Facing increased pressure to improve retention and graduation rates, engineering departments are increasingly scrutinizing whether they are getting their desired outcomes from core mathematics coursework. Since mathematics courses are a significant source of attrition and many engineering faculty are unhappy with students' mathematical abilities, more engineering departments are increasingly looking at drastic options of taking students out of mathematics courses and teaching students mathematics themselves. To mitigate this trend, it may be valuable to better understand what engineering faculty hope students learn from their mathematics coursework. When engineering faculty explain why they require these high-failure prerequisites, many claim that "mathematical maturity", not calculus skill, is the desired outcome of completing the core math sequence of courses. To better understand what engineering faculty mean by "mathematical maturity", we conducted a qualitative thematic analysis of how 27 engineering faculty members define "mathematical maturity". We found that these engineering faculty believed that the mathematically mature student would have strong mathematical modeling skills supported by the ability to extract meaning from symbols and the ability to use computational tools as needed. Faculty frequently lamented that students had underdeveloped epistemic beliefs that undermined their modeling skills, thinking that mathematics is unrelated to the real world and has little practical value. They attributed these dysfunctional epistemic beliefs to their perception that mathematics is too often taught without genuine physical context and realistic examples. We suggest potential avenues for reform that will allow mathematics departments to better serve their client departments in engineering and thus retain control of their courses.