1. 1 You will need two blank pieces of A4 paper, and something else to write on Outer & Inner Tasks: on being clear about what a mathematical task is supposed to offer students John Mason Agder Sept 2009 The Open University Maths Dept University of Oxford Dept of Education These slides will be on my website for your use by the end of the weekend

2. 2 Didactic Contract: teacher perspective  What do you think students need to do in order to learn mathematics? –Complete tasks? –Practice to perfection? –Engage in activity, then withdraw from the action and consider the effects? (Simon & Tzur) (Re-view, Re-flect, Look Back, Re-Construct …)

3. 3 Didactic Contract: student perspective  What would your students say they needed to do in order to learn mathematics? –Do (most of) most of the tasks … get ‘the answers’  Students expect the teacher to ask/tell them things to do  As a result of which, requisite learning will presumably take place –Is this a reasonable theory?

4. 4 Ideas  Tasks -> activity -> experience –but need to learn from the experience  Actions on objects (physical, virtual, mental, symbolic) –But what matters is the effects: efficiency & super methods  Explaining or accounting for phenomena  Task Affordances: –Possibilities, constraints and attunements –Outer – Inner – Meta aspects of tasks –Opportunity for learning from experience?

5. 5 Inner, Outer & Meta Aspects of Tasks  Inner: –what themes encountered, what powers used?  Outer: –what actually asked to do? –not always what teacher imagines or intends!  Meta: –propensities that might come to surface  Working on educating awareness –so as to enable actions in the future

6. 6 Kites

7. 7 Outer, Inner & Meta Tasks  Outer –Folding paper; discerning elements; recognising relationships  Inner –Recognising elements; perceiving properties; reasoning on the basis of agreed properties –Property of A4 paper –Anticipating; Taste of ‘surprise’  Meta –Guessing -> conjecturing –Waiting to be told what to do(?)

8. 8 Virtual Phenomena Point on Circle Tangent through Pt Chord to end of diameter Reflect tangent in chord What do you expect will happen as point on circle moves?

9. 9 Reflections Dimensions of possible variation: - Point on circle - Diameter  chord - Tangent  fixed angle Ranges of permissible change: - Anywhere on circle - Any chord (what about tangent?) - Any angle (0? obtuse?) Undoing or Reversing - If a curve has ‘this’ property’, must it be a circle?

10. 10 Inner & Meta Task Point on Circle Tangent through Pt Chord to end of diameter Reflect tangent in chord What do you expect will happen as point on circle moves? Outer Task - detect, express, explain invariance Inner Task - add relevant elements - experience angle chasing (reasoning) - use of angle theorems - experience theme of invariance in the midst of change - experience DofPV & RofPCh Meta Task - Keeping Going - What do I Know? What do I Want? - Surprise?

11. 11 Areal I increase the longer side by 20% and decrease the shorter side by 20% I decrease the longer side by 20% and increase the shorter side by 20% Which has the larger area? To increase by 20%, multiply by 1.2.8To decrease by 20%, I have a rectangle

12. 12 Outer, Inner & Meta Tasks  Outer –Calculating with percentages  Inner –Percentages as multiplicative relationships –‘seeing’ via a diagram and algebraically  Meta –Using different ways of (re)presenting –Encountering aversion to diagrams/arithmetic/percentages

13. 13 Reading a Diagram: Seeing As … x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + x(1-x) + (1-x) 2 (1-z)xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

14. 14 Outer, Inner & Meta Tasks  Outer –Discerning details; expressing relationships  Inner –Encountering ‘algebra’ –Experiencing generalisation –Seeing the same thing in two (or more) different ways; exploiting  Meta –Suppressing immediate reactions; diving in too deeply too quickly

15. 15 Watch What Happens Can you see … 1 + 2 + 2 + 2 + 2 4x1 + 4x3 + 4x5 + 4x7 + 4x9 Say What You Saw 1 + 3 + 5 + … + 2n+1 = (n+1) 2 1 + 3 + 5 + … + 2n+1 = (n+1) 2 (1 + 9) 2 = 4 x 5 2 (1 + 2n+1) 2 = 4(n+1) 2 (1 + 2n+1) 2 = 4(n+1) 2 1 + 3 + 5 + 7 + 9 = 5 2 4(1 + 3 + 5 + … + 2n+1) = 4(n+1) 2 4(1 + 3 + 5 + … + 2n+1) = 4(n+1) 2

16. 16 More Or Less Altitude & Area Draw a scalene triangle moresameless more same less are a altitude Same alt more area more alt same area more alt more area less alt more area less alt less area more alt less area same alt less area less alt same area

17. 17 More Or Less Percent & Value 50% of 40 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

18. 18 More Or Less … A journey of 200 Km for 4 hrs averages 50Km/hr mor e sam e less mor e sam e less 50Km/hr 4 hrs Same Distance speed time 50Km/hr 4 hrs mor e sam e less mor e sam e less Less Distance speed time 50Km/hr 4 hrs mor e sam e less mor e sam e less More Distance speed time !!!

19. 19 Outer, Inner & Meta Task  Outer –Fill in chart  Inner –Making choices –Constructing own objects –Encountering multiplicative reasoning  Meta –Tendency to accept first idea that comes –Desire for tidiness, completeness

20. 20 Tasks are …  … only things to do  Activity is … –what results from attempting tasks –What generates experience  What matters is what you do with that experience  Outer, Inner & Meta aspects of tasks  Affordances, Constraints and Attunements

21. 21 To Pursue Further  Google Mathemapedia (a wikipedia for mathematics education)  http:// mcs.open.ac.uk/jhm3 –Where you will find these slides and Notes about tomorrow’s Distributed Task  Designing and Using Mathematical Tasks (Tarquin)