1 Necessary Movements of Attention 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? 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 … x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + 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  Angle  Length  Fraction  Regular shape  Dimenaion  Scale  Dy/dx  Transformation  Graph  Unknown; variable; function  Equivalence  Difference  Perimeter  identity Equation Denominator numerator Prime Factor

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 = 1 3  = 2 3  = 3 3

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