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Student Conjectures in Geometry

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

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  1. Student Conjectures in Geometry PME 2000 Anderson Norton University of Georgia

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

  3. 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

  4. 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?

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

  6. Method • Three high school geometry students • van Hiele interview • Five 45-minute teaching experiments • Geometer’s Sketchpad

  7. Videotapes from last three sessions Notes from each session Highlighted conjectures Cross-case comparison Data & analysis

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

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

  10. Results: Two patterns for conjecture • Abduction • Repeated assimilation (perceptual judgement)

  11. 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

  12. 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

  13. Fitting the pattern of action Scheme Expected Result Perceived Situation Activity Perturbation!

  14. 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)

  15. 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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