Mathematical thinking is difficult to define precisely but most authors
agree that the following are important aspects of it: conjecturing,
reasoning and proving, making connections, abstraction, generalization
and specialization. In order to develop mathematically, it is necessary for
learners of mathematics not only to master new mathematical content but
also to develop these skills. However, undergraduate courses in
Mathematics tend to be described in terms of the mathematical content
and techniques students should master and theorems they should be able
to prove. It would appear from such descriptions that students are
expected to pick up the skills of (advanced) mathematical thinking as a
by-product. Moreover, recent studies have shown that many sets of
mathematical tasks produced for students at the secondary-tertiary
transition emphasize lower level skills, such as memorization and the
routine application of algorithms or procedures. In this paper we will
consider some suggestions from the literature as to how mathematical
thinking might be specifically fostered in students, through the use of
different types of mathematical tasks. Efforts were made to interpret these
recommendations in the context of a first undergraduate course in
Calculus, on which large numbers of students may be enrolled. This itself
constrains to some extent the activities in which the teachers and learners
can engage. The tasks referred to here are set as homework problems on
which students may work individually or collaboratively. We will report
preliminary feedback from the students with whom such tasks were
trialled, describing the students’ reactions to these types of tasks and their
understanding of the purposes of the tasks
