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Committees

Arts and Sciences Learning Objectives

Mathematics

  1. Basic Understanding of Higher Mathematics:
     
    1. Working understanding of basic insights and methods in a broad variety of mathematical areas, including but not limited to calculus, linear algebra, real analysis and abstract algebra
       
    2. Working understanding of basic insights, methods and overarching ideas in one or more important subfields of mathematics, such as probability / statistics, partial differential equations, numerical linear algebra / numerical analysis, geometry /topology, number theory and complex analysis.
       
    3. Clear understanding of the role of hypotheses, conclusions, counterexamples and other forms of mathematical evidence in the development and formulation of mathematical ideas
       
    4. The ability to define and use concepts by their properties alone, rather than by explicit construction or calculation
       
    5. Using mathematics to think about problems outside of mathematics

    These learning outcomes are embodied in our course offerings, and other requirements for the bachelor degree in Mathematics.
     

  2. Written Communication:
     
    1. Ability to state and understand definitions and theorems with all requisite precision
       
    2. Efficient and coherent presentation of arguments in the form of proofs

    Again, these are judged first in course work. The first exposure to these usually comes in honors calculus, discrete mathematics, or linear algebra, and continues for the remainder of the bachelors degree program.
     

  3. Oral Communication:
     
    1. Clear explanation of key ideas and general strategies
       
    2. Motivation of underlying issues
       
    3. Defense of ideas and arguments before an audience
       
    4. Thinking on one's feet; fielding questions
       
  4. Research Skills:
     
    1. Searching the literature, and organizing information from disparate sources
       
    2. Reading and understanding definitions and statements of results
       
    3. Absorbing arguments and applying them in practice

    Evaluation of these skills is judged from class presentations, defense of the senior thesis, etc.
     

  5. Production Skills:
     
    1. Computer literacy

    This is judged by having the student take a course involving the use of a reasonably sophisticated computer language, some senior theses, etc.
     

  6. Problem Solving Skills:
     
    1. The ability to solve problems involving undergraduate mathematics, including the ability to formulate and present a mathematical proof.

  7. This is judged in classes where the principal skill that is tested is constructive problem solving, and practiced in the various teams that are fielded by the department for mathematical competitions.

Learning Objectives for Ph.D. Program in Mathematics

  1. Theoretical Understanding:
    • Working understanding of basic insights and methods in a broad variety of mathematical areas
    • Clear understanding of key hypotheses and conclusions
    • Synthesis of formal theory into a comprehensive picture of mathematical phenomena
    • Application of general theory to specific examples
    • Sharpening of intuition through appropriate counterexamples

    As a first step, these learning outcomes are judged in the context of courses. They are further judged in qualifying exams in three areas, each an hour-long oral examination in front of a committee of two faculty.

  2. Written Communication:
    • Clear and precise formulation of definitions and theorems
    • Efficient and coherent presentation of arguments in the form of proofs and/or data
    • Effective motivation of formal theory
    • Organization of individual results into a coherent theory

    Again, these are judged first in course work. Final evaluation is carried out by the dissertation committee, consisting of three faculty from the department and one outside member, in judging the dissertation.

  3. Oral Communication:
    • Explanation of key ideas and general strategies
    • Motivation of underlying issues
    • Clear oral presentation of arguments
    • Effective use of visual media (e.g., blackboard, computer screen)
    • Defense of ideas and arguments before an audience
    • Thinking on one's feet; fielding questions

    Some of this is judged from oral presentations in courses, further evaluation occurs during qualifying examinations and subsequently in the candidacy examination (a one-hour lecture by the student prior to embarking on dissertation research, on the general proposed area of study), and then in the oral disseration defense, consisting of a public lecture followed by an oral examination by the thesis committee.

  4. Research Skills (using resources):
    • Searching the literature
    • Reading and understanding definitions and statements of results
    • Organizing information from disparate sources
    • Absorbing arguments and applying them in practice
    • Reading mathematical exposition in a foreign language

    Evaluation of these skills is judged from the presentation at the candidacy examination and the dissertation defense. Foreign language skill is evaluated in a language exam consisting of translating a page or two of mathematics in French, German or Russian, administered within the mathematics department.

  5. Production Skills:
    • Making the transition from consumer of mathematics to producer of mathematics
    • Developing intuition about a new problem or situation
    • Using intuition to create models or formal arguments
    • Understanding the role of examples
    • Organizing arguments into a coherent presentation
    • Computer literacy

    This is above all else the role of the dissertation, and is judged by the dissertation committee on the basis of the written dissertation and the presentation at the dissertation defense.

  6. Teaching:
    • Classroom and blackboard techniques
    • Organization and presentation of mathematics at a level appropriate to the audience
    • Testing students

    Every student is required to teach at least one course in the department before being awarded the Ph.D. degree.

Learning Objectives for Masters' Program in Mathematics

  1. Theoretical Understanding:
    • Working understanding of basic insights and methods in a broad variety of mathematical areas
    • Clear understanding of key hypotheses and conclusions
    • Synthesis of formal theory into a comprehensive picture of phenomena
    • Application of general theory to specific examples
    • Sharpening of intuition through appropriate counterexamples

    As a first step, these learning outcomes are judged in the context of courses. Students in the Masters' program are required to take nine courses, subject to a distribution requirement to insure exposure to a broad variety of fields.

  2. Written Communication:
    • Clear and precise formulation of definitions and theorems
    • Efficient and coherent presentation of arguments in the form of proofs and/or data
    • Organization of individual results into a coherent theory

    Again, these are judged first in course work.

  3. Oral Communication:
    • Explanation of key ideas and general strategies
    • Motivation of underlying issues
    • Clear oral presentation of arguments
    • Effective use of visual media (e.g., blackboard, computer screen)
    • Defense of ideas and arguments before an audience
    • Thinking on one's feet; fielding questions

    Some of this is judged from oral presentations in courses. After completing course requirements, students have two options for completing the Masters' degree requirements. One is to pass three oral examinations in different areas of mathematics; the other is to write and defend a Masters' thesis, usually a survey of a research area or an inā€depth study of a selected research publication. The first option is generally taken by students continuing to doctoral studies. The second option involves an oral defense of the thesis in a public lecture, followed by questions from a committee of two faculty members.

  4. Research Skills (using resources):
    • Searching the literature
    • Reading and understanding definitions and statements of
    • Organizing information from disparate sources

    For students following the standard Masters' thesis path to the degree, this is the main substance of writing the thesis. The thesis is read and judged by a committee of three faculty members before being approved for submission.

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