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Overview and Learning Goals

Overview

The mathematics major welcomes students from a broad range of backgrounds and driven by a diverse set of interests. It is designed with one common purpose: the pursuit of mathematics as a fundamental human endeavor with the ability to describe both the concrete and abstract aspects of the world around us. The power and utility of mathematics comes from its analytical approach to problem-solving. Our courses and requirements develop the ability to reason from hypothesis to conclusion, to analyze and solve quantitative problems, and to clearly and precisely articulate the underlying thought process. Students are encouraged to explore a broad range of courses offered by the mathematics department. It is central to our mission as a research and educational department to offer support and encouragement so that each member of our community can succeed and thrive.

Learning Goals

Understanding Higher Mathematics

1. Understanding the need toÌýtransitionÌýfrom a procedural/computational understanding of mathematics to a broaderÌýconceptual understandingÌýencompassing logical reasoning, generalization, abstraction, and formal proof.
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2. Understanding theÌýcore of mathematical culture: the value and validity of careful reasoning, of precise definition, and close argument. Understanding mathematics as a growing body of knowledge, driven by creativity, a search for fundamental structure and interrelationships, and a methodology that is both powerful and intellectually compelling.
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3. Understanding the fundamentals ofÌýmathematical reasoning and logical argument, including the role of hypotheses, conclusions, counterexamples, and other forms of mathematical evidence in the development and formulation of mathematical ideas.
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4. Understanding the methods and fundamental role of mathematics inÌýmodeling and solutionÌýof critical real-world challenges in science and social science.
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5. Understanding the basic insights and methods of a broad variety of mathematical areas. All students of mathematics must achieve such understanding inÌýcalculus, naïve set and function theory, and linear algebra; and, ideally, will further achieve such understanding inÌýprobability and statistics, differential equations, analysis, and algebra.
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6. Understanding in greater depth of at least one important subfield of mathematics such as abstract algebra, real analysis, geometry, topology, statistics, optimization, modeling, numerical methods, and dynamical systems.

Skills Required for Effective Use of Mathematical KnowledgeÌý

1. Problem-solving—to develop confidence in one’s ability to tackle difficult problems in both theoretical and applied mathematics; to translate between intuitive understandings and formal definitions and proofs; to formulate precise and relevant conjectures based on examples and counterexamples; to prove or disprove conjectures; to learn from failure; and to realize solutions are often multi-staged and require creativity, time, and patience.
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2. Modeling—to interactively construct, modify, and analyze mathematical models of systems encountered in the natural and social sciences; to assess a model's accuracy and usefulness; and to draw contextual conclusions from them.
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3. Technology—to recognize and appreciate the important role of technology in mathematical work, and to achieve proficiency with the technological tools of most value in one’s chosen area of concentration.
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4. Data and Observation—to be cognizant of the uses of data and empirical observation in forming mathematical and statistical models, providing context for their use, and establishing their limits.
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5. Presentation—to produce clear, precise, motivated, and well-organized expositions, in both written and oral form, using precise reasoning and genuine analysis.
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6. Mathematical Literature—to know how to effectively search the mathematical literature and how to appropriately combine and organize information from a variety of sources.

Options for Majoring or Minoring in the Department

Students may elect to major in mathematics, the computer science and mathematics interdisciplinary major, the mathematics and economics interdisciplinary major, the mathematics and education interdisciplinary major, or to coordinate a major in mathematics with digital and computational studies, education, or environmental studies. Students pursuing coordinate or interdisciplinary majors may not normally elect a second major. Non-majors may elect to minor in mathematics.

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