### Mathematics

## Mathematics

The mathematics major is designed to prepare students for graduate work in mathematics, for teaching mathematics in secondary school, and for a variety of careers in mathematics-related fields in government, business, or industry.

All students are expected to learn methods and techniques of problem solving and to develop facility in the mathematical mode of thinking. They are expected to become acquainted with the major areas of current interest in mathematics, with the great achievements of the past, and with the fundamental problems of number, space, and infinity.

The mathematics minor is designed for all interested students, but it is particularly suited for students majoring in any of the sciences (including computer science) or economics, as well as students interested in pursuing the MAT Program after graduation. The minor will introduce the student to calculus, linear algebra, and the foundations of mathematics.

**Math Learning Outcomes
**

Upon graduation with a major in mathematics, students will

- demonstrate mastery of the core concepts in algebra and analysis.
- understand the convincing nature of mathematical proof.
- be able to write clear, coherent, logical proofs, including proofs by contradiction and mathematical induction.
- give clear and organized written and verbal explanations of mathematical ideas.
- approach and solve mathematical problems independently.
- explain and critique mathematical reasoning through speaking and writing in a precise and articulate manner.
- be prepared for post-graduate studies in mathematics and mathematics-related fields.
- be able to apply critical thinking skills outside of a mathematical context.

#### DEGREE REQUIREMENTS FOR THE MAJOR

To earn a bachelor of arts degree with a major in mathematics, a student must satisfy the following minimum requirements:

- General College Requirements (see “Curriculum” section).
- Required Mathematics Courses (10 courses):
- MATH 151: Calculus I
- MATH 152: Calculus II
- MATH 255: Vector Calculus
- MATH 256: Linear Algebra
- MATH 281: Foundations of Mathematics
- MATH 312: Differential Equations
- MATH 321: Algebra I
- MATH 322: Algebra II
- MATH 351: Analysis I
- MATH 352: Analysis II

- All students must select one of the following three options as the capstone experience of their education:
- St. Mary’s Project in Mathematics (eight credits)*
- One senior-level mathematics** course and a Senior Project in Mathematics: MATH 495 (four credits)
- Two senior-level mathematics** courses
- * The requirement may also be satisfied by completing a St. Mary’s Project in another area. If a student wishes to do a project in another area, the approval of the department must be secured in advance.
- ** Senior-level mathematics courses carry the designation “MATH 4xx”.

- Students must earn a grade of C- or better in all courses listed in items 2. and 3. above and maintain an overall GPA of 2.0 or better in these required courses.

Students who are interested in graduate studies in theoretical mathematics should add at least two senior-level courses in theoretical mathematics to their schedules. It is also recommended that all students majoring in mathematics develop a proficiency in programming during their studies. This may be obtained by taking the course COSC 120. Acquiring skills with a mathematics software package such as Maple or Mathematica is desirable.

#### DEGREE REQUIREMENTS FOR THE MINOR

- Required Courses (5 courses)
- MATH 151: Calculus I
- MATH 152: Calculus II
- MATH 255: Vector Calculus
- MATH 256: Linear Algebra
- MATH 281: Foundations of Mathematics

- Students must complete the required 5 courses, earn a grade of C- or better in each course taken to fulfill the minor, and the cumulative grade-point average of courses used to satisfy the minor must be at least 2.0.

#### TEACHER CERTIFICATION IN MATHEMATICS

A Master of Arts in Teaching program is available at St. Mary’s College of Maryland after completion of the baccalaureate degree. Students who are interested in becoming teachers should contact the chair of the Department of Educational Studies or an education adviser in their major field of study for suggested coursework in educational studies, and their specific major. These consultations should take place during the first semester of the sophomore year. It is recommended that such students take statistics (MATH 221).

#### FACULTY

Casey Douglas, Sanford Ganzell (department chair), Susan Goldstine, Alan Jamieson, Lindsay Jamieson, Emek Köse, David Kung, Alexander Meadows, Simon Read, Ivan Sterling

#### MATHEMATICS COURSES (MATH)

##### MATH 111. Precalculus (4F)

Functions and graphs. Transformations, compositions, inverses, and combinations of functions. Exponentials and logarithms. Trigonometric functions and their inverses. Polynomial and Rational functions. This course is designed to prepare students for further studies in mathematics and the sciences; in particular, for an in-depth study of calculus. The course does not satisfy the Core Curriculum requirement in Mathematics.

##### MATH 131. Survey of Mathematics (4E)

This course will include study of both theoretical and applied aspects of mathematics. Topics will vary from section to section and may include the following: number systems, mathematical modeling, Euclidean and non-Euclidean geometry, projective geometry, group theory, graph theory, mathematical logic, sets and infinity, topology, the concepts of calculus, and the history of mathematics. The course is recommended for students of the liberal arts who wish to obtain a general view of contemporary mathematics. MATH 131 satisfies the Core Curriculum requirement in Mathematics.

##### MATH 151, 152. Calculus I, II (4E)

The differential and integral calculus of functions of one variable: limits and continuity, the derivative, curve sketching, applications of the derivative, indefinite integrals and differential equations, definite integrals and the fundamental theorem, integration methods, applications of the integral, the convergence of sequences and series, power series, Taylor’s theorem and analytic functions, polar coordinates and parametric equations. MATH 151 satisfies the Core Curriculum requirement in Mathematics. Prerequisite: Familiarity with high school trigonometry is expected. *MATH 151 is a prerequisite for MATH 152*.

##### MATH 161. Math for Teachers I (4F)

The foundations of arithmetical reasoning including general problem-solving skills; sets and operations; the use of manipulatives to model arithmetic; arithmetic in other bases; standard, alternative and invented algorithms; fractions and proportional reasoning; basic number theory. Student-centered pedagogies will be modeled and discussed.

##### MATH 162. Math for Teachers II (4S)

Geometry (including constructions and proofs), tessellations and tilings of the plane, polyhedra, measurement, basic probability and statistics. Student-centered pedagogies will be modeled and discussed. *(MATH 161 is not a prerequisite for this course.)*

##### MATH 181. Emerging Scholars Program (1E)

Supplemental problem-solving workshop for calculus (MATH 151, 152) students in the Emerging Scholars Program. Enrollment by permission of instructor. May be repeated for credit.

##### MATH 200. Discrete Mathematics (4S)

Set theory, elementary logic, sequences and mathematical induction, functions and relations, counting techniques, matrix theory, graphs and trees. MATH 200 satisfies the Core Curriculum requirement in Mathematics. MATH 200 assumes more mathematical preparation than MATH 131.

##### MATH 201. Psychological Statistics (4E)

The analysis of experimental data, including data from both laboratory and natural settings. Parametric analysis through two-way analysis of variance and nonparametric statistics. This course is cross-listed as PSYC 201.

##### MATH 221. Introduction to Statistics (4S)

Introduction to the concepts and methods of statistics, including descriptive statistics (measures of central tendency, dispersion and shape, as well as data organization), probability theory, probability distributions, confidence intervals, hypothesis testing, types of error, correlation and regression, and analysis of variance. Computer software which provides statistical capabilities is used to apply the concepts covered to realistic data sets from the biological and/or social sciences.

##### MATH 255. Vector Calculus (4E)

The differential and integral calculus of scalar and vector-valued functions in one and several variables. *Prerequisite: MATH 152*.

##### MATH 256. Linear Algebra (4E)

Vectors in the plane and in space, vector spaces, linear transformations, matrices and determinants, systems of linear equations, characteristic values and vectors, inner product spaces and orthogonality. *Prerequisites: MATH 255; or MATH 152 and permission of the instructor*.

##### MATH 281. Foundations of Mathematics (4E)

Mathematical logic; proof techniques and proof writing; set theory (including Cantor’s theory of the infinite); relations and functions; theoretical foundations of number systems including the natural numbers, integers, rationals, reals, and complex numbers. *Prerequisite: MATH 152*.

##### MATH 293. Field Studies in Mathematics Education (1-4,E)

This course provides experience in a school setting for students seeking teacher certification and for others interested in learning more about the nature of the school, the nature of children, the nature of mathematics education, and about teaching/learning processes within school settings. Students may take at most two of the following courses for a total of up to four credit-hours: ILCC 293, ILCS 293, IlLCF 293, ILCG 293, EDUC 293, MATH 293. *Prerequisite: MATH 256; or MATH 152 and permission of the instructor*.

##### MATH 312. Differential Equations (4S)

Solution methods for first-order differential equations; existence and uniqueness theorems; solutions of second-order linear differential equations; power series methods; Laplace transformations; applications. *Prerequisite: MATH 256; or MATH 152 and permission of instructor*.

##### MATH 321, 322. Algebra I, II (4F, 4S)

A study of abstract algebraical systems and the mappings that preserve their structure: groups, rings, fields, and vector spaces; homomorphisms and isomorphisms. Credit is allowed for MATH 321 without registration for MATH 322. *Prerequisite for MATH 321: MATH 281. Prerequisite for MATH 322: MATH 321*.

##### MATH 351, 352. Analysis I, II (4F, 4S)

The real number system, metric spaces, compactness and connectedness, convergence and summability, limits and continuity, measure and integration. Credit is allowed for MATH 351 without registration for MATH 352. *Prerequisite for MATH 351: MATH 281. Prerequisite for MATH 352: MATH 351*.

##### MATH 391. Putnam Seminar (1F)

Preparation for the Putnam Exam, an annual math competition held in December. Topics include general problem-solving strategies and previous exam problems which typically integrate knowledge from different areas of mathematics. May be repeated for credit.

##### MATH 392. General Problem Solving (1S)

Problem-solving methods in higher mathematics, with an emphasis on how different strategies are used across different areas of math. May be repeated for credit.

##### MATH 411. Partial Differential Equations (4AF)

Solution methods for basic partial differential equations, with a detailed study of the heat and wave equations. Topics include Fourier series solutions, integral transform methods, numerical methods for elliptic, parabolic, and hyperbolic equations. *Prerequisite: MATH 312; or MATH 152 and permission of the instructor*.

##### MATH 421. Combinatorics (4AF)

Topics may include the following: permutations, combinations, partitions, counting principles, generating functions, partially ordered sets, designs and codes, graphs and trees, planarity, networks, Hamiltonian cycles, Eulerian tours, combinatorial designs, games of complete information, asymptotic methods, combinatorial existence theorems, and Ramsey theory. *Prerequisites: MATH 281 or permission of the instructor*.

##### MATH 451. Complex Analysis (4AS)

Complex numbers and functions, differentiability, integration, Cauchy theory, power series, and analytic continuation. *Prerequisite: MATH 281*.

##### MATH 461. Topology (4AS)

Topological spaces, separation axioms, compactness and connectedness, continuity, metrizability, an introduction to algebraic topology. *Prerequisite: MATH 281*.

##### MATH 481. Topics in Applied Mathematics (4S)

An in-depth study of an important field in applied mathematics. A detailed course description will be available in the online “Schedule of Classes” before registration. May be repeated for credit if the topic is not repetitive. *Prerequisite: Consent of the instructor*.

##### MATH 482. Topics in Theoretical Mathematics (4F)

A rigorous study of an important field of theoretical mathematics. A detailed course description will be available in the online “Schedule of Classes” before registration. May be repeated for credit if the topic is not repetitive. *Prerequisite: Consent of the instructor*.

##### MATH 493, 494. St. Mary’s Project in Mathematics (1 - 8E)

The St. Mary’s Project in mathematics is one of the culminating experiences in the mathematics major. It usually is completed in the two semesters of the student’s senior year. The project draws on and extends knowledge, analytical skills, and creative thought developed through previous work in an area or areas of mathematics or mathematics education. Usually, it is initiated by the student; however, the student may peruse lists of project ideas developed by the mathematics faculty or draw on other sources. The student shall select a faculty mentor and a topic with the advice of the department chair. A project proposal must be submitted, identifying the area to be explored and the methods of inquiry to be used. While working on the project, the student should learn a significant amount of mathematics beyond that learned in previous course work. Upon completion, the project shall be presented to the public in a way agreed upon by the student, the mentor, and the department chair. *Prerequisite: Consent of mentor and department chair*.

##### MATH 495. Senior Project in Mathematics (4E)

Together with a 400-level mathematics course, the Senior Project in mathematics can be a component of the capstone experience in the major. Normally, a student will complete the project during the senior year. It draws on previous course work and study and should expand the student’s horizon in mathematics and develop his or her thinking skills. The idea should come from the student, but lists of project ideas developed by the mathematics faculty are available, and other sources may be used. The student shall select a faculty mentor and a topic with the advice of the department chair. A project proposal must be submitted, identifying the area to be explored and the methods of inquiry to be used. While working on the project, the student should learn a significant amount of mathematics beyond that learned in previous course work. Upon completion, the project shall be presented to the public in a way agreed upon by the student, the mentor, and the department chair. May be repeated for credit if the topic is not repetitive. *Prerequisite: Consent of mentor and department chair*.

##### MATH 398, 498. Off-Campus Internship (4-16E)

A variety of off-campus learning opportunities can be arranged through the Career Development Center. The off-campus internship is an individually designed experience that allows the student to explore the relationship between learning in the classroom and the practical application of knowledge in everyday work situations. *Prerequisites: Admission to the Internship Program and approval of the academic adviser and department chair. (See “Internships” under “Academic Policies” section.) Credit/no credit grading*.

##### MATH 199, 299, 399, 499. Independent Study (1-4E)

This course consists of an independent creative or research project designed by the student and supervised by a mathematics faculty member. The nature of the project, the schedule for accomplishment, and the means of evaluation must be formalized in a learning contract prior to registration. May be repeated for credit if the topic is not repetitive. (See “Independent Study” under “Academic Policies” section.)