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Unformatted text preview: Thinking Mathematically
and
and
Learning Mathematics
Learning
Mathematically
John Mason
Greenwich
Oct 2008
1 Conjecturing Atmosphere
Everything said is said in order to
Everything consider modifications that may be
needed
needed
Those who ‘know’ support those who are
Those
unsure by holding back or by asking
revealing questions
revealing 2 Up & Down Sums 1+3+5+3+ 1 = 22 + 32 = 3x4+1 1 + 3 + … + (2n–1) + … + 3 + 1
= 3 (n–1)2 + n2 = n (2n–2) + 1 One More
What numbers are one more than the
product of four consecutive integers?
product Let a and b be any two numbers, one of them
even. Then ab/2 more than the product of any
number, a more than it, b more than it and a+b
more than it, is a perfect square, of the number
squared plus a+b times the number plus ab/2
squared.
4 Remainders of the Day
Write down a number that leaves a reminder of 1 when
Write
divided by 3
divided
and another
and another
Choose two simple numbers of this type and multiply
Choose
What is special about the ‘1’?
What
them together:
what remainder does it leave when divided by 3?
what
Why?
What is special
What
about the ‘3’?
What is special
about the ‘1’? 5 Primality
What is the second positive nonprime
What after 1 in the system of numbers of the
form 1+3n?
form
100 = 10 x 10 = 4 x 25
What does this say about primes in the
What
multiplicative system of numbers of the
form 1 +3n?
form
What is special about the ‘3’? 6 InterRootal Distances
Sketch a quadratic for which the interrootal
Sketch
distance is 2.
distance
and another
and another
How much freedom do you have?
What are the dimensions of possible variation
What
and the ranges of permissible change?
and
If it is claimed that [1, 2, 3, 3, 4, 6] are the interrootal distances of a quartic, how would you
rootal
check?
check?
7 Bag Constructions (1)
Here there are three bags. If
Here
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.
two For four bags, what is the least
number of objects to meet the same
constraint?
constraint?
For four bags, what is the least
number of colours to meet the
same constraint?
same
8 17 objects
3 colours Bag Constructions (2)
Here there are 3 bags and two
Here
objects.
objects.
There are [0,1,2;2] objects in
There
the bags with 2 altogether
the
Given a sequence like [2,4,5,5;6]
Given
or [1,1,3,3;6] how can you tell if
there is a corresponding set of
bags?
bags?
9 Statisticality
write down five numbers whose mean is
write 5
and whose mode is 6
and whose median is 4 10 ZigZags
Sketch the graph of y = x – 1
Sketch
1
Sketch the graph of y =  x  1  2
Sketch
Sketch the graph of
Sketch
y =   x – 1 – 2 – 3
What sorts of zigzags can you make, and not
What
make?
make?
Characterise all the zigzags you can make
Characterise
using sequences of absolute values like this.
using
11 Towards the Blanc Mange function 12 Reading Graphs 13 Examples
Of what is x an example?
Of what is y = x2 and example?
Of
– y = b + (x – a)2 ? 14 Functional Imagining
Imagine a parabola
Now imagine another one
Now
the other way up.
the
Now put them in two planes
Now
at right angles to each
other.
other.
Make the maximum of the
Make
downward parabola be on
the upward parabola
the
15 Now sweep your downward
Now
parabola along the upward
parabola so that you get a
surface
surface MGA 16 Getti
Man
Artic
Getti
Man Powers
Specialising & Generalising
Conjecturing & Convincing
Imagining & Expressing
Ordering & Classifying Distinguishing & Connecting
Assenting & Asserting 17 Themes
Doing & Undoing
Invariance Amidst Change
Freedom & Constraint
Extending & Restricting Meaning 18 Teaching Trap
19 Learning Trap Doing for the learners what
they can already do for
themselves
themselves
Teacher Lust:
Teacher
– desire that the learner
learn
learn
– desire that the learner
desire
appreciate and understand
– Expectation that learner
Expectation
will go beyond the tasks as
set
set
– allowing personal
allowing
excitement to drive
behaviour
behaviour Expecting the teacher to do
for you what you can already
do for yourself
do
Learner Lust:
Learner
– desire that the teacher
teach
teach
– desire that learning will be
desire
easy
easy
– expectation that ‘dong the
expectation
tasks’ will produce
learning
learning
– allowing personal
reluctance/uncertainty to
drive behaviour
drive 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
Emotions
(affect)
(affect) Body (enaction)
Habits
Practices 21 Structure of a Topic
Language Patterns
& prior Skills Root Questions
predispositions Standard Confusions
& Obstacles Different Contexts in
which likely to arise;
dispositions Techniques &
Incantations Aw
ar
ss en e Emoti
on
Only Emotion is
Only
Harnessable
Harnessable
Only Behaviour is
Only
Only Awareness is
Only
Trainable
Trainable
Educable
Educable
o
o
v
vii
ha
ha
Be ur
Be ur 22 Imagery/Senseof/Awareness;
Connections 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
j.h.mason@open.ac.uk
25 Modes of interaction
Expounding
Explaining
Exploring
Examining
Exercising
Expressing Teacher
Student
Content
Expounding Student
Content
Teacher
Examining Teacher
Content
Student Student
Teacher
Content Explaining Exploring Content
Teacher
Student
Expressing Content
Student
Teacher
Exercising ...
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 Winter '08
 Staff
 Math

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