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)

Adolescent learning and secondary mathematics

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 … 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

Affordances of exemplification tasks

… 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

x

- 2

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

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)

Enlargement

Affordances of comparing methods

• identify supermethods

• informed choice is empowering

• knowing limitations is empowering

• understand why we have algorithms

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

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

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

6 + 7 + 8 = 21

Affordances of enquiry tasks:

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

2

6

3

2

212

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

…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 ….

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

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 …??

Raising Achievement in Secondary Mathematics Watson (Open University Press)

Mathematics as a Constructive Activity Watson & Mason (Erlbaum)