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Necessary MovementsNecessary Movementsofof

AttentionAttention

John MasonJohn Mason

ATMATM

March 2009March 2009

2

CalculateCalculate

€

10000×10004−10002×999810000×10000−10001×9999

€

€

1234321234321×2468642468641−12343212343201234321234320×2468642468641 + 1234321234321

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ComparisonsComparisons

€

10000×10004−10002×999810000×10000−10001×9999

€

1234321234321×2468642468641−12343212343201234321234320×2468642468641 + 1234321234321

What is the same and what different about 7, 14, 21?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

AttentionAttention

MacroMacro– LocusLocus– FocusFocus– MultiplicityMultiplicity

MicroMicro– Holding Wholes (gazing)Holding Wholes (gazing)– Discerning DetailsDiscerning Details– Recognising Recognising RelationshipsRelationships

– Perceiving PropertiesPerceiving Properties– Reasoning (solely on the Reasoning (solely on the basis of agreed basis of agreed properties)properties) EducableEducable

– Additive Additive Multiplicative comparison Multiplicative comparison– Discrete Discrete Continuous Continuous– Reacting Reacting Responding (rules Responding (rules tools) tools)– ‘‘just is’ just is’ social social abductive & abductive & deductive reasoningdeductive reasoning

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Odd One OutOdd One Out

Which is the odd one out?Which is the odd one out?

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Seeing AsSeeing As

✎ Raise your hand when you Raise your hand when you can see the diagram as can see the diagram as illustratingillustrating1/3 of something1/3 of something

1 : 21 : 2

✎ What else can you ‘see What else can you ‘see as’?as’?

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Square CountSquare Count

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BaggedBagged

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?

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Triangle CountTriangle Count

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RegionalRegional

Arrange the three coloured Arrange the three coloured regions in order of arearegions in order of area

11

Reading a Diagram: Seeing As …Reading a Diagram: Seeing As …

a

a

x3 + x(1–x) + (1-x)3

x2 + (1-x)2

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)

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TopicsTopics

CountingCounting Angle MeasureAngle Measure Ratio (Thales)Ratio (Thales) TrigonometryTrigonometry FunctionFunction

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Gelett BurgessGelett Burgess

Remarkable truly, is Art!See — Elliptical wheels on a Cart!It looks very fairIn the Picture up there;But imagine the Ride when you start!

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Stressing & IgnoringStressing & Ignoring

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ConstructsConstructs

AAnglengle LLengthength FFractionraction RRegular shapeegular shape DDimenaionimenaion SScalecale DDy/dxy/dx TTransformationransformation GGraphraph UUnknown; variable; nknown; variable;

functionfunction EEquivalencequivalence DDifferenceifference PPerimetererimeter identityidentity

EquationDenominator numeratorPrimeFactor

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Length-Angle ShiftsLength-Angle Shifts

What 2D shapes have the property What 2D shapes have the property that there is a straight line that there is a straight line that cuts them into two pieces that cuts them into two pieces each mathematically similar to each mathematically similar to the original?the original?

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Symbol DecodingSymbol Decoding

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2k+ 1( )k= j( )

j=1

n−1

∑

j +1( )j=1

n−1

∑

∑ =n3 1 = 11 = 133

3 + 5 = 23 + 5 = 233

7 + 9 + 11 = 7 + 9 + 11 = 3333

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Geometric MultiplicationGeometric Multiplication

Use Thales Theorem to calculate, Use Thales Theorem to calculate, given x, ygiven x, y, , and zand z;;

xxy/zy/z x/yzx/yz 1/x+1/y1/x+1/y