Download Flotur Plenary

Developing Mathematical Thinking
John Mason
Flötur, Selfoss
Sept 2008
1
Some Throat Clearing
What
you get from this session will
be what you notice happening inside
you
Everything said is to be treated as a
conjecture, and tested in your
experience
If you don’t engage in my tasks, you
will get nothing!
2
Getting Going
If
the difference of two numbers is even,
then their product is the difference of two
squares
Specialising
in order to
(re)generalise
3
How often do you arrange
for your students
to use this
power for themselves?
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?

4
17 objects
3 colours
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.

5
Bag Constructions (3)
Here
there are 3 bags and
two objects.
There are [0,1,2;2] objects in
the bags and 2 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?
6
QuickTime™ and a
Animation decompressor
are needed to see this picture.
7
Triangle Count
9
Attention
Holding Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the basis of agreed
properties
10
Doing & Undoing
What
operation undoes ‘adding 3’?
What operation undoes ‘subtracting 4’?
What operation undoes
‘subtracting from 7’?
What are the analogues for multiplication?
What undoes multiplying by 3?
What undoes dividing by 2?
What undoes multiplying by 3/2?
What undoes dividing by 3/2?
11
Tunja Sequences
-1 x -1 – 1 = -2 x 0
0 x 0 – 1 = -1 x 1
1x1–1=
2x2–1=
3x3–1=
4x4–1=
12
With
the
Grain
0x2
1x3
2 x 4 Across the Grain
3x5
Magic Square Reasoning
2
6
7
2
1
5
9
8
Sum(
13
3
) – Sum(
4
What other
configurations
like this
give one sum
equal to another?
Try to describe
them in words
) =0
More Magic Square Reasoning
Sum(
14
) – Sum(
) =0
Map Drawing Problem
 Two
people both have a copy of the same map of
Iceland.
 One uses Reykjavik as the centre for a scaling by a
factor of 1/3
 One uses Akureyri as the centre for a scaling by a factor
of 1/3
 What is the same, and what is different about the maps
they draw?
15
Some Mathematical Powers
Imagining
& Expressing
Specialising & Generalising
Conjecturing & Convincing
Stressing & Ignoring
Ordering & Characterising
Seeing Sameness & Seeing
Difference
Assenting & Asserting
16
Some Mathematical Themes
Doing
and Undoing
Invariance in the midst of Change
Freedom & Constraint
17
Structure of the Psyche
Awareness (cognition)
Imagery
Will
Emotions
(affect)
Body (enaction)
Habits
Practices
18
Structure of a Topic
Language Patterns
& prior Skills
Imagery/Senseof/Awareness; Connections
Root Questions
predispositions
Different Contexts in which
likely to arise;
dispositions
Standard Confusions
& Obstacles
Techniques & Incantations
Emotion
Only Emotion is Harnessable
Only Awareness is Educable
19
Only Behaviour is Trainable
For More Details
Thinkers (ATM, Derby)
Questions & Prompts for Mathematical Thinking
Secondary & Primary versions (ATM, Derby)
Mathematics as a Constructive Activity (Erlbaum)
Structured Variation Grids
Studies in Algebraic Thinking
Other Publications
This and other presentations
http://mcs.open.ac.uk/jhm3j
[email protected]
20