1. 1 Learning to Think and Reason Geometrically: The roles of monologue, dialogue, and multilogue and the Structure of Attention 2007 John Mason SMC

2. 2 Outline  Some assumptions  Some tasks  Some reflections

3. 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. 4 Triangular Count How many triangles? What is the same and what different about them? What variations are possible? More segments? remove-right-angle? …?

5. 5 What did you need to do with your attention? Parallelism How many angles do you need to know to work out all the angles?

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

7. 7 Kite-Fold

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

9. 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. 10 How are AP and AB related now? Eddy’s Theorem O B A P O A B P O’ How are AP and AB related? AP ABAB OO’ OA =

11. 11 Vecten

12. 12 Cosine Law

13. 13 Sine Law

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

15. 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. 16 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 yourself doing with your attention? What did you notice about the role of monologue, dialogue, and multilogue for your own understanding? What can you imagine yourself trying in the future? What struck you about today?