Using Counterexamples, Puzzles and Provocations in the Teaching and Learning of Calculus
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This paper describes author’s experiences in both teaching with and research on counterexamples, puzzles and provocations in calculus as a pedagogical strategy. The results of several experimental studies with students and teachers/lecturers of calculus are presented and discussed. Examples of incorrect statements (to be disproved by counterexamples), selected puzzles and provocations offered to the participants of the studies are provided. The overwhelming statistics from those studies showed that both groups – students and lecturers – were very positive about using counterexamples, puzzles and provocations in teaching and learning of calculus. They found the strategy to be effective in the sense that it could help to: deepen conceptual understanding; reduce or eliminate common misconceptions; advance one’s mathematical thinking beyond the merely procedural or algorithmic; enhance critical thinking skills; expand a student’s ‘example space’ of functions with interesting properties; make teaching/learning more active and creative.