The problem addressed in this qualitative single case study is low mathematical achievement levels among students in grades K-4 in rural Vermont schools, based on district-wide and school-based progress monitoring assessments and Vermont's Smarter Balanced Assessment scores. The purpose of this qualitative single case study was to explore the types of professional development needed to foster high-quality mathematical instructional coaching by K-4 mathematical instructional coaches in a rural Vermont school district. The theoretical framework guiding this study was andragogy, the method and practice of teaching adult learners. Qualitative data were collected from a sample of (n=24) teachers and (n=3) mathematics instructional coaches through an online survey, semi-structured interviews, and focus groups. Study findings identified the types of professional development K-4 mathematical instructional coaches perceived as necessary to strengthen the instructional coaching model, recommending professional development in mathematical pedagogy, adult learning theory, and instructional coaching. The findings also identified the skills that K-4 teachers in a rural Vermont school district perceived as necessary to improve instructional coaching practices, including understanding student development, understanding adult learning theory, strong mathematical pedagogy, building collaborative relationships, mentoring, leadership, and being data-driven. Recommendations for practice include offering professional development for coaches that focuses on developing deeper mathematical pedagogy skills, understanding adult learning theory, skills in mentoring and leadership, as well as interpreting, analyzing, and understanding student mathematical data. Recommendations for future research include extending the research to other grade levels and other geographic contexts. [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://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]