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
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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
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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!
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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 11
1 1 1  1 1  11
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
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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
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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

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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
33
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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