1. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 1 Effective Mathematics Teaching & Learning Educating Awareness & Training Behaviour Through Harnessing Emotions Exeter Sept 03

2. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 2  Learning is maximally effective when: v all aspects of psyche are involved v learners are active (doing, construing) v learners are using their natural powers  Teaching is maximally effective when: v ethos is mathematical v being math’l with & in front of learners v evident caring: for learners; for maths

3. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 3 Write down two numbers which sum to ten Write down another two numbers which sum to ten … and another two numbers which sum to ten What did you notice?

4. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 4 Write down two numbers which sum to ten What can change, and still some feature is preserved? Change the TEN Change SUM Change TWO Change NUMBERS

5. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 5 Dimensions-of-Possible-Variation label for a collection of questions every learner can ask for themselves … What can change and still … is preserved? What is the Range-Of-Permissible-Change in each case? This applies to every task & to every concept, even to how tasks are presented

6. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 6 I have written down two numbers which sum to ONE. I square the larger and add the smaller (A); I square the smaller and add the larger (B): Which of my two answers A & B will be the biggest? Make a conjecture !! What did you notice?

7. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 7 Depicting 1 1

8. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 8 Depicting 1 1

9. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 9 Depicting 1 1

10. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 10 Depicting 1 1

11. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 11 Depicting 1 1

12. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 12 Depicting 1 1

13. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 13 Depicting 1 1

14. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 14 Depicting 1 1

15. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 15 Generalising 1 1 The product of the largest of each pair + the smallest of one pair is the same as What if they sum to something else? S S The product of the smallest of each pair + the other largest

16. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 16 Relation to Curriculum Generalisation is NOT something to be taught, but a power to be developed and invoked … in EVERY lesson A lesson without the opportunity for learners to generalise is NOT a mathematics lesson!

17. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 17 Imagining & Expressing (communicating) Specialising & Generalising Conjecturing & Convincing (reasoning) Organising & Characterising

18. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 18 Invariance in the Midst of Change Freedom & Constraint Doing & Undoing Extending & Contracting Meaning

19. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 19  Learning is maximally effective when: v all aspects of psyche are involved  Awareness, behaviour, emotions v learners are active (doing, construing)  doing ≠ construing  making choices; experiencing creative energy flow v learners are using their natural powers

20. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 20 Learners’ Theory:  If I complete the tasks I am set then learning will (presumably) take place  Assenting ––> Asserting, Anticipating  Following ––> Formulating  Taking initiative; Making choices Learners’ Development:

21. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 21 What is the point of teaching if there is little learning? Major pressure … obligation to cover everything … difficult to get through in time available … ask myself ‘who is covering the syllabus? Is it the learners, or just me?’ Sometimes not much is written down; when learners become creative, misconceptions and confusions surface and life gets messy; building sites are messy places From Malcolm Swan (Nottingham)

22. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 22  Teaching is maximally effective when: v ethos is mathematical  conjecturing atmosphere  exposure to themes & heuristics v being math’l with & in front of learners  displaying use of powers  struggling sometimes; exploring together v evident caring:  for learners;  for mathematics (mathematical thinking)

23. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 23 Teaching Traps  Doing for learners what they can already do for themselves  doing ≠ construing; working through ≠ working on  The more clearly I specify behaviour sought from learners, the easier it is for them to display it without generating it for themselves

24. Conjectures Thinking Dimensions- of-Possible- Variation Powers Themes Learning Tensions Teaching 24 Perhaps the greatest of all pedagogical fallacies is the notion that a person learns only the particular thing being studied at the time. Collateral learning … may be and often is much more important the the (actual) lesson. John Dewey We are wise to create systems for spin-offs rather than for pay-offs. Bill Brookes