Solving Mathematical Problems
Mathematically
John Mason
ATM-MA
Cornwall
May2009
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Assumptions
What
you get from this session will be largely
what you notice happening for you
If you do not participate, I guarantee you will
get nothing!
I assume a conjecturing atmosphere
– Everything said has to be tested in experience
– If you know and are certain, then think and listen;
– If you are not sure, then take opportunities to try to
express your thinking
Learning
is a maturation process, and so
invisible
– It can be promoted by pausing and withdrawing from
the immediate action in order to get an overview
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Kites
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Visiting
 People
set off at the same time to walk between
two towns, some in each direction. They all
meet at noon. They have lunch together for an
hour, then carry on as before; some reach their
destination at 5pm, and the others at 7:15 pm.
 At what time did they all set out?
distance
• Let the speeds be v and w, and the
time before noon be h
• The am distances are vh and wh
• The pm distances are 4v and
25w/4
• So wh = 4v and vh = 25w/4
• Whence h2 = 25 and h = 5
• So they set out at 7:00 am
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wh
vh
h
12:00
4:00
6:1
time
Triangle Count
5
Max-Min
6
2
5
6
8
3
2
4
1
7
7
6
1
2
9
4
6
8
9
5
8
9
8
2
5
9
7
2
1
9
8
3
7
1
9
6
9
Max-Min
In
a rectangular array of numbers,
calculate
– The maximum value in each row, and then the
minimum of these
– The minimum in each column and then the
maximum of these
How
do these relate to each other?
What about interchanging rows and
columns?
What about the mean of the maxima of
each row, and the maximum of the means
of each column?
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Imagining & Expressing
Sliding Round and Round
Imagine a triangle
Imagine a second
triangle
with one vertex on the
edge of the first triangle
Now always keeping the
second triangle just
touching the first, slide it
round the first.
NO Rotating Allowed!
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Get a sense of
the freedom available
‘just touching’
What is the length of the locus
of the centroid of the sliding triangle?
Up & Down Sums
1+3+5+3+ 1
22 + 3 2
=
=
3x4+1
See
generality
through a
particular
Generalise!
1 + 3 + … + (2n–1) + … + 3 + 1
=
9
(n–1)2 + n2
= n (2n–2) + 1
Interlude on Tasks
 tasks
are for initiating activity;
 through engaging in activity people have the
opportunity to experience things, particularly
mathematical actions and their effects.
 Teachers have intentions when they choose tasks:
– it is not that the task itself will promote learning, but that the
inner task will involve the directing or re-directing of attention
– and perhaps the internalising or integrating of actions
previously dependent on being triggered by some outside
agency (ZPD).
 It
is valuable, even necessary to draw back from the
action and to become aware of the actions and their
effects (utility)
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Differences
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1 1 1
1  1 1
7 6 42
2 1 2
1 11
1 1 1  1 1  11
3 2 6
8 7 56 6 24 4 8
Anticipating
1  1 1  1  1
Generalising
4 3 12 2 4
Rehearsing
1 1 1
5 4 20
Checking
1  1 1  11 1 1  1  1
Organising
6 5 30 2 3 3 6 4 12
Powers
Am
I stimulating learners to use their own
powers, or am I abusing their powers by
trying to do things for them?
–
–
–
–
–
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To imagine & to express
To specialise & to generalise
To conjecture & to convince
To stress & to ignore
To extend & to restrict
Reflections
Much
of mathematics can be seen
as studying actions on objects
Frequently it helps to ask yourself
what actions leave some relationship
invariant; often this is what is
studied mathematically
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More Resources
Questions
& Prompts for Mathematical
Thinking
(ATM Derby: primary & secondary versions)
Thinkers (ATM Derby)
Mathematics as a Constructive Activity
(Erlbaum)
Designing & Using Mathematical Tasks
(Tarquin)
http: //mcs.open.ac.uk/jhm3
j.h.mason @ open.ac.uk
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