1 John Mason IMEC9 Sept 2007 Using Theoretical Constructs to Inform Teaching.
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Transcript of 1 John Mason IMEC9 Sept 2007 Using Theoretical Constructs to Inform Teaching.
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John MasonJohn Mason
IMEC9IMEC9
Sept 2007Sept 2007
Using Theoretical Constructs to Inform Teaching
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OutlineOutline
Teaching MathematicsTeaching Mathematics– Tasks, activities, experience, reflectionTasks, activities, experience, reflection
Teaching People To Teach Teaching People To Teach MathematicsMathematics– ConsistencyConsistency– Awareness of the role ofAwareness of the role of
Tasks, activities, experience, reflectionTasks, activities, experience, reflection
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My MethodsMy Methods
ExperientialExperiential
What you get from this session isWhat you get from this session iswhat you noticewhat you notice
happens inside you you, and how happens inside you you, and how you relate that to your own situationyou relate that to your own situation
ReflectionReflection– Linking to theoriesLinking to theories– Preparing to notice more carefully in futurePreparing to notice more carefully in future– Brief-but-vivid accountsBrief-but-vivid accounts
The canal may not itself drink, The canal may not itself drink, but it performs the function but it performs the function
of conveying water to the thirstyof conveying water to the thirsty
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One SumOne Sum
I have two numbers which sum to 1I have two numbers which sum to 1 Which will be larger:Which will be larger:
The square of The square of the larger added the larger added to the smaller?to the smaller?
The square of The square of the smaller the smaller
added added to the larger?to the larger?
Don’t calculate!!!Conjecture!
Only then Check!
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One Sum DiagramsOne Sum Diagrams
a
1
1
1-aa2
(1-a)2
a
Anticipating,not waiting
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DecimalDecimal
Write down a decimal number Write down a decimal number between 2 and 3between 2 and 3
and which does NOT use the digit and which does NOT use the digit 55
which DOES use the digit 7which DOES use the digit 7 and which is as close to 5/2 as and which is as close to 5/2 as
possiblepossible
2.47
2.4972.479
2.499…972.479…9
2.49…979…
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Difference of TwoDifference of Two
Write down two numbers Write down two numbers which differ by 2which differ by 2
And another pairAnd another pair And another pairAnd another pair And another pair which obscure And another pair which obscure
the fact that the difference is 2the fact that the difference is 2
9999 & 10001
Fractions?
Decimals?
Negatives?
… ?
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How often do you set tasks for themwhere they need to do this?
Do you encourage your learners to do this?
What did you do first?
CharacterisingCharacterising What numbers can be two What numbers can be two
more than the sum of four more than the sum of four consecutive whole numbers?consecutive whole numbers?
What numbers can be one more thanthe product of four consecutive numbers?
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Sketchy GraphsSketchy Graphs
Sketch the graphs of a pair of straight lines Sketch the graphs of a pair of straight lines whose whose y-y-intercepts differ by 2intercepts differ by 2
Sketch the graphs of a pair of straight linesSketch the graphs of a pair of straight lineswhose whose xx-intercepts differ by 2-intercepts differ by 2
Sketch the graphs of a pair of straight linesSketch the graphs of a pair of straight lineswhose slopes differ by 2whose slopes differ by 2
Sketch the graphs of a pair of straight linesSketch the graphs of a pair of straight linesmeeting all three conditionsmeeting all three conditions
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More Or Less Altitude & AreaMore Or Less Altitude & AreaMore Or Less Altitude & AreaMore Or Less Altitude & Area
Draw a scalene triangle
more
same
less
moresameless
areaaltitud
e
Same altmore area
more altsame area
more altmore area
less altmore area
less altless area
more altless area
same altless area
less altsame area
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More Or Less Area & More Or Less Area & PerimeterPerimeterMore Or Less Area & More Or Less Area & PerimeterPerimeter
Draw a rectangle
more
same
less
moresameless
areaperimet
er
Same perimmore area
more perimsame area
more perimmore area
less perimmore area
less perimless area
more perimless area
same perimless area
less perimsame area
When can it be done? When can it not be done?
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Omar KhayamOmar Khayam
Myself when young did eagerly frequentDoctor and Saint, and heard great ArgumentAbout it and about: but evermoreCame out by the same Door as in I went
Pursuing knowledge in childhood we riseUntil we become masterful and wiseBut if we look through the disguiseWe see the ties of worldly lies
In childhood we strove to go to school,Our turn to teach, joyous as a ruleThe end of the story is sad and cruelFrom dust we came, and gone with winds cool.
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MGA & DTRMGA & DTR
aa
Getting-a-sense-ofManipulatingGetting-a-sense-ofArticulatingManipulatingDoing Doing
Talking Talking RecordinRecordin
gg
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PowersPowers
Specialising & GeneralisingSpecialising & Generalising Conjecturing & ConvincingConjecturing & Convincing Imagining & ExpressingImagining & Expressing Ordering & ClassifyingOrdering & Classifying Distinguishing & ConnectingDistinguishing & Connecting Assenting & AssertingAssenting & Asserting
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ThemesThemes
Doing & UndoingDoing & Undoing Invariance Amidst ChangeInvariance Amidst Change Freedom & ConstraintFreedom & Constraint Extending & Restricting Extending & Restricting
MeaningMeaning
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ProtasesProtases
A sequence of experiences does not add up to
an experience of that sequence
One thing we do not often learn from experience,
is that we do not often learn from experience alone
Habit forming can be habit forming
Absence of evidenceis NOT
evidence of absence
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Implicit ContractImplicit Contract
If learners ‘do’ the tasks they are set, If learners ‘do’ the tasks they are set, then they will ‘learn’ what is requiredthen they will ‘learn’ what is required– Contrat didactiqueContrat didactique
The more clearly and specifically the The more clearly and specifically the teacher specifies the behaviour sought, teacher specifies the behaviour sought, the easier it is for learners to display the easier it is for learners to display that behaviour that behaviour withoutwithout encountering mathematics, encountering mathematics, withoutwithout thinking mathematically thinking mathematically– Didactic tensionDidactic tension
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Task & ActivityTask & Activity A A tasktask is what an author publishes, is what an author publishes,
what a teacher intends, what a teacher intends, what learners undertake to attempt. what learners undertake to attempt.– These are often very differentThese are often very different
What happens is What happens is activityactivity Teaching happens in the interaction Teaching happens in the interaction
made possible by activity: performing made possible by activity: performing familiar actions in new ways to make familiar actions in new ways to make new actionsnew actions
Learning happens through reflection and Learning happens through reflection and integrating new actions into functioningintegrating new actions into functioning
Teaching takes place in timeLearning takes place over time
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Worlds of ExperienceWorlds of Experience
Material
World
World of
Symbols
Inner World
of imager
y
enactive iconic symbolic
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Worlds, MGA, DTRWorlds, MGA, DTR
Enactive-Iconic-SymbolicEnactive-Iconic-Symbolic– Three modes; three worldsThree modes; three worlds
ManipulatingManipulating––Getting-a-sense-Getting-a-sense-ofof––ArticulatingArticulating
DoingDoing––TalkingTalking––RecordingRecording
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Further ReferenceFurther Reference
MathempediaMathempedia ( (http://www.ncetm.org.ukhttp://www.ncetm.org.uk))
Fundamental Constructs in Fundamental Constructs in Mathematics EducationMathematics Education,, RoutledgeFalmer, London (2004).RoutledgeFalmer, London (2004).
Designing and Using Designing and Using Mathematical TasksMathematical Tasks. St. Albans: . St. Albans: Tarquin.Tarquin.
[email protected]://mcs.open.ac.uk/
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