Paradoxically, traditional ways of teaching are, for most students, causing precisely the
opposite effect that university mathematicians desire. The sheer difficulty and volume of
material to be covered in a university mathematics degree makes it difficult for students
to cope with formal mathematical content in a limited time. But does this mean that we
must accept the status quo of a huge formal syllabus with widespread rote learning, or is
it not possible to modify courses to allow students to develop ways of thinking more
mathematically? The acquisition of a wide repertoire of advanced mathematical thinking
is a challenge which now faces university mathematicians. Is it a challenge which will be
accepted?