Promoting Mathematical Thinking
From Teaching Procedures To
Thinking Mathematically:
Making Use
of Students’ Natural Powers
John Mason
Gothenberg
Nov 30 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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3
Tasks promote Activity;
Activity involves Aactions;
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
Responsible teaching
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Articulate about justifying choices of
–
–
–
–
4
tasks
ways of initiating mathematical thinking
ways of sustaining mathematical thinking
ways of concluding mathematical thinking
Learning (Mathematics)
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What is Avaibale to be learned
(what is varying and in what ways)
What Actions are Initiated
What Dispositions are Evoked
What Powers are called upon
My Way of Working
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Phenomenological-Experiential
– Try to generate an experience,
– draw attention to it
– label it in some way
6
One More Than
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What numbers can be presented as one more than
the product of four consecutive numbers?
One natural response is to use algebra
(if that is confidence-inspiring)
– But that runs into obstacles

One natural response is to try some specific
examples…
– In order to locate a relationship that might be an instance of
a property!
Specialising
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Generalising
From Thomas Lingefjård
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Given the numbers 1, 3, 4, and 6 - try to construct all
numbesr from 20 to 30 by simple arithmetic (addition,
subtraction, multiplication and division). No other way
of combining or using numbers as power of is
allowed. For instance: 1*6*3 + 4 = 22. In every
calculation, all four digits must be present.
Try to find a number which consists of 769 digits, the
sum of all the digits is 3693, every pair of
consecutive digits is either a multiple of 17 or of 23
and all multiples of 17 or 23 in two digits is in the
number.
More or Less grids
Perimeter
Area
More
Same
Less
More
Same
Less
9
With as little change as possible from the original!
Counting Out

In a selection ‘game’ you start at the left and count
forwards and backwards until you get to a specified
number (say 37). Which object will you end on?
A
B
C
D
E
1
2
3
4
5
9
8
7
6
10
…
If that object is elimated, you start again from the ‘next’. Which
object is the last one left?
10
Substitution Pattern Generating
W –> WB
B –> W
How many squares will there be?
How many white squares will there be?
How many blue squares will there be?
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Substitution Relationships
W
WBB
WBB BW BW
⬆ ⬆ ⬆ ⬆⬆ ⬆⬆
WBB BW BW BW WBB BW WBB
⬆
⬆
⬆
⬆
⬆
⬆
⬆
WBB BW BW BW WBB BW WBB BW WBB WBB BW BW BW WBB WBB BW BW
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Gasket Sequences
13
Sundaram’s Sieve
16
27
38
49
60
71
82
13
22
31
40
49
58
67
10
17
24
31
38
45
52
7
12
17
22
27
32
37
4
7
10
13
16
19
22
Claim: N will appear in the table iff 2N + 1 is composite
What number will appear in the Rth row and the Cth column?
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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 to the
left.
ThOANs
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Think of a number between 0 and 10
Add six
Multiply by the first number you 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
20
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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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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What’s The Difference?
–
=
First, add one to each
First,
add one to the larger and subtract
one from the smaller
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What then
would be
the difference?
What could
be varied?
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)?
Understanding Division
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234234 is divisible by 13 and 7 and 11;
What is the remainder on dividing 23423426 by 13?
By 7? By 11?
Make up your own!
More or Less grids
Perimeter
Area
More
Same
Less
More
Same
Less
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With as little change as possible from the original!
Put your hand up when you can see …
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Something that is 3/5 of something else
Something that is 2/5 of something else
Something that is 2/3 of something else
Something that is 5/3 of something else
What other fraction-actions can you see?
How did your
attention shift?
Put your hand up when you can see …
Something that is 1/4 – 1/5
of something else
Did you look for
something that is 1/4 of something else
and for
something that is 1/5 of the same thing?
What did you have to do with
your attention?
Can you generalise?
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1
1
1
- =
c -1 c c ( c -1)
1 1 c-r
- =
r c
rc
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 speeds
time
distance
d
d
t1 =
t2 =
2v1
2v2
d
d
t =
+
2v1 2v2
29
t
t
d1 = v1 d2 = v2
2
2
2d
t=
v1 + v2
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?
30
One Sum Diagrams
1
(1- )2
1
2
1-
Anticipating,
not waiting
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Reading a Diagram
x3 + x(1–x) + (1-x)3
x2z + x(1-x) + (1-x)2(1-z)
xyz + (1-x)y + (1-x)(1-y)(1-z)
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x2 + (1-x)2
xz + (1-x)(1-z)
yz + (1-x)(1-z)
Outer & Inner Tasks
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Outer Task
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Inner Task
–
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–
–
–
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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?
Imagining
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Basis of Geometric Thinking
Basis of Anticipating
Basis of ‘Realising’
Basis of Accessing & Enriching Example Spaces
Basis of Planning
Geometric Images
ATM
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Powers
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Every child that gets to school has already displayed
the power to
–
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–
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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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Mathematical Themes
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Doing & Undoing
Invariance in the midst of change
Freedom & Constraint
Restricting & Expanding Meaning
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
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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
Attention
/
/
/
/
/
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Holding Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the basis of properties
Motivation
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Motivation is not a thing
– Sense of gap or disturbance
– Appropriate challenge + Trust in teacher
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Phenomena to explain using mathematics
Mathematical phenomena to explain & appreciate
The Problem about Problem Solving
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It is not simply a Friday afternoon entertainment
It is not a ‘thing’ you (or the students) do
It is an orientation to learning and doing mathematics
Change of Vocabulary:
– Teaching using exploration as one mode of interaction
among many
– ‘teaching Investigatively’
– Using Stdeunts’ Powers to teach Mathematics
– …
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Pedagogic Strategies & Didactic Tactics
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In how many different ways can you …
Do as many exercises as you need to do in order to be
able to do any uestion of this type
– Construct an easy, hard, peculiar, general question of this type
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What is the same and what different about …
If this is the answer, what questions of this type would
give the same answer?
What sorts of answers can you get to questions of this
type?
Presentation
– Particular  General
– General –> Particular –> Re-Generalise
– Partly General –> Particular –> Re-Generalise
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Follow Up
mcs.open.ac.uk/jhm3
j.h.mason @ open.ac.uk
Thinking Mathematically (new edition)
Designing and Using Mathematical Tasks (Tarquin)
Questions and Prompts … (from ATM)
Thinkers (from ATM)
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