1

John MasonJohn Mason

IMEC9IMEC9

Sept 2007Sept 2007

Using Theoretical Constructs to Inform Teaching

2

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

3

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

4

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!

5

One Sum DiagramsOne Sum Diagrams

a

1

1

1-aa2

(1-a)2

a

Anticipating,not waiting

6

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…

7

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?

… ?

8

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?

9

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

10

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

11

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?

12

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.

13

MGA & DTRMGA & DTR

aa

Getting-a-sense-ofManipulatingGetting-a-sense-ofArticulatingManipulatingDoing Doing

Talking Talking RecordinRecordin

gg

14

PowersPowers

Specialising & GeneralisingSpecialising & Generalising Conjecturing & ConvincingConjecturing & Convincing Imagining & ExpressingImagining & Expressing Ordering & ClassifyingOrdering & Classifying Distinguishing & ConnectingDistinguishing & Connecting Assenting & AssertingAssenting & Asserting

15

ThemesThemes

Doing & UndoingDoing & Undoing Invariance Amidst ChangeInvariance Amidst Change Freedom & ConstraintFreedom & Constraint Extending & Restricting Extending & Restricting

MeaningMeaning

16

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

17

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

18

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

19

Worlds of ExperienceWorlds of Experience

Material

World

World of

Symbols

Inner World

of imager

y

enactive iconic symbolic

20

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

21

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.

J.H.Mason@open.ac.ukhttp://mcs.open.ac.uk/

jhm3