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Getting Children to Make Mathematical Use

of their Natural Powers

The Open UniversityMaths Dept University of Oxford

Dept of Education

Promoting Mathematical Thinking

John Mason‘Powers’

Norfolk Mathematics ConferenceNorwich

Nov 28 2012

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Conjectures

Everything said here today is a conjecture … to be tested in your experience

The best way to sensitise yourself to learners …… is to experience parallel phenomena yourself

So, what you get from this session is what you notice happening inside you!

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Tasks

Tasks promote Activity; Activity involves Actions; Actions generate Experience;

– but one thing we don’t learn from experience is that we don’t often learn from experience alone

It is not the task that is rich …– but whether it is used richly

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Memory

Rhythms Counting

1 2 3 42 3 4 53 4 5 64 5 6 7

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Glimpsed

Say What You Saw Sketch what you think you

saw Compare with what others

drew How did you go about it?

Draw What You SawSay What You See

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More or Less grids

More Same

Less

More

Same

LessPerimeter

Area

With as little change as possible from the original!

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Circle Round a Square Imagine a Square Now imagine a circle in the same plane as the

square, so that the two are touching at a single point

Now imagine the circle rolling around the outside of the square, always staying in touch

Pay attention to the centre of the circle as it rolls What is the path the centre takes, and how long is

it?

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Numberline Movements Imagine you are standing on a number line

somewhere facing the positive direction.(Make a note of where you are!)

Go forward three steps; Now go backwards 5 steps Now turn through 180° Go backwards 3 steps Go forwards 1 step You should be back where you started but facing in

the negative direction.

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ThOANs

Think of a number between 0 and 10 Add six Multiply by the number you first thought of Add 4 Subtract twice the number you first thought of Take the square root (positive!) subtract the number you first thought of You (and everybody else) are left with 2!

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Ride & Tie Imagine that you and a friend have a single

horse (bicycle) and that you both want to get to a town some distance away.

In common with folks in the 17th century, one of you sets off on the horse while the other walks. At some point the first dismounts, ties the horse and walks on. When you get to the horse you mount and ride on past your friend. Then you too tie the horse and walk on…

Supposing you both ride faster than you walk but at different speeds, how do you decide when and where to tie the horse so that you both arrive at your destination at the same time?

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Ride & Tie Imagine, then draw a diagram!

Does the diagram make sense (meet the constraints)?

Seeking Relationships

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Two Journeys Which journey over the same distance at two

different speeds takes longer:– One in which both halves of the distance are done at

the specified speeds– One in which both halves of the time taken are done

at the specified speedsdistance time

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Named Ratios

Now take a named ratio (eg density) and recast this task in that language

Which mass made up of two densities has the larger volume:– One in which both halves of the mass have the fixed

densities– One in which both halves of the volume have the

same densities?

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Elastic Multiplication Make a mark about 1 cm from each end of

your elastic … this is your thumbnail mark

Make a mark half way between your thumbnail marks

Make marks one-third and two-thirds of the way between your thumbnail marks

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Counter Scaling

Someone has placed 5 counters side-by-side in a line

Someone else has made a similar line with 5 counters but with one counter-width space between counters.

By what factor has the length of the original line been scaled?

How many counters would be needed so that the scale factor was 15/8?

“Fence-post Reasoning”Generalise!

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Outer & Inner Tasks

Outer Task– What author imagines– What teacher intends– What students construe– What students actually do

Inner Task– What powers might be used?– What themes might be encountered?– What connections might be made?– What reasoning might be called upon?– What personal dispositions might be challenged?

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Powers Every child that gets to school has already

displayed the power to– imagine & express– specialise & generalise– conjecture & convince– organise and categorise

The question is …– are they being prompted to use and develop those

powers?– or are those powers being usurped by text, worksheets

and ethos? In each lesson, does every child in the class

have an opportunity to use (and develop) one or more powers?

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Problem Solving Skills

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Reflection Tasks promote activity; activity involves

actions; actions generate experience; – but one thing we don’t learn from experience is that

we don’t often learn from experience alone It is not the task that is rich

– but the way the task is used Teachers can guide and direct learner

attention What are teachers attending to?

– Powers– Themes– Heuristics– The nature of their own attention

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Mathematical Powers

Those with Mathematical Powers are Super Heroes!

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Captain CC

I have the power of

convincing I can prove what I think to others

I have the power of

conjecture I can say

what I think will happen

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Follow Up j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 Presentations Thinking Mathematically (Pearson) Questions & Prompts (ATM) Learning & Doing Mathematics (Tarquin) Developing Thinking in Algebra (Sage)