# 1 Generalisation: Fostering & Supporting Algebraic Thinking John Mason Trondheim Oct 2007.

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1 Generalisation: Fostering & Supporting Algebraic Thinking John Mason Trondheim Oct 2007

2 Assumptions  Generalisation lies at the very core of mathematics and mathematical thinking  A lesson without the opportunity for learners to generalise … is not a mathematics lesson!

3 What’s The Difference? What could be varied? –= First, add one to each First, add one to the larger and subtract one from the smaller What then would be the difference?

4 Think Of A Number (Thoan)  intrigues adolescents  Displays power over numbers  Introduces a device for dealing with as-yet- unknown numbers

5 Four Consecutives  Write down four consecutive numbers and add them up  and another  Now be more extreme!  What is the same, and what is different about your answers? + 1 + 2 + 3 + 6 4

6 Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Classifying & Characterising  Fixing & Changing  Stressing & Ignoring  Attending & Intending

7 Pattern Continuation … …

8 Experiencing Generalisation  Pleasure in use of powers; disposition: affective generalisation (Helen Drury)  Going across the grain: cognitive generalisation  Going with the grain: enactive generalisation

9 Raise Your Hand When You See … Something which is 2/5 of something; 3/4 of something; 5/2 of something; 4/3 of something; 3/4 of 2/5 of something; 3/4 of 4/3 of something; 1 ÷ 2/5 of something; 1 ÷ 3/4 of something

10 CopperPlate Multiplication

11 Paper Folding Shape?

12 What Would Happen If …?  The tap wasn’t turned off  It never rained  The power went off  A nearby stream flooded  You kept on cutting a piece of paper in half …………

13 One More  What numbers are one more than the sum of four consecutive integers? 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.  What numbers are one more than the product of four consecutive integers?

14 Perforations How many holes for a sheet of r rows and c columns of stamps? If someone claimed there were 228 perforations in a sheet, how could you check?

15

16 Consecutive Sums Say What You See

17 Worlds of Experience Material World World of Symbol s Inner World of imagery enactiveiconicsymbolic

18 Remainders of the Day (1)  Write down a number which when you subtract 1 is divisible by 5  and another  Write down one which you think no-one else here will write down.

19 Remainders of the Day (2)  Write down a number which when you subtract 1 is divisible by 2  and when you subtract 1 from the quotient, the result is divisible by 3  and when you subtract 1 from that quotient the result is divisible by 4  Why must any such number be divisible by 3?

20 Remainders of the Day (3)  Write down a number which is 1 more than a multiple of 2  and which is 2 more than a multiple of 3  and which is 3 more than a multiple of 4  … … … …

21 Remainders of the Day (4)  Write down a number which is 1 more than a multiple of 2  and 1 more than a multiple of 3  and 1 more than a multiple of 4  … … … …

22 Four Odd Sums

24 Cutting Chocolate Bars  In how few cuts can you separate the bar into its pieces?  You can only cut one piece at a time!

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