Promoting Mathematical Thinking
Getting Children
to Make Mathematical Use
of their
Natural Powers
John Mason
‘Powers’
Norfolk Mathematics Conference
Norwich
Nov 28 2012
The Open University
Maths Dept
1
University of Oxford
Dept of Education
Conjectures
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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!
Tasks
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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
Memory
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Rhythms
Counting
1234
2345
3456
4567
4
Glimpsed
Say
DrawWhat
WhatYou
YouSee
Saw
5
 Say What You Saw
 Sketch what you think you saw
 Compare with what others drew
 How did you go about it?
More or Less grids
Perimeter
Area
More
Same
Less
More
Same
Less
6
With as little change as possible from the original!
Circle Round a Square
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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?
Numberline Movements
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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.
ThOANs
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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!
Ride & Tie
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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?
Ride & Tie
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Imagine, then draw a diagram!
Seeking
Relationships
/
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Does the diagram make sense
(meet the constraints)?
Two Journeys
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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 speeds
time
distance
d
d
t1 =
t2 =
2v1
2v2
d
d
t =
+
2v1 2v2
12
t
t
d1 = v1 d2 = v2
2
2
2d
t=
v1 + v2
Named Ratios
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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
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
Make a mark about 1 cm from each end of your
elastic … this is your thumbnail mark
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Make a mark half way between your thumbnail marks
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Make marks one-third and two-thirds of the way
between your thumbnail marks
Counter Scaling
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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
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Outer Task
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Inner Task
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What author imagines
What teacher intends
What students construe
What students actually do
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?
Powers
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Every child that gets to school has already displayed
the power to
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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?
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In each lesson, does every child in the class have an
opportunity to use (and develop) one or more powers?
Problem Solving Skills
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Reflection
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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
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It is not the task that is rich
– but the way the task is used
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Teachers can guide and direct learner attention
What are teachers attending to?
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Powers
Themes
Heuristics
The nature of their own attention
Mathematical Powers
Those with Mathematical Powers are Super Heroes!
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Captain CC
I have the
power of
conjecture I
can say what
I think will
happen
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I have the
power of
convincing I
can prove
what I think
to others
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Follow Up
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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)