Transcript Document
The Open University
Maths Dept
University of Oxford
Dept of Education
Getting Students to Take Initiative
when
Learning & Doing Mathematics
John Mason
Oslo
Jan 2009
1
Do You Know Any Students Who
…
do
the minimum to get through a
lesson?
wait to be told what to do?
finish quickly and then mess around?
are content to assent to what is said
and done, but rarely assert
mathematically?
2
Fraction Construction
Write
down two numbers that differ
by 3/7
and another pair
and another pair
And another pair that make the
difference as obscure as possible
4
Decimal Construction
Write
down
– A decimal number between 3 and 4
– that does not use the digit 5
– and that does use the digit 7
– and that is as close to 7/2 as possible
5
Line Construction
Write
down the equation of a straight
line that passes through the point (1,0)
and another
and another
Write them all down!
6
More Line Constructions
Sketch
the graph of two straight
lines whose
– x-intercepts differ by 2; and another …
– y-intercepts differ by 2; and another …
– slopes differ by 2; and another …
Sketch
the graph of two straight
lines meeting all three constraints
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Max-Min
In
a rectangular array of numbers,
calculate
– The maximum value in each row, and then the
minimum of these
– The minimum in each column and then the
maximum of these
How
do these relate to each other?
What about interchanging rows and
columns?
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Raise your hand when you can see
…
Something which is 2/5 of something
Something which is 3/5 of something
Something which is 2/3 of something
– What others can you see?
Something which is 2/5 of 5/3 of something;
3/5 of 5/3 of something;
Something which is 2/5 of 5/3 of something;
– What part is it of your whole?
Something which is 1/3 of 3/5 of something;
– What part is it of your whole?
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Something which is 5/3 of 3/5 of something
Something which is 2/3 of 3/2 of something
Getting Others To See …
1/4 – 1/5 = 1/20
1/4 – 1/20 = 1/5
1/5 – 1/20 = 1/4
1/a – 1/b = ?
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Doing & Undoing
What
operation undoes ‘adding 3’?
What operation undoes ‘subtracting 4’?
What operation undoes
‘subtracting from 7’?
What are the analogues for multiplication?
What undoes multiplying by 3?
What undoes dividing by 2?
What undoes dividing by 3/2?
What undoes multiplying by 3/2?
Now do it piecemeal!
What undoes ‘dividing into 12’?
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Remainder Construction
Write
down a number that leaves a
remainder of 1 on dividing by 3
and another
and another
Write down two, multiply them together,
and find the remainder on dividing by 3
What is special
about the ‘3’?
What is special
about the ‘1’?
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Distributed Examples
Write
down a number that leaves a
remainder of 1 when divided by 7
Now write down one which is easy to see
leaves a remainder of 1 on dividing by 7
Multiply by your number by the number of
someone sitting beside you
Does the product have the same
property?
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Remainders of the Day
Write down a number which when
you subtract 1 is divisible by 2
and when you subtract 1 from that
quotient, the result is divisible by 3
and when you subtract 1 from that
quotient the result is divisible by 4
Why must any such number be
divisible by 3?
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Remainders of the Day
Write down a number which is 1 more
than a multiple of 2
and which is 2 more than a multiple
of 3
and which is 3 more than a multiple
of 4
…
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Making Sense of the World
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More Or Less Whole & Part
? of 35 is 21
Part
more
more
3/4 of 40
is 30
same
6/7 of 35
is 30
less
4/5 of 30
is 24
Whole
17
same
3/5 of 35
is 21
less
0
Difference
Divisions
1
2
4–2=4÷2
1
2
–3=4
5
1
3
–4=5
÷1 4
6
1
4
–5=6
1
÷
4
7
1
5
–6=7
÷1 6
4
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1
÷
2
3
1
–
(-1)
=
-2
1
–0=1
-1
1
oops
1
–2=3
1
1
0
÷-2(-1)
1
÷-1 oops
1
÷
1
2
3
How does this fit in?
3
5
5
Going with the grain
Going across the grain
Differences
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1 1 1
1 1 1
7 6 42
2 1 2
1 11
1 1 1 1 1 11
3 2 6
8 7 56 6 24 4 8
Anticipating
1 1 1 1 1
Generalising
4 3 12 2 4
Rehearsing
1 1 1
5 4 20
Checking
1 1 1 1 1 1 1 1 1
Organising
6 5 30 2 3 3 6 4 12
Up & Down Sums
1+3+5+3+ 1
22 + 3 2
=
=
3x4+1
See
generality
through a
particular
Generalise!
1 + 3 + … + (2n–1) + … + 3 + 1
=
20
(n–1)2 + n2
= n (2n–2) + 1
Kites
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Reacting & Responding
Do
you know any students who jump at
the first idea that comes to mind?
Do you know any students who react
negatively when challenged by something
unfamiliar?
Assenting ––> Asserting
– conjecturing, trying, reasoning, …
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When Do You Take Initiative?
When
you are interested, engaged,
involved
When you have a stake in getting
something finished
When you are surprised or intrigued
When something is or becomes
‘real’ for you
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When is Real-ity
Sense
of purpose (engagement)
Sense of utility (present or future)
Use of own powers
24
Strategies
Learners
Making Significant
Mathematical Choices
– Learner Constructed Examples of
Mathematical Objects
– Learner Constructed Examples of
Exercises
– Learners deciding which exercises
need doing
– Distributed example construction
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ZPD
When
students are ready to shift
from
– Reacting to cues and triggers
– to initiating actions for themselves
Scaffolding
& Fading
– Directed, prompted, spontaneous
use of strategies, powers, concepts,
techniques
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Task Design
expert awareness is converted into
instruction in behaviour
– transposition didactique
27
Task & Activity
A task is what an author publishes,
what a teacher intends,
what learners undertake to attempt.
– These are often very different
What happens is activity
Teaching happens in the interaction
occasioned by activity
28
Teaching takes place in time
Learning takes place over time
Tasks
Learners encounter variation
Learners build up example spaces
Learners rehearse other techniques
while exploring
Learners encounter disturbances
and surprises
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Purpose & Utility
Ainley & Pratt
whose purposes?
whose utility?
– mathematics is useful
planning from objectives leads to dull
lessons;
planning from tasks may mean avoidance
of mathematical ideas, thinking, etc.
Issue:
how much do you tell learner in advance?
– Inner and outer aspects of tasks
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Teacher Aims and Goals
students to …
– make use of their powers
– experience mathematical themes
– encounter mathematical concepts,
topics
– develop facility and fluency with
techniques
– use technical terms to express their
conjectures and understandings
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Learner Aims & Goals
As learners, to
– do as little as necessary to complete
tasks adequately
– attract as little (or as much) attention as
possible
– be stimulated, inspired, engaged
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Task Dimensions
How initiated
– in silence; through phenomenon (shown or
imagined)
How sustained
– Group discussion; distributed tasks;
individual
How concluded
How structured
– Simple to complex; Particular to general
– Complex simplified; General specialised
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MGA
Getti ng-a-sense-of
Manipul ating
Ar ticulating
Getti ng-a-sense-of
Manipul ating
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Reflection
What
did you notice happening for
you mathematically?
What might you be able to use in an
upcoming lesson?
Imagine yourself in the future, using
or developing or exploring
something you have experienced
this morning!
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More Resources
Questions
& Prompts for Mathematical
Thinking
(ATM Derby: primary & secondary versions)
Thinkers (ATM Derby)
Mathematics as a Constructive Activity
(Erlbaum)
Designing & Using Mathematical Tasks
(Tarquin)
http: //mcs.open.ac.uk/jhm3
j.h.mason @ open.ac.uk
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