Eliminating counterexamples: A Grade 8 student's learning trajectory for contrapositive proving

- Contrapositive argumentation is accessible to Grade 8 students.
- The eliminating counterexamples conception is useful.
- Intermediate conceptions are identified.
- A learning trajectory is developed.

This study uses a teaching experiment and retrospective analysis to develop a learning trajectory for improving a Grade 8 student's ability to construct, critique, and validate contrapositive arguments. The study is predicated on the hypothesis that adolescents perform poorly on contrapositive reasoning tasks because they lack sufficient ways of justifying contrapositive argumentation as a viable mode of argumentation. By studying a student's actions and comments as she develops, critiques, and validates not-the-conclusion-implies-the-conditions-are-impossible arguments for conditional claims, a promising learning trajectory for contrapositive argumentation is developed. The student's learning trajectory demonstrates how a conception of contrapositive proving as eliminating counterexamples can be useful in developing, critiquing, and validating contrapositive arguments.