Using counter-examples in teaching calculus
The paper deals with a practical issue encountered by many lecturers teaching first-year university Calculus. A big proportion of students seem to be able to find correct solutions to test and exam questions using familiar steps and procedures. Yet they lack deep conceptual understanding of the underlying theorems and sometimes have misconceptions. In order to reduce or eliminate misconceptions, and for deeper understanding of the concepts involved, the students were given the incorrect mathematical statements and were asked to construct counter-examples to disprove the statements. More than 600 students from 10 universities in different countries were questioned regarding their attitudes towards the method of using counter-examples for eliminating misconceptions and deeper conceptual understanding. The vast majority of the students reported that the method was very effective and made learning mathematics more challenging, interesting and creative.