Transcript + 1

Thinking Mathematically
and
Learning Mathematics
Mathematically
John Mason
St Patrick’s College
Dublin
Feb 2010
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Conjecturing Atmosphere
Everything
said is said in order to
consider modifications that may be
needed
Those who ‘know’ support those
who are unsure by holding back or
by asking revealing questions
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Up & Down Sums
1+3+5+3+ 1
22 + 3 2
=
=
3x4+1
1 + 3 + … + (2n–1) + … + 3 + 1
=
3
(n–1)2 + n2
= n (2n–2) + 1
Doing & Undoing
Whenever
you find you can ‘do’
something, ask yourself how to
‘undo’ it.
– If doing is ‘subtract from 100’, what is
the undoing?
– If undoing is ‘divide 120 by’, what is the
undoing?
– If doing is find the roots of a
polynomial, what is the undoing?
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Reading Graphs
5
Remainders of the Day
 Write
down a number that leaves a reminder of 1
when divided by 3
 and another
 and another
 Choose two simple numbers of this type and
multiply them together:
what remainder does it leave when divided by 3?
 Why?
What is
special
about
the ‘3’?
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What is
special
about
the ‘1’?
Primality
What
is the second positive non-prime
after 1 in the system of numbers of the
form 1+3n?
100 = 10 x 10 = 4 x 25
What does this say about primes in
the multiplicative system of numbers
of the form 1 +3n?
What is special about the ‘3’?
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Undoing Special Cases
  e
d e
x
x
solves f '  f
dx
1
what solves f ' 
?
f
what solves f '  f 2 ?
…
8
what else?
MGA
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Powers
Specialising
& Generalising
Conjecturing
Imagining
Ordering
& Convincing
& Expressing
& Classifying
Distinguishing
Assenting
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& Connecting
& Asserting
Themes
Doing
& Undoing
Invariance
Freedom
& Constraint
Extending
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Amidst Change
& Restricting Meaning
Teaching Trap
Learning Trap
 Expecting the teacher to
Doing for the learners
do for you what you can
what they can already do
for themselves
already do for yourself
 Teacher Lust:
 Learner Lust:
– desire that the learner
– desire that the teacher
learn
tell me what to do
– desire that the learner
– desire that learning will
appreciate and
be easy
understand
– expectation that ‘dong
– Expectation that learner
the tasks’ will produce
will go beyond the tasks
learning
as set
– allowing personal
– allowing personal
excitement to drive
reluctance/uncertainty
behaviour
to drive behaviour
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
Didactic Tension
The more clearly I indicate
the behaviour sought from learners,
the less likely they are to
generate that behaviour for themselves
(Guy Brousseau)
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Didactic Transposition
Expert awareness
is transposed/transformed into
instruction in behaviour
(Yves Chevellard)
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More Ideas
For Students
(1998) Learning & Doing Mathematics (Second revised edition),
QED Books, York.
(1982). Thinking Mathematically, Addison Wesley, London
For Lecturers
(2002) Mathematics Teaching Practice: a guide for university
and college lecturers, Horwood Publishing, Chichester.
(2008). Counter Examples in Calculus. College Press, London.
http://mcs.open.ac.uk/jhm3
[email protected]
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