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

2. 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?

3. 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?

4. 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

5. 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 )

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

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

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

9. 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

10. 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