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Demonstration of the use of variation to scaffold abstract thinking

This study explores the use of variation to scaffold abstract thinking in mathematics education. It focuses on inductive reasoning and relational reasoning to develop structural insight. The study also generalizes for different number and times table grids, introducing new question types and variations. The role of variation in generating examples for inductive reasoning is highlighted, along with the use of outcomes as new objects for further variations.

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Demonstration of the use of variation to scaffold abstract thinking

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

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

  3. Generalise for 100 number grid

  4. Generalise for another number

  5. Generalise for any number: variables and parameters

  6. What new kinds of question can be asked and why?

  7. 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)?

  8. Generalise for a times table grid

  9. What new kinds of question can be asked and why?

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

  11. 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)

  12. 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)

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

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