The purpose of this study is to explore elementary educators' perceptions regarding what it means to conceptually understand mathematics, the emphasis teachers place on utilizing evidence-based teaching processes, as identified as effective by National Council of Teachers of Mathematics (NCTM), and how these perceptions influence their instructional choices. The theoretical framework includes Lev Vygotsky's theory of Zone of Proximal Development. The methodological type will be a qualitative collective case study consisting of 8 participants, which include classroom teachers of grades 2-5. The TIMSS (Trends in International Mathematics and Science Study) conducted every four years, has highlighted gaps in mathematics performance in the United States as compared to some other countries; however, the TIMMS has found an increase in the average math scores of fourth graders in the United States from 2003 and 2007 as well as from 2011 and 2015 which may be tied to shifting standards that highlight more comprehensive conceptual understanding in the area of mathematics and teaching practices that better lend themselves to these standards. Teacher perception of the standards and strategies specifically delineated within the New York State Next Generation Mathematics Learning Standards (2017) may tie directly to the successful implementation of NCTM's teaching processes and strategies that support a more robust conceptual understanding of mathematics among students. This qualitative collective case study examines the mathematics beliefs and practices of eight teachers in grades 2-5. The findings demonstrate elementary teachers' use of problem-solving, reasoning and proof, communication, connections, and representations as means by which mathematic content is accessed by students. Teachers' pre-service training, prior work experience (if applicable), teaching experience, professional development, and self-efficacy in teaching mathematics are factors that may impact their instructional choices. [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.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]