The purpose of this three-paper manuscript dissertation is to add value to the limited knowledge base of research surrounding the quality of teacher-child interactions in pre-k mathematics contexts. The first paper, based on an extensive review of literature, presents a theoretical basis for using five-frames to support children's development of Gelman and Gallistel's (1978) "how-to-count" principles (one-to-one-correspondence, stable-order, and cardinality) and also describes ways that five-frames can be leveraged to expose children to basic operations situated in the context of direct modeling activities. This manuscript concludes by discussing the developmental and instructional implications of using five-frames in pre-k mathematics contexts. Manuscript two presents results of an in-depth study of the quality of teacher-child interactions in pre-k (as measured by the Classroom Assessment Scoring System--CLASS, and the Classroom Observation of Early Mathematics-Environment and Teaching--COEMET) across five teachers facilitating mathematics activities from the "MyTeachingPartner Math-Science (MTP-MS)" curriculum. This paper presents descriptive statistics for both CLASS and COEMET independently (considering quality across contextual factors such as group setting, mathematical domain, and activity type) and also investigates correlational data between the measures. Results suggest that the two measures are highly correlated and that the qualities of teacher-child interactions are not statistically different across contextual factors. The final manuscript explores interactions focusing on pre-k students' acquisition of two-digit place value understanding across four "MyTeachingPartner Math" activities involving color-coded ten-frame representations. In this qualitative analysis, we give particular focus to the challenges faced by students as well as consider the specific instructional strategies employed by two high quality mathematics teachers to scaffold basic place value understanding. Results of this study suggest pre-k students' understanding of "leftovers" (i.e., ones place) emerges before the tens place and that many face challenges with the concept of unitizing. Results also indicate high quality pre-k teachers typically use leading questions to leverage multiple representations to effectively scaffold students' emerging understanding. [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.]