The purpose of this dissertation was to study how teachers and school leaders perceived a specific set of classroom math innovations, and how those innovations impacted instruction in relation to the Georgia Vision Project (GVP) standards and recommendations. This was a qualitative study conducted in two GVP districts. The participants in the study were five elementary teachers, two school administrators, and two district leaders. The participants were interviewed to gain an understanding of their perceptions of recent math innovations. The innovations included (a) math instruction using manipulatives (such as counting objects and puzzles) that utilize the Concrete Representational Abstract (CRA) model, which engages students to conceive from the concrete to the abstract; (b) differentiation through flexible student grouping; (c) information about how different subgroups of students learn mathematics; and (d) math professional learning. Previous research had focused on these innovations separately. However, no research study had grouped these innovations together to see how teachers perceived them within the context of a math classroom, and how teachers implemented them in their classrooms in order to increase student achievement. This qualitative case study included schoolteacher and educational leader interviews, observations, and artifacts. The two districts in the study were high performing in the area of mathematics. The results indicated that schoolteachers and educational leaders could not directly relate the math innovations to student success and, moreover, to the GVP standards and recommendations. During the study all GVP standards were analyzed at varying levels. The study primarily focused on the teaching and learning standard, which was a significant initiative for both districts. Both districts had varying levels of implementation concerning the innovations in the study: (a) use of manipulatives, (b) differentiation in classrooms, and (c) professional learning. All participants referenced the innovations as a part of their instruction, but could not directly relate the innovations beneficial to the success of the students.

Teaching algebra-based concepts to students with learning disabilities: The effects of preteaching using a gradual instructional sequence by Sarah Jean Watt An Abstract of a thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Teaching and Learning Special Education) in the Graduate College of The University of Iowa May 2013 Thesis Supervisor: Associate Professor William J. Therrien Research to identify validated instructional approaches to teach math to students with LD and those at-risk for failure in both core and supplemental instructional settings is necessary to assist teachers in closing the achievement gaps that exist across the country. The concrete-to-representational-to-abstract instructional sequence (CRA) has been identified through the literature as a promising approach to teaching students with and without math difficulties (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Cass, Cates, Smith, & Jackson (e.g. CSA), 2003; Flores, 2010). The CRA sequence transitions students from the use of concrete manipulatives to abstract symbols through the use of explicit instruction to increase computational and conceptual understanding. The main purpose of this study was to assess the effects of preteaching essential pre-algebra skills on the overall algebra achievement scores for students with disabilities and those at-risk for failure in math. Specifically the study examined the following research questions: (1) What are the effects of preteaching math units using the CRA instructional sequence on the algebra achievement of students with LD and those at risk for math failure? (2) What are the effects of preteaching math units using the CRA instructional sequence on the transfer of algebra-based skills of students with LD and those at risk for math failure to the general education setting? (3) What are the effects of preteaching math units using the CRA instructional sequence on the maintenance of algebra-based skills for students with LD and those at risk for math failure? Summary of Study Design and Findings Thirty-two students enrolled in one of four 6th grade classrooms across two elementary schools participated in this study. Sixth grade students who currently receive tier 2 or tier 3 supplemental and intensive instruction in math; and those identified as having a math learning disability will be participants. A treatment and control, pre/post experimental design was used to examine the effect of the intervention on students' math achievement. The intervention was replicated across two math units related to teaching algebra-based concepts: Solving Equations and Fractions. The treatment condition consisted of a combination of preteaching and the use of the CRA instructional sequence. Prior to each unit, Solving Equations and Fractions, researchers pretaught students 3 essential prerequisite skills necessary for success in the upcoming unit, at the concrete, representational, and abstract levels of learning. Each preteaching session lasted for ten days, 30 minutes each day. Immediate, delayed, and follow-up measures were used to support the examination of the research questions and hypotheses. Overall findings indicate that the combination of preteaching using the CRA gradual sequence is effective at improving the overall algebra performance for students with disabilities.

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Approximately 5 to 14% of school age children are affected by mathematics learning disabilities. With the implementation of inclusion, many of these children are now being educated in the regular education class- room setting and may require additional support to be successful in algebra. Therefore, teachers need to know what interventions are available to them to facilitate the algebraic learning of students with mathemat- ics learning disabilities. This systematic literature review aims to identify, and critically analyze, interventions that could be used when teaching algebra to these students. The five included articles focused on interven- tions that can be used in algebra, specifically when solving equations. In the analysis of the five studies two types of interventions emerged: the concrete-representational-abstract model and graphic organizers. The concrete-representational-abstract model seems to show it can be used successfully in a variety of scenarios involving solving equations. The use of graphic organizers also seems to be helpful when teaching higher- level algebra content that may be difficult to represent concretely. This review discovered many practical implications for teachers. Namely, that the concrete-representational-abstract model of intervention is easy to implement, effective over short periods of time and appears to positively influence the achievement of all students in an inclusive classroom setting. The graphic organizer showed similar results in that it is easy to implement and appears to improve all students’ learning. This review provided a good starting point for teachers to identify interventions that could be useful in algebra; however, more research still needs to be done. Future research is suggested in inclusive classroom settings where the general education teacher is the instructor and also on higher-level algebra concepts.

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