This study explored 78 elementary school principals' perceptions of classroom mathematics instruction in an effort to build understanding of the professional vision (Goodwin, 1994) of elementary school principals as it relates to mathematics instruction. This study also tested the theory of Leadership Content Knowledge (Stein & Nelson, 2003) which proposes that principal instructional leadership in mathematics is, in part, dependent upon content knowledge in mathematics. The principals were all current public school principals in the state of Florida and participated in a year-long, content-specific professional development series in mathematics and science for school leaders. The research questions in this exploratory study were (1) What is the relationship between elementary principals' knowledge in mathematics and their level of expertise in classroom observation of elementary mathematics instruction? (2) What elements of mathematics instruction are consistently interpreted similarly or dissimilarly between mathematics specialists and elementary school principals? (3) What do elementary school principals commonly notice or look for when observing mathematics instruction? The study employed a mixed-methods design to explore the correlation between mathematics knowledge and observation expertise as well as to begin to catalogue the appreciative system and terms used by principals to build a general picture of the professional vision of school principals. Data sources included the Mathematics Knowledge for Teaching survey (Hill & Ball, 2004), the Reformed Teaching Observation Protocol (Piburn et al., 2000; Sawada et al., 2000), and written responses to open-ended prompts designed to elicit principal perceptions of classroom instruction in mathematics. All 78 participants completed the surveys after observing three videos of classroom mathematics instruction. Written responses were analyzed and used to determine commonly noticed aspects of mathematics instruction as well as insights into principal recommendations for instructional improvement. A set of codes were generated from the written data, applied to the data, tested for reliability, and used to describe the relative frequency, consistency, and dependency of principal remarks. During the course of this study, a method for measuring the observation expertise of individual principals was developed. This study found no significant correlation between principals' mathematics knowledge and their observation expertise. When particular surface-level, observable features such as group work and high levels of apparent student engagement appeared in the video, principals had highly consistent interpretations of mathematics instruction. Mathematics specialists had highly consistent interpretations of traditional instruction, but there was less agreement between mathematics specialists regarding more reform-oriented instruction. This study also found that elementary school principals have a positive impression of reform-oriented instruction in mathematics and a negative impression of traditional instruction. Regarding what principals notice in mathematics instruction, some of the central topics that principals are looking for in classroom instruction include student engagement, group work (by students), the use of manipulatives, the use of vocabulary by both teachers and students, occasional checks for student understanding, physical and emotional safety, and teacher-student rapport. Principal recommendations for instructional improvement often included reference to group work and manipulatives without reference to specific ways that manipulatives may illuminate mathematical ideas or ways to structure group work for increased learning. Results of this study will guide professional development for school principals as instructional leaders in mathematics. Results also suggest further development of methods for rigorous study of the interaction of variables within the theory of Leadership Content Knowledge. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]