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Promoting Mathematical Thinking
Reasoning Reasonably
in Mathematics
The Open University
Maths Dept
1
John Mason
Schools Network
Warwick
June 2012
University of Oxford
Dept of Education
Outline
 Some
Tasks to engage in
 On which to reflect
 In a conjecturing atmosphere
– Everything said must be tested in your experience
 Make
Contact with
– Natural powers used to reason mathematically
– Mathematical themes
– Pedagogic choices
 Straw
Poll: please raise your hand if you
– Teach proof or reasoning …
…to previously high attaining students
…to previously low attaining students
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Secret Places
Homage to Tom O’Brien (1938 – 2010)
 One
of the places around the table is a
secret place.
 If you click near a place, the colour will
tell you whether you are hot or cold:
– Hot means that the secret place is there
or else one place either side
– Cold means that it is at least two places
away either side
What is your
best strategy
to locate the
secret place?
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Reflexive Stance
 What
did you notice yourself doing?
 What actions were effective?
 What actions were not effective?
4
Magic Square Reasoning
2
2
6
7
2
1
5
9
8
3
What other
configurations
like this
give one sum
equal to another?
Try to describe
them in words
4
Any colour-symmetric
arrangement?
Sum(
5
) – Sum(
)
=0
More Magic Square Reasoning
Sum(
6
) – Sum(
) =0
Reflexive Stance
 What
did you notice yourself doing?
 What actions were effective?
 What actions were ineffective?
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Convincing & Justifying
 Justifying
actions
– Social aspect
…convincing yourself
…convincing a friend
…convincing a sceptic
– Learning to be sceptical of other’s reasoning
– Learning to be sceptical of your own reasoning
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Four Consecutives
 Write
down four consecutive numbers
and add them up
 and another
 and another
 Now be more extreme!
 What is the same, and what is
different about your answers?
 What numbers can be expressed as
the sum of four consecutive
numbers?
Express 424 as a sum of four
consecutive numbers
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+1
+2
+3
4
+6
Doing & Undoing
Carpet Theorem
How are the red and blue areas related?
10
Doug French’s Fractional Parts
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What’s the same and what different?
What can be varied?
Construct your own
What is the ‘whole’?
Square Sums
Imagine a triangle …
Imagine the circumcentre;
Drop perpendiculars to the
edges;
On each half of each edge,
put a square outwards, one
yellow, one cyan.
More generally …
Does
The sum of the Yellow
areas = the sum of the
cyan areas
12
???!!?
?
Polygon Perimeter Projections
 Imagine
a square
 Imagine a point traversing the perimeter of the square
 Imagine projections of that point onto the x and y axes
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14
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Withdrawing from the Action
 What
actions did you carry out?
 What mde you think of those?
 Which ones were effective (for possible use in the future)?
 Did I
–
–
–
–
–
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Gaze (holding wholes)?
Discern Details
Recognise Relationships in the particular?
Perceive Properties as instantiated
Reason on the basis of properties?
Eyeball Reasoning
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Square Deductions
3b-3a
a+3b
3a+b
a b
2a+b
a+2b
a+b
3
1
1
9
2 3
7
18
5
8
3(3b-3a) = 3a+b
12a = 8b
So 3a = 2b
For an overall square
4a + 4b = 2a + 5b
So 2a = b
For n squares upper left
n(3b - 3a) = 3a + b
So 3a(n + 1) = b(3n - 1)
But not also 2a = b
Square Deductions
x+3y
3x+y
x
2x+y
19
y
x+y
x+2y
To be consistent,
3x + y + x = x + 3y + y
So 3x = 3y
Rectangle Deduction
(x+7y)/4
Suppose the
rectangles are
two by one
(x+7y)/8
(7x+y)/4
x
(3x+y)/2
(3x+y)/4
y
(x+3y)/2
(x+y)/2
x+y
(x+3y)/4
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28
40
(x+7y)/8 + y = (7x + y)/4 + x
19
15
21
13
21x = 13y
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17
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Wason’s cards
 Each
card has a letter on one side and a numeral on
the other.
 Which 2 cards must be turned over in order to verify
that
“on the back of a vowel
there is always an even number”?
A
2
B
3
Which cards must be turned over to verify that
“on the back of a red vowel there is a blue even number”
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Diamond Multiplication
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Seven Circles
How many different
angles can you
discern, using only
the red points?
How do you know
you have them all?
How many
different
quadrilaterals?
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Bag Constructions (1)
 Here
there are three bags. If you
compare any two of them, there is
exactly one colour for which the
difference in the numbers of that
colour in the two bags is exactly 1.
For four bags, what is the least
number of objects to meet the same
constraint?
 For four bags, what is the least
number of colours to meet the same
constraint?

17 objects
3 colours
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Reflective Stance
 What
–
–
–
–
Checking?
Interpreting?
Specialising?
Being systematic?
 What
–
–
–
–
–
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did you notice yourself doing?
forms of attention did you notice?
Gazing (holding wholes)?
Discerning Details?
Recognising Relationships?
Perceiving Properties?
Reasoning on the basis of Properties?
Bag Constructions (2)
 For
b bags, how few objects can you
use so that each pair of bags has the
property that there are exactly two
colours for which the difference in
the numbers of that colour in the two
bags is exactly 1.
Construct four bags such that for
each pair, there is just one colour for
which the total number of that colour in
the two bags is 3.

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Bag Constructions (3)
 Here
there are 3 bags and 2 objects.
 [0,1,2;2]means that there are 0, 1 and 2
objects in three bags and 3 objects
altogether
 Given a sequence like [2,4,5,5;6] or
[1,1,3,3;6] how can you tell if there is a
corresponding set of bags?
 In how many different ways can you put
k objects in b bags?
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Reasoning Types
 Logical
Deduction
– Arithmetic: reasoning into the unknown
– Algebra: reasoning from the unknown
 Empirical
(needs justification)
 Exhaustion of cases
 Contradiction
Issue is often
 (Induction)
what can I assume?
what can I use?
what do I know?
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Attention
Holding Wholes (gazing)
Discerning Details (discriminating)
Recognising Relationships (in a situation)
Perceiving properties (as being instantiated)
Reasoning On The Basis of Agreed Properties
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Powers
Imagining
& Expressing
Specialisng & Generalising
– Stressing & Ignoring
Conjecturing
& Convincing
(what is the status of what you say?)
Organising & Characterising
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Themes
Doing
and Undoing
Invariance in the Midst of Change
Freedom & Constraint
Extending & Restricting
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Follow Up
mcs.open.ac.uk
/ jhm3 … presentations; applets;
j.h.mason @ open.ac.uk
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