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: – Discrete statement– Conscious, though not necessarily explicitly

stated– Uncertain, and the conjecturer is concerned

about its validity

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

• The student experiences a perturbing phenomenon, P

• However, P would be a logical consequence of A.

• Therefore, the student adopts A conjecturally

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

Graham

• White, middle-class male

• Computer game design

• Poor classroom performance

• Box and shearing properties

• Lines of symmetry

• Disowning conjectures

DB

C

A

A'

C'

Diane

• White, middle-class female

• Softball player

• Performed well in class

• Unafraid to share thoughts

• Self-monitoring, reflective

• Strong informal deductive skills

BA

D

C

E

F G

HI

Results: Two patterns for conjecture

• Abduction

• Repeated assimilation (perceptual judgement)

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

• 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

PerceivedSituation Activity

Expected Result

Perturbation!

Scheme

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...

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