Transcript Fostering & Sustaining Math`l Th`g Leicester

Fostering and Sustaining
Mathematical Thinking
ATM-MA meeting
Leicester
Nov 2008
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John Mason
Open University & University of
Oxford
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 informative questions
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Conjectures
The
richness of mathematical tasks does NOT
lie in the tasks themselves
NOR does it lie in the format of interactions
It DOES lie in
– Opportunities afforded matching student
attunements
– Possibilities for use of student powers including
variations and extensions
– Access to encountering mathematical themes
It
DEPENDS on teachers’ ‘being’, manifested in
– Teacher-Learners relationships
– Teacher’s mathematical awareness
– Working milieu
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More Conjectures
The
richness of learners’ mathematical
experience depends on
– Opportunities to use and develop their
own powers
– Opportunities to make significant
mathematical choices
– Being in the presence of mathematical
awareness
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Pattern Continuation
…
…
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Children’s Copied Patterns
model
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4.1 yrs
Marina Papic MERGA 30 2007
Children’s Own Patterns
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
5.0 yrs
5.1 yrs
5.4 yrs
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Marina Papic MERGA 30 2007
Four Consecutives
What numbers arise as
sum
one more than the product of
4 consecutive numbers?
n,
n + a,
n+b,
n + (a+b)
So natural to specialise!
Try some examples …
in order to see what is going on
… to re-generalise
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Triangle Count
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Up & Down Sums
1+3+5+3+ 1
22 + 3 2
=
=
3x4+1
1 + 3 + … + (2n–1) + … + 3 + 1
=
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(n–1)2 + n2
= n (2n–2) + 1
What’s The Difference?
–
=
First, add one to each
First,
add one to the larger and
subtract one from the smaller
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What then
would be
the difference?
What could
be varied?
What’s The Ratio?
÷
=
First, multiply each by
3
First,
multiply the larger by 2 and
divide the smaller by 3
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Which will be
What is the ratio? ?
larger?
What could
be varied?
Specific Centre of Gravity
When a can of fizzy drink is full, the centre
of gravity is in the middle of the can;
When the can is empty, the centre of
gravity is in the middle of the can (it is a
mathematical can!)
How does the centre of gravity of the can
move as liquid is taken out, and when is it
at its lowest point?
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Averaged Speed
At
some construction on a motorway, the
sign said speed limit 50; average speed
calculated
I notice that for a specified number of
minutes I was going 60. For how long do I
have to go at 30? At 35?
I notice that for a specified distance I was
going 60. How far do I have to go at 30? At
35?
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Averaged Speed Graphs
Know:
distance
50mph
D
=V
T
D+d
=w
T+t
d =v
t
d
D
Also know:
w, T, V, v
time
TV + tv = w(T + t)
T
t
Want: t
t(w – v) = V – w
Also know:
w, D, V, v
D+d
=
w
w
d( 1 – 1
)
w v
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d
)
v
D
+
V
Want: d
D + d = w(
D
+
V
d
v
= D( 1 – )1
w
V
Remainders of the Day
 Write
down a number that leaves a reminder of 1
when divided by 3
 and another
 and another
 Multiply two of these numbers together:
what remainder does it leave when divided by 3?
 Why?
 What is special about the ‘3’?
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Primality
What
does ‘prime’ mean in the system of
numbers leaving a remainder of 1 when
divided by 3?
What are the first three positive nonprimes 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’?
What is special about the ‘1’?
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Fractional
What
can be said about fractions
where numerator and denominator
both leave a remainder of 1 when
divided by 3?
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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?

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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?
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Magic Square Reasoning
2
6
7
2
1
5
9
8
Sum(
21
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(
22
) – Sum(
) =0
Composite Doing & Undoing
I
am thinking of a number
I add 8 and the answer is 13.
I add 8 then multiply by 2;
the answer is 26.
I add 8, multiply by 2, subtract 5;
the answer is 21.
I add 8, multiply by 2, subtract 5, divide by 3.
The answer is 7
HOW do you turn +8, x2, -5, ÷3 answer 7
into a solution?

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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?
Two different expressions!
What undoes dividing by 3/2?
Two different expressions!
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Fractional Increase and Decrease
1
1
(1 + ) (1 – )
3
2
2
2
(1 + ) (1 – )
5
7
a
(1 + ) (1 –
b
25
=1
By how much do I have to
decrease in order to undo an
increase by one-half?
=1
By how much do I have to
increase in order to undo a
decrease by two-sevenths?
) =1
Additive & Multiplicative Perspectives
What
is the relation between the
numbers of squares of the two
colours?
Difference of 2, one is 2 more:
additive
Ratio of 3 to 5; one is five thirds the
other etc.:
multiplicative
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Raise your hand when you can
see
 Something
which is 2/5 of something
 Something which is 3/5 of something
 Something which is 2/3 of something
– What others can you see?
 Something which is 1/3 of 3/5 of something
 Something which is 3/5 of 1/3 of something
 Something which is 2/5 of 5/2 of something
 Something which is 1 ÷ 2/5 of something
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What fractions can you ‘see’?
What
relationships between
fractions can you see?
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Attention
Holding Wholes (gazing)
Discerning Details
Recognising Relationships
Perceiving Properties
Reasoning on the basis of agreed
properties
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Reflections
 Initiating Activity
 Sustaining
Activity
 Transcending Activity
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Conjectures
The
richness of mathematical tasks does NOT
lie in the tasks themselves
NOR does it lie in the format of interactions
It DOES lie in
– Opportunities afforded matching student
attunements
– Possibilities for use of student powers including
variations and extensions
– Access to encountering mathematical themes
It
DEPENDS on teachers’ ‘being’, manifested in
– Teacher-Learners relationships
– Teacher’s mathematical awareness
– Working milieu
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More Conjectures
The
richness of learners’ mathematical
experience depends on
– Opportunities to use and develop their
own powers
– Opportunities to make significant
mathematical choices
– Being in the presence of mathematical
awareness
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Powers
Specialising
& Generalising
Conjecturing
Imagining
Ordering
& Convincing
& Expressing
& Classifying
Distinguishing
Assenting
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& Connecting
& Asserting
Themes
Doing
& Undoing
Invariance
Freedom
& Constraint
Extending
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Amidst Change
& Restricting Meaning
For More Stimulation
Starting
Points (ATM)
Thinking Mathematically (Pearson)
MA
NRICH website
These slides available on
http://mcs.open.ac.uk/jhm3 [workshops]
[email protected]
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