Committees
Arts and Sciences Learning Objectives
Mathematics
 Basic Understanding of Higher Mathematics:
 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
 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.
 Clear understanding of the role of hypotheses, conclusions,
counterexamples and other forms of mathematical evidence in the
development and formulation of mathematical ideas
 The ability to define and use concepts by their properties alone,
rather than by explicit construction or calculation
 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.
 Written Communication:
 Ability to state and understand definitions and theorems with all
requisite precision
 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.
 Oral Communication:
 Clear explanation of key ideas and general strategies
 Motivation of underlying issues
 Defense of ideas and arguments before an audience
 Thinking on one's feet; fielding questions
 Research Skills:
 Searching the literature, and organizing information from disparate
sources
 Reading and understanding definitions and statements of results
 Absorbing arguments and applying them in practice
Evaluation of these skills is judged from class presentations, defense
of the senior thesis, etc.
 Production Skills:
 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.
 Problem Solving Skills:
 The ability to solve problems involving undergraduate mathematics,
including the ability to formulate and present a mathematical proof.
 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
 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 hourlong oral
examination in front of a committee of two faculty.
 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.
 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 onehour 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.
 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.
 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.
 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
 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.
 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.
 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 inādepth 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.
 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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