1 Joined Up Reflections on Drawing to a Close John Mason April 2008

2 Conjecturing Atmosphere  Everything said, is said in order to consider modifications that may be needed  Those who ‘know’ support those who are unsure, either by holding back or by asking informative questions  Those who are unsure take the opportunity to try to express, so as to clarify their thinking

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4 Experience …  One thing we don’t seem to learn from experience … … is that we don’t often seem to learn from experience alone!  A succession of experiences does not add up to an experience of that succession

5 ‘Joining up’ Prompts  What would you like to work on as a result of experiences this week? Tell someone near you, or write down something that will remind you Tell someone near you, or write down something that will remind you  Looking back, what has surprised or struck you this week? Tell someone near you, or write down something that will remind you Tell someone near you, or write down something that will remind you

6 Worlds of Experience Material World World of Symbol s Inner World of imagery enactive iconic symbolic roots trunkleaves Worlds during Modelling and Problem-Solving Aspects of Conceptual Development Modes of Attending Epistemological Stances

7 What Teachers Can Do  aim to be mathematical with and in front of learners  aim to do for learners only what they cannot yet do for themselves  focus on provoking learners to –use and develop their (mathematical) powers –encounter (mathematical) themes & heuristics –learn about themselves (inner & outer tasks) –make mathematically significant choices  direct attention, guide energies

8 Perimeter & Area: case study  Often confused  Measured differently  Different things to attend to Different ways to attend –‘Seeing’ how shapes are built up (areas) –‘Seeing’ edges

9 How Much Information?  How few rectangles needed to compose it?  Design a rectilinear region requiring –3 lengths to find the perimeter and –8 lengths to find the area  How few rectangles needed to compose it?  How much information about lengths do you need in order to work out –the perimeter? –the area?

10 More Or Less Perimeter & Area moresameless more same less are a Perimeter same perim more area more perim same area more perim more area less perim more area less perim less area more perim less area same perim less area less perim same area Draw a rectilinear figure which requires at least 4 rectangles in any decomposition

11 John Donne ( ) No task is an island, complete unto itself; every task is a piece of the whole, a part of a domain … heretofore unrecognised as a mathematics teacher!

12 Two-bit Perimeters 2a+2b What perimeters are possible using only 2 bits of information? a b

13 Two-bit Perimeters 4a+2b What perimeters are possible using only 2 bits of information? a b

14 Two-bit Perimeters 6a+2b What perimeters are possible using only 2 bits of information? a b

15 Two-bit Perimeters 6a+4b What perimeters are possible using only 2 bits of information? a b

16 Principal Foci of a lesson  core awarenesses underlying topics  familiar actions which need challenging, developing, extending  generating reflection through drawing out of immersion in activity  getting learners to make significant choices  prompting learners to use and develop their natural powers … all involve directing learners’ attention

17 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

18 Task Domains  Dimensions-of-possible-variation (what can change without method or approach changing)  Ranges-of-permissible-change (over what range can things change)  Ways of presenting tasks  Ways of interacting during activity  Ways of concluding activity

19 Some Mathematical Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Stressing & Ignoring  Organising & Characterising  Assenting & Asserting

20 Some Mathematical Themes  Doing and Undoing  Invariance in the midst of Change  Freedom & Constraint  Extending & Restricting (meaning, focus of attention, special cases, …)

21 Authority, Stances, Grounds & Warrants  Statements are true because … –They ‘work’ in the material world (empirical) –They just ‘are’ (absolute) –Teacher (Adult, text, internet, peer) ‘says so’ (social) –Mathematical Reasoning (structure)  Mathematics is important because … –Utility in the material world –Historical-cultural product –Domain for experiencing shifts in warrants –Domain for the development of natural powers etc.

22 Energies Automatic Sensitive Conscious Creative food, air, sunlight enactive cognitive Experienced as J. G. Bennett affective

23 Alfred Lord Tennyson: Ulysses I am a part of all that I have met; Yet all experience is an arch wherethro' Gleams that untravell'd world whose margin fades Forever and forever when I move. How dull it is to pause, to make an end, … … And this gray spirit yearning in desire To follow knowledge like a sinking star, Beyond the utmost bound of human thought.