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To confirm the deepest thing in our students is the educators special privilege. It demands that we see in the failures of adolescence and its confusions, the possibility of something untangled, clear, directed (Barbara Windle)

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Adolescent learning and secondary mathematics Anne Watson University of Oxford Sherbrooke, May 2008

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Closer Find a number which is closer to 3/8 than it is to 3/16 … and another

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More … and another Make up a linear equation in x whose solution is 5 … and another … and another, but this one must be VERY different from the previous one

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Affordances of exemplification tasks … and another Awareness of example spaces Awareness of dimensions of variation Awareness of ranges of change

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Comparing equivalent objects How many ways can you find to express the number of dots in this diagram?

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Affordances of comparison How many ways …? Equivalent representations Transformation between representations Arguments about completeness

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Grid multiplication x+ 3 x - 2

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Surds/irrationals Use grid multiplication to find a pair of numbers like a + b which, when multiplied, have no irrational bits c d a b

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Affordances of construction tasks: to learn how to enquire to solve problems in ad hoc fashion to extend and enrich personal example space to understand properties and structure (stronger mathematical activity)

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Enlargement

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Affordances of comparing methods identify supermethods informed choice is empowering knowing limitations is empowering understand why we have algorithms

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Adolescence is about … identity belonging being heard being in charge being supported reorganising neural pathways in frontal cortex feeling powerful understanding the world negotiating authority arguing in ways which make adults listen sex

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Adolescent learning is progress from ad hoc to abstract from imagined fantasy to imagined actuality from intuitive notions to scientific notions from empirical approaches to reasoned approaches

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Mathematics learning is progress from ad hoc to abstract from imagination to abstraction from intuitive notions to scientific notions from empirical approaches to reasoned approaches

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Consecutive sums = = = 21

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Affordances of enquiry tasks: Choice; action (agency) Conjectures; perspectives (identity) Ownership (empowerment; identity) Discussion (collaboration) Reflection Changes in mathematical activity??

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The fallacy of choice Choice does not necessarily lead to stronger mathematical activity

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Fallacy of reflection: to validate and assess work to evaluate personal effort to evaluate strength of procedures, working methods and results to identify structure, abstractions, relations, properties (stronger mathematical activity)

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Possible shifts in mental activity due to teacher intervention in consecutive sums Discrete – continuous Additive - multiplicative Rules – tools Procedure – meaning Example – generalisation Perceptual – conceptual Operations – inverses Pattern – relationship Relationship – properties Conjecture – proof Results – reflection on results Result – reflection on procedure/method Inductive – deductive Other ….

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Multiplicative relationships

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x 2 = 24 x 3 = 24 e x = 24

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Multiplicative relationships

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Multiplicative relationships

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xy = 24 x = 24/ y y = 24/ x What is the same/different about the last two?

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Multiplicative relationships What two numbers multiply to give 24? …and another What three numbers multiply to give 24? What number squared gives 24?

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Problematic aspects of secondary mathematics probability proportion & ratio non-linear sequences symbolic representation proving things adding fractions….. understanding limits using algebraic relationships reasoning from properties …

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What shifts are needed to learn secondary mathematics? Discrete – continuous Additive - multiplicative Rules – tools Procedure – meaning Example – generalisation Perceptual – conceptual Operations – inverses Pattern – relationship Relationship – properties Conjecture – proof Results – reflection on results Result – reflection on procedure/method Inductive – deductive Other ….

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Adolescent actualisation in mathematics identity as active thinker belonging to the class being heard by the teacher understanding the world negotiating the authority of the teacher through mathematics being able to argue mathematically in ways which make adults listen

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Adolescent actualisation in mathematics being in charge of personal example space being supported by inherent sense of mathematics feeling powerful by being able to generate mathematics being helped to make explicit shifts of conceptualisation sex …??

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Raising Achievement in Secondary Mathematics Watson (Open University Press) Mathematics as a Constructive Activity Watson & Mason (Erlbaum)

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