Prior research in mathematics education confirms the significance of representations as an integral part of curriculum standards. The effective teaching of representations is related to teachers' content and pedagogical knowledge. A teacher must know the salient features of the mathematical concept as well as the attributes of various representations to be able to explicitly demonstrate connections among them. The role of the teacher does not end after introducing the representation, as s/he needs to focus on students' sense-making, justifications, argumentation, and reasoning during the learning process, for which the teacher needs to students' conceptions as they engage in using representations. In addition to knowledge, beliefs also significantly impact teachers' instructional decisions. If teachers believe that representations are useful for understanding mathematics, they tend to use the representation to support the development of student understanding of the mathematical idea (Izsak, 2008). However, if teachers regard representations simply as visual aids, they might use representations only to show answers. Said differently, both knowledge and beliefs determine teachers' instructional orientation. Consequently, this dissertation research attends to both cognitive and psychological constructs by addressing the following overarching question: "To what extent do elementary in-service teachers' knowledge and beliefs about using representations explain their instructional use of drawn models in the mathematics classroom?" This question is addressed with three studies, the first two of which are designed to assess the validity of data gathered through two newly developed instruments: Assessing the Structure of Knowledge in Teaching Mathematics (ASKTM)-Representations (Study 1) and Instructional-Orientation for Drawn Models (Study 2). Current standards for educational and psychological testing were used to examine the validity evidence. Data were collected from a random sample of elementary in-service teachers from a geographically and economically diverse range of schools in a Midwestern state. Each of these two validation studies includes data from two different subsamples. For Study 1, data come from two sources: survey data from 1,849 in-service teachers who responded to the Qualtrics survey in 2018 and 30 interviews with the teachers who responded to these survey items. For Study 2, information was gathered from 484 in-service teachers completing the Qualtrics survey in 2019. Both quantitative (descriptive, item-analysis, and confirmatory factor analysis) and qualitative (feedback from context-experts, thematic coding of think-aloud interview data) methods were used in Study 1 and Study 2. Overall, the instruments evaluated in Study 1 and Study 2 were found to be sound tools to gain insight into teachers' knowledge of representations (Study 1) and instructional orientation for drawn models (Study 2) while teaching fractions and decimals. For Study 3, a sample of 484 in-service elementary-grade teachers was administered a survey that comprised an assessment of their knowledge about representations; an instructional use component to capture how often, with which instructional orientation, and why the teachers used drawn models in their instruction; and a beliefs component that measured the extent to which the teachers' beliefs were related to the significance of using drawn models in students' learning. The results of the survey indicate that both knowledge and beliefs are significant predictors in determining teachers' instructional orientation for mathematical representations. These findings assert a need to focus on both cognitive and psychological aspects to understand and address teachers' instructional orientation related to representation use. [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.]