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Motivating formal geometry
Anne WatsonMathsfest
Cork2012
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
Adolescenceidentitybelongingbeing heardbeing in chargebeing supported
feeling powerfulunderstanding the worldnegotiating authorityarguing in ways which make
adults listen
Shifts of focus in mathematics for adolescents
generalities - examplesmaking change - thinking about mechanismsmaking change - undoing changemaking change - reflecting on the resultsfollowing rules - using toolsdifferent points of view - representationsrepresenting - transforminginduction - deductionusing 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
he
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)
www.atm.org.ukThinkersQuestions & Prompts for Mathematical Thinking
Institute of Mathematics Pedagogy 2013mcs/open.ac.uk/jhm3 (Mason, Swan, Watson)
