Algebraic Thinking: Exploring Elementary Teachers’ Perceptions And Vertical Alignment In The K-8 Curriculum

The transition from arithmetic in elementary school to algebra in the middle grades often presents challenges to students as they encounter new experiences and content in mathematics classrooms which require them to think algebraically. It is widely agreed that students can benefit from early exposure to algebraic ideas within the context of mathematics already taught at the elementary level and that this exposure will aid them in their understanding of algebra in later grades. The opportunity to develop algebraic thinking skills at the elementary level relies heavily upon the ways in which teachers understand algebraic thinking and how they choose to develop these skills within their classrooms. This paper documents the design of a study which aimed to explore elementary teachers’ perceptions of algebraic thinking in terms of its presence in the North Carolina Standard Course of Study for grades K-5, difficulties encountered during the implementation of the study, and an exploration of the vertical alignment of selected algebra concepts throughout the current North Carolina Standard Course of Study for grades K-8

Analyzing common algebra-related misconceptions and errors of middle school students.

The purpose of this study was to examine common algebra-related misconceptions and errors of middle school students. In recent years, success in Algebra I is often considered the mathematics gateway to graduation from high school and success beyond. Therefore, preparation for algebra in the middle grades is essential to student success in Algebra I and high school. This study examines the following research question: What common algebra-related misconceptions and errors exist among students in grades six and eight as identified on student responses on an annual statewide standardized assessment? In this study, qualitative document analysis of existing data was used in order to analyze sixth- and eighth-grade student responses on a statewide standardized assessment. Secondary data sources consisted of Algebra I student responses which were also analyzed qualitatively using document analysis and follow-up interviews with key informants. The primary analysis indicated that (l) numerous misconceptions and errors identified in the review of literature were present on both the sixth- and eighth-grade open-responses; (2) basic computational errors with whole numbers (a secondary skill), were found consistently throughout the sixth- and eighth-grade open-responses; (3) a greater number of misconceptions and errors identified in the review of the literature were present on the eighth-grade items than were found on the sixth-grade items; (4) students often lost points for reasons other than mathematical misconceptions or errors; and (5) some refinement and reorganization of Welder\u27s (2007) framework could prove beneficial when using the framework for data analytic purposes. The results of this study provided information about the common misconceptions and errors students possess on prerequisite algebra skills. The findings revealed common algebra misconceptions and trends that can help guide instruction for middle school mathematics teachers. The findings have direct implications for classroom practice and further confirm the need for strong and knowledgeable teachers of mathematics at the elementary and middle grades. The researcher suggests that schools, both in the state whose standardized assessment was examined as well as other states, use this information to help build awareness of common prerequisite algebra-related misconceptions and errors in elementary and middle grades mathematics teachers

Facilitating high school student success through READ 180: Analysis of program impact using measures of academic progress (MAP)

In response to its failure to meet state mandated proficiency standards in reading and mathematics over the past three years, a rural, Title I high school (LS) in South Carolina purchased and implemented the commercially available literacy program READ 180 (R180) for the 2008-2009 academic year. While previous research reported by Scholastic, Incorporated (R180) had provided support for the use of R180 in improving literacy, these studies have been criticized recently for their lack of comparable control groups, experimenter bias and lack of data from other content areas such as mathematics. The purpose of this study was to determine the relative effectiveness of R180 in improving reading and math performance when compared with traditional high school English course instruction in a group of ninth grade students at LS. The theoretical framework for this study was based on Vygotsky\u27s cognitive developmental theory which emphasizes the role of language in learning in all content areas. A group of below average reading ability students was assigned by LS to the R180 instructional class while a second group of average ability students was assigned to the traditional English course (TRAD). Both groups were pre and post tested in reading and math using the state-sponsored Measures of Academic Progress (MAP) standardized achievement test. Dependent samples t-tests and Analysis of Covariance were used to analyze the data. The results indicated statistically significant improvements in both math and reading scores for the TRAD group but not for the R180 group. This study has implications for positive social change in the form of independent, empirically-based data to both inform the administration of LS in future decision making regarding funding for the very costly R180 program as well as contributing to the overall database on R180\u27s effectiveness

Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part I: template-based generic programming

An approach for incorporating embedded simulation and analysis capabilities in complex simulation codes through template-based generic programming is presented. This approach relies on templating and operator overloading within the C++ language to transform a given calculation into one that can compute a variety of additional quantities that are necessary for many state-of-the-art simulation and analysis algorithms. An approach for incorporating these ideas into complex simulation codes through general graph-based assembly is also presented. These ideas have been implemented within a set of packages in the Trilinos framework and are demonstrated on a simple problem from chemical engineering

WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction

WavePacket is an open-source program package for numeric simulations in quantum dynamics. It can solve time-independent or time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows, e.g., to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semi-classical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. Being highly versatile and offering visualization of quantum dynamics 'on the fly', WavePacket is well suited for teaching or research projects in atomic, molecular and optical physics as well as in physical or theoretical chemistry. Building on the previous Part I which dealt with closed quantum systems and discrete variable representations, the present Part II focuses on the dynamics of open quantum systems, with Lindblad operators modeling dissipation and dephasing. This part also describes the WavePacket function for optimal control of quantum dynamics, building on rapid monotonically convergent iteration methods. Furthermore, two different approaches to dimension reduction implemented in WavePacket are documented here. In the first one, a balancing transformation based on the concepts of controllability and observability Gramians is used to identify states that are neither well controllable nor well observable. Those states are either truncated or averaged out. In the other approach, the H2-error for a given reduced dimensionality is minimized by H2 optimal model reduction techniques, utilizing a bilinear iterative rational Krylov algorithm

Image 100 procedures manual development: Applications system library definition and Image 100 software definition

An outline for an Image 100 procedures manual for Earth Resources Program image analysis was developed which sets forth guidelines that provide a basis for the preparation and updating of an Image 100 Procedures Manual. The scope of the outline was limited to definition of general features of a procedures manual together with special features of an interactive system. Computer programs were identified which should be implemented as part of an applications oriented library for the system

dPV: An End-to-End Differentiable Solar-Cell Simulator

We introduce dPV, an end-to-end differentiable photovoltaic (PV) cell simulator based on the drift-diffusion model and Beer-Lambert law for optical absorption. dPV is programmed in Python using JAX, an automatic differentiation (AD) library for scientific computing. Using AD coupled with the implicit function theorem, dPV computes the power conversion efficiency (PCE) of an input PV design as well as the derivative of the PCE with respect to any input parameters, all within comparable time of solving the forward problem. We show an example of perovskite solar-cell optimization and multi-parameter discovery, and compare results with random search and finite differences. The simulator can be integrated with optimization algorithms and neural networks, opening up possibilities for data-efficient optimization and parameter discovery

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed