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Necessary Movements of Attention

Necessary Movements of Attention. John Mason ATM March 2009. Calculate. Comparisons. What is the same and what different about 7, 14, 21?. What was the same and what different about the two tasks? ‘difference’ associated with subtraction; N o word for multiplicative comparison?

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Necessary Movements of Attention

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  1. Necessary MovementsofAttention John Mason ATM March 2009

  2. Calculate

  3. Comparisons What is the same and what different about 7, 14, 21? • What was the same and what different about the two tasks? • ‘difference’ associated with subtraction; • No word for multiplicative comparison? • So how do we educate awareness of a useful shift of attention to multiplicative comparison? • When do we use multiplicative-comparison naturally?

  4. Attention • Macro • Locus • Focus • Multiplicity • Micro • Holding Wholes (gazing) • Discerning Details • Recognising Relationships • Perceiving Properties • Reasoning (solely on the basis of agreed properties) • Educable • Additive  Multiplicative comparison • Discrete  Continuous • Reacting  Responding (rules  tools) • ‘just is’  social  abductive & deductive reasoning

  5. Odd One Out • Which is the odd one out?

  6. Seeing As • Raise your hand when you can see the diagram as illustrating 1/3 of something 1 : 2 • What else can you ‘see as’?

  7. Square Count

  8. Bagged The number of counters in a bag is deemed to be the total number of counters in all the bags contained in that bag. The ‘bag-depth’ of a bag is the maximum number of bags within bags within bags … in that bag. For what numbers of counters is it possible to have a bag containing that many counters, subject to the constraint that each bag contains exactly one more counter than its bag depth?

  9. Triangle Count

  10. Regional • Arrange the three coloured regions in order of area

  11. Reading a Diagram: Seeing As … x2 + (1-x)2 x3 + x(1–x) + (1-x)3 x2z + x(1-x) + (1-x)2(1-z) xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

  12. Topics • Counting • Angle Measure • Ratio (Thales) • Trigonometry • Function

  13. Gelett Burgess Remarkable truly, is Art! See — Elliptical wheels on a Cart! It looks very fair In the Picture up there; But imagine the Ride when you start!

  14. Stressing & Ignoring

  15. Constructs Equation Denominator numerator Prime Factor • Angle • Length • Fraction • Regular shape • Dimenaion • Scale • Dy/dx • Transformation • Graph • Unknown; variable; function • Equivalence • Difference • Perimeter • identity

  16. Length-Angle Shifts • What 2D shapes have the property that there is a straight line that cuts them into two pieces each mathematically similar to the original?

  17. Symbol Decoding • 1 = 13 • 3 + 5 = 23 • 7 + 9 + 11 = 33

  18. Geometric Multiplication • Use Thales Theorem to calculate, given x, y, and z; • xy/z • x/yz • 1/x+1/y

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