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You will need two blank pieces of A4 paper, and something else to write on

Outer & Inner Tasks:Outer & Inner Tasks:on being clear about on being clear about

what a mathematical task what a mathematical task is supposed to offer studentsis supposed to offer students

John MasonJohn Mason

AgderAgder

Sept 2009Sept 2009

The Open UniversityMaths Dept University of Oxford

Dept of Education

These slides will be on my website for your use by the end of the weekend

These slides will be on my website for your use by the end of the weekend

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Didactic Contract: teacher Didactic Contract: teacher perspectiveperspective

What do you think students need to What do you think students need to do in order to learn mathematics?do in order to learn mathematics?– Complete tasks?Complete tasks?– Practice to perfection?Practice to perfection?– Engage in activity, then withdraw from Engage in activity, then withdraw from

the action and consider the effects? the action and consider the effects? (Simon & Tzur)(Simon & Tzur)(Re-view, Re-flect, Look Back, Re-(Re-view, Re-flect, Look Back, Re-Construct …)Construct …)

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Didactic Contract: student Didactic Contract: student perspectiveperspective

What would your students say they What would your students say they needed to do in order to learn needed to do in order to learn mathematics?mathematics?– Do (most of) most of the tasks … get ‘the Do (most of) most of the tasks … get ‘the

answers’answers’ Students expect the teacher to Students expect the teacher to

ask/tell them things to doask/tell them things to do As a result of which, requisite As a result of which, requisite

learning will presumably take placelearning will presumably take place– Is this a reasonable theory?Is this a reasonable theory?

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IdeasIdeas

Tasks -> activity -> experienceTasks -> activity -> experience– but need to learn from the experiencebut need to learn from the experience

Actions on objects (physical, virtual, Actions on objects (physical, virtual, mental, symbolic)mental, symbolic)– But what matters is the effects: But what matters is the effects:

efficiency & super methods efficiency & super methods Explaining or accounting for Explaining or accounting for

phenomenaphenomena Task Affordances:Task Affordances:

– Possibilities, constraints and attunementsPossibilities, constraints and attunements– Outer – Inner – Meta aspects of tasksOuter – Inner – Meta aspects of tasks– Opportunity for learning from experience?Opportunity for learning from experience?

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Inner, Outer & Meta Aspects of Inner, Outer & Meta Aspects of TasksTasks

Inner: Inner: – what themes encountered, what powers what themes encountered, what powers

used?used? Outer:Outer:

– what actually asked to do?what actually asked to do?– not always what teacher imagines or not always what teacher imagines or

intends!intends! Meta:Meta:

– propensities that might come to surfacepropensities that might come to surface Working on educating awarenessWorking on educating awareness

– so as to enable actions in the futureso as to enable actions in the future

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KitesKites

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Outer, Inner & Meta TasksOuter, Inner & Meta Tasks

OuterOuter– Folding paper; discerning elements; Folding paper; discerning elements;

recognising relationshipsrecognising relationships InnerInner

– Recognising elements; perceiving Recognising elements; perceiving properties; reasoning on the basis of properties; reasoning on the basis of agreed propertiesagreed properties

– Property of A4 paperProperty of A4 paper– Anticipating; Taste of ‘surprise’Anticipating; Taste of ‘surprise’

MetaMeta– Guessing -> conjecturingGuessing -> conjecturing– Waiting to be told what to do(?)Waiting to be told what to do(?)

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Virtual PhenomenaVirtual Phenomena

Point on CircleTangent through PtChord to end of diameterReflect tangent in chord

What do you expect will happen

as point on circle moves?

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ReflectionsReflections

Dimensions of possible variation:- Point on circle- Diameter chord- Tangent fixed angle

Ranges of permissible change:- Anywhere on circle- Any chord (what about tangent?)- Any angle (0? obtuse?)

Undoing or Reversing- If a curve has ‘this’ property’, must it be a circle?

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Inner & Meta TaskInner & Meta Task

Point on CircleTangent through PtChord to end of diameterReflect tangent in chord

What do you expect will happen

as point on circle moves?

Outer Task- detect, express, explain invariance

Inner Task- add relevant elements- experience angle chasing (reasoning)- use of angle theorems- experience theme of invariance in the midst of change- experience DofPV & RofPCh

Meta Task- Keeping Going- What do I Know? What do I Want?- Surprise?

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ArealAreal

I increase the longer side by 20%and decrease the shorter side by 20%

I decrease the longer side by 20%and increase the shorter side by 20%Which has the larger

area?

To increase by 20%,

multiply bymultiply by

1.2

.8To decrease by 20%,

I have a rectangleI have a rectangle

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Outer, Inner & Meta TasksOuter, Inner & Meta Tasks

OuterOuter– Calculating with percentagesCalculating with percentages

InnerInner– Percentages as multiplicative Percentages as multiplicative

relationshipsrelationships– ‘‘seeing’ via a diagram and algebraicallyseeing’ via a diagram and algebraically

MetaMeta– Using different ways of (re)presentingUsing different ways of (re)presenting– Encountering aversion to Encountering aversion to

diagrams/arithmetic/percentages diagrams/arithmetic/percentages

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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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Outer, Inner & Meta TasksOuter, Inner & Meta Tasks

OuterOuter– Discerning details; expressing Discerning details; expressing

relationshipsrelationships InnerInner

– Encountering ‘algebra’Encountering ‘algebra’– Experiencing generalisationExperiencing generalisation– Seeing the same thing in two (or more) Seeing the same thing in two (or more)

different ways; exploitingdifferent ways; exploiting MetaMeta

– Suppressing immediate reactions; Suppressing immediate reactions; diving in too deeply too quicklydiving in too deeply too quickly

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More Or Less Altitude & AreaMore Or Less Altitude & Area

Draw a scalene triangle

more

same

less

moresameless

areaaltitud

e

Same altmore area

more altsame area

more altmore area

less altmore area

less altless area

more altless area

same altless area

less altsame area

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More Or Less Percent & ValueMore Or Less Percent & Value

50% of 40 is 20

more

same

less

moresameless

% of

Value

50% of 40 is 20

50% of 60 is 3040% of 60 is 24

60% of 60 is 36

40% of 30 is 12

60% of 30 is 20

40% of 50 is 20

40% of 40 is 16

50% of 30 is 15

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More Or Less …More Or Less …

A journey of 200 Km for 4 hrs averages 50Km/hr

more

same

less

moresameless

50Km/hr4 hrs

Same Distance

speedtime

50Km/hr4 hrs

more

same

less

moresameless

Less Distance

speedtime

50Km/hr4 hrs

more

same

less

moresameless

More Distance

speedtime

!!!

!!!

!!!

!!! !!!

!!!

!!! !!!

!!!

!!!

!!!

!!!

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Outer, Inner & Meta TaskOuter, Inner & Meta Task

OuterOuter– Fill in chartFill in chart

InnerInner– Making choicesMaking choices– Constructing own objectsConstructing own objects– Encountering multiplicative reasoningEncountering multiplicative reasoning

MetaMeta– Tendency to accept first idea that comesTendency to accept first idea that comes– Desire for tidiness, completenessDesire for tidiness, completeness

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Tasks are …Tasks are …

… … only things to doonly things to do Activity is …Activity is …

– what results from attempting taskswhat results from attempting tasks– What generates experienceWhat generates experience

What matters is what you do with What matters is what you do with that experiencethat experience

Outer, Inner & Meta aspects of tasksOuter, Inner & Meta aspects of tasks Affordances, Constraints and Affordances, Constraints and

AttunementsAttunements

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To Pursue FurtherTo Pursue Further

Google Google MathemapediaMathemapedia (a wikipedia for mathematics (a wikipedia for mathematics education)education)

http:// mcs.open.ac.uk/jhm3http:// mcs.open.ac.uk/jhm3– Where you will find these slides and Where you will find these slides and

Notes about tomorrow’s Distributed Task Notes about tomorrow’s Distributed Task Designing and Using Mathematical Designing and Using Mathematical

Tasks (Tarquin)Tasks (Tarquin)