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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
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