1 while you are waiting: warm up 1: in a certain club there are 47 people altogether, of whom 31 are...
TRANSCRIPT
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While you are waiting:While you are waiting:
Warm Up 1:in a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both?
Warm Up 2:in a certain club there are 28 people. There are 14 poets, 11 painters and 15 musicians; there are 22 who are either poets or painters or both, 21 who are either painters or musicians or bothand 23 who are either musicians or poets or both.How many people are all three: poets, painters and musicians?
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ResolvingResolving
31
47 total
2947–31
47–29
31–(47–29)29–(47–31)
poets painters
In a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both?
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14
11
15 musicians
poets painters
28 total
23 musicians or painters
21 poets or musicians
22 poets or painters
28–23 28–21
28–22
14+15–21
11+15–23
(14+15-21) + (14+11-22) + (11+15-23) – (28– ((28-23) + (28-22) + (28-21)) 2
14+11–22
In a certain club there are 28 people. There are 14 poets, 11 painters and 15 musicians; there are 22 who are either poets or painters or both, 21 who are either painters or musicians or bothand 23 who are either musicians or poets or both.How many people are all three: poets, painters and musicians?
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Learning & TeachingLearning & TeachingJohn MasonJohn Mason
AIMSSEC Jan 2010AIMSSEC Jan 2010
The Open UniversityMaths Dept University of Oxford
Dept of Education
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ReflectionsReflections
Re-entering a situationRe-entering a situationre-constructing what you noticedre-constructing what you noticedpreparing to act in the futurepreparing to act in the future
Doing some actions; Doing some actions; Talking about those actions; Talking about those actions; Recording those actions; Recording those actions; reflecting on the effects of those actionsreflecting on the effects of those actions
Do–Talk–RecordDo–Talk–Record
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ReflectionsReflections
Do you recall being invited to talk to a Do you recall being invited to talk to a neighbour (talking in pairs)?neighbour (talking in pairs)?– Getting you Getting you assertingasserting (taking initiative, (taking initiative,
conjecturing)conjecturing)– Not simply Not simply assentingassenting to what others say to what others say
Do you recall being asked to stop and Do you recall being asked to stop and reflect on what you had been doing?reflect on what you had been doing?– Tasks ––> Activity ––> ExperienceTasks ––> Activity ––> Experience– BUT withdrawing from the action and BUT withdrawing from the action and
considering the effects of those actions, considering the effects of those actions, the way you use your powers, is vital in the way you use your powers, is vital in order to learn from that experience.order to learn from that experience.
Re-entering a situationre-constructing what you noticedpreparing to act in the future
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Finding The QuestionFinding The Question
You are given cardsYou are given cards1 2 3 4 5 6
Arrange the cards to make the Arrange the cards to make the calculation true:calculation true:
4 1 3 2 6 5148 =
–
What might this do for learners?What could I change in the mathematics of the task so that through practice learners integrate that experience?What could I change in the format of the task? Why might I?
What might this do for learners?What could I change in the mathematics of the task so that through practice learners integrate that experience?What could I change in the format of the task? Why might I?
Re-entering a situationre-constructing what you noticedpreparing to act in the future
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Worlds of MathematicsWorlds of Mathematics
Enactive: doing, Enactive: doing, manipulatingmanipulating
Iconic: imaginingIconic: imagining Symbolic: presenting with Symbolic: presenting with
symbolssymbols
DoingTalkingRecording
Manipulating …Getting a sense of …Articulating …
Relationships ––> Properties
… what?
why?
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Revealing ShapesRevealing Shapes
The coloured shapes on the right are somewhere in the white cells on the left.By clicking on a shape and then on a cell, you can test your conjectures as to which shape is where.
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ReflectionsReflections
What actions were you called upon to What actions were you called upon to perform?perform?
What was the effect on you?What was the effect on you? What does the task offer learners?What does the task offer learners? What can be changed in the content of What can be changed in the content of
the task?the task? What can be changed in the format of What can be changed in the format of
the task?the task?
Re-entering a situationre-constructing what you noticedpreparing to act in the future
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Addition Table Completions Addition Table Completions
totals
How many entries do you need in order to fill out the table?
27 5
1 60
-9
totals
How many entries do you need in order to fill out a table with r rows and c columns altogether?
What can you predict already?
69
-36
184515
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ReflectionsReflections
What actions were you called upon to What actions were you called upon to perform?perform?
What was the effect on you?What was the effect on you? What does the task offer learners?What does the task offer learners? What can be changed in the content of What can be changed in the content of
the task?the task? What can be changed in the format of What can be changed in the format of
the task?the task?
Re-entering a situationre-constructing what you noticedpreparing to act in the future
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Reconstructing Reconstructing Multiplication TablesMultiplication Tables
x 2 5 7
3 6 15
21 5 1
025
35 6 1
230
42
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Generalising by SpecialisingGeneralising by Specialising
x b/s c/s
sa/b
a bc/a
s b c
x 50/s
60/s
12s/25
24 125
s 50 60
What about fractions?
What about quadratics?
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ReflectionsReflections
What actions were you called upon to What actions were you called upon to perform?perform?
What was the effect on you?What was the effect on you? What does the task offer learners?What does the task offer learners? What can be changed in the content of What can be changed in the content of
the task?the task? What can be changed in the format of What can be changed in the format of
the task?the task?
Re-entering a situationre-constructing what you noticedpreparing to act in the future
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Mill-Dienes-Bloor AnimationsMill-Dienes-Bloor Animations
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ReflectionsReflections
What actions were you called upon to What actions were you called upon to perform?perform?
What was the effect on you?What was the effect on you? What does the task offer learners?What does the task offer learners? What can be changed in the content of What can be changed in the content of
the task?the task? What can be changed in the format of What can be changed in the format of
the task?the task?
Re-entering a situationre-constructing what you noticedpreparing to act in the future
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Magic Square ReasoningMagic Square Reasoning
51 9
2
4
6
8 3
7
– = 0Sum( ) Sum( )
Try to describethem in words
What other configurations
like thisgive one sum
equal to another?
2
2
Any colour-Any colour-symmetric symmetric
arrangement?arrangement?
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More Magic Square ReasoningMore Magic Square Reasoning
– = 0Sum( ) Sum( )
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ReflectionsReflections
What actions were you called upon to What actions were you called upon to perform?perform?
What was the effect on you?What was the effect on you? What does the task offer learners?What does the task offer learners? What can be changed in the content of What can be changed in the content of
the task?the task? What can be changed in the format of What can be changed in the format of
the task?the task?
Re-entering a situationre-constructing what you noticedpreparing to act in the future
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Worlds of MathematicsWorlds of Mathematics
Enactive: doing, Enactive: doing, manipulatingmanipulating
Iconic: imaginingIconic: imagining Symbolic: presenting with Symbolic: presenting with
symbolssymbols
DoingTalkingRecording
Manipulating …Getting a sense of …Articulating …
Relationships ––> Properties
… what?
why?
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My best Wishes …My best Wishes …
may you continue to enjoy your teaching more may you continue to enjoy your teaching more and moreand more
JohnM
http://mcs.open.ac.uk/jhm3http://mcs.open.ac.uk/jhm3