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Student Conjectures in Geometry. PME 2000 Anderson Norton University of Georgia. …the gods have certainty, whereas to us as men conjecture (only is possible). Alcmaeon. Conjecture. Conjecere: “ to throw together ” An idea formed in experience that satisfies the following properties:

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student conjectures in geometry

Student Conjectures in Geometry

PME 2000

Anderson Norton

University of Georgia

conjecture
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
research questions
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?
abduction
Abduction
  • The student experiences a perturbing phenomenon, P
  • However, P would be a logical consequence of A.
  • Therefore, the student adopts A conjecturally
method
Method
  • Three high school geometry students
  • van Hiele interview
  • Five 45-minute teaching experiments
  • Geometer’s Sketchpad
data analysis
Videotapes from last three sessions

Notes from each session

Highlighted conjectures

Cross-case comparison

Data & analysis
graham
Graham
  • White, middle-class male
  • Computer game design
  • Poor classroom performance
  • Box and shearing properties
  • Lines of symmetry
  • Disowning conjectures
diane
Diane
  • White, middle-class female
  • Softball player
  • Performed well in class
  • Unafraid to share thoughts
  • Self-monitoring, reflective
  • Strong informal deductive skills
results two patterns for conjecture
Results: Two patterns for conjecture
  • Abduction
  • Repeated assimilation (perceptual judgement)
abduction1
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
repeated assimilation
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
fitting the pattern of action
Fitting the pattern of action

Scheme

Expected

Result

Perceived

Situation

Activity

Perturbation!

limitations and suggestions
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)
closing words
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

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