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

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

8. 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

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

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

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

12. Grid multiplication x
+ 3
- 2

13. 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

14. 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)

15. Enlargement

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

19. 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

20. 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

21. 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

22. Consecutive sums 1 + 2 + 3 + 4 + 5 + 6 = 21 10 + 11 = 21
= 21

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

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

25. 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)

26. 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 ….

27. Multiplicative relationships

28. Multiplicative relationships

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

30. Multiplicative relationships24 12 2 2 6 3 2

31. Multiplicative relationships

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

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

34. 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 …

35. 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 ….

36. 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

37. 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 …??

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