# 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,

## Presentation on theme: "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,"— Presentation transcript:

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)

Anne Watson University of Oxford Sherbrooke, May 2008

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

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

… and another Awareness of example spaces Awareness of dimensions of variation Awareness of ranges of change

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

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

Grid multiplication x + 3 - 2

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

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)

Enlargement

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

being supported reorganising neural pathways in frontal cortex feeling powerful understanding the world negotiating authority arguing in ways which make adults listen sex

from ad hoc to abstract from imagined fantasy to imagined actuality from intuitive notions to ‘scientific’ notions from empirical approaches to reasoned approaches

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

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

Choice; action (agency) Conjectures; perspectives (identity) Ownership (empowerment; identity) Discussion (collaboration) Reflection Changes in mathematical activity??

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

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)

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 ….

Multiplicative relationships

Multiplicative relationships

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

Multiplicative relationships
24 12 2 2 6 3 2

Multiplicative relationships

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

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

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 …

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 ….