1. Motivating formal geometry Anne Watson Cork 2012

2. Shifts (Watson: work in progress) Methods: from proximal, ad hoc, and sensory and procedural methods of solution to abstract concepts Reasoning: from inductive learning of structure to understanding and reasoning about abstract relations Focus of responses: to focusing on properties instead of visible characteristics - verbal and kinaesthetic socialised responses to sensory stimuli are often inadequate for abstract tasks Representations:from ideas that can be modelled iconically to those that can only be represented symbolically

3. Shifts (van Hiele levels of understanding) Visualise, seeing whole things Analyse, describing, same/different Abstraction, distinctions, relationships between parts Informal deduction, generalising, identifying properties Rigour, formal deduction, properties as new objects

4. Shifts (mentioned by Cuoco et al. but not explicitly – my analysis) Between generalities and examples From looking at change to looking at change mechanisms (functions) Between various points of view Between deduction and induction Between domains of meaning and extreme values as sources of structural knowledge

5. Adolescence  identity  belonging  being heard  being in charge  being supported  feeling powerful  understanding the world  negotiating authority  arguing in ways which make adults listen

6. Shifts of focus in mathematics  generalities - examples  making change - thinking about mechanisms  making change - undoing change  making change - reflecting on the results  following rules - using tools  different points of view - representations  representing - transforming  induction - deduction  using domains of meaning - using extreme values

7. Proof as collaborative game Is it true that the radius of the inscribed circle of a 3,4,5 triangle has to be 1?

8. Constructions Cunning constructions Artful additions Genius drawing

9. Finally Area of triangle is the sum of the areas of three triangles, each with base a side of the 3,4,5 triangle and height is the radius of the inscribed circle

10. Fantasy world rules and moves Rulekeepers Movers Consequencers Prompters M

11. f d e d g h e Consider this diagram, which is part of the full tessellation:

12. Mystery clues Bob the Banker is facing up to Peter the People’s Investigator Bob claims he had (only) three bags of other people’s banknotes; he has given it all away as exactly equal amounts to each of three charities. Bob remembers that the three totals in the three bags were consecutive numbers Peter the People’s Investigator wants to know if this is possible

13. ... the People’s Investigator searches for clues

14. anne.watson@education.ox.ac.uk www.atm.org.uk Thinkers Questions and Prompts for Mathematical Thinking Institute of Mathematics Pedagogy 2013