## Courses ofInstruction

### Mathematics

**Chair of the Department of Mathematics and Computer Science**

Yan Hao, Associate Professor, Chair

**Mathematics Faculty **

Jocelyn Bell, Associate Professor

Jennifer Biermann, Associate Professor

Jonathan Forde, Professor

Yan Hao, Associate Professor

Erika L. C. King, Associate Professor

Joseph Rusinko, Associate Professor

**Computer Science Faculty **

Stina Bridgeman, Associate Professor

Chris Fietkiewicz, Assistant Professor

Hanqing Hu, Assistant Professor

**Emeritus Faculty**David Belding

Irving Bentson

Carol Critchlow

Kevin Mitchell

Ann Oaks

John Vaughn

Mathematics has always been one of the core subjects of a liberal arts education because it promotes rigorous thinking and problem-solving ability. Many students who major in mathematics go on to graduate school or to work in related professions. For other students, mathematics is popular as a second major or as a minor in combination with another major from any of the Colleges’ academic divisions.

To meet the challenges, opportunities, and responsibilities encountered after graduation, the Department of Mathematics and Computer Science offers the opportunity to explore both theoretical and applied mathematics, and mathematics majors are encouraged to obtain a broad but firm foundation in the discipline. Majors acquire skill in the use of mathematical methods for dealing with problems from a variety of disciplines and complement these tools with some training in computer science.

Students are supported in this challenging course of study by strong student-faculty relationships, extensive office hours and an open departmental culture. The knowledge and abilities that Mathematics majors and minors learn in individual classes are woven together in an integrative capstone experience, as well as in the opportunities to conduct research in collaboration with faculty members in their areas of expertise, such as mathematical biology, graph theory, topology, and abstract algebra.

**Mission Statement**

As the logic and language of mathematics are integral to disciplines ranging from science and technology to art and philosophy, the skills of mathematics are essential to a liberal arts education and a variety of future careers.* *The mission of the mathematics program at HWS is to help students develop the habits of thought necessary to understand, analyze, and communicate complex ideas, and to instill in our students the willingness and confidence to explore and risk failure to make new discoveries.

**Offerings**

The Department of Mathematics and Computer Science offers two majors in mathematics (B.A. and B.S.), and a minor in mathematics. In addition to the specific courses listed below, other courses may be approved by the department for credit toward the major. To be counted toward the major or minor, all courses must be passed with a grade of C- or better; credit/no credit courses cannot be taken toward the major or minor.

**Mathematics Major (B.A.)**

*disciplinary, 11 courses**Learning Objectives:*

- Construct valid logical arguments and analyze the reasoning of others.
- Implement a variety of mathematical structures to model and analyze complex problems.
- Communicate mathematical ideas clearly, in verbal and visual form, by using appropriate mathematical terminology and notation.
- Develop the ability to independently read and understand written mathematics.

*Requirements:*

MATH 135, MATH 204, and MATH 232; CPSC 124; either MATH 331 or MATH 375; MATH 471; two additional mathematics courses at the 200-level or above; one additional mathematics courses at the 300-level or above; and two additional courses chosen from mathematics (MATH 131 and above) and computer science (CPSC 220 and above). Completion of an honors project and presentation of the thesis at the Senior Symposium may be substituted for MATH 471.

**Mathematics Major (B.S.)**

*disciplinary, 15 courses**Learning Objectives:*

- Construct valid logical arguments and analyze the reasoning of others.
- Implement a variety of mathematical structures to model and analyze complex problems.
- Communicate mathematical ideas clearly, in verbal and visual form, by using appropriate mathematical terminology and notation.
- Develop the ability to independently read and understand written mathematics.

*Requirements:*

MATH 135, MATH 204, MATH 232, MATH 331, and MATH 375; MATH 471; CPSC 124; three additional mathematics courses at the 200-level or above; one additional mathematics courses at the 300-level or above; one additional computer science course (CPSC 220 and above); and three additional courses in the Natural Science Division that count towards the major in their respective departments. Completion of an honors project and presentation of the thesis at the Senior Symposium may be substituted for MATH 471.

**Mathematics Minor**

*disciplinary, 5 courses**Requirements:*

Five mathematics courses at or above MATH 131, at least one of which is 300-level or above.

**Course Descriptions**

**MATH 100 Elementary Functions** Intended for students who plan to continue in the calculus sequence, this course involves the study of basic functions: polynomial, rational, exponential, logarithmic, and trigonometric. Topics include a review of the real number system, equations and inequalities, graphing techniques, and applications of functions. A problem-solving lab is an integral part of the course. Permission of instructor is required. This course does not count toward the major or minor in mathematics. (Offered annually)

**MATH 110 Discovering in Mathematics** A study of selected topics dealing with the nature of mathematics, this course has an emphasis on the origins of mathematics, and a focus on mathematics as a creative endeavor. This course does not count toward the major or minor in mathematics. (Offered each semester)

**MATH 114 Mathematics for Informed Citizenship** This course explores the uses and abuses of numbers in a wide variety of areas. The modern world is built of numbers. In science, medicine, business, politics, and even culture, numbers are used to bolster claims and debunk conventional wisdom. A deeper understanding of the mathematics behind these arguments can help us determine what to trust and when to doubt, teach us how to weigh the risks versus rewards, and allow us to come to grapple with the vast scale of the universe and the national debt. Mathematical topics will include randomness, basic statistics, linear regression, inference and nonlinearity. An emphasis is placed on critical engagement with numerical evidence and mathematical thinking as deployed in the culture at large. This course has a significant writing component.

**MATH 115 Foundations of School Math** Students will study the mathematical foundations of elementary school mathematics. This course will develop a student's abilities to reason mathematically, to solve mathematical problems, and to communicate mathematical ideas effectively. Primary attention will be devoted to mathematical reasoning in areas drawn from number systems and algebraic structures, number theory, algebra and geometry, probability and statistics, and discrete mathematics. Students will gain a deeper understanding of the nature and structure of mathematics, and more specifically of how elementary school mathematics is embedded within the broader discipline of mathematics. Prerequisite: must be in the Teacher Education Program pursuing certification to teach in an elementary school setting. (Offered alternate years)

**MATH 130 Calculus I** This course offers a standard introduction to the concepts and techniques of the differential calculus of functions of one variable. A problem-solving lab is an integral part of the course. This course does not count towards the major in mathematics. Prerequisite: Satisfactory performance on the department's placement exam, or MATH 100. (Offered each semester)

**MATH 131 Calculus II** This course is a continuation of the topics covered in MATH 130 with an emphasis on integral calculus, sequences, and series. A problem-solving lab is an integral part of the course. Prerequisite: MATH 130 or permission of the instructor. (Offered each semester)

**MATH 135 First Steps Into Advanced Math** This course emphasizes the process of mathematical reasoning, discovery, and argument. It aims to acquaint students with the nature of mathematics as a creative endeavor, demonstrates the methods and structure of mathematical proof, and focuses on the development of problem-solving skills. Specific topics covered vary from year to year. MATH 135 is required for the major and minor in mathematics. Prerequisite: MATH 131, MATH 232 or permission of the instructor. (Offered annually)

**MATH 204 Linear Algebra** This course is an introduction to the concepts and methods of linear algebra. Among the most important topics are general vector spaces and their subspaces, linear independence, spanning and basis sets, solution space for systems of linear equations, and linear transformations and their matrix representations. It is designed to develop an appreciation for the process of mathematical abstraction and the creation of a mathematical theory. Prerequisites: MATH 131 or MATH 232, and MATH 135 strongly suggested, or permission of the instructor. Required for the major in mathematics. (Offered annually)

**MATH 214 Applications of Linear Algebra** A continuation of linear algebra with an emphasis on applications. Among the important topics are eigenvalues and eigenvectors, diagonalization, and linear programming theory. The course explores how the concepts of linear algebra are applied in various areas, such as, graph theory, game theory, differential equations, Markov chains, and least squares approximation. Prerequisite: MATH 204. (Offered alternate years)

**MATH 232 Multivariable Calculus** A study of the concepts and techniques of the calculus of functions of several variables, this course is required for the major in mathematics. Prerequisite: MATH 131. (Offered annually)

**MATH 237 Differential Equations** This course is an introduction to the theory, solution techniques, and applications of ordinary differential equations. Models illustrating applications in the physical and social sciences are investigated. The mathematical theory of linear differential equations is explored in depth. Prerequisites: MATH 232 and 204, or permission of the instructor. MATH 204 may be taken concurrently. (Offered annually)

**MATH 278 Number Theory** This course couples reason and imagination to consider a number of theoretical problems, some solved and some unsolved. Topics include divisibility, primes, congruences, number theoretic functions, primitive roots, quadratic residues, and quadratic reciprocity, with additional topics selected from perfect numbers, Fermat's Theorem, sums of squares, and Fibonacci numbers. Prerequisites: MATH 131 and MATH 204, or permission of the instructor. (Offered alternate years)

**MATH 313 Graph Theory** A graph is an ordered pair (V,E) where V is a set of elements called vertices and E is a set of unordered pairs of elements of V called edges. This simple definition can be used to model many ideas and applications. While many of the earliest records of graph theory relate to the studies of strategies of games such as chess, mathematicians realized that graph theory is powerful well beyond the realm of recreational activity. In this class, we will begin by exploring the basic structures of graphs including connectivity, subgraphs, isomorphisms and trees. Then we will investigate some of the major results in areas of graph theory such as traversability, coloring and planarity. Course projects may also research other areas such as independence, domination and matching. Prerequisites: MATH 135 and MATH 204. (Offered every third year)

**MATH 331 Foundations Analysis I** This course offers a careful treatment of the definitions and major theorems regarding limits, continuity, differentiability, integrability, sequences, and series for functions of a single variable. Prerequisites: MATH 135 and MATH 204. (Offered annually)

**MATH 350 Probability** This is an introductory course in probability with an emphasis on the development of the student's ability to solve problems and build models. Topics include discrete and continuous probability, random variables, density functions, distributions, the Law of Large Numbers, and the Central Limit Theorem. Prerequisite: MATH 232 or permission of instructor. (Offered alternate years)

**MATH 351 Mathematical Statistics** This is a course in the basic mathematical theory of statistics. It includes the theory of estimation, hypothesis testing, and linear models, and, if time permits, a brief introduction to one or more further topics in statistics (e.g., nonparametric statistics, decision theory, experimental design). In conjunction with an investigation of the mathematical theory, attention is paid to the intuitive understanding of the use and limitations of statistical procedures in applied problems. Students are encouraged to investigate a topic of their own choosing in statistics. Prerequisite: MATH 350. (Offered alternate years)

**MATH 353 Mathematical Models** This course investigates a variety of mathematical models from economics, biology, and the social sciences. In the course of studying these models, such mathematical topics as difference equations, eigenvalues, dynamic systems, and stability are developed. This course emphasizes the involvement of students through the construction and investigation of models on their own. Prerequisites: MATH 204 and MATH 237, or permission of the instructor. (Offered alternate years)

**MATH 375 Abstract Algebra I** This course studies abstract algebraic systems such as groups, examples of which are abundant throughout mathematics. It attempts to understand the process of mathematical abstraction, the formulation of algebraic axiom systems, and the development of an abstract theory from these axiom systems. An important objective of the course is mastery of the reasoning characteristic of abstract mathematics. Prerequisites: MATH 135 and MATH 204, or permission of the instructor. (Offered annually)

**MATH 436 Topology** This course covers the fundamentals of point set topology, starting from axioms that define a topological space. Topics typically include: continuity, connectedness, compactness, metric spaces, product spaces, and separation axioms. Prerequisite: MATH 331 or permission of the instructor. (Offered every third year)

**MATH 471 Mathematics Capstone** While the subject matter varies, the writing enriched capstone seminar addresses an advanced topic in mathematics. The development of the topic draws on students' previous course work and helps consolidate their earlier learning. Students are active participants, presenting material to one another in both oral and written form, and conducting individual research on related questions or analyzing a specific mathematical concept from multiple perspectives. This course may be repeated and count toward the major more than once. Prerequisite: Complete MATH 331 or MATH 375 with a C- or above, and have either senior status or permission of the instructor. (Offered annually)

**MATH 495****/496**** Honors**

**MATH 499 Mathematics Internship**

Courses offered occasionally or as demand warrants:

MATH 332 Foundations of Analysis II

MATH 360 Foundations of Geometry

MATH 371 Topics in Mathematics

MATH 376 Abstract Algebra II

MATH 446 Real Analysis

MATH 448 Complex Analysis