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Learning to Think and Reason Geometrically:

Learning to Think and Reason Geometrically:. The roles of monologue, dialogue, and multilogue and the Structure of Attention. John Mason SMC. 2007. Outline. Some assumptions Some tasks Some reflections. Some assumptions.

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Learning to Think and Reason Geometrically:

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  1. Learning to Think and ReasonGeometrically: The roles of monologue, dialogue, and multilogueand the Structure of Attention John MasonSMC 2007

  2. Outline • Some assumptions • Some tasks • Some reflections

  3. Some assumptions • Learners have displayed all necessary powers to reason geometrically • They may not be familiar with directing their attention geometrically • Geometrical Reasoning is about determining what must, may, or cannot be true in general situations

  4. Triangular Count How many triangles? What is the same andwhat different aboutthem? What variations are possible? More segments? remove-right-angle? …?

  5. Parallelism How manyangles do youneed to knowto work out allthe angles? What did you need to do withyour attention?

  6. 7 Circles Using ONLY intersections of circles How many different triangles can you find? How many different quadrilaterals can you find? What are you doing with your attention?

  7. Kite-Fold

  8. 9-Pin Triangles How many triangles? How many different triangles? What do you need to do with your attention? Geoff Faux

  9. Rectangular Layout • Say What You See! • Arrange some sheets of paper similarly, and continue • Conjecture; explain; vary Phenomenon ––> exploration; exercise; Extending awareness; Training behaviour;Harnessing emotions

  10. A A O’ P O B AP OO’ P = O AB B OA Eddy’s Theorem How are AP and ABrelated? How are AP and ABrelated now?

  11. Vecten

  12. Cosine Law

  13. Sine Law

  14. Attention • Gazing at wholes • Discerning details • Recognising relationships • Perceiving properties • Reasoning on the basis of properties

  15. Roles • Roles of Others • As audience: (their presence mediates between you and content: seeking clarity) • As source: (they initiate exposition, they direct your attention; you enter their world) • As resource: (you initiate, they listen; they enter your world) • As expert: (you use them to validate your own criteria concerning your learning) • Roles of Self • Integrating others’ ideas (practices, foci and structure of attention) into current situation and into possible actions for the future.

  16. What struck you about today? What can you imagine yourself trying in the future? Some Reflections • Notice the geometrical term: • It requires movement out of the current space into a space of one higher dimension in order to achieve it What did you notice about the role of monologue, dialogue, and multilogue for your own understanding? What did you notice yourself doing with your attention?

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