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?
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?
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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
23 Slope Reading
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!