Teaching and aligning upper level mathematics classes to Michigan high school content expectations in an alternative education setting : approaches and issues

After teaching regular education secondary mathematics for seven years, I accepted a position in an alternative education high school. Over the next four years, the State of Michigan adopted new graduation requirements phasing in a mandate for all students to complete Geometry and Algebra 2 courses. Since many of my students were already struggling in Algebra 1, getting them through Geometry and Algebra 2 seemed like a daunting task. To better instruct my students, I wanted to know how other teachers in similar situations were addressing the new High School Content Expectations (HSCEs) in upper level mathematics. This study examines how thoroughly alternative education teachers in Michigan are addressing the HSCEs in their courses, what approaches they have found most effective, and what issues are preventing teachers and schools from successfully implementing the HSCEs. Twenty-six alternative high school educators completed an online survey that included a variety of questions regarding school characteristics, curriculum alignment, implementation approaches and issues. Follow-up phone interviews were conducted with four of these participants. The survey responses were used to categorize schools as successful, unsuccessful, and neutral schools in terms of meeting the HSCEs. Responses from schools in each category were compared to identify common approaches and issues among them and to identify significant differences between school groups. Data analysis showed that successful schools taught more of the HSCEs through a variety of instructional approaches, with an emphasis on varying the ways students learned the material. Individualized instruction was frequently mentioned by successful schools and was strikingly absent from unsuccessful school responses. The main obstacle to successful implementation of the HSCEs identified in the study was gaps in student knowledge. This caused pace of instruction to also be a significant issue. School representatives were fairly united against the belief that the Algebra 2 graduation requirement was appropriate for all alternative education students. Possible implications of these findings are discussed.

SPATIAL UNDERSTANDING OF MATRIX INVERSION FOR INVERSE FORCE ESTIMATION

A basic approach to study a NVH problem is to break down the system in three basic elements – source, path and receiver. While the receiver (response) and the transfer path can be measured, it is difficult to measure the source (forces) acting on the system. It becomes necessary to predict these forces to know how they influence the responses. This requires inverting the transfer path. Singular Value Decomposition (SVD) method is used to decompose the transfer path matrix into its principle components which is required for the inversion. The usual approach to force prediction requires rejecting the small singular values obtained during SVD by setting a threshold, as these small values dominate the inverse matrix. This assumption of the threshold may be subjected to rejecting important singular values severely affecting force prediction. The new approach discussed in this report looks at the column space of the transfer path matrix which is the basis for the predicted response. The response participation is an indication of how the small singular values influence the force participation. The ability to accurately reconstruct the response vector is important to establish a confidence in force vector prediction. The goal of this report is to suggest a solution that is mathematically feasible, physically meaningful, and numerically more efficient through examples. This understanding adds new insight to the effects of current code and how to apply algorithms and understanding to new codes.

CAN CHINA USE ALTERNATIVE ENERGIES INSTEAD OF COAL TO PROVIDE MORE ELECTRICITY BY 2030?

Following the rapid growth of China's economy, energy consumption, especially electricity consumption of China, has made a huge increase in the past 30 years. Since China has been using coal as the major energy source to produce electricity during these years, environmental problems have become more and more serious. The research question for this paper is: "Can China use alternative energies instead of coal to produce more electricity in 2030?" Hydro power, nuclear power, natural gas, wind power and solar power are considered as the possible and most popular alternative energies for the current situation of China. To answer the research question above, there are two things to know: How much is the total electricity consumption in China by 2030? And how much electricity can the alternative energies provide in China by 2030? For a more reliable forecast, an econometric model using the Ordinary Least Squares Method is established on this paper to predict the total electricity consumption by 2030. The predicted electricity coming from alternative energy sources by 2030 in China can be calculated from the existing literature. The research results of this paper are analyzed under a reference scenario and a max tech scenario. In the reference scenario, the combination of the alternative energies can provide 47.71% of the total electricity consumption by 2030. In the max tech scenario, it provides 57.96% of the total electricity consumption by 2030. These results are important not only because they indicate the government's long term goal is reachable, but also implies that the natural environment of China could have an inspiring future.