1 Wherein lies the Richness of Mathematical Tasks? John Mason Windsor & Datchett Feb 2008

2 Conjectures  The richness of mathematical tasks does NOT lie in the task itself  NOR does it lie in the format of interactions  It DOES lie in the teacher’s ‘being’, manifested in –teacher-learners relationships –Teacher’s mathematical awareness

3 More Conjectures  The richness of learners’ mathematical experience depends on –Opportunities to use and develop their own powers –Opportunities to make significant mathematical choices –Being in the presence of mathematical awareness

4 Conjecturing Atmosphere  Everything said is said in order to consider modifications that may be needed  Those who ‘know’ support those who are unsure by holding back or by asking informative questions

5 What is changing and what is invariant? Some Galileo Sum Ratios ,,,, … What is the same and what is different? A single task is of little interest! What variations & extensions are possible?

6 Differences Anticipating Generalising Rehearsing Checking Organising

7 Some Sums = Generalise Justify Watch What You Do Say What You See = = =

8 Remainders of the Day (1)  Write down a number which when you subtract 1 is divisible by 5  and another  Write down one which you think no- one else here will write down.

9 Remainders of the Day (2)  Write down a number which when you subtract 1 is divisible by 2  and when you subtract 1 from the quotient, the result is divisible by 3  and when you subtract 1 from that quotient the result is divisible by 4  Why must any such number be divisible by 3?

10 Remainders of the Day (3)  Write down a number which is 1 more than a multiple of 2  and which is 2 more than a multiple of 3  and which is 3 more than a multiple of 4 …………

11 Remainders of the Day (4)  Write down a number which is 1 more than a multiple of 2  and 1 more than a multiple of 3  and 1 more than a multiple of 4 …………

12 Consecutive Sums Say What You See

13 More Or Less Percent & Value 50% of something is 20 moresameless more same less % of Value 50% of 40 is 20 50% of 60 is 30 40% of 60 is 24 60% of 60 is 36 40% of 30 is 12 60% of 30 is 20 40% of 50 is 20 40% of 40 is 16 50% of 30 is 15

14 More Or Less Rectangles & Area moresameless more same fewer 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 into rectangles

15 More Or Less Whole & Part ? of 35 is 21 moresameless more same less Whole Part 3/5 of 35 is 21 3/4 of 28 is 21 6/7 of 35 is 30 3/5 of 40 is 24

16 Algebra Readings Say What You See Express symbolically

17 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

18 Worlds of Experience Material World World of Symbol s Inner World of imagery enactiveiconicsymbolic

19 Principal Foci  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

20 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

21 Some Mathematical Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Stressing & Ignoring  Organising & Characterising

22 Some Mathematical Themes  Doing and Undoing  Invariance in the midst of Change  Freedom & Constraint