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Learning to Learning to Think and ReasonThink and Reason

Geometrically:Geometrically:

The roles of The roles of monologue, dialogue, and monologue, dialogue, and

multiloguemultilogueand the and the

Structure of AttentionStructure of Attention

2007

John MasonSMC

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OutlineOutline

Some assumptionsSome assumptions Some tasksSome tasks Some reflectionsSome reflections

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Some assumptionsSome assumptions

Learners have displayed all Learners have displayed all necessary powers to reason necessary powers to reason geometricallygeometrically

They may not be familiar with They may not be familiar with directing their attention directing their attention geometricallygeometrically

Geometrical Reasoning is about Geometrical Reasoning is about determining what determining what mustmust, , maymay, or , or cannotcannot be true in general situations be true in general situations

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Triangular CountTriangular Count

How many triangles?

What is the same andwhat different about

them?

What variations are possible?

More segments?remove-right-angle?

…?

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What did you need to do withyour attention?

ParallelismParallelism

a

How manyangles do youneed to knowto work out all

the angles?

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How many different quadrilaterals can you find?

7 Circles7 Circles Using ONLY intersections of circles

How many different triangles can you find?

What are you doing with

your attention?

a

7

Kite-FoldKite-Fold

8

What do you need to do with your attention?

9-Pin Triangles9-Pin Triangles

a

a

a

Geoff Faux

How many triangles?

How many different triangles?

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Rectangular LayoutRectangular Layout

Say What You See!Say What You See! Arrange some sheets of paper Arrange some sheets of paper

similarly, and continuesimilarly, and continue Conjecture; explain; varyConjecture; explain; vary

Phenomenon ––> exploration; exercise;

Extending awareness; Training behaviour;Harnessing emotions

a

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How are AP and ABrelated now?

Eddy’s TheoremEddy’s Theorem

O

B

A

P

O

A

BP

O’

How are AP and ABrelated?

APAB

OO’OA

=

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VectenVecten

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Cosine LawCosine Law

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Sine LawSine Law

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AttentionAttention

Gazing at wholesGazing at wholes Discerning detailsDiscerning details Recognising relationshipsRecognising relationships Perceiving propertiesPerceiving properties Reasoning on the basis of Reasoning on the basis of

propertiesproperties

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RolesRoles Roles of OthersRoles of Others

– As audience: As audience: (their presence mediates between (their presence mediates between you and content: seeking clarity)you and content: seeking clarity)

– As source: As source: (they initiate exposition, they direct (they initiate exposition, they direct your attention; you enter their world)your attention; you enter their world)

– As resource: As resource: (you initiate, they listen; they enter (you initiate, they listen; they enter your world)your world)

– As expert: As expert: (you use them to validate your own (you use them to validate your own criteria concerning your learning)criteria concerning your learning)

Roles of SelfRoles of Self– Integrating others’ ideas (practices, foci and Integrating others’ ideas (practices, foci and

structure of attention) into current situation and structure of attention) into current situation and into possible actions for the future.into possible actions for the future.

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Some ReflectionsSome Reflections Notice the geometrical term:Notice the geometrical term:

– It requires movement out of the current It requires movement out of the current space into a space of one higher space into a space of one higher dimension in order to achieve itdimension in order to achieve it

What did you notice yourself doing

with your attention?

What did you notice about the role of monologue, dialogue,

and multilogue for your own understanding?

What can you imagine yourself trying in the future?

What struck you about today?