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### On the Structure of Attention

Knowledge Starts with Pre-ConceptionsSEMT’03 Prague

1234567890

3 798 652 401

1 234 567 890

0.379 865 240 1

0,379 865 240 1

0.123 456 789 0

0,123 456 789 0

Structure of Attention: SEMT’03 Prague

Pythagoras

Structure of Attention: SEMT’03 Prague

Attention can be placed:

on a whole, a unit; an entity

on distinctions, on boundaries; discerning foreground from background

on relationships among aspects or features discerned

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

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

Structure of Attention: SEMT’03 Prague

Art & Mathematics

Theo van Doesburg

Arithmetic composition 1 (1930)see FLM 21 (2) & 22 (3)

Dimensions-of-Possible-Variation

Range-of-Permissible-Change

Structure of Attention: SEMT’03 Prague

Geometry

a

= 1

b

Dimensions-of-Possible-Variation

a = b

a – b = 0

Range-of-Permissible-Change

Structure of Attention: SEMT’03 Prague

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 Prague

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 Prague

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 Prague

Fractions

1

6 + 4

1

1 + 1

=

=

2

12 + 8

2 + 2

2

:

12

3

:

25

5

1

1

1

1

6 + 4

1

7 + 5

=

=

12 + 8

2

14 + 20

12

3

3

:

2

2

4

25

5

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

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

Structure of Attention: SEMT’03 Prague

Exercises

1

1

2

4

1

6

1

5

3

7

7

3

12

11

11

12

illustrate (using appropriately chosen oblongs or circles)

a)

b)

,

,

,

,

,

,

Structure of Attention: SEMT’03 Prague

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.

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

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 Prague

Exercises

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

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

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

Structure of Attention: SEMT’03 Prague

Springboard (U.K.)

1/2

3/4

4/5

1/5

1/5

What fraction of these shapes is shaded?

Shade in this fraction of each shape.

Structure of Attention: SEMT’03 Prague

World Class Tests (U.K.)

5

17

3

7

1

1

5

7

5

12

13

40

7

18

36

5

6

12

1

1

1

1

subdividing

+

+

2

2

3

3

measuring

1

3

1

1

+

subdividing

+

counting

4

9

4

6

The large rectangle is divided into smaller squares

What fraction of the large rectangle is shaded?

Structure of Attention: SEMT’03 Prague

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

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 Prague

Summary

- Attention can be
- 1: unitary 2: discerning 3: relating 4: property-forming 5: characterising
- Dimensions-of-Possible-Variation &Range-of-Permissible-Change

Structure of Attention: SEMT’03 Prague

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