Junior and Senior Mathematics Courses Required for Mathematics Majors
Text: Foundations of Higher Mathematics 3rd ed. by Fletcher and Patty, PWS Publishers
Course Description: This course is intended to be an introduction to mathematical rigor and proof. A specific goal is to help prepare students for the demands of standard junior-senior courses in analysis and algebra. The approach is strictly hands-on. In this class we concentrate on the "language" that all mathematicians use, namely, proofs. This will strengthen our ability to understand and communicate mathematics in all its various forms. More specifically, we will study basic techniques for formulating and writing mathematical proofs. Throughout the course complete proofs are given that illustrate proof techniques and are intended to serve as models. The course is organized in a manner which parallels the content of the text. We cannot discuss every specific topic in the text, but the course is designed so that your understanding of the material depends heavily on studying the text and doing the drill problems and outside assignments. We start with a review of logic and the language of proofs, and then a review of sets. We then study types of proofs including mathematical induction, relations and orders, functions, a little combinatorics, a look at countable and uncountable sets, and finish with topics chosen from introductory group theory and the foundations of calculus. At this writing, there are a series of 'labs' to go with this class. These are extensive documents in Mathematica which are designed to help you learn some mathematics by doing mathematics.
Text: Discrete Mathematics and its Applications by Kenneth Rosen, 4th ed., 1999, McGraw Hill
Course Objectives: The goals of a course in discrete mathematics are several. Discrete mathematics deals with processes that are made up of individual steps as opposed to the calculus, which studies processes which change continuously. Topics studied will be chosen from graph and tree theory, modern applications of number theory, combinatorics, algorithms and their analysis, Boolean algebra, logic, modeling computations and the design of computing machines. Our author interweaves five themes throughout the text: mathematical reasoning, combinatorial analysis, discrete structures, applications and modeling, and algorithmic thinking. This course will emphasize modern applications that are being used in today's world. We will study enough of the subject matter of each topic to be able to understand and appreciate some applications of it. We will do some work on computers using the program Mathematica and the graphing calculator (TI family).
Pre-requisite and Text: Math 334 – continuation using the same text
Course Material: We will begin with a brief review of the early chapters of the text, and a detailed review of chapter 7. The main material for this course will be chapters 8 – 14 plus some supplementary materials. I hope for this course to be a synthesis of a general statistics course and a traditional calculus-based course in that I want you to walk away understanding the concepts (general stats course) as well as the computations (traditional calc-based course). We will use the computer for many of the computations – mainly SPSS, but possibly Excel and Mathematica as well. We will meet in the computer lab when it is appropriate.
Text: Applied Numerical Analysis by Gerald & Wheatley, 6th ed., by Addison-Wesley, 1999.
Course Description: This course will branch beyond the problems we have encountered in previous courses, which, believe it or not, were fairly easily solved by exact techniques. We will examine approximation techniques for numerically solving real-world problems arising in engineering, the physical and life sciences, the social sciences, and computer science. My goal for the course is that you will understand how to apply these techniques, when they are appropriate, and how to interpret the results. We will use Mathematica quite frequently in this course, rather than spending time learning a traditional programming language. To expand our knowledge of Mathematica and its capabilities, we will usually spend one day a week (Thursday) in the Science Building Computer lab (Room 147).
Pre-Requisite: Math 301 (Foundations of Mathematics)
Texts: Contemporary Abstract Algebra by Joseph Gallian (Heath, 4th ed, 1998) (new ed. coming Aug 2001) and Geometry and Its Applications by Walter Meyer (Academic Press, 1999)
Course Description and Objectives: This course covers the traditional topics from abstract algebra, including groups, rings, integral domains, fields, and homomorphic and isomorphic relationships, as well as standard topics from geometry, including axiomatic systems in both Euclidean and Non-Euclidean geometries and transformational geometry with vectors and matrices, The focus for the class is the contemporary applications of the concepts presented, together with the weaving together of geometric and algebraic themes. Applications include, robotics, boolean algebra, symmetry groups, coding, adjacency matrices, the global positioning system (GPS),etc. By the end of this course, students should be able to write proofs, do computations, and carry out applications of the topics covered. They should see the beauty in the mathematical structures and their use “…for understanding the 21st century world that is unfolding around us.” (Meyer)
Text: Mathematical Modeling, 2nd Edition, by Mark Meerschaert (Academic Press)
Course Description: This course is the capstone of your mathematical tenure at Carroll. On our journey this semester, we will explore the application of mathematics to a variety of real-world problems. We will not only use much of the mathematics you have learned in your other courses; we will also tie together and extend it as well. We will focus on the application of applied mathematics to solving optimization problems, focusing extensively on the modeling process, effectively solving models, and sensitivity analysis of models. Our textbook will provide a basis from which we can begin our explorations. We focus most of our attention on projects selected from my experiences as both an engineer and operations research analyst, from COMAP Mathematical Competition in Modeling problems, and from UMAP Modules. We will meet on a regular basis in the computer lab and use Mathematica, Excel, and GAMS (and possibly other software) for many of the projects.
Required course only for prospective secondary mathematics teachers
Text: Journey Through Genius by William Dunham, © 1990 by Wiley and by Penguin Books
Course Overview: This is a required course for mathematics secondary-education majors. It is intended to give students an insight into some of the great masterpieces of mathematics, as seen in their historical contexts. Mathematics is an ever-growing discipline in which new ideas are built upon the old. Developing an understanding of the individuals who were the creators of mathematics helps one better appreciate their creations.
Students select a topic that involves mathematics and their cognate concentration. They research the topic and work with one or more faculty members in exploring the application they have chosen. A formal write-up and presentation is required at the end of the semester. More research and a longer and more formal paper is required for those who choose to do this as an Honors Thesis.