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Thinking Mathematically
and
Learning Mathematics
Mathematically
John Mason
Greenwich
Oct 2008
1
Conjecturing Atmosphere
Everything
said is said in order to
consider modifications that may be
needed
Those who ‘know’ support those
who are unsure by holding back or
by asking revealing questions
2
Up & Down Sums
1+3+5+3+ 1
22 + 3 2
=
=
3x4+1
1 + 3 + … + (2n–1) + … + 3 + 1
=
3
(n–1)2 + n2
= n (2n–2) + 1
Remainders of the Day
 Write
down a number that leaves a reminder of 1
when divided by 3
 and another
 and another
 Choose two simple numbers of this type and
What is special about the ‘1’?
multiply them together:
what remainder does it leave when divided by 3?
 Why?
 What is special
about the ‘3’?
What is special
about the ‘1’?
5
Primality
What
is the second positive non-prime
after 1 in the system of numbers of the
form 1+3n?
100 = 10 x 10 = 4 x 25
What does this say about primes in
the multiplicative system of numbers
of the form 1 +3n?
What is special about the ‘3’?
6
Inter-Rootal Distances
Sketch
a quadratic for which the interrootal distance is 2.
and another
and another
How much freedom do you have?
What are the dimensions of possible
variation and the ranges of permissible
change?
If it is claimed that [1, 2, 3, 3, 4, 6] are the
inter-rootal distances of a quartic, how
would you check?
7
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?

8
17 objects
3 colours
Bag Constructions (2)
Here
there are 3 bags and
two objects.
There are [0,1,2;2] objects in
the bags with 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?
9
Statisticality
write
down five numbers whose
mean is 5
and whose mode is 6
and whose median is 4
10
ZigZags
the graph of y = |x – 1|
Sketch the graph of y = | |x - 1| - 2|
Sketch the graph of
y = | | |x – 1| – 2| – 3|
What sorts of zigzags can you make, and
not make?
Characterise all the zigzags you can make
using sequences of absolute values like
this.
Sketch
11
Towards the Blanc Mange function
12
Reading Graphs
13
Examples
Of
what is |x| an example?
Of what is y = x2 and example?
– y = b + (x – a)2 ?
14
Functional Imagining
Imagine
a parabola
Now imagine another
one the other way up.
Now put them in two
planes at right angles to
each other.
Make the maximum of
the downward parabola
be on the upward
parabola
Now sweep your downward
15
parabola along the upward
parabola so that you get a
surface
MGA
Getting-a-sense-of
Manipulating
Articulating
Getting-a-sense-of
Manipulating
16
Powers
Specialising
& Generalising
Conjecturing
Imagining
Ordering
& Convincing
& Expressing
& Classifying
Distinguishing
Assenting
17
& Connecting
& Asserting
Themes
Doing
& Undoing
Invariance
Freedom
& Constraint
Extending
18
Amidst Change
& Restricting Meaning
Teaching Trap
Learning Trap
 Expecting the teacher to
Doing for the learners
do for you what you can
what they can already do
for themselves
already do for yourself
 Teacher Lust:
 Learner Lust:
– desire that the learner
– desire that the teacher
learn
teach
– desire that the learner
– desire that learning will
appreciate and
be easy
understand
– expectation that ‘dong
– Expectation that learner
the tasks’ will produce
will go beyond the tasks
learning
as set
– allowing personal
– allowing personal
excitement to drive
reluctance/uncertainty
behaviour
to drive behaviour
19

Human Psyche
Training
Behaviour
Educating Awareness
Harnessing Emotion
Who does these?
– Teacher?
– Teacher with learners?
– Learners!
20
Structure of the Psyche
Awareness (cognition)
Imagery
Will
Emotions
(affect)
Body (enaction)
Habits
Practices
21
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
22
Only Behaviour is Trainable
Didactic Tension
The more clearly I indicate
the behaviour sought from learners,
the less likely they are to
generate that behaviour for themselves
(Guy Brousseau)
23
Didactic Transposition
Expert awareness
is transposed/transformed into
instruction in behaviour
(Yves Chevellard)
24
More Ideas
For Students
(1998) Learning & Doing Mathematics (Second revised edition),
QED Books, York.
(1982). Thinking Mathematically, Addison Wesley, London
For Lecturers
(2002) Mathematics Teaching Practice: a guide for university
and college lecturers, Horwood Publishing, Chichester.
(2008). Counter Examples in Calculus. College Press, London.
http://mcs.open.ac.uk/jhm3
[email protected]
25
Modes of interaction
Expounding
Explaining
Exploring
Examining
Exercising
Expressing
Teacher
Student
Content
Expounding
Teacher
Content
Student
Explaining
Student
Content
Teacher
Examining
Student
Teacher
Content
Exploring
Content
Teacher
Student
Expressing
Content
Student
Teacher
Exercising