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.

**Supporting outcomes:**

(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

adept at

(a) developing conjectures, constructing correct proofs and refuting faulty ones.

**Supporting outcomes:**

(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.

**Supporting outcomes:**

(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.

**Supporting outcomes:**

(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

adequately.

**Supporting outcomes: **

** **(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

degrees.

(4) Students should be able to communicate mathematics clearly to others.

**Supporting outcomes:**

(a) Students should demonstrate the ability to write clear, well-explained solutions to

homework problems.

(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

professors.

(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.