Adolescence and secondary mathematics: shifts of perspective

Anne Watson December 2008

Adolescence: social identity belonging being heard being in charge being supported

feeling powerful understanding the

world negotiating authority arguing in ways which

make adults listen

Adolescence: emotional

Self-concept, motivation, engagement etc. In all school subjects there is more

difference between students in these aspects than between classes and schools

BUT in maths, there is significant difference between classes in orientation, self-handicapping, disengagement, enjoyment of the subject, aspirations, and teacher-student relationships – significantly higher than in any other subject

Adolescence: the brain

Massive reorganisation of neural networks in parts which organise interactions, making sense of social situations, relating to the world

What the reorganisation IS or DOES no one yet knows – but it does seem to be associated with perception, interaction and talk

Adolescence: the mind

Acceleration of development of social and intellectual capabilities: Focusing on salient factors/editing out irrelevant

factors Comparing relationships Dealing with conflicting situations Retracing steps of argument Chunking/objectifying/abstracting Unambiguous classification Comparing across classifications Anticipation/ imagining reality Extending ideas of similarity beyond the visual

Focusing on salient factors/editing out irrelevant factors

Propensity to generalise from what is available

May over-generalise; generalise irrelevant features if they don’t know what is relevant

Comparing relationships

Comparing differences and ratios Comparing outcomes of operations Reasoning about relationships

rather than objects and quantities

Dealing with conflicting situations

Extending old ideas to new meanings

Reorganising earlier understandings Redefining

Retracing steps of argument

Can review arguments Can reapply arguments Can reverse arguments

Chunking/objectifying/abstracting

Building new concepts from old Using ‘new’ language with meaning Results of old procedures being new

objects

Unambiguous classification

Be precise about classification Need to resolve ambiguity Return to class inclusion

Comparing across classifications

Sameness and difference as raw material for new ideas, or for distinguishing between old ideas

Anticipation/ imagining reality

Extend beyond available range of application

Extend beyond visual representations

Turn imagined action into other representations

Extending ideas of similarity beyond the visual

Focus on properties, not appearance Focus on process and mechanisms

rather than visual output

Focusing on salient factors/editing out irrelevant factors

Assuming all graphs go through the origin; assuming all rectangles are parallel to edge of pages

Teaching: choose range of examples

Comparing relationships

Rates of change; distributive law (order of operations); equations as objects

Teaching: focus on relationship as object; focus on structure of expressions

Dealing with conflicting situations

Multiplication and addition do not ‘make things bigger’; ‘more digits’ does not mean ‘bigger number’

Teaching: recognise conflicts (not errors) and give time to discuss new meanings

Retracing steps of argument

Inverse operations; express reasoning; refine reasoning (proof)

Teaching: encourage expressing and retracing arguments; ask students to re-work worked examples; inner language

Chunking/objectifying/abstracting

Number as a product of prime factors

Equation as the ‘name’ of a function Ratio as a new arithmetical object

Unambiguous classification

Sort out names of shapes - inclusion and exclusion; proportions in shapes and proportional relationships; discrete v. continuous

Teaching: use technical terms in talk; relate words and classifications; deal with ambiguity

Comparing across classifications

Compare linear graphs to proportional functions; compare sine to cosine; compare ‘regular’ to ‘symmetrical’

Teaching: use ‘same/different’ as frequent classroom tool

Anticipation/ imagining reality

What will happen when x = 0? What will happen when n becomes very big? What will happen when the wheel turns through 360°? What sort of function might fit this data?

Teaching: encourage conjecture; focus on the power of special examples; change representations

Extending ideas of similarity beyond the visual

What is the same about all pentagons in all orientations? What is the difference between bar charts and histograms?

Teaching: talk about properties and the difference between what you see and what you know

Adolescent learning/ mathematics learning

from ad hoc and visual reponses to abstract ideas and prediction

from imagined fantasy to imagined actuality with constraints and consequences

from intuitive notions to ‘scientific’ notions

from empirical approaches to reasoned approaches

from doing to controlling

Key ‘learnable-teachable’ shifts in secondary mathematics

Discrete – continuous Additive – multiplicative -

exponential Procedures as rules –

procedures as tools Examples– generalisations Perceptions – conceptions Operations & inverses-

structures and relations

Reading signs – reading meaning

Patterns – properties Assumptions of linearity-

thinking about variation Getting results – reflection on

method and results Inductive/empirical reasoning

– deductive reasoning

Synthesis of research on how children learn mathematics (Nuffield)

Bryant, Nunes, Watson

Watch this space ….

Watson (2006) Raising Achievement in Secondary Mathematics (Open University Press)

Watson & Mason (2006) Mathematics as a Constructive Activity (Erlbaum)

anne.watson@education.ox.ac.uk

www.cmtp.co.uk