Adolescence and secondary
mathematics: shifts of
perspective
Anne Watson
December 2008
Adolescence: social
identity
 feeling powerful
 belonging
 understanding the
world
 being heard
 being in charge  negotiating authority
 being supported 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:






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
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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

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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

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Discrete – continuous
Additive – multiplicative exponential

Procedures as rules –

procedures as tools
Examples– generalisations
Perceptions – conceptions
Operations & inversesstructures and relations 
Reading signs – reading
meaning
Patterns – properties
Assumptions of linearitythinking 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)

anne.watson@education.ox.ac.uk

www.cmtp.co.uk
Mathematics as a Constructive
Activity (Erlbaum)