Transcript Adolescent learning and secondary mathematics Anne Watson

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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
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Find a number which is closer to 3/8 than it
is to 3/16
… and another
… and another
More ‘… and another’
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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
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… and another
• Awareness of example spaces
• Awareness of dimensions of variation
• Awareness of ranges of change
Comparing equivalent objects
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How many ways can you find to express the
number of dots in this diagram?
Affordances of comparison
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How many ways …?
• Equivalent representations
• Transformation between representations
• Arguments about completeness
Grid multiplication
x
x
-2
+3
Surds/irrationals
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Use grid multiplication to find a pair of
numbers like a + √b which, when multiplied,
have no irrational bits
a
c
√d
√b
Affordances of construction
tasks:
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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 …
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identity
belonging
being heard
being in charge
being supported
reorganising neural
pathways in frontal
cortex
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feeling powerful
understanding the
world
negotiating authority
arguing in ways which
make adults listen
sex
Adolescent learning is progress
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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
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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:
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Choice; action (agency)
Conjectures; perspectives (identity)
Ownership (empowerment; identity)
Discussion (collaboration)
Reflection
Changes in mathematical activity??
The fallacy of choice
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Choice does not necessarily lead to
stronger mathematical activity
Fallacy of reflection:
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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’
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Discrete – continuous
Additive - multiplicative
Rules – tools
Procedure – meaning
Example – generalisation
Perceptual – conceptual
Operations – inverses
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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
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24
2
12
2
6
3
2
Multiplicative relationships
Multiplicative relationships
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xy = 24
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x = 24/ y
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y = 24/ x
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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
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probability
proportion & ratio
non-linear sequences
symbolic
representation
proving things
adding fractions…..
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understanding limits
using algebraic
relationships
reasoning from
properties …
What shifts are needed to learn
secondary mathematics?
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Discrete – continuous
Additive - multiplicative
Rules – tools
Procedure – meaning
Example – generalisation
Perceptual – conceptual
Operations – inverses
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Pattern – relationship
Relationship – properties
Conjecture – proof
Results – reflection on
results
Result – reflection on
procedure/method
Inductive – deductive
Other ….
Adolescent actualisation in
mathematics
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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
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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)