If not, what yes?

Abstract

This article presents an instructional approach to constructing discovery-oriented activities. The cornerstone of the approach is a systematically asked question ‘If a mathematical statement under consideration is plausible, but wrong anyway, how can one fix it?’ or, in brief, ‘If not, what yes?’ The approach is illustrated with examples from calculus and geometry. It is argued that the ‘If not, what yes?’ approach facilitates conjecturing and proving, constructing meaningful examples and counterexamples and has a potential for creating learning situations, in which responsibility for achieving desirable mathematical results is devolved from an instructor to the learners.

Keywords:

• discovery-oriented activities
• problem posing
• what-if-not
• examples and counterexamples
• calculus
• geometry