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On the Structure of Attention

On the Structure of Attention. Knowledge Starts with Pre-Conceptions SEMT’03 Prague. What Do You See?. How do you see it? What are you attending to?. NO Number Words!. NO Number Words!. What did you see?. 8. 16. 24. 40. 32. 48. 56. 64. 72. 80. Sum Complexity.

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On the Structure of Attention

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  1. On the Structure of Attention Knowledge Starts with Pre-ConceptionsSEMT’03 Prague

  2. What Do You See? • How do you see it? • What are you attending to? Structure of Attention: SEMT’03 Prague

  3. NO Number Words! NO Number Words! What did you see? Structure of Attention: SEMT’03 Prague

  4. 8 16 24 40 32 48 56 64 72 80 Structure of Attention: SEMT’03 Prague

  5. Sum Complexity What on earth … ? Structure of Attention: SEMT’03 Prague

  6. 3798652401 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

  7. Pythagoras Structure of Attention: SEMT’03 Prague

  8. 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

  9. 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

  10. Geometry a = 1 b Dimensions-of-Possible-Variation a = b a – b = 0 Range-of-Permissible-Change Structure of Attention: SEMT’03 Prague

  11. DofPV applied c a d b a = b c d Structure of Attention: SEMT’03 Prague

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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.

  21. 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

  22. 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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