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Fundamental ConstructsFundamental ConstructsUnderpinningUnderpinning

Pedagogic ActionsPedagogic Actionsin Mathematics Classroomsin Mathematics Classrooms

John Mason March 2009John Mason March 2009

The Open UniversityMaths Dept University of Oxford

Dept of Education

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OutlineOutline

Raise some pedagogic questionsRaise some pedagogic questions Engage in some mathematical Engage in some mathematical

thinkingthinking Use this experience to engage with Use this experience to engage with

those questionsthose questions

If you fail to prepare for your

surface,prepare for your

surface to fail

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Teaching takes place in timeLearning takes place over time

Learning & DoingLearning & Doing

What do learners need to do in What do learners need to do in order to learn mathematics?order to learn mathematics? What do they think they need to do?What do they think they need to do?

What are mathematical tasks for?What are mathematical tasks for? WWhat do learners think they are for?hat do learners think they are for?

Doing ≠ Construing

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Doing & Undoing AdditivelyDoing & Undoing Additively

What operation undoes What operation undoes ‘‘adding 3adding 3’?’?‘‘subtracting 4subtracting 4’?’?‘‘adding 3 then subtracting 4adding 3 then subtracting 4’?’?‘‘subtracting from 7subtracting from 7’?’?‘‘subtracting from 11 then subtracting from 11 then subtracting from 7subtracting from 7’?’?

(11 - )

7 - ) (7 - )11 - )

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Doing & Undoing Doing & Undoing MultiplicativelyMultiplicatively

What are the analogues for What are the analogues for multiplication?multiplication? What undoes multiplying by 3?What undoes multiplying by 3? What undoes dividing by 2?What undoes dividing by 2? What undoes dividing by 3/2?What undoes dividing by 3/2? What undoes multiplying by 3/2?What undoes multiplying by 3/2?

Now do it piecemeal!Now do it piecemeal! What undoes ‘dividing into 12’?What undoes ‘dividing into 12’?

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ReflectionReflection Doing & Undoing (mathematical theme)Doing & Undoing (mathematical theme) Don’t need particulars as test-bedDon’t need particulars as test-bed Recognising relationships but then Recognising relationships but then

perceiving them as propertiesperceiving them as properties Dimensions-of-Possible-VariationDimensions-of-Possible-Variation

Range-of-Permissible-ChangeRange-of-Permissible-Change Relationship between adding & Relationship between adding &

subtracting; between multiplying & subtracting; between multiplying & dividingdividing

You can work things out for yourselfYou can work things out for yourself Importance of listening to what is said Importance of listening to what is said

and seeing it in several different waysand seeing it in several different ways Worksheet-itisWorksheet-itis

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Some ConstructsSome Constructs Outer, Inner & Meta Task(s)Outer, Inner & Meta Task(s) Didactic transpositionDidactic transposition

– EExpert awareness xpert awareness instructions in instructions in behaviourbehaviour

Didactic contractDidactic contract Didactic tensionDidactic tension

– The more clearly the teacher specifies the The more clearly the teacher specifies the behaviourbehaviour sought, sought,the easier it is for learners to display that the easier it is for learners to display that behaviour without generating it from and behaviour without generating it from and for themselvesfor themselves

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Similarly Shapely CutsSimilarly Shapely Cuts What planar shapes have the property What planar shapes have the property

that they can be cut by a straight line into that they can be cut by a straight line into two pieces both similar to the original?two pieces both similar to the original?

Just ask for similar to

each other?

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ReflectionReflection Breaking away from the familiarBreaking away from the familiar SSwitching from edges to angles and back witching from edges to angles and back

to edges (choosing what to attend to)to edges (choosing what to attend to) Mathematical similarity: angles & ratiosMathematical similarity: angles & ratios Asking ”what are the possibilities?”Asking ”what are the possibilities?”

(analysis by cases)(analysis by cases) ReasoningReasoning Acknowledging ignorance (Mary Boole)Acknowledging ignorance (Mary Boole) Manipulating familiar diagrams in fresh Manipulating familiar diagrams in fresh

wayway ZPD: acting for yourself rather than in ZPD: acting for yourself rather than in

reaction to cue/instructionreaction to cue/instruction

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Magic Square ReasoningMagic Square Reasoning

51 9

2

4

6

8 3

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– = 0Sum( ) Sum( )

Try to describethem in words

What other configurations

like thisgive one sum

equal to another?

2

2

Any colour-Any colour-symmetric symmetric

arrangement?arrangement?

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More Magic Square ReasoningMore Magic Square Reasoning

– = 0Sum( ) Sum( )

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ReflectionReflection

What are the inner tasks?What are the inner tasks?Invariance in the midst of changeInvariance in the midst of change

Movements of attention:Movements of attention:Discerning detailsDiscerning detailsRecognising relationshipsRecognising relationshipsPerceiving these as propertiesPerceiving these as propertiesReasoning with unknown entities based Reasoning with unknown entities based

on agreed propertieson agreed properties Doing & UndoingDoing & Undoing Dealing with unspecified-unknown Dealing with unspecified-unknown

numbersnumbers

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Leibniz’s TriangleLeibniz’s Triangle

1

2

1

2

1

3

1

6

1

3

1

4

1

5

1

1

4

1

12

1

12

1

20

1

5

1

20

1

30

1

60

1

30

1

6

1

30

1

60

1

6

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ReflectionReflection

Movements of attentionMovements of attention– Discerning detailsDiscerning details– Recognising relationshipsRecognising relationships– Perceiving propertiesPerceiving properties– Reasoning on the basis of agreed Reasoning on the basis of agreed

propertiesproperties InfinityInfinity Connections (Pascal’s triangle)Connections (Pascal’s triangle)

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MGA, DTR & Worlds of MGA, DTR & Worlds of ExperienceExperience

Doing–Talking–Recording3 Worlds:

Enactive–Iconic–Symbolic

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VariationVariation

Dimensions-of-possible-variationDimensions-of-possible-variationRange-of-permissible-changeRange-of-permissible-change

Invariance in the midst of changeInvariance in the midst of change

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What are tasks for?What are tasks for?

Tasks generate activityTasks generate activity AActivity provides experience of engaging ctivity provides experience of engaging

in (mathematical) actionsin (mathematical) actions Inner task is Inner task is ……

– WWhat concepts & themes expected to hat concepts & themes expected to encounter; encounter;

– what actions expected to modify or extendwhat actions expected to modify or extend– WWhat actions to internalise for selfhat actions to internalise for self

IIn order to learn from experience, it is n order to learn from experience, it is necessary to withdraw from immersion in necessary to withdraw from immersion in actionaction– RReflection on and reconstruction of highlightseflection on and reconstruction of highlights

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Implicit Theories & Implicit Theories & ConstructsConstructs

worthy of worthy of CritiqueCritique

Doing = LearningDoing = Learning IIf I get the answers, I must be learningf I get the answers, I must be learning

The muscle metaphorThe muscle metaphor– KKeep exercising and eventually you can do eep exercising and eventually you can do

itit The Collective HypothesisThe Collective Hypothesis

– TTalking produces learningalking produces learning The Jacobs Staircase metaphorThe Jacobs Staircase metaphor

– LLearning progresses steadily and earning progresses steadily and uniformlyuniformly

Worksheets are necessary:Worksheets are necessary:– FFor managing the classroomor managing the classroom– FFor record keeping as evidence of activityor record keeping as evidence of activity– FFor learningor learning

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Darwininian MetaphorDarwininian Metaphor

Development when the Development when the organism and the organism and the environment are mutually environment are mutually challenging and when there challenging and when there are sufficient mutations to are sufficient mutations to provide variationprovide variation– EExcessive challenge leads to xcessive challenge leads to

loss of speciesloss of species– Inadequate challenge leads to Inadequate challenge leads to

loss of flexibility loss of flexibility

Birmingham moths

LearnersTeachersInstitutio

ns

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Maintaining Complexity

Taking Account of the Whole Taking Account of the Whole PsychePsyche

Enaction – Cognition – AffectEnaction – Cognition – Affect Behaviour – Awareness – EmotionBehaviour – Awareness – Emotion Doing – Noticing – FeelingDoing – Noticing – Feeling

Change ≠ doing differentlyDeveloping = enhancing and enriching being

Being mathematical with and in front of Being mathematical with and in front of learnerslearners

sso that they experience o that they experience what it is like what it is like bbeing mathematicaleing mathematical

Being

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For Access to Fundamental For Access to Fundamental ConstructsConstructs

NCETM website (Mathemapedia)NCETM website (Mathemapedia) Fundamental Constructs in Fundamental Constructs in

Mathematics Education Mathematics Education (RoutledgeFalmer)(RoutledgeFalmer)