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Making Use of Making Use of Students’ Natural PowersStudents’ Natural Powersto Think Mathematicallyto Think Mathematically
John MasonJohn Mason
GrahamstownGrahamstown
May 2009May 2009
The Open UniversityMaths Dept University of Oxford
Dept of Education
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Some SumsSome Sums
4 + 5 + 6 =9 + 10 + 11 + 1216
Generalise
Justify
Watch What You Do
Say What You See
1 + 2 =3
7 + 8= 13 + 14 + 15
17 + 18 + 19 + 20+ = 21 + 22 + 23 + 24
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Consecutive SumsConsecutive Sums
Say What You See
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CopperPlate CopperPlate CalculationsCalculations
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DifferenceDifferenceDivisionsDivisions
4 – 2 = 4 ÷ 2
4 – 3 = 4 ÷ 312
12
5 – 4 = 5 ÷ 413
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6 – 5 = 6 ÷ 514
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7 – 6 = 7 ÷ 615
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3 – 2 = 3 ÷ 211
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0 – (-1) = 0 ÷ (-1)
1-2
1-2
2 1oops
1 – 0 = 1 ÷ oops1-1
1-1
How does this fit in?
Going with the grain
Going across the grain
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Leibniz’s TriangleLeibniz’s Triangle
1
2
1
2
1
3
1
6
1
3
1
4
1
5
1
1
4
1
12
1
12
1
20
1
5
1
20
1
30
1
60
1
30
1
6
1
30
1
60
1
6
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Remainders of the Day (1)Remainders of the Day (1)
Write down a number which Write down a number which when you subtract 1 is divisible when you subtract 1 is divisible by 5by 5
and anotherand another and anotherand another Write down one which you Write down one which you
think no-one else here will think no-one else here will write down.write down.
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Remainders of the Day (2)Remainders of the Day (2)
Write down a number which when Write down a number which when you subtract 1 is divisible by 2you subtract 1 is divisible by 2
and when you subtract 1 from the and when you subtract 1 from the quotient, the result is divisible by quotient, the result is divisible by 33
and when you subtract 1 from that and when you subtract 1 from that quotient the result is divisible by 4quotient the result is divisible by 4
Why must any such number be Why must any such number be divisible by 3? divisible by 3?
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Remainders of the Day (3)Remainders of the Day (3)
Write down a number which is Write down a number which is 1 more than a multiple of 21 more than a multiple of 2
and which is 2 more than a and which is 2 more than a multiple of 3multiple of 3
and which is 3 more than a and which is 3 more than a multiple of 4multiple of 4
… …
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Remainders of the Day (4)Remainders of the Day (4)
Write down a number which is Write down a number which is 1 more than a multiple of 21 more than a multiple of 2
and 1 more than a multiple of and 1 more than a multiple of 33
and 1 more than a multiple of and 1 more than a multiple of 44
… …
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AssumptionsAssumptions What you get from this session will be largely What you get from this session will be largely
what you notice happening for youwhat you notice happening for you If you do not participate, I guarantee you will If you do not participate, I guarantee you will
get nothing!get nothing! I assume a conjecturing atmosphereI assume a conjecturing atmosphere
– Everything said has to be tested in experienceEverything said has to be tested in experience– If you know and are certain, then think and listen;If you know and are certain, then think and listen;– If you are not sure, then take opportunities to try If you are not sure, then take opportunities to try
to express your thinkingto express your thinking Learning is a maturation process, and so Learning is a maturation process, and so
invisibleinvisible– It can be promoted by pausing and withdrawing It can be promoted by pausing and withdrawing
from the immediate action in order to get an from the immediate action in order to get an overviewoverview
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Triangle CountTriangle Count
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Max-MinMax-Min
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
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Max-MinMax-Min
In a rectangular array of numbers, In a rectangular array of numbers, calculate calculate – The maximum value in each row, and The maximum value in each row, and
then the minimum of thesethen the minimum of these– The minimum in each column and then The minimum in each column and then
the maximum of thesethe maximum of these How do these relate to each other?How do these relate to each other? What about interchanging rows and What about interchanging rows and
columns?columns? What about the mean of the maxima What about the mean of the maxima
of each row, and the maximum of of each row, and the maximum of the means of each column?the means of each column?
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Up & Down SumsUp & Down Sums
1 + 3 + 5 + 3 + 1
3 x 4 + 122 + 32
1 + 3 + … + (2n–1) + … + 3 + 1
==
n (2n–2) + 1 (n–1)2 + n2 ==
Generalise!See
generalitythrough aparticular
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DifferencesDifferences
17=16−142
AnticipatingGeneralising
Rehearsing
Checking
Organising
18=17−156
=16−124
=14−18
13=12−16
14=13−112
=12−14
15=14−120
16=15−130
=12−13=13−16=14− 112
12=11−12
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PowersPowers
Am I stimulating learners to use Am I stimulating learners to use their own powers, or am I abusing their own powers, or am I abusing their powers by trying to do things their powers by trying to do things for them?for them?– To imagine & to expressTo imagine & to express– To specialise & to generaliseTo specialise & to generalise– To conjecture & to convinceTo conjecture & to convince– To stress & to ignoreTo stress & to ignore– To extend & to restrictTo extend & to restrict
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ReflectionsReflections
Much of mathematics can be Much of mathematics can be seen as studying actions on seen as studying actions on objectsobjects
Frequently it helps to ask Frequently it helps to ask yourself what actions leave yourself what actions leave some relationship invariant; some relationship invariant; often this is what is studied often this is what is studied mathematicallymathematically
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More ResourcesMore Resources
Questions & Prompts for Mathematical Questions & Prompts for Mathematical Thinking Thinking ((ATM Derby: primary & secondary ATM Derby: primary & secondary versions)versions)Thinkers (Thinkers (ATM Derby)ATM Derby)Mathematics as a Constructive Activity Mathematics as a Constructive Activity (Erlbaum)(Erlbaum)Designing & Using Mathematical Tasks Designing & Using Mathematical Tasks (Tarquin)(Tarquin)http: //http: //mcs.open.ac.uk/jhm3mcs.open.ac.uk/jhm3j.h.mason @ open.ac.ukj.h.mason @ open.ac.uk
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1 2
345
6
7 8 9 10
11
12
13
18
19
20
21 22 23 24 25 26
27
28
29
30
3132
14151617
3334353637
38
39
40
41
42
43 44 45 46 47 48 49 50
1
4
9
16
25
49
36
21
1
2 3 4
5
6789
101112
13
18 19 20
21
22
23
242526272829
303132
14 15 16 17
33
34
35
36 37 38 39 40 41 42 43 44
45
46
47
48
49
50
64
81
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Gasket SequencesGasket Sequences
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PerforationsPerforations
How many holes for a sheet of
r rows and c columnsof stamps?
If someone claimedthere were 228 perforations
in a sheet, how could you check?