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Noticing:Noticing:the key to teaching, learningthe key to teaching, learning

and researchand research

John MasonJohn Mason

Modena & NapoliModena & Napoli20072007

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It is only after you come to know the surface of things

that you venture to see what is underneath;But the surface of things is inexhaustible

(Italo Calvino)

He who loves practice without theory is like the sailor

who boards ship without a rudder and compass and never knows where he may cast. Practice always rests on good theory.

(Leonardo Da Vinci)

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Say What You SeeSay What You See

a

a

4

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Say What You SeeSay What You See

5

aaa

30361635542428402863483642364232455681497254635160119905

24206478979654CopperPlate CopperPlate

MultiplicationMultiplication

aaa

30361635542428402863483642364232455681497254635160119905

79654242064789

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Giacomo Candido 1871-1941Giacomo Candido 1871-1941What does it say?What does it say?

aa

(x + y)4x2y2(x + y)2x2y22xyx2y22xy2xy=

=+[x2 + y2 + (x + y)2]2

2[x4 + y4 + (x + y)4]

=

+

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ReflectionsReflections

What did you do with your What did you do with your attention?attention?– Hold wholes (gazing)?Hold wholes (gazing)?– Discern details?Discern details?– Recognise relationships?Recognise relationships?– Perceive properties?Perceive properties?– Reason on the basis of Reason on the basis of

properties?properties?

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What Can a Teacher Do for What Can a Teacher Do for Learners?Learners?

Direct attentionDirect attention Display mathematical Display mathematical behaviourbehaviour Reinforce behaviourReinforce behaviour Get out of the wayGet out of the way

Stress (and consequently ignore)Stress (and consequently ignore) Amplify & Edit (what learners say and Amplify & Edit (what learners say and

do)do)

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In Order to Direct Learner In Order to Direct Learner AttentionAttention

I need to know what I attend to myselfI need to know what I attend to myself– So, that I can direct learners attentionSo, that I can direct learners attention

I need to know I need to know howhow I am attending I am attending– What can I do to prompt learners to What can I do to prompt learners to

attend to the same things in similar ways?attend to the same things in similar ways?

Noticing

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What does What does ‘develop/research teaching’ ‘develop/research teaching’ mean?mean?

Assemble different strategies & Assemble different strategies & tactics, different actions to tactics, different actions to initiateinitiate

Watch out for opportunities to use Watch out for opportunities to use themthem

Observe what happensObserve what happens

Noticing

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Grid MovementGrid Movement

7

?

+3

-3

x2÷2

((7+3)x2)+3

is a path from 7 to ‘?’.

What expression represents the reverse of this path?

What values can ‘?’ have:- if only + and x are used- if exactly one - and one ÷ are used, with as many + & x as necessaryWhat about other cells?Does any cell have 0? -7?Does any other cell have 7?

Characterise ALL the possible values that can appear in a cell

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Number SpiralNumber Spiral

1 2

345

6

7 8 9 10

11

12

13

18

19

20

21 22 23 24 25 26

27

28

29

30

3132

14151617

3334353637

38

39

40

41

42

43 44 45 46 47 48 49 50

1

4

9

16

25

49

36

13

1

2 3 4

5

6789

101112

13

18 19 20

21

22

23

242526272829

303132

14 15 16 17

33

34

35

36 37 38 39 40 41 42 43 44

45

46

47

48

49

50

64

81

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Tunja Display (1)Tunja Display (1)

2x2 - 2 - 2 = 1x1 - 13x2 - 3 - 2 = 2x1 - 1

4x2 - 4 - 2 = 3x1 - 1

5x2 - 5 - 2 = 4x1 - 1

…

3x3 - 3 - 3 = 2x2 - 1

4x3 - 4 - 3 = 3x2 - 1

5x3 - 5 - 3 = 4x2 - 1

…

Generalise!

Run Backwards

1x2 - 1 - 2 = 0x3 - 1

0x2 - 0 - 2 = (-1)x3 - 1

(-1)x2 - (-1) - 2 = (-2)x3 - 1

2x3 - 2 - 3 = 1x2 - 1

1x3 - 1 - 3 = 0x2 - 1

0x3 - 0 - 3 = (-1)x2 - 1

……

…

(-1)x3 - (-1) - 3 = (-2)x2 - 1

…

…

…

…

…

……

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Tunja Display (2)Tunja Display (2)

4x3x2 - 2x3 - 4x2 = 4x3 - 24x4x2 - 2x4 - 4x2 = 6x3 - 2

4x5x2 - 2x5 - 4x2 = 8x3 - 2

4x6x2 - 2x6 - 4x2 = 10x3 - 2…

4x3x3 - 2x3 - 4x3 = 4x5 - 24x4x3 - 2x4 - 4x3 = 6x5 - 2

4x5x3 - 2x5 - 4x3 = 8x5 - 2

4x6x3 - 2x6 - 4x3 = 10x5 - 2…

Generalise!

Run Backwards

4x2x2 - 2x2 - 4x2 = 2x3 - 2

4x1x2 - 2x1 - 4x2 = 0x3 - 2

4x0x2 - 2x0 - 4x2 = (-2)x3 - 2

4x(-1)x2 - 2x(-1) - 4x2 = (-4)x3 - 2

…

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Remainders of the Day (1)Remainders of the Day (1)

Write down a number which Write down a number which when you subtract 1 is divisible when you subtract 1 is divisible by 5by 5

and anotherand another and anotherand another Write down one which you think Write down one which you think

no-one else here will write down.no-one else here will write down.

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Remainders of the Day (2)Remainders of the Day (2)

Write down a number which when Write down a number which when you subtract 1 is divisible by 2you subtract 1 is divisible by 2

and when you subtract 1 from the and when you subtract 1 from the quotient, the result is divisible by quotient, the result is divisible by 33

and when you subtract 1 from that and when you subtract 1 from that quotient the result is divisible by 4quotient the result is divisible by 4

Why must any such number be Why must any such number be divisible by 3? divisible by 3?

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Remainders of the Day (3)Remainders of the Day (3)

Write down a number which is Write down a number which is 1 more than a multiple of 21 more than a multiple of 2

and which is 2 more than a and which is 2 more than a multiple of 3multiple of 3

and which is 3 more than a and which is 3 more than a multiple of 4multiple of 4

… …

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Remainders of the Day (4)Remainders of the Day (4)

Write down a number which is Write down a number which is 1 more than a multiple of 21 more than a multiple of 2

and 1 more than a multiple of and 1 more than a multiple of 33

and 1 more than a multiple of and 1 more than a multiple of 44

… …

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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

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More Magic Square More Magic Square ReasoningReasoning

– = 0Sum( ) Sum( )

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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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AttentionAttention

Gazing at wholesGazing at wholes Discerning detailsDiscerning details Recognising relationshipsRecognising relationships Perceiving propertiesPerceiving properties Reasoning on the basis of Reasoning on the basis of

propertiesproperties

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ObservationObservation

A name or label immediately shapes A name or label immediately shapes what is seen and what is what is seen and what is

recalledrecalled (Frederick Bartlett 1932)(Frederick Bartlett 1932)

Observation is theory ladenObservation is theory laden (George Hanson 1958)(George Hanson 1958)

We want our theories to be as fact We want our theories to be as fact laden as our facts are theory laden as our facts are theory

ladenladen ((NelsonNelson Goodman 1978)Goodman 1978)

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What are the significant What are the significant products of research?products of research?

Transformations of the ‘being’ of the Transformations of the ‘being’ of the researchers researchers

Increased sensitivity to notice what was Increased sensitivity to notice what was previously not noticedpreviously not noticed

Refined vocabulary for discussing, Refined vocabulary for discussing, discerning and analysingdiscerning and analysing

Awareness which informs future choices Awareness which informs future choices of actionof action SelfSelf OthersOthers

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Accounts-of & Accounting-Accounts-of & Accounting-forfor

Accounts-of: brief-but-vivid accountsAccounts-of: brief-but-vivid accountsReduce-remove theorising, judgement, Reduce-remove theorising, judgement,

excuses, evaluations, justificationsexcuses, evaluations, justifications

I cannot evaluate your analysis I cannot evaluate your analysis if I cannot distinguish it if I cannot distinguish it

from the data itselffrom the data itself

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Reporting DataReporting Data

“They couldn’t …”

“They can’t”

“They didn’t display evidence of …”

“They don’t display evidence of …”

Accounting For Account of

“I didn’t detect evidence of …”

“I don’t detect evidence of …”

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Precision ConjecturePrecision Conjecture

The more precisely the data is The more precisely the data is specified,specified,

the more we learn about the more we learn about the researcher’s sensitivities to the researcher’s sensitivities to

noticenotice

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ProtasesProtases

I cannot change others;I cannot change others;I I cancan work at changing myself work at changing myself

To express is to over stressTo express is to over stress

One thing we do not seem to learn from One thing we do not seem to learn from experience, experience, is that we do not often learn from is that we do not often learn from experience aloneexperience alone

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Natural EpistemologyNatural Epistemology

Noticing – Marking – RecordingNoticing – Marking – Recording ConjecturingConjecturing Resonance seekingResonance seeking

– with own experiencewith own experience– with otherswith others

Validity found in Validity found in – use by local community of practiceuse by local community of practice– own future practices informedown future practices informed

Avoid the teaching of Avoid the teaching of speculators,speculators,

whose judgements are whose judgements are not confirmed not confirmed by experience. by experience. (Leonardo Da Vinci)(Leonardo Da Vinci)

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SpectingSpecting

InterspectiveInterspective

Extrasp

ective

Extrasp

ective

IInnttrroossppeeccttiivvee

IntraspectiveIntraspective

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Essence of Discipline of Essence of Discipline of NoticingNoticing

Systematic ReflectionSystematic Reflection– Past (accounts-of not accounting-for)Past (accounts-of not accounting-for)

Preparing & NoticingPreparing & Noticing– For Future & PresentFor Future & Present

Recognising ChoicesRecognising Choices– Could-have & Could-be Could-have & Could-be

(not should have or should be)(not should have or should be) ValidatingValidating

– for Self & with Othersfor Self & with Others

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Recognising Choices

DistinguishingChoices

AccumulatingAlternatives

Identifying & labelling

Validating with OthersDescribingMoments

RefiningExercises

Systematic ReflectionKeeping Accounts

SeekingThreads

Preparing & NoticingImagining

PossibilitiesNoticing

Possibilities

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Interwoven WorldsInterwoven Worlds

OwnOwn world of experience world of experience

TryingTrying

ReflectingReflecting

Seeking Seeking resonanceresonance

with otherswith others

Colleague'sColleague's world of experience world of experience

World of observations World of observations & theories & theories

RecognisingRecognising Possibilities Possibilities

ExpressingExpressingPreparing Preparing

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The universe is a mirrorThe universe is a mirrorin which we can contemplate in which we can contemplate

onlyonlywhat we have learned what we have learned

to know about ourselvesto know about ourselves(Italo Calvino)(Italo Calvino)

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Developing Thinking in Algebra (Sage 2005)

Researching Your Own Practice using the Discipline of Noticing (Routledge 2002)

Fundamental Constructs in Mathematucs Education (Routledge 2005)

John MasonJohn Masonj.h.mason @ open.ac.ukj.h.mason @ open.ac.ukhttp://mcs.open.ac.uk/jhm3http://mcs.open.ac.uk/jhm3