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Student Conjectures in GeometryPowerPoint Presentation

Student Conjectures in Geometry

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

Alcmaeon

Conjecture (only is possible)

- 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 (only is possible)

- 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 (only is possible)

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

Method (only is possible)

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

Videotapes from last three sessions (only is possible)

Notes from each session

Highlighted conjectures

Cross-case comparison

Data & analysisGraham (only is possible)

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

Diane (only is possible)

- 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 (only is possible)

- Abduction
- Repeated assimilation (perceptual judgement)

Abduction (only is possible)

- 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 (only is possible)

- 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 (only is possible)

Scheme

Expected

Result

Perceived

Situation

Activity

Perturbation!

Limitations and Suggestions (only is possible)

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... (only is possible)

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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