Learning to Think and Reason Geometrically:
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1 Learning to Learning to Think and Reason Think and Reason Geometrically: Geometrically: The roles of The roles of monologue, dialogue, and monologue, dialogue, and multilogue multilogue and the and the Structure of Attention Structure of Attention 2007 John Mason SMC
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Learning to Think and Reason Geometrically:. The roles of monologue, dialogue, and multilogue and the Structure of Attention. John Mason SMC. 2007. Outline. Some assumptions Some tasks Some reflections. Some assumptions. - PowerPoint PPT Presentation
Transcript of Learning to Think and Reason Geometrically:
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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
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Kite-FoldKite-Fold
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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?
