Responsive, Reflective & Responsible teaching

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Responsive, Reflective & Responsible teaching

John MasonAIMSSECACE Yr 2Jan 2013

The Open UniversityMaths Dept University of Oxford

Dept of Education

Promoting Mathematical Thinking

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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.

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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

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Reflective Teaching Learning from experience What could have been different?Should –>

Could

Do this at the end of a

lessonwhile 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 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

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Responsible Teaching

Able to justify choices of– Intentions (mathematical)– Tasks– Interventions– Pedagogic strategies

Requires the development of a vocabularyfor talking about pedagogic intentions andchoices!

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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?

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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

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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?

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Issues Arising

Choice of numbers Choice of wording Choice of setting:

– actual objects; drawings; symbols

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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?

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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

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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?

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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 -46 4 8 23 1 5 -11 -1 3 -3

Stuck? Speciali

se!

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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

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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?

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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

cd

efge-(a-b)

a be ?

a b c defg

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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?

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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 -1 93 -1 -2 -6-2 0 3 5

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Some FrameworksDoing – 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?

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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

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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

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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