Effective Mathematics Teaching & Learning

1. Effective Mathematics Teaching & Learning Educating Awareness & Training Behaviour Through Harnessing Emotions Exeter Sept 03

2. Learning is maximally effective when: • all aspects of psyche are involved • learners are active (doing, construing) • learners are using their natural powers • Teaching is maximally effective when: • ethos is mathematical • being math’l with & in front of learners • evident caring: for learners; for maths

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. 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. 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-Changein each case? • This applies • to every task & to every concept, even to how tasks are presented

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?

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15. 1 S 1 S Generalising The product of the largest of each pair + the smallest of one pair is the same as The product of the smallest of each pair + the other largest What if they sum to something else?

16. Relation to Curriculum A lesson without the opportunity for learners to generalise is NOT a mathematics lesson! Generalisation is NOT something to be taught, but a power to be developed and invoked … in EVERY lesson

17. Imagining & Expressing (communicating) Specialising & Generalising Conjecturing & Convincing (reasoning) Organising & Characterising

18. Invariance in the Midst of Change Freedom & Constraint Doing & Undoing Extending & Contracting Meaning

19. Learning is maximally effective when: • all aspects of psyche are involved • Awareness, behaviour, emotions • learners are active (doing, construing) • doing ≠ construing • making choices; experiencing creative energy flow • learners are using their natural powers

20. Learners’ Theory: • If I complete the tasks I am set then learning will (presumably) take place Learners’ Development: • Assenting ––> Asserting, Anticipating • Following ––> Formulating • Taking initiative; Making choices

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. Teaching is maximally effective when: • ethos is mathematical • conjecturing atmosphere • exposure to themes & heuristics • being math’l with & in front of learners • displaying use of powers • struggling sometimes; exploring together • evident caring: • for learners; • for mathematics (mathematical thinking)

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. Perhaps the greatest of all pedagogical fallaciesis 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