Report

Promoting Mathematical Thinking Responsive, Reflective & Responsible teaching John Mason AIMSSEC ACE Yr 2 Jan 2013 The Open University Maths Dept 1 University of Oxford Dept of Education Ways of Working 2 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. 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 3 Trying not to do for students what they can alredy do for themselves Reflective Teaching Learning from experience What could have been different? Should –> Could 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 Do this at the end of a lesson while students are making a note of what they thought the lesson was about! 4 Responsible Teaching Able to justify choices of – – – – 5 Intentions (mathematical) Tasks Interventions Pedagogic strategies Requires the development of a vocabulary for talking about pedagogic intentions and choices! 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? 6 Reflection & Justification (Mathematical) Powers used? – Imagining and Expressing; Specialising & Generalising; Conjecturing & Convincing; – Being Systematic – Making records Themes Encountered – – – – 7 Seeking Relationships Invariance in the midst of change Freedom & Constraint Doing & Undoing 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? 8 Issues Arising Choice of numbers Choice of wording Choice of setting: – actual objects; drawings; symbols 9 31: a game for two players 10 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? 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 – – – – 11 Seeking Relationships Invariance in the midst of change Freedom & Constraint Doing & Undoing 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? 12 Selective Sums Cover up one entry from each row and each column. Add up the remaining numbers. The answer is (always) the same! Why? Stuck? Specialise! 13 0 -2 2 -4 6 4 8 2 3 1 5 -1 1 -1 3 -3 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? – – – – 14 Seeking Relationships Invariance in the midst of change Freedom & Constraint Doing & Undoing 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? 15 Selective Sums How much freedom of choice do you have when making up your own? a b d e b a b f e ? e-(a-b) g Opportunity to generalise 16 c Opportunity to quantify freedom of choice c d Selective Sums Variation 17 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? 5 6 -1 3 2 3 -1 6 1 3 -5 6 1 6 -2 3 5 3 1 2 3 2 2 3 4 3 1 6 7 6 1 3 Chequered Selective Sums 18 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? 2 -5 -3 -6 4 -1 9 0 3 -1 -2 -6 -2 0 3 5 Some Frameworks Doing – Talking – Recording (DTR) (MGA) See – Experience – Master (SEM) Enactive – Iconic – Symbolic Material – Mental–Symbols (EIS) 19 Specialise … in order to locate structural Stuck? What do I know? relationships … then re-Generalise for yourself What do I want? 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 20 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 21 Follow Up 22 j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 These slides and the Hand Outs will be on Memory Sticks & Moodle