1. 1 Responsive, Reflective & Responsible teaching John Mason AIMSSEC ACE Yr 2 Jan 2013 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2. 2 Ways of Working  Everything said here today is a conjecture  It is uttered so it can be thought about and modified if necessary  What you get from this session will mostly be what you notice happening inside you … how you use your mathematical powers.

3. 3 Responsive Teaching  Responding to student’s needs –Class as a whole –Particular students  Listening to Students  Giving them time –to think, –to experiment –to conjecture  Supporting them to –Modify their conjecture  Trying not to do for students what they can alredy do for themselves

4. 4 Reflective Teaching  Learning from experience  What could have been different? Should –> Could Do this at the end of a lesson while students are making a note of what they thought the lesson was about!  Imagining yourself in the future, acting in some way that you would prefer instead of some habit that has developed  Making a note at the end of the lesson of ONE thing that struck you, that stood out, about the lesson

5. 5 Responsible Teaching  Able to justify choices of –Intentions (mathematical) –Tasks –Interventions –Pedagogic strategies  Requires the development of a vocabulary for talking about pedagogic intentions and choices!

6. 6 Set Ratios  In how many different ways can you place 17 objects so that there are equal numbers of objects in each of two sets?  What about requiring that there be twice as many in the left set as in the right set?  What about requiring that the ratio of the numbers in the left set to the right set is 3 : 2?  What is the largest number of objects that CANNOT be placed in the two sets in any way so that the ratio is 5 : 2? What can be varied?

7. 7 Reflection & Justification (Mathematical)  Powers used? –Imagining and Expressing; Specialising & Generalising; Conjecturing & Convincing; –Being Systematic –Making records  Themes Encountered –Seeking Relationships –Invariance in the midst of change –Freedom & Constraint –Doing & Undoing

8. 8 Reflection & Justification (Task Format)  Why 17 objects to be placed? –What follow-up was missing? –What about 18? (opportunity for ‘same and different’)  Confusion between ‘left set’ and ‘left part of diagram’!!!  Something available if some finish first part quickly  How was work sustained?  How was work brought to a conclusion? –Conjectures? –Something not fully resolved? –Opportunity to reflect back over the event?

9. 9 Issues Arising  Choice of numbers  Choice of wording  Choice of setting: –actual objects; drawings; symbols

10. 10 31: a game for two players  At each move the player chooses a whole number of cubes from 1 to 5 and adds them to a common pile.  The first person to get the total number of cubes in the common pile to be 31, wins.  What is your (best) strategy?

11. 11 Reflection & Justification (Mathematical)  Topic –Adding; choosing and predicting –Reasoning backwards from 31  Powers used? –Imagining and Expressing; Specialising & Generalising; Conjecturing & Convincing; –Being Systematic –Making records  Themes Encountered –Seeking Relationships –Invariance in the midst of change –Freedom & Constraint –Doing & Undoing

12. 12 Reflection & Justification (Task Format)  Did you use cubes?  Confusion???  How was work sustained?  How was work brought to a conclusion? –Conjectures? –Something not fully resolved? –Opportunity to reflect back over the event?

13. 13 Selective Sums  Cover up one entry from each row and each column. Add up the remaining numbers.  The answer is (always) the same!  Why? 0 -2 2 -4 6 4 8 2 3 1 5 1 3 -3 Stuck? Specialise!

14. 14 Reflection & Justification (Mathematical)  Topic Reviewed or Met? –Practicing addition & subtraction (whole numbers, integers, fractions, even decimals) –Making choices with constraints  Powers used? –Imagining and Expressing; Specialising & Generalising; Conjecturing & Convincing; –Being Systematic –Making records  Themes Encountered? –Seeking Relationships –Invariance in the midst of change –Freedom & Constraint –Doing & Undoing

15. 15 Reflection & Justification (Task Format)  Why objects, not simply imagining or using pencil?  Confusion???  Something available if some finish first-part quickly?  How was work sustained?  How was work brought to a conclusion? –Conjectures? –Something not fully resolved? –Opportunity to reflect back over the event?

16. 16 Selective Sums Opportunity to generalise Opportunity to quantify freedom of choice  How much freedom of choice do you have when making up your own?ab c d e f g e-(a-b) ab e ? a b c d e f g

17. 17 Selective Sums Variation  Choose a number s from 1, 2, 3  Select s numbers from each row and column (cover up 4–s numbers from each row and column)  Add up all the selected numbers  Why is it always the same?

18. 18 Chequered Selective Sums  Choose one cell in each row and column.  Add the entries in the dark shaded cells and subtract the entries in the light shaded cells.  What properties make the answer invariant?  What property is sufficient to make the answer invariant? 0 2 -5 -3 -6 4 9 3 -2 -6 -2 0 3 5

19. 19 Some Frameworks Doing – Talking – Recording (DTR) Enactive – Iconic – Symbolic Material – Mental–Symbols (EIS) See – Experience – Master (SEM) (MGA) Specialise … in order to locate structural relationships … then re-Generalise for yourself What do I know? What do I want? Stuck?

20. 20 Issues Arising  Choice of numbers  Choice of wording  Choice of setting: –actual objects; drawings; symbols  Opportunities for Students to –Make significant mathematical choices –Use their own powers –Reflect on what has been effective for them

21. 21 Responsible Reflection!  What did you notice for yourself?  What has struck you from this session?  What would you like to try out or evelop?  Imagine yourself working on that for yourself –Modifying something to use in your situation –Trying something out –Reflecting on what was effective

22. 22 Follow Up   j.h.mason @ open.ac.uk   mcs.open.ac.uk/jhm3   These slides and the Hand Outs will be on Memory Sticks & Moodle