Teachers in the United States rate their lives better than all other occupation groups, trailing only physicians. However, as a nation we face a shortage of qualified math and science teachers. Because students often turn to faculty members for career advice, it is important that faculty hold accurate and positive perceptions of grade 7-12 math and science teaching. To help identify faculty perceptions of the teaching career, an instrument known as the Perceptions of Teaching as a Profession in Higher Education (PTaP.HE) (Pronounced P-Taffy) was developed. In this work a reduced-basis factor analysis was performed on the PTaP.HE and eight underlying factors were identified. These factors provide insight into faculty thinking about grade 7-12 teaching. When survey data are analyzed by these underlying factors, it is found that faculty perceive themselves and their departments as supportive of those who want to pursue grade 7-12 math or science teaching. However, their perceptions of grade 7-12 math and science teaching are sometimes internally inconsistent, with some perceptions being positive and others negative. This work highlights the importance of sharing accurate information about the teaching career with faculty members, so they can in turn share accurate information with their students.