Promoting Mathematical Thinking
Responsive, Reflective
& Responsible teaching
John Mason
AIMSSEC
ACE Yr 2
Jan 2013
The Open University
Maths Dept
1
University of Oxford
Dept of Education
Ways of Working
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2
Everything said here today is a conjecture
It is uttered so it can be thought about and modified if
necessary
What you get from this session will mostly be what
you notice happening inside you … how you use
your mathematical powers.
Responsive Teaching

Responding to student’s needs
– Class as a whole
– Particular students
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Listening to Students
Giving them time
– to think,
– to experiment
– to conjecture
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Supporting them to
– Modify their conjecture
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3
Trying not to do for students what they can alredy do
for themselves
Reflective Teaching
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Learning from experience
What could have been different?
Should –> Could
Imagining yourself in the future,
acting in some way that you would prefer
instead of some habit that has developed
Making a note at the end of the lesson
of ONE thing that struck you, that stood out, about
the lesson
Do this at the end
of a lesson
while students are
making a note of what
they thought the
lesson was about!
4
Responsible Teaching
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Able to justify choices of
–
–
–
–
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5
Intentions (mathematical)
Tasks
Interventions
Pedagogic strategies
Requires the development of a vocabulary
for talking about pedagogic intentions and
choices!
Set Ratios
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In how many different ways can you place 17 objects so
that there are equal numbers of objects in each of two
sets?
What about requiring that there be twice as many in the
left set as in the right set?
What about requiring that the ratio of the numbers in the
left set to the right set is 3 : 2?
What is the largest number of objects that CANNOT be
placed in the two sets in any way so that the ratio is 5 : 2?
What can
be varied?
6
Reflection & Justification (Mathematical)
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Powers used?
– Imagining and Expressing; Specialising & Generalising;
Conjecturing & Convincing;
– Being Systematic
– Making records
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Themes Encountered
–
–
–
–
7
Seeking Relationships
Invariance in the midst of change
Freedom & Constraint
Doing & Undoing
Reflection & Justification (Task Format)
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Why 17 objects to be placed?
– What follow-up was missing?
– What about 18? (opportunity for ‘same and different’)
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Confusion between ‘left set’ and ‘left part of
diagram’!!!
Something available if some finish first part quickly
How was work sustained?
How was work brought to a conclusion?
– Conjectures?
– Something not fully resolved?
– Opportunity to reflect back over the event?
8
Issues Arising
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Choice of numbers
Choice of wording
Choice of setting:
– actual objects; drawings; symbols
9
31: a game for two players
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10
At each move the player chooses a whole number of
cubes from 1 to 5 and adds them to a common pile.
The first person to get the total number of cubes in the
common pile to be 31, wins.
What is your (best) strategy?
Reflection & Justification (Mathematical)
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Topic
– Adding; choosing and predicting
– Reasoning backwards from 31
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Powers used?
– Imagining and Expressing; Specialising & Generalising;
Conjecturing & Convincing;
– Being Systematic
– Making records
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Themes Encountered
–
–
–
–
11
Seeking Relationships
Invariance in the midst of change
Freedom & Constraint
Doing & Undoing
Reflection & Justification (Task Format)
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Did you use cubes?
Confusion???
How was work sustained?
How was work brought to a conclusion?
– Conjectures?
– Something not fully resolved?
– Opportunity to reflect back over the event?
12
Selective Sums
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Cover up one entry from each row
and each column. Add up the
remaining numbers.
The answer is (always) the same!
Why?
Stuck?
Specialise!
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0
-2
2
-4
6
4
8
2
3
1
5
-1
1
-1
3
-3
Reflection & Justification (Mathematical)
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Topic Reviewed or Met?
– Practicing addition & subtraction (whole numbers, integers,
fractions, even decimals)
– Making choices with constraints
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Powers used?
– Imagining and Expressing; Specialising & Generalising;
Conjecturing & Convincing;
– Being Systematic
– Making records
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Themes Encountered?
–
–
–
–
14
Seeking Relationships
Invariance in the midst of change
Freedom & Constraint
Doing & Undoing
Reflection & Justification (Task Format)
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Why objects, not simply imagining or using pencil?
Confusion???
Something available if some finish first-part quickly?
How was work sustained?
How was work brought to a conclusion?
– Conjectures?
– Something not fully resolved?
– Opportunity to reflect back over the event?
15
Selective Sums
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How much freedom of choice do
you have when making up your
own?
a
b
d
e b
a
b
f
e
?
e-(a-b)
g
Opportunity to generalise
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c
Opportunity to quantify
freedom of choice
c
d
Selective Sums Variation
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Choose a number s
from 1, 2, 3
Select s numbers from
each row and column
(cover up 4–s numbers
from each row and
column)
Add up all the selected
numbers
Why is it always the
same?
5
6
-1
3
2
3
-1
6
1
3
-5
6
1
6
-2
3
5
3
1
2
3
2
2
3
4
3
1
6
7
6
1
3
Chequered Selective Sums
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Choose one cell in each row and
column.
Add the entries in the dark
shaded cells and subtract the
entries in the light shaded cells.
What properties make the
answer invariant?
What property is sufficient to
make the answer invariant?
2
-5 -3
-6 4
-1 9
0
3
-1 -2 -6
-2
0
3
5
Some Frameworks
Doing – Talking – Recording
(DTR)
(MGA)
See – Experience – Master
(SEM)
Enactive – Iconic – Symbolic
Material – Mental–Symbols
(EIS)
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Specialise …
in order to locate structural
Stuck?
What do I know?
relationships …
then re-Generalise for yourself
What do I want?
Issues Arising
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Choice of numbers
Choice of wording
Choice of setting:
– actual objects; drawings; symbols
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Opportunities for Students to
– Make significant mathematical choices
– Use their own powers
– Reflect on what has been effective for them
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Responsible Reflection!
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What did you notice for yourself?
What has struck you from this session?
What would you like to try out or evelop?
Imagine yourself working on that for yourself
– Modifying something to use in your situation
– Trying something out
– Reflecting on what was effective
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Follow Up
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j.h.mason @ open.ac.uk
mcs.open.ac.uk/jhm3
These slides and the Hand Outs will be on Memory
Sticks & Moodle