Promoting Mathematical Thinking
Reasoning
Mathematically
John Mason
Mathsfest
Cork
Oct 2012
The Open University
Maths Dept
1
University of Oxford
Dept of Education
Specific Aims for Ordinary Level
 an
understanding of mathematical concepts and of their
relationships
 confidence and competence in basic skills
 the ability to solve problems
 an introduction to the idea of logical argument
 appreciation both of the intrinsic interest of
mathematics and of its usefulness and efficiency for
formulating and solving problems
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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!
3
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
 Something more is required
4
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
within one place either way
– Cold means that it is at least two
places away
What is your
best strategy
to locate the
secret place?
5
Counting Out
a selection ‘game’ you start at the left and count
forwards and backwards until you get to a specified
number (say 37 or 177). Which object will you end on?
 In
A
B
C
D
E
1
2
3
4
5
9
8
7
6
10
…
If that object is eliminated, you start again from the ‘next’.
Which object is the last one left?
6
Alternating Square Sums
 Imagine
a triangle
 Imagine a point inside the triangle
 Drop perpendiculars to the three
sides of the triangle
 Each side of the triangle comprises
two segments
 On each segment of each edge,
construct a square
 Alternately
colour the squares yellow and cyan around the
triangle
 Conjecture:
the sum of the areas of the yellow squares is
the sum of the areas of the cyan squares.
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 For
what hexagons is this the case?
Selective Sums
 Add
up any 4 entries, one taken
from each row and each column.
 The answer is (always) 6
 Why?
0
-2
2
-4
6
4
8
2
3
1
5
-1
1
-1
3
-3
Example of (use of) permutations
Example of seeking invariant relationships
Example of focusing on actions preserving
an invariance
Opportunity to generalise
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Selective Sums
 Add
up any 4 entries, one
taken from each row and
each column.
 Is the answer always the
same?
 Why?
9
5
6
-1
3
2
3
-1
6
1
3
-5
6
1
6
-2
3
5
3
1
2
3
2
2
3
4
3
1
6
7
6
1
3
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 makes the
answer invariant?
 What property is sufficient to
make the answer invariant?
10
2
-5 -3
-6 4
-1 9
0
3
-1 -2 -6
-2
0
3
5
Circles in Circles
How are the
red and yellow
areas related?
red
orange
yellow
11
Carpet Theorems
 In
a room there are two carpets whose combined area
is the area of the room.
– The area of overlap is the area of floor uncovered
 In a room there are two carpets. They are moved so as
to change the amount of overlap.
– The change in the area of overlap is the change in
area of uncovered floor
12
Rectangular Room with 2 Carpets
How are the red and blue areas related?
13
Perimeter Projections
 The
red point traverses the
quadrilateral
 The vertical movement of the
red point is tracked.
 What shape is the graph?
 Given
a graphical track of the
vertical movement and the
horizontal movement,
 What is the shape of the
polygon?
14
Square Deduction
Could these all be
squares?
15
Square Deduction: tracking arithmetic
(3x3+4)/3
3x4-3x3
3+3x4
3x3+4
3 4
2x3+4 3+4
3+2x4
Track the 3 and the 4:
Replace the 3 by a and the 4 by b
16
Square Deduction: acknowledging ignorance
(3a+b)/3
3b-3a
a+3b
3a+b
a b
2a+b
a+b
a+2b
3a+b = 3(3b-3a)
12a = 8b
So 3a = 2b
For an overall square
4a + 4b = 2a + 5b
So 2a = b
3
1
1
9
7
17
2 3
5
8
For n squares upper left
n(3b - 3a) = 3a + b
So 3a(n + 1) = b(3n - 1)
But not also 2a = b
Reflection
 What
aspects of reasoning…
– Stood out for you?
– Involved some struggle
 What actions …
– Did you undertake?
– Were ineffective (why?)
– Were effective (why?)
18
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
 What matters more than the particular answer is …
– how do you know?
– what can you vary and still the same approach works?
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Reminder
 Do–Talk–Record
– Generating need to communicate
 Provoking Engagement
– Surprise
– Challenge (trust)
 Promoting Mathematical thinking
– How do you know? …
– Why must …?
 Active Students
– Constructing
– Extending
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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
Questions and Prompts … (Primary version from ATM)
Thinkers
Institute of Mathematical Pedagogy
August 6-9 2013
mcs.open.ac.uk/jhm3
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