Generalisation Fostering Supporting Algebraic Thinking John Mason Trondheim
Generalisation: Fostering & Supporting Algebraic Thinking John Mason Trondheim Oct 2007 1
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! 2
What’s The Difference? – = First, add one to each First, add one to the larger and subtract one from the smaller 3 What then would be the difference? What could be varied?
Think Of A Number (Thoan) intrigues adolescents / Displays power over numbers / Introduces a device for dealing with as-yetunknown numbers / 4
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? 5 +1 +2 +3 4 +6
Powers /Imagining & Expressing /Specialising & Generalising /Conjecturing & Convincing /Classifying & Characterising /Fixing & Changing /Stressing & Ignoring /Attending & Intending 6
Pattern Continuation … … 7
Experiencing Generalisation /Going with the grain: enactive generalisation /Going across the grain: cognitive generalisation /Pleasure in use of powers; disposition: affective generalisation (Helen Drury) 8
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 9
Copper. Plate Multiplication 10
Paper Folding Shape? 11
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 /… 12
One More What numbers are one more than the sum of four consecutive integers? / What numbers are one more than the product 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. 13
Perforations How many holes for a sheet of r rows and c columns of stamps? 14 If someone claimed there were 228 perforations in a sheet, how could you check?
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Consecutive Sums Say What You See 16 Say What You See
Worlds of Experience Inner World of imagery Material World enactive 17 iconic World of Symbol s symbolic
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. 18
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? / 19
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 /… / 20
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 /… / 21
Four Odd Sums 22
Slope Reading 23
Cutting Chocolate Bars /In how few cuts can you separate the bar into its pieces? /You can only cut one piece at a time! 24
4 3 4 2 41 4 4 2 1 20 4 5 2 2 7 4 6 2 3 8 4 7 2 4 99 4 0 39 19 6 1 2 18 5 4 3 38 17 16 15 14 37 36 35 34 33 48 2 25 5 1 0 1 1 1 2 1 3 3 2 49 4 9 2 6 2 7 2 8 2 9 3 0 3 1 5 0
64 36 37 38 39 35 14 15 16 34 1 3 1 2 3 1 33 3 2 1 1 3 0 2 9 18 4 2 19 4 3 20 3 4 2 1 2 2 2 3 2 4 4 0 17 41 2 1 0 9 1 5 8 7 6 2 8 2 7 2 6 2 5 4 4 4 5 4 6 4 7 48 4 9 5 0 81
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