Browsing by Subject "Multiple representations"
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Item Multiple representations and rate of change: the nature of diverse Students' Initial Understanding.(2010-02) Whitney, Stephanie RaeAccess to quality mathematics content and instruction has been equated as a civil right (Moses & Cobb, 2001), as it is a gatekeeper to higher education and lucrative careers. Unfortunately, access to empowering mathematics is not a reality for many urban youth. Data show an achievement gap between Black, Hispanic and American Indian students and their peers which, in turn, result in unequal access to education and economic opportunities (NAEP, 2007; U.S. Department of Education, 1997). This evaluative case study focused on eight racially diverse high school students from an urban charter school who were studying Algebra 1 in classroom contexts that were student-centered and discourse-based. The study took place after a sequence of six lessons that introduced representations of linear functions through geometric patterns. Pairs of students participated in a series of interview tasks, largely modeled from problems in their curriculum Algebra Connections (Dietiker, Kysh, Sallee, Hoey, 2005), that focused on their understanding of representations of functions (tables, graphs, equations and context), the embodiment of rate of change in the representations and translating among the representations. The study sought to understand the nature of the students’ initial understandings of representations and translations including their natural language when describing observations, the perspectives from which they approached the tasks (explicit or recursive) and how their thinking compares to literature in the field. The students were able to translate linear functions directly between any two representations and many were able to extend their thinking to novel problems involving non-linear functions.Item Professional development and teacher change: teachers' practices and beliefs about using multiple representations in teaching Mathematics(2013-07) Park, Mi SunThis study explores mathematics teachers' beliefs about using multiple representations, such as manipulative, pictorial, symbolic, language, and realistic representations that are described the Lesh translation model. Using a qualitative approach, the teachers' beliefs about the helpfulness and hindrances of using each representation described in the Lesh translation model were clarified. Teachers' belief changes were also explored, before and after participating in a yearlong professional development program. Furthermore, this study investigated the teachers' reflections on teaching mathematics based on their beliefs during the program. Results indicated that the teachers' beliefs were mostly in line with the Lesh translation model. Changes in the teachers' beliefs were also observed after participating in the program. These changed beliefs led teachers to change their practices in the classroom. The teachers mostly used pictorial, symbolic, language, and realistic representations during the three observations periods. Using translations within and between these types, especially pictorial and realistic representations, were varied in the three classroom observations. On the other hand, the teachers used few physical and virtual manipulatives, even though they claimed to have positive beliefs about using them in the classroom. That is, the teachers' beliefs concerning the hindrances of using manipulatives prevented them from using these representations in the classroom. Furthermore, changing teachers' practices required more time after their beliefs were changed. In future studies, researchers could focus on a longitudinal study of the interactions between beliefs and practices. In addition, research could explore more effective activities that involve physical manipulatives or virtual manipulatives, which reduce any hindrances that were identified in this study for K-12th grade students.