Finnish mathematics teacher students' skills and tendencies to use informal and formal reasoning in the case of derivative

The arguments constructed in mathematical reasoning may be either formal or informal: They may be based either on definitions, axioms and previously proven theorems or on concrete interpretations of mathematical concepts and situations. In addition, arguments may be superficial or deep. Results shared in this paper are from three different studies in which both students’ skills to produce informal and formal arguments and their tendencies to choose between informal and formal reasoning in problem solving situations were studied. The students in all these studies were Finnish high school pre-service mathematics teachers, and the data was collected by using a written test and videotaped interviews. The tasks used were about the concept of derivative. Results of the studies indicated that the students’ skills to produce informal and formal arguments were dependent on each other. The difference between the levels of these skills was not significant, but several students had a tendency to avoid the use of the formal definition of derivative, which led to difficulties in problem solving situations. However, this tendency could not be explained by the students’ inadequate skills to handle the definition of derivativ