The results from the 2007 Trends in International Mathematics and Science Study (TIMSS) revealed that the scores of fourth grade students in the 90th percentile showed no measurable change from the results of the first TIMSS assessment in 1995. Contrarily, the scores of the lowest performing fourth graders showed statistically significant improvement between these two tests. Therefore, a better understanding of the ways in which mathematically promising students are instructed, the characteristics these students exhibit through their mathematical behavior and the form of expression they use to reveal these characteristics of talent is warranted. The following study uses qualitative methods to examine the mathematical behavior of promising students and the instructional strategies of teachers in heterogeneous classrooms. The study is comprised of three manuscripts each which look at a different aspect of this topic (1) the relationship of teacher mathematical content knowledge and her preparedness to develop mathematical talent, (2) the abilities of teachers with varying levels of content knowledge to differentiate instruction and (3) portraits of mathematically promising student work when experiencing a problem-based learning curriculum. The findings indicate that mathematical content knowledge plays a key role in the types of instructional decisions teachers make when choosing mathematical tasks as well as the quality of discourse and pacing in the lesson. In addition, mathematically promising students exhibited several characteristics identified in research literature as traits of mathematical talent and expressed these characteristics in six different ways including verbally, through illustrations, numerically, kinesthetically, with technology, and through writing. Furthermore, many students were identified as having mathematical promise as a result of being offered the opportunities to express their mathematical thinking in this variety of ways and the majority of these students come from underrepresented groups in regard to gifted and talented identification. The implications from this study show that increasing the level of elementary teacher content knowledge, and providing students with rich problem-based learning curriculum that allows for multiple modes of expression will increase the number of students identified as mathematically promising as well as provide these students with the type of quality instruction that will develop their mathematical abilities. [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.]