changeInvariance in the midst of change Doing & UndoingDoing & Undoing Freedom & ConstraintFreedom & Constraint Extending & Restricting MeaningExtending & Restricting Meaning

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RepriseReprise

What is rich about a task?What is rich about a task?– The task format?The task format?– The task content?The task content?– The way of working on the task?The way of working on the task?– The outer, inner or meta aspects?The outer, inner or meta aspects?– Correspondence between:Correspondence between:

intended, enacted & experiencedintended, enacted & experienced

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Further ReadingFurther Reading

Mason, J. & Johnston-Wilder, S. (2006 2Mason, J. & Johnston-Wilder, S. (2006 2ndnd edition). edition). Designing and Using Mathematical Designing and Using Mathematical TasksTasks. St. Albans: Tarquin.. St. Albans: Tarquin.

Prestage, S. & Perks, P. 2001, Prestage, S. & Perks, P. 2001, Adapting and Adapting and Extending Secondary Mathematics Activities: Extending Secondary Mathematics Activities: new tasks for oldnew tasks for old, Fulton, London., Fulton, London.

Mason, J. & Johnston-Wilder, S. (2004). Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematics Fundamental Constructs in Mathematics EducationEducation, RoutledgeFalmer, London., RoutledgeFalmer, London.

Mason, J. 2002, Mason, J. 2002, Mathematics Teaching Mathematics Teaching Practice: a guide for university and college Practice: a guide for university and college lecturerslecturers, Horwood Publishing, Chichester, Horwood Publishing, Chichester