Rethinking geometrical knowledge

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Re-thinking geometrical knowledge

• Professor Dave Pratt
• Institute of Education, University of London

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How does the introduction of new technologies
change?

• The way mathematical ideas are represented
• What is learned
• What can be learned
• How it is learned

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The case of geometry

• What is geometry?
• What do we know about learning geometry?
• How might software change the learning of
  geometry?

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What is Geometry?
Dual Nature
Axiomatic System Properties Relationships Axioms
Logic Deduction
Science of Space Seeing Exploring Induction Surp
rise Beauty
?
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Paradox

• Geometry is related to reality more than other
  areas of maths yet also more starkly a discipline
  of mind and deduction

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Brief history of geometry in schools

• In past geometry defended in school
  curriculumdiscipline in geometrical argument
  is a training in general accuracy (MA 1963, p8)
• Findings from studies of learning geometry
• hard to follow chains of deductive reasoning
• hard to construct chains of deductive reasoning
• Process or content?
• deductivity not taught as reinvention
  Freudenthal, 1973

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Removing the shackles of Euclid (Mathematics
Teaching, 1981-3

• Children do classification (e.g. of quads)
• Discuss interpretations (struggle for
  definitions, what is a diagonal?)
• Visualise, investigate (what if?) predict (rotate
  a circle round a diameter)
• Can this agenda be more effectively realised with
  digital technologies?

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Van Hiele levels Assessing geometrical knowledge

• Level 1 RecognitionThis is a triangle
• Level 2 Analysisrectangles have equal
  diagonals
• Level 3 Informal DeductionEvery square is a
  rectangle
• Level 4 Formal DeductionProve the angle sum
  of a triangle is 180

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Van Hiele levels

• Developed through analysis of learning when
  taught by conventional methods.
• Do this hierarchical structure necessarily still
  apply when digital technologies are used?

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Difficulties in learning geometry

• Particularity of diagrams
• Ambiguity of diagrams
• Prototype phenomenon
• Misconceptions
• Angle
• Subset/set
• Reflections

• Can digital technology help to support more
  sophisticated geometrical knowledge?

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Fischbein

• Figural elementsquares look like this
• Conceptual elementsquares have four equal side
  and four equal angles
• Fischbein seeks a fusion of the figural and the
  conceptual figural concepts
• Can digital technology be used to support the
  fusion of figural concepts?

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Tools and learning

• Can we change way the object is constructed?
• Can we change the way the object is described
• Can we add to the focus on the figural to
  emphasise a focus on properties and how those
  properties are connected?

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Alice Hansens approach

• Quads

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Introducing Dynamic Geometry
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Challenge Make a square
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Notion of messing up

• Can your square be messed up?
• Is it a figure or is it a drawing? (construction)
• What happens if you delete part of the square?
  (functional dependence)

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Attending to properties

• Can you construct a rhombus?
• (Hint consider its diagonals)

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Advantages and disadvantagesAubels Theorem
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Advantages and disadvantages

• How was the software helpful?
• What could not be achieved through the software?
• Discuss any advantages and any disadvantages of
  using dynamic geometry software to learn about
  this theorem.

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Main Initial Features
Figure
Drawing
distinguish
Theoretical object Properties Cant be messed up
Actual object Looks OK Perception
in menus explicit construction of
properties feedback!! Experimental (testing
hypotheses)