Learning to Think and Reason Geometrically:

1
Learning to Think and ReasonGeometrically

• The roles of monologue, dialogue, and
  multilogueand the Structure of Attention

John MasonSMC
2007
2
Outline

• Some assumptions
• Some tasks
• Some reflections

3
Some assumptions

• Learners have displayed all necessary powers to
  reason geometrically
• They may not be familiar with directing their
  attention geometrically
• Geometrical Reasoning is about determining what
  must, may, or cannot be true in general situations

4
Triangular Count
How many triangles?
What is the same andwhat different aboutthem?
What variations are possible?
More segments? remove-right-angle? ?
5
Parallelism
How manyangles do youneed to knowto work out
allthe angles?
What did you need to do withyour attention?
6
7 Circles
Using ONLY intersections of circles
How many different triangles can you find?
How many different quadrilaterals can you find?
What are you doing with your attention?
7
Kite-Fold
8
9-Pin Triangles
How many triangles?
How many different triangles?
What do you need to do with your attention?
Geoff Faux
9
Rectangular Layout

• Say What You See!
• Arrange some sheets of paper similarly, and
  continue
• Conjecture explain vary

Phenomenon gt exploration exercise Extending
awareness Training behaviourHarnessing
emotions
10
Eddys Theorem
How are AP and ABrelated now?
How are AP and ABrelated?
11
Vecten
12
Cosine Law
13
Sine Law
14
Attention

• Gazing at wholes
• Discerning details
• Recognising relationships
• Perceiving properties
• Reasoning on the basis of properties

15
Roles

• Roles of Others
• As audience (their presence mediates between you
  and content seeking clarity)
• As source (they initiate exposition, they direct
  your attention you enter their world)
• As resource (you initiate, they listen they
  enter your world)
• As expert (you use them to validate your own
  criteria concerning your learning)
• Roles of Self
• Integrating others ideas (practices, foci and
  structure of attention) into current situation
  and into possible actions for the future.

16
Some Reflections

• Notice the geometrical term
• It requires movement out of the current space
  into a space of one higher dimension in order to
  achieve it

What did you notice about the role of monologue,
dialogue, and multilogue for your own
understanding?
What did you notice yourself doing with your
attention?