1 teaching as the directing of attention and learning as the transformation of attention 2007 john...

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Teaching as Teaching as The Directing of AttentionThe Directing of Attention

andandLearning as Learning as

the Transformation of Attentionthe Transformation of Attention

2007

John MasonStowarzyszenia Nauczycieli

MatematykiMuszyna

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Outline: teaching using attentionOutline: teaching using attention

Directing Attention Towards:Directing Attention Towards:– stressing, focusingstressing, focusing– confrontingconfronting

Directing Attention Away:Directing Attention Away:– de-emphasising through exploringde-emphasising through exploring

(integration through (integration through subordination)subordination)

Structure of AttentionStructure of Attention

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Directing Attention TowardsDirecting Attention Towards

4

What’s The Difference?What’s The Difference?

What could be varied?

– =

First, add one to eachFirst, add one to the larger and subtract one from the smaller

What differencewould that make

to the differences?

What differencewould that make

to the differences?

Directing Attention Towards

s are in ... s are in ... 1133

one fifth?one fifth?


How many ... are in ...?How many ... are in ...?

How many How many How many How many

s are in s are in 1122

1 ?1 ?2 ?2 ?3 ?3 ?4 ?4 ?blah ?blah ?oop ?oop ?... ?... ?

What are you doing?What are you doing?What are you doing?What are you doing?

one third?one third?2233

??



5

fraction which divides into bothfraction which divides into both

an integral number of times. an integral number of times.

fraction which divides into bothfraction which divides into both

an integral number of times. an integral number of times.

Find a … Find a …

14141515

10102121

andand

and anotherand another


find the largest.find the largest.

6 Directing Attention Towards

Find a ...Find a ...

fraction which bothfraction which both

divide into an integral number of times.divide into an integral number of times.

fraction which bothfraction which both

divide into an integral number of times.divide into an integral number of times.

14141515

10102121

anandd



find the smallest.find the smallest.


7

8

More doesn’t always go with More doesn’t always go with MoreMoreand Less doesn’t always go and Less doesn’t always go with Lesswith Less Draw a (non-square) rectangleDraw a (non-square) rectangle

– Now adjust it so that the area Now adjust it so that the area goes up while the perimetergoes up while the perimetergoes upstays the samegoes down

Adjust = cut &/or pasteAdjust = alter rectangle

AreaPerimeter

Up

Same

Down

UpSam

eDown

Dina Tirosh 2007

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Directing Attention AwayDirecting Attention Away(ignoring, de-emphasising; (ignoring, de-emphasising; exploring)exploring)

In how many different ways In how many different ways can a unit fraction be written can a unit fraction be written as the difference of two unit as the difference of two unit fractions?fractions?

In how many different ways In how many different ways can you decompose a given can you decompose a given rectilinear region into non-rectilinear region into non-overlapping rectangles?overlapping rectangles?

Directing Attention Away

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Learner Choice: Learner Choice: Breadth & Depth Breadth & Depth ExercisesExercises3y = 12 – 2x x = 3

2x + 3y = 127x – y = 19

2x + 3y = 1221x – 3y = 57

2x + 3y = 1223x = 69

3y = 13 – 2x x = 5

2x + 3y = 139x – 6y = 39

2x + 3y = 1311x – 3y = 52

2x + 3y = 1313x = 65

2y = 14 + 5x x = -2

-5x + 2y = 14 8x - y = -18

-5x + 2y = 1416x – 2y = -36

-5x + 2y = 14 11x = -22

5y = 14 + 3x x = 2

-3x + 5y = 14 4x + y = 12

-3x + 5y = 1420x + 5y = 60

-3x + 5y = 14 23x = 46

(Dave Herring 2007)

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Using AttentionUsing Attention

True or FalseTrue or False What do you doWhat do you doto decide?to decide?

What do you doWhat do you doto decide?to decide?

What might 8yrWhat might 8yrolds do to olds do to decide?decide?

What might 8yrWhat might 8yrolds do to olds do to decide?decide?

218

+711

=711

+218

154

+75

=194

+25

52 +73 - 73 =52

Marta Mollina (Granada)Marta Mollina (Granada)

127

+53

=57

+2+23

5 13 - 5 13 = 26

12 + 11 = 11 + 1212 + 11 = 11 + 12

51 + 51 = 50 + 5251 + 51 = 50 + 52

27 + 48 - 48 = 2727 + 48 - 48 = 27

257 - 34 = 257 - 30 - 4257 - 34 = 257 - 30 - 4

137 - 137 = 5137 - 137 = 512Directing Attention

Towards

Raise Your Hand When You See ... Raise Your Hand When You See ...

something which is 2/5 of something; something which is 2/5 of something; 3/5 of something;3/5 of something;5/2 of something; 5/3 of something;5/2 of something; 5/3 of something;2/5 of 5/2 of something; 3/5 of 5/3 of 2/5 of 5/2 of something; 3/5 of 5/3 of something;something;5/2 of 2/5 of something; 5/3 of 3/5 of 5/2 of 2/5 of something; 5/3 of 3/5 of something;something;1 ÷ 2/5 of something; 1 ÷ 3/5 of 1 ÷ 2/5 of something; 1 ÷ 3/5 of something;something;

2/5 ÷ 3/5 of something?2/5 ÷ 3/5 of something?


something which is 2/5 of something; something which is 2/5 of something; 3/5 of something;3/5 of something;5/2 of something; 5/3 of something;5/2 of something; 5/3 of something;2/5 of 5/2 of something; 3/5 of 5/3 of 2/5 of 5/2 of something; 3/5 of 5/3 of something;something;5/2 of 2/5 of something; 5/3 of 3/5 of 5/2 of 2/5 of something; 5/3 of 3/5 of something;something;1 ÷ 2/5 of something; 1 ÷ 3/5 of 1 ÷ 2/5 of something; 1 ÷ 3/5 of something;something;



1313Pat Thompson (2004)Pat Thompson (2004)Directing Attention

Towards

Ride & TieRide & Tie

Which graph makes sense, and why?Which graph makes sense, and why?Which graph makes sense, and why?Which graph makes sense, and why?

What is the same, and what different What is the same, and what different about ...about ...

What is the same, and what different What is the same, and what different about ...about ...

14Directing Attention Towards

Structures of AttentionStructures of AttentionMacro StructuresMacro Structures locus, focus, multiplicitylocus, focus, multiplicity

Micro StructureMicro Structure Holding Wholes (gazing)Holding Wholes (gazing) Discerning DetailsDiscerning Details Recognising Relationships Recognising Relationships

(situated; particular)(situated; particular) Perceiving Properties (general)Perceiving Properties (general) Reasoning on the basis of stated Reasoning on the basis of stated

propertiesproperties

Macro StructuresMacro Structures locus, focus, multiplicitylocus, focus, multiplicity




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Structures of AttentionStructures of AttentionMacro StructuresMacro Structures locus, focus, multiplicitylocus, focus, multiplicity




Macro StructuresMacro Structures locus, focus, multiplicitylocus, focus, multiplicity




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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( )


QuestionsQuestions

Why is directing attention important?Why is directing attention important?

Need for confluence between learner and Need for confluence between learner and teacher attentionteacher attention

Being aware of the structure and focus of Being aware of the structure and focus of your own attention is necessary if you are to your own attention is necessary if you are to direct the attention of learners appropriatelydirect the attention of learners appropriately

Being aware of the importance of attention Being aware of the importance of attention can alert you to reasons for what learners say can alert you to reasons for what learners say and doand do

Why is directing attention important?Why is directing attention important?

Need for confluence between learner and Need for confluence between learner and teacher attentionteacher attention

Being aware of the structure and focus of Being aware of the structure and focus of your own attention is necessary if you are to your own attention is necessary if you are to direct the attention of learners appropriatelydirect the attention of learners appropriately

Being aware of the importance of attention Being aware of the importance of attention can alert you to reasons for what learners say can alert you to reasons for what learners say and doand do

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SummarySummary

How is attention directed?•Towards (stressing, focusing)Towards (stressing, focusing)

•Away (ignoring, de-emphasising)Away (ignoring, de-emphasising)

Strategies for Directing AttentionSay What You SeeWatch What You DoIn How Many Different Ways Can You ...?Show me that you see ...What is the same, and what is different about … ?

• Making Use of the Structure of Attention

How is attention directed?•Towards (stressing, focusing)Towards (stressing, focusing)

•Away (ignoring, de-emphasising)Away (ignoring, de-emphasising)

Strategies for Directing AttentionSay What You SeeWatch What You DoIn How Many Different Ways Can You ...?Show me that you see ...What is the same, and what is different about … ?

• Making Use of the Structure of Attention

1 teaching as the directing of attention and learning as the transformation of attention 2007 john...

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