Motivating formal geometry Anne Watson Mathsfest Cork 2012

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

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

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

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

Shifts of focus in mathematics for adolescents  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

Proof as collaborative game Is it true that the radius of the inscribed circle of a 3,4,5 triangle has to be 1? (the audience had all the information necessary on secret notes and had to shout them out when they thought they would be helpful)

Constructions Cunning constructions Artful additions Genius drawing (a phrase coined by a 13 year old student about constructions)

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

Fantasy world rules and moves Rulekeepers ( members of the audience had statements in envelopes they could use to give moves, or rules (these are the axioms), or state consequences (theorems) to build up a fantasy world) Movers Consequencers Prompters (these people had the word ‘why?’ in their envelopes) M ( the master mathematician who could intervene to keep things on track and move them along)

f d e d g h e Consider this diagram, which is part of the full tessellation (this was the fantasy world that was built; from it you can prove many elementary theorems about angles, triangles and parallel lines)):

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

... the People’s Investigator searches for clues (the clues were pieces of squared paper that could be put together to show that the sum of three consecutive numbers is a product of three)

Thinkers Questions & Prompts for Mathematical Thinking Institute of Mathematics Pedagogy 2013 mcs/open.ac.uk/jhm3 (Mason, Swan, Watson)