Learning Goals and Outcomes for the Math Major
Overall goals for this major:
(1) Our students should develop the capacity for thoughtful engagement with quantitative,
geometric, and logical reasoning.
(a) Students should demonstrate proficiency in expressing well-posed calculus “word prob-
lems” mathematically, and solving them.
(b) Students should be able to create and interpret graphs in calculus.
(c) Further along in the major, students should be able to use mathematical language
logically and precisely, and translate expressions involving quantifiers and logic from
English into mathematical expression.
(2) As they progress through our curriculum, students are expected to become increasingly
(a) developing conjectures, constructing correct proofs and refuting faulty ones.
(i) Students should demonstrate proficency at detecting and describing simple pat-
terns in collections of related mathematical examples. Examples: Find a formula that describes a sequence. Articulate the steps used to solve an integral, and use
the same method to solve a similar one.
(ii) Students should be able to construct basic proofs of propositions by unpacking
definitions of terms, making simple logical connections, and drawing conclusions.
For example: the proof that the square of every even number is also even.
(iii) Students should be able to execute more advanced techniques of proof such as
dividing into exhaustive cases, proof by contradiction, and proof by induction,
and be able decide in which situations each is appropriate.
(iv) Students should be able to construct more advanced and intricate proofs that
rely on previously proved theorems and techniques such as those described above. (v) Students should demonstrate the ability to examine a flawed mathematical ar-
gument and clearly explain why the argument is flawed.
(b) using mathematical or statistical models to understand complex real-world phenomena.
(i) Students should show familiarity with several basic models and be able to deter-
mine when they are appropriate.
(ii) Students should be able to work with real data and use them to perform estima-
tions, make predictions, and come to decisions.
(iii) Students should know how to interpret the results of the model in the context of
the original phenomena and communicate their conclusions to nonexperts.
(c) working with axiomatically-defined structures.
(i) Students should demonstrate the ability to decide whether a near-example sat-
isfies the definition of a vector space, and to explain what needs to be verified to determine whether a subset of a vector space is a subspace.
(ii) Given an axiomatization of the integers or the real numbers, students should be
able to construct proofs of fundamental properties of these systems.
(iii) Further along in the major, students should demonstrate the ability to reason
axiomatically about groups, topological spaces, geometries, etc. and prove basic
theorems about these structures.
(3) Students who are bound for graduate school in the mathematical sciences should be prepared
(a) Students who apply to appropriately selective programs should gain admission.
(b) Alumni who participate in such programs should self-report that the level of prepara-
tion they received was appropriate, given F&M’s status as a liberal arts college.
(c) Alumni who enter graduate programs in the mathematical sciences should finish their
(4) Students should be able to communicate mathematics clearly to others.
(a) Students should demonstrate the ability to write clear, well-explained solutions to
(b) Students should demonstrate the ability to use appropriate tools (a chalkboard, a
computer screen, etc.) to communicate mathematics “in person” to their peers and
(c) Students should be able to use appropriate computer software (e.g. LATEX, or Word)
to prepare documents with mathematical or technical content.
(d) Students should show the ability to include an appropriate level of detail and back-
ground when communicating with different audiences.
Additionally, reflecting the math department’s service role for the rest of the College: Students
who take courses in our Department should gain an understanding of mathematical content that fits well into the overall academic curiculum at F&M.