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)  