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1 The Role of Attention in Teaching & Learning Mathematics John Mason Stockholm Biennalen 2008

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2 Phenomena Have you ever been perplexed by children’s struggle to learn to reason mathematically? Have you ever suddenly realised that learners were ‘seeing’ something quite different to what you were ‘seeing’? Have you ever wondered why students make unexpected errors?

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3 Outline Some mathematical tasks through which you can try to trap movements of your attention Some reflection on the structure and movement of attention How this might affect teaching?

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4 Early Multiplication 934 3 6 0 9 2 7 1 2 1 6 0 3 0 4 0 9 1 2 4 1 3 293 6 7 2 Treviso and Pacioli Multiplications Historical Topics for the Mathematics Classroom, NCTM p134. 2 7 0 9 1 2 0 9 0 3 0 4 3 6 1 2 1 6 4 1 3 276 2 9 3 93 4

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5 Decoding 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 )

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6 Squares Green area = 2 x Beige area Triangles equal in area O P X Q Y

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7 States of Attention Holding Wholes (Gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the Basis of Properties (deducing from definitions)

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8 Movement of Attention Attention moves –Spontaneously –Reactively –Responsively NOT sequential RARELY stationary

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9 Attention & Learning Lesson actions Tasks --> Activity --> Experience --> basis for sense-making Participation in three worlds: –Physical (enactive; doing) –Mental (iconic; imagining) –Verbal-Symbolic (symbolic; recording) Sense-making requires drawing attention OUT of immersion in tasks

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10 Attention & Teaching Immersing learners in situations long enough for their attention to ‘move about’ When explaining something –Become aware of structure of own attention; be more overt When listening to learners –Try to locate structure and movements of their attention When analysing learner errors –Consider whether their attention is not as expected When promoting reasoning –Be explicit about properties assumed

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1 Asking Questions in order to promote Mathematical Reasoning John Mason East London June 2010 The Open University Maths Dept University of Oxford Dept.

1 Asking Questions in order to promote Mathematical Reasoning John Mason East London June 2010 The Open University Maths Dept University of Oxford Dept.

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