Department Learning Goals
Understanding Higher Mathematics
- Understanding the need to transition from a procedural/computational understanding of mathematics to a broader conceptual understanding encompassing logical reasoning, generalization, abstraction, and formal proof.
- Understanding the core 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.
- Understanding the fundamentals of mathematical 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.
- Understanding the methods and fundamental role of mathematics in modeling and solution of critical real world challenges in science and social science.
- Understanding the basic insights and methods of a broad variety of mathematical areas. All students of mathematics must achieve such understanding in calculus, naïve set & function theory, and linear algebra, and ideally will further achieve such understanding in probability & statistics, differential equations, analysis, and algebra.
- Understanding in greater depth 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 Knowledge.
- 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.
- Modeling—to iteratively construct, modify and analyze mathematical models of systems encountered in the natural and social sciences, to assess a models' accuracy and usefulness, and to draw contextual conclusions from them.
- 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.
- 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.
- Presentation—to produce clear, precise, motivated, and well-organized expositions, in both written and oral form, using precise reasoning and genuine analysis.
- Oral Argumentation—to think on one’s feet, to field questions, and to defend mathematical ideas and arguments before a range of audiences.
- Mathematical Literature—to know how to effectively search the mathematical literature and how to appropriately combine and organize information from a variety of sources.