Projects in math education

Includes bibliographical references (leaves 48-49)This research fulfills a requirement for the Masters in Mathematics for Educational Careers Option. The objective of this research is to analyze the process and the mathematics by which teachers can introduce graduate level mathematic topics in earlier courses. This thesis is based on topics studied in Algebra and Number Theory, Linear Algebra and Geometry, Concepts of Analysis, Discrete Mathematics, and Probability and Statistics. I included research papers from Algebra and Number Theory, Linear Algebra and Geometry, and Discrete Mathematics and the summaries of the research papers from Concepts of Analysis and Probability and Statistics. The first research paper for Linear Algebra and Geometry titled "Camera Projections and Projective Geometry" was inspired by a geometry other than Euclidean geometry. Students are not often introduced to non-Euclidean geometry until upper-division college courses. My intent was to demonstrate projective geometry as a geometry that is experienced every day. The next research paper for Algebra and Number Theory titled "Factorization of Polynomials in Mathematica" was inspired by the process by which calculators and mathematics software do calculations. In particular, my research paper investigates the mathematics behind the "Factor" command in Mathematica. The last research paper for Discrete Mathematics titled 11 2- and 3-Colorable Maps with Many Pairs of Neighboring Regions" was inspired by the 4-colors theorem which states that every planar map of connected countries can be colored using four colors in such a way that countries with a common border (not just a point) receive different colors. The concept of coloring a map is understood at an early age, but the mathematics behind this theorem has a long history and continues to be developed. The summary for Concepts of Analysis titled "Calculus in the High School Classroom" breifly describes seven projects that can be used to enhance a calculus course. The summary for Probability and Statistics titled "Democratic Votes Against 2000 Census Data" looks for a correlation between median age, owner-occupied housing units, percentage of adults with bachelor's degrees or higher, first language other than English, median household income, density per square mile of land (population), and race (white) against the 2000 U.S. Census to the voting patterns of 10 randomly chosen states in the 2004 presidential election