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Responsive, Reflective & Responsible teaching

The Open University Maths Dept. University of Oxford Dept of Education. Promoting Mathematical Thinking. Responsive, Reflective & Responsible teaching. John Mason AIMSSEC ACE Yr 2 Jan 2013. Ways of Working. Everything said here today is a conjecture

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Responsive, Reflective & Responsible teaching

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

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

  5. Responsible Teaching • Able to justify choices of • Intentions (mathematical) • Tasks • Interventions • Pedagogic strategies • Requires the development of a vocabularyfor talking about pedagogic intentions andchoices!

  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. 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. 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. Issues Arising • Choice of numbers • Choice of wording • Choice of setting: • actual objects; drawings; symbols

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

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

  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. 2 -5 0 -3 Chequered Selective Sums 4 -1 -6 9 -6 -1 -2 3 • 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 3 -2 5

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

  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. 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. 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

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