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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

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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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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?