Student Conjectures in Geometry

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Student Conjectures in Geometry

• PME 2000
• Anderson Norton
• University of Georgia

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the gods have certainty, whereas to us as men
conjecture (only is possible)
Alcmaeon
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Conjecture

• Conjecere to throw together
• An idea formed in experience that satisfies the
  following properties
• Discrete statement
• Conscious, though not necessarily explicitly
  stated
• Uncertain, and the conjecturer is concerned about
  its validity

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Research questions

• What is the nature of conjecture in geometry?
• What is the nature of plausible reasoning
  supporting these conjectures?
• How might the roles of conjecture and plausible
  reasoning be fitted in a larger theory of
  learning?

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Abduction

• The student experiences a perturbing phenomenon,
  P
• However, P would be a logical consequence of A.
• Therefore, the student adopts A conjecturally

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Method

• Three high school geometry students
• van Hiele interview
• Five 45-minute teaching experiments
• Geometers Sketchpad

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Data analysis

• Videotapes from last three sessions
• Notes from each session
• Highlighted conjectures
• Cross-case comparison

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Graham

• White, middle-class male
• Computer game design
• Poor classroom performance
• Box and shearing properties
• Lines of symmetry
• Disowning conjectures

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Diane

• White, middle-class female
• Softball player
• Performed well in class
• Unafraid to share thoughts
• Self-monitoring, reflective
• Strong informal deductive skills

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Results Two patterns for conjecture

• Abduction
• Repeated assimilation (perceptual judgement)

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Abduction

• Based on experience (for Graham, largely
  experience with computer graphics)
• Difficult to distinguish from perceptual
  judgement
• Adopting a conjecture that can be easily
  checked/refuted
• Relates the surprising result of a scheme to an
  unusual key property of the situation that is
  least common to experience

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Repeated assimilation

• Perceptual judgement and assimilation
• Based on experience (for Diane, largely classroom
  experience)
• Once again, relies upon the recognition of a
  previously ignored key property that is uncommon
  in experience
• The role of reflection (self-monitor) in creating
  further perturbation

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Fitting the pattern of action
Scheme
Expected Result
Perceived Situation
Activity
Perturbation!
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Limitations and Suggestions

• In order to better understand conjecture
• 1) We need to describe the function(s) they serve
  in the self-regulation of schemes
• 2) We need to understand the restrictions and
  advantages of the particular environment (e.g.
  students using GSP)
• 3) Logical analysis may not ever describe the
  formation of conjecture (see 1)

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Closing words...
Self control is the character which distinguishes
reasoning from the processes by which perceptual
judgements are formed, and self-control of any
kind is purely inhibitory. It originates nothing.
Peirce