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Demonstration of the use of variation to scaffold abstract thinking Anne Watson ICMI Study 22 Oxford 2013

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Principles Inductive reasoning (pattern) -> structural insight Relational reasoning (covariation) -> structural insight

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Generalise for 100 number grid

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1234567 891011121314 15161718192021 22232425262728 29303132333435 36373839404142 43444546474849 Generalise for another number

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12345678 910111213141516 1718192021222324 2526272829303132 3334353637383940 4142434445464748 4950515253545556 5758596061626364 Generalise for any number: variables and parameters

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What new kinds of question can be asked and why?

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New question-types On an 9-by-9 grid my tetramino covers 8 and 18. Guess my tetramino. What tetramino, on what grid, would cover the numbers 25 and 32? What tetramino, on what grid, could cover cells (m-1) and (m+7)?

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Generalise for a times table grid

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What new kinds of question can be asked and why?

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New question-types What is the smallest ‘omino’ that will cover cells (n + 1, m – 11) and (n -3, m + 1)?

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813202940536885104 1318253445587390109 2025324152658097116 29344150617489106125 404552617285100117136 535865748598113130149 68738089100113128145164 859097106117130145162181 104109116125136149164181200

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Variations and their affordances Shape and orientation (comparable examples) Position on grid (generalisations on one grid) Size of number grid (generalisations with grid size as parameter) Object: grid-shape as ‘new’ compound object to be acted upon (abstraction as a new object-action) Nature of number grid (focus on variables to generalise a familiar relation) Unfamiliar number grid (focus on relations between variables)

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Role of variation Awareness of variation as generating examples for inductive reasoning Using outcomes of inductive reasoning as new objects for new variations Twin roles of presenting variation and directing questions (cf. also the paper by Hart in Theme C)

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ATM resources mcs.open.ac.uk/jhm3 (applets & animations)

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