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1 Learning to Think and to Reason Algebraically and the Structure of Attention 2007 John Mason SMC

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2 Outline Some assumptions Some tasks Some reflections

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3 Some assumptions A lesson without the opportunity for learners to generalise is not a mathematics lesson Learners come to lessons with natural powers to make sense Our job is to direct their attention appropriately and effectively

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4 Grid Sums In how many different ways can you work out a value for the square with a ‘?’ only using addition? 7 ? ? To move to the right one cell you add 3. To move up one cell you add 2. Using exactly two subtractions?

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5 Grid Movement 7 ? +3-3 x2 ÷2 ((7+3)x2)+3 is a path from 7 to ‘?’. What expression represents the reverse of this path? What values can ‘?’ have: - if only + and x are used - if exactly one - and one ÷ are used, with as many + & x as necessary What about other cells? Does any cell have 0? -7? Does any other cell have 7? Characterise ALL the possible values that can appear in a cell

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6 Varying & Generalising What are the dimensions of possible variation? What is the range of permissible change within each dimension of variation? You only understand more if you extend the example space or the scope of generality

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7 Number Line Movements Rotate it about the point 5 through 180° –where does 3 end up? –…–…–…–… Now rotate it again bit about the point -2. Now where does the original 3 end up? Generalise! Imagine a number line

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12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 1 4 9 16 25 49 36 Number Spirals

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12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 64 81

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10 CopperPlate Multiplication

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11 Four Odd Sums

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12 Tunja Sequences 1 x 1 – 1 = 2 x 2 – 1 = 3 x 3 – 1 = 4 x 4 – 1 = 0 x 2 1 x 3 2 x 4 3 x 5 0 x 0 – 1 =-1 x 1 -1 x -1 – 1 =-2 x 0 Across the Grain With the Grain

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13 Tunja Display (1) 2x2 - 2 - 2 = 1x1 - 1 3x2 - 3 - 2 = 2x1 - 1 4x2 - 4 - 2 = 3x1 - 1 5x2 - 5 - 2 = 4x1 - 1 … 3x3 - 3 - 3 = 2x2 - 1 4x3 - 4 - 3 = 3x2 - 1 5x3 - 5 - 3 = 4x2 - 1 … Generalise! Run Backwards 1x2 - 1 - 2 = 0x3 - 1 0x2 - 0 - 2 = (-1)x3 - 1 (-1)x2 - (-1) - 2 = (-2)x3 - 1 2x3 - 2 - 3 = 1x2 - 1 1x3 - 1 - 3 = 0x2 - 1 0x3 - 0 - 3 = (-1)x2 - 1 … … … (-1)x3 - (-1) - 3 = (-2)x2 - 1 … … … … … ……

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14 Tunja Display (2) 4x3x2 - 2x3 - 4x2 = 4x3 - 2 4x4x2 - 2x4 - 4x2 = 6x3 - 2 4x5x2 - 2x5 - 4x2 = 8x3 - 2 4x6x2 - 2x6 - 4x2 = 10x3 - 2 … 4x3x3 - 2x3 - 4x3 = 4x5 - 2 4x4x3 - 2x4 - 4x3 = 6x5 - 2 4x5x3 - 2x5 - 4x3 = 8x5 - 2 4x6x3 - 2x6 - 4x3 = 10x5 - 2 … Generalise! Run Backwards 4x2x2 - 2x2 - 4x2 = 2x3 - 2 4x1x2 - 2x1 - 4x2 = 0x3 - 2 4x0x2 - 2x0 - 4x2 = (-2)x3 - 2 4x(-1)x2 - 2x(-1) - 4x2 = (-4)x3 - 2 …

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15 Structured Variation Grids Generalisations in two dimensions Available free at http://mcs.open.ac.uk/jhm3

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16 One More What numbers are one more than the product of four consecutive integers? What numbers are one more than the product of four consecutive integers? Let a and b be any two numbers, at least 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.

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17 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.

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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?

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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 … … … …

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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 … … … …

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21 Magic Square Reasoning 519 2 4 6 83 7 –= 0Sum( ) Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2

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22 More Magic Square Reasoning –= 0Sum( )Sum( )

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23 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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24 Gasket Sequences

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25 Toughy 12345678

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26 Powers Specialising & Generalising Conjecturing & Convincing Imagining & Expressing Ordering & Classifying Distinguishing & Connecting Assenting & Asserting

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27 Themes Doing & Undoing Invariance Amidst Change Freedom & Constraint Extending & Restricting Meaning

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28 Some Reflections Notice the geometrical term: –It requires movement out of the current space into a space of one higher dimension in order to achieve it

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29 Attention Gazing at wholes Discerning details Recognising relationships Perceiving properties Reasoning on the basis of properties

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30 John Mason J.h.mason @ open.ac.uk http://mcs.open.ac.uk/jhm3 Developing Thinking in Algebra (Sage 2005)

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