With the rapid development of artificial intelligence, work and leisure is expected to change, and human creative competence is expected to be increasingly important, partly for the workplace and the economy, but also for thriving and well-being. There is some interest in fostering creative competence in mathematics, where it is seen as relating largely to mathematical problem solving, when solutions are not matters of routine, at least for the student. Early studies, however, indicate little impact on teaching, which has tended to adhere to past practices without creative competence being an explicit goal or provision for achieving it. A belief that such a competence will grow undirected and unaided, or that it is only for the gifted student goes against current views of creative potential. Such beliefs could present obstacles to change, yet seem to have received little explicit exploration. This study aimed to identify some of the obstacles to fostering mathematics undergraduates' creative thinking arising from tutors' notions of mathematical creativity and its place in education. The views of twenty-two UK university mathematics tutors were collected and collated by phenomenographic analysis, a qualitative method designed to identify categories of belief. This showed latent potential problems arising from differences in the tutors' beliefs about creativity (e.g. creativity being in the process or the product), and about its origin (e.g. creativity arising from nature or nurture). Adding these to students' beliefs increases reservations about a readiness to foster students' creative competences in mathematics. Ways of overcoming these obstacles are suggested. We felt it essential that mathematicians have a common understanding of mathematical creativity, and that certain myths about creative abilities are corrected. These understandings need to be shared with students, and provision made for fostering their creative thinking at a macro level (e.g. courses and workshops) and a micro level (e.g. rubrics to guide thinking). Collaborative creative thinking is also valued in the workplace. It would add to students' assets if this provision included opportunities to practise it.