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The Role of AttentionThe Role of Attentionin Teaching & Learning in Teaching & Learning

MathematicsMathematics

John MasonJohn Mason

StockholmStockholm

Biennalen 2008Biennalen 2008

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PhenomenaPhenomena

Have you ever been perplexed by Have you ever been perplexed by children’s struggle to learn to children’s struggle to learn to reason mathematically?reason mathematically?

Have you ever suddenly realised Have you ever suddenly realised that learners were ‘seeing’ that learners were ‘seeing’ something quite different to what something quite different to what you were ‘seeing’?you were ‘seeing’?

Have you ever wondered why Have you ever wondered why students make unexpected students make unexpected errors?errors?

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OutlineOutline

Some mathematical tasksSome mathematical tasksthrough which you can try to through which you can try to trap movements of your trap movements of your attentionattention

Some reflection on the structure Some reflection on the structure and movement of attentionand movement of attention

How this might affect teaching?How this might affect teaching?

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Early MultiplicationEarly Multiplication

9 3 4

36

09

27

12

16

03

04

09

12

4

1

3

2 9 3

6

7

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Treviso and Pacioli MultiplicationsHistorical Topics for the Mathematics Classroom, NCTM p134.

27

09

12

09

03

04

36

12

16 4

1

3

2 7 6

2

9

3

9 3 4

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Decoding a CalculationDecoding a Calculation

The calculation comes from an Arabic manuscript Hindu Reckoning written by Kushyar ibn-Lebban about 1000 C.E. (quoted in NCTM 1969

p133)

aa

562548254795478683983983983948 79 86

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SquaresSquares

Green area = 2 x Beige area

Triangles equal in area

O

P

X

QY

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States of AttentionStates of Attention

Holding Wholes (Gazing)Holding Wholes (Gazing)

Discerning DetailsDiscerning Details

Recognising RelationshipsRecognising Relationships

Perceiving PropertiesPerceiving Properties

Reasoning on the Basis of Reasoning on the Basis of PropertiesProperties(deducing from definitions)(deducing from definitions)

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Movement of AttentionMovement of Attention

Attention moves Attention moves – SpontaneouslySpontaneously– ReactivelyReactively– ResponsivelyResponsively

NOT sequentialNOT sequential RARELY stationaryRARELY stationary

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Attention & LearningAttention & Learning

Lesson actionsLesson actionsTasks --> Activity --> Experience --> Tasks --> Activity --> Experience -->

basis for sense-makingbasis for sense-making Participation in three worlds:Participation in three worlds:

– Physical (enactive; doing)Physical (enactive; doing)– Mental (iconic; imagining)Mental (iconic; imagining)– Verbal-Symbolic (symbolic; recording)Verbal-Symbolic (symbolic; recording)

Sense-making requires drawing Sense-making requires drawing attention OUT of immersion in tasksattention OUT of immersion in tasks

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Attention & TeachingAttention & Teaching Immersing learners in situations Immersing learners in situations

long enough for their attention to long enough for their attention to ‘move about’‘move about’

When explaining somethingWhen explaining something– Become aware of structure of own Become aware of structure of own

attention; be more overtattention; be more overt When listening to learnersWhen listening to learners

– Try to locate structure and movements Try to locate structure and movements of their attentionof their attention

When analysing learner errorsWhen analysing learner errors– Consider whether their attention is not Consider whether their attention is not

as expectedas expected When promoting reasoningWhen promoting reasoning

– Be explicit about properties assumedBe explicit about properties assumed