A Study of the Effects of Teachers' Knowledge on the NCTM and New York State Mathematics Standards with Students in Grades 7- 12.
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Edge detection using parallel ant colony optimization with Hadoop MapReduce: implementation and scalabilityThe Ant Colony Optimization (ACO) is a popular optimization algorithm that finds use in multiple application areas. Though not among the common uses of this algorithm, edge detection in image analysis is a very functional application of this meta-heuristic. To improve the edge detection capabilities, the inherent parallel nature of the ACO method can be combined with the distributed computing framework provided by the Hadoop/Map-Reduce infrastructure. The latter provides a simple, scalable and fault-tolerant distributed processing paradigm that has been popular in industry and the academic community. In this thesis, we explore the Elastic MapReduce service provided by Amazon Web Services to implement ACO algorithm for edge detection in images, and study its scalability and effectiveness by standard metrics. In addition, we demonstrate a filtering technique to reduce the noisy background of images to achieve significant improvement in the accuracy of edge detection.
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A Curriculum Project on Expressions and Equations in Mathematics 7 Aligned to the New York State Common Core and Learning StandardsAs our nation is in a time of educational reform, a curriculum project on Expressions and Equations in the Mathematics 7 curriculum is appropriate as it is aligned to the Common Core State Standards. The created study of Expressions and Equations in Mathematics 7 addresses the New York State Standards from the Expressions and Equations Domain and includes student learning objectives that demonstrate understanding of solving equations through a modeling approach. Studies have concluded that when students are able to touch and manipulate mathematics, they learn best (Raymond & Leinenbach, 2000). A table of contents is provided for easy access to each lesson plan, worksheets, and assessments needed for the 9-day unit plan. In addition, each lesson plan includes the goal for the lesson, the instructional outcomes, and the New York State Common Core Learning Standards that are addressed. The validity of this curriculum project was assessed by the author of the unit plan as the created lessons were implemented in the author’s classroom in October 2014. Furthermore, the validity of the curriculum project was measured by two cooperating veteran teachers that critiqued the unit plan though a questionnaire. The paradigm shift from the NCTM standards to the Common Core State Standards provides opportunities for educators to re-design their teaching in the classroom; this curriculum project on Expressions and Equations is created to be a resource to aid in this shift.
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A Curriculum Project on Incorporating Proofs in Lessons to Deepen Students’ Understanding of Mathematical ConceptsThis curriculum project provides secondary mathematics teachers with lessons that incorporate proofs to deepen student understanding of mathematical concepts through mathematical reasoning and discourse. The use of proofs in secondary math is essential to prepare students for college mathematics expectations. Mathematical proofs are difficult for students because it is not something that they often see throughout their secondary education. The Constructivist Learning Theory states that students learn by building on prior knowledge to create new understanding, thus each lesson in this curriculum project begins with a review section to connect prior learning to new concepts. Likewise, the use of the Growth Mindset and grit, also used in the curriculum, can encourage students to believe in their mathematical abilities. Encouraging students to attempt questions or proofs without providing immediate support can send a message that they are capable of meeting high expectations. Within the curriculum project there are four lessons: Introduction to Proofs; Quadratic Formula; Properties of Paralellograms; and Remainder Theorem. The last three lessons align with the curriculum for Algebra 1, Geometry, and Algebra 2 respectively. The first lesson was designed to introduce proofs before they are applied in any of these courses.
