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Adolescent Learning in Secondary Mathematics: Affordances and Progress

This article explores the affordances of different teaching tasks in adolescent learning of secondary mathematics, such as exemplification, comparison, construction, and enquiry tasks. It discusses how these tasks can support the progress from ad hoc to abstract reasoning and from intuitive notions to scientific concepts. The article also examines the shifts in mental activity that can occur with teacher intervention and the potential problematic aspects of secondary mathematics. Overall, it highlights the importance of adolescent actualization in mathematics and the role of the educator in facilitating this process.

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Adolescent Learning in Secondary Mathematics: Affordances and Progress

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

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

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

  4. 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

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

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

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

  8. x + 3 x - 2 Grid multiplication

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

  10. 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)

  11. Enlargement

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

  13. 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 Adolescence is about …

  14. 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

  15. 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

  16. Consecutive sums 1 + 2 + 3 + 4 + 5 + 6 = 21 10 + 11 = 21 6 + 7 + 8 = 21

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

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

  19. 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)

  20. 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 …. Possible shifts in mental activity due to teacher intervention in ‘consecutive sums’

  21. Multiplicative relationships

  22. Multiplicative relationships

  23. Multiplicative relationships x 2 = 24 x 3 = 24 e x = 24

  24. Multiplicative relationships 24 2 12 2 6 3 2

  25. Multiplicative relationships

  26. Multiplicative relationships • xy = 24 • x = 24/ y • y = 24/ x • What is the same/different about the last two?

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

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

  29. 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 …. What shifts are needed to learn secondary mathematics?

  30. 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

  31. 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 …??

  32. Raising Achievement in Secondary MathematicsWatson (Open University Press) • Mathematics as a Constructive ActivityWatson & Mason (Erlbaum)

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