On the Structure of Attention

On the Structure of Attention

Knowledge Starts with Pre-Conceptions

SEMT’03 Prague

Structure of Attention: SEMT’03 Prague2

What Do You See?

How do you see it?

What are you attending to?

Structure of Attention: SEMT’03 Prague3

What did you see?NO Number Words!

NO Number Words!

Structure of Attention: SEMT’03 Prague4

aa

8

16

24

32 40

48

56

64

72 80

Structure of Attention: SEMT’03 Prague5

€

2 k + 1( )

k = j

j = 1

n − 1

∑

j

j = 1

n

∑

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

− 1

∑= n

3

What on earth … ?Sum Complexity

Structure of Attention: SEMT’03 Prague6

3798652401 1234567890

0.379 865 240 10,379 865 240 1

3 798 652 401 1 234 567 890

0.123 456 789 00,123 456 789 0

Structure of Attention: SEMT’03 Prague7

Pythagoras

32 + 42 = 52

52 + 122 = 132

132 + 842 = 852

32 + 42 = 52

52 + 122 = 132

72 + 242 = 252

92 + 402 = 412

Structure of Attention: SEMT’03 Prague8

Attention can be placed:

a

on relationships among aspects or features discerned

a

on relationships as properties which objects ‘like it’ might have

a

on properties as formalised and stated independently of any particular objects; as axioms from which deductions can be made

a

on a whole, a unit; an entity

on distinctions, on boundaries; discerning foreground

from background

a

Structure of Attention: SEMT’03 Prague9

a

Theo van Doesburg

Arithmetic composition 1 (1930)

see FLM 21 (2) & 22 (3)

Dimensions-of-Possible-Variation

Range-of-Permissible-

Change

Art & Mathematics

Structure of Attention: SEMT’03 Prague10

Geometry

a = ba – b = 0

ab

= 1

Dimensions-of-Possible-Variation

Range-of-Permissible-

Change

Structure of Attention: SEMT’03 Prague11

DofPV applied

a

b

cd

ab

cd=

Structure of Attention: SEMT’03 Prague12

All-But-Equilateral

Imagine a quadrilateral which is all-but-equilateral:all sides except at most one are equal

Imagine a pentagon which is all-but-equilateral:all sides except at most one are equal

Imagine a … which is all-but-equilateral:all sides except at most one are equal

Structure of Attention: SEMT’03 Prague13

Listening to Children

Pessia & DIna

Ruti, Ronit, Ronit & Malka

Chantal & Louise

Marie

We notice what we are prepared to notice,what strikes us as unexpected or out of the ordinary‘We often see what we want to see’

Structure of Attention: SEMT’03 Prague14

Fractions

What must I attend to, and how, to interpret fraction diagrams?

What must I attend to, and how, to use fraction diagrams?

Structure of Attention: SEMT’03 Prague15

Fractions

6 + 412 + 8

= 12

7 + 514 + 20 =

13

1

2

1

4

1225

35

: 1225

35

::

What must I attend to, and how, to interpret fraction notation?

What must I attend to, and how, to use fraction notation?

6 + 412 + 8

1

2

1

2

= 1 + 12 + 2

= 12

Structure of Attention: SEMT’03 Prague16

Exercises

QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.

514 illustrate (using appropriately chosen oblongs or circles)

a)

b)

17

13

23

47

111

112

511

612

, , ,

, , ,

Structure of Attention: SEMT’03 Prague17

Exercises

509. Granny prepared 1 litre of chocolate and she divided it into three in such a way that in each of the mugs there was the same amount.

a) What proportion of the chocolate that Granny had prepared was in each of the mugs?

b) When Betka was washing the dishes she filled a litre lemonade bottle and then she poured all this water into the two mugs. This completely filled them. Had the mugs been filled to the brim with chocolate?

Structure of Attention: SEMT’03 Prague18

Exercises

QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.

510. 28 sportsmen participated in a ski competition but only 21 of them made it to the finish.

b) what proportion of the whole number of participants did not finish?

a) what proportion of the whole number of participants finished the competition?

Structure of Attention: SEMT’03 Prague19

Springboard (U.K.)What fraction of these shapes is shaded?

Shade in this fraction of each shape.

4/5

3/4 1/2

1/51/5

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World Class Tests (U.K.)The large rectangle is divided into smaller squares

What fraction of the large rectangle is shaded?

Pool, P. 2003, What to do …, Maths Teaching 182 p42-44.

subdividingcounting

measuring 14

39

+

513

1740

37

718

136

15

56

712

512

subdividing 12

13

+

14

16

+

12

13

+

Structure of Attention: SEMT’03 Prague21

Revision Puzzle (1933)

The shaded parts of this figure are squares.How many squares can you count in it?

(the answer is not 3)How many oblongs can you count in it?

(the answer is more than 10)

Structure of Attention: SEMT’03 Prague22

Summary

Attention can be1: unitary 2: discerning 3: relating 4: property-forming 5: characterising

Dimensions-of-Possible-Variation &Range-of-Permissible-Change