Ongoing research in physics education has demonstrated that physics majors often do not develop a working knowledge of basic concepts in mechanics, even after standard instruction in upper-level mechanics courses. A central goal of this work has been to explore the ways in which students make--or do not make--appropriate connections between physics concepts and the more sophisticated mathematics (e.g., differential equations, vector calculus) that they are expected to use. Many of the difficulties that students typically encounter suggest deeply-seated alternate conceptions, while others suggest the presence of loosely or spontaneously connected intuitions. Analysis of results from pretests (ungraded quizzes), written exams, and informal classroom observations are presented to illustrate specific examples of naïve intuitions and related difficulties exhibited by the students. Also presented are examples of instructional strategies that appear to be effective in addressing these difficulties.