Carol Schumacher has been a member of the Kenyon faculty since the fall of 1988. She received her BA from Hendrix College and her Ph.D. in mathematics from the University of Texas at Austin. She has served three terms as chair of the department of mathematics and statistics and is the recipient of Kenyon's Trustee Teaching Award. She is currently serving as chair of the faculty. She is the author of *Closer and Closer: Introducing Real Analysis* and *Chapter Zero: Fundamental Notions of Abstract Mathematics*, 2E.

Schumacher is active in the Mathematical Association of America and recently co-chaired the committee that wrote the *2015 CUPM Curriculum Guide for Majors in the Mathematical Sciences*. In recent years she has been invited to address the Project NExT fellows at their summer workshop and has also given invited addresses at the Legacy of R.L. Moore conference.

Schumacher regularly teaches calculus, foundations and real analysis I and II. In recent years she has also taught differential…

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Carol Schumacher has been a member of the Kenyon faculty since the fall of 1988. She received her BA from Hendrix College and her Ph.D. in mathematics from the University of Texas at Austin. She has served three terms as chair of the department of mathematics and statistics and is the recipient of Kenyon's Trustee Teaching Award. She is currently serving as chair of the faculty. She is the author of *Closer and Closer: Introducing Real Analysis* and *Chapter Zero: Fundamental Notions of Abstract Mathematics*, 2E.

Schumacher is active in the Mathematical Association of America and recently co-chaired the committee that wrote the *2015 CUPM Curriculum Guide for Majors in the Mathematical Sciences*. In recent years she has been invited to address the Project NExT fellows at their summer workshop and has also given invited addresses at the Legacy of R.L. Moore conference.

Schumacher regularly teaches calculus, foundations and real analysis I and II. In recent years she has also taught differential equations, linear algebra II, geometry and a non-majors course called "The Art of Mathematics."

### Education

1989 — Doctor of Philosophy from Univ Texas Austin*

1982 — Bachelor of Arts from Hendrix College

### Courses Recently Taught

MATH 111

## Calculus I

#### MATH 111

The first in a three-semester calculus sequence, this course covers the basic ideas of differential calculus. Differential calculus is concerned primarily with the fundamental problem of determining instantaneous rates of change. In this course we will study instantaneous rates of change from both a qualitative geometric and a quantitative analytic perspective. We will cover in detail the underlying theory, techniques and applications of the derivative. The problem of anti-differentiation, identifying quantities given their rates of change, also will be introduced. The course will conclude by relating the process of anti-differentiation to the problem of finding the area beneath curves, thus providing an intuitive link between differential calculus and integral calculus. Those who have had a year of high school calculus but do not have advanced placement credit for MATH 111 should take the calculus placement exam to determine whether they are ready for MATH 112. Students who have .5 unit of credit for calculus may not receive credit for MATH 111. Prerequisite: solid grounding in algebra, trigonometry and elementary functions.

MATH 112

## Calculus II

#### MATH 112

The second in a three-semester calculus sequence, this course has two primary foci. The first is integration, including techniques of integration, numerical methods and applications of integration. This study leads into the analysis of differential equations by separation of variables, Euler's method and slope fields. The second focus is the notion of convergence, as manifested in improper integrals, sequences and series, particularly Taylor series. Prerequisite: MATH 111 or AP score of 4 or 5 on Calculus AB exam or an AB subscore of 4 or 5 on the Calculus BC exam or permission of instructor. Offered every semester.

MATH 191

## ST: Art of Mathematics

#### MATH 191

MATH 213

## Calculus III

#### MATH 213

The third in a three-semester calculus sequence, this course examines differentiation and integration in three dimensions. Topics of study include functions of more than one variable, vectors and vector algebra, partial derivatives, optimization and multiple integrals. Some of the following topics from vector calculus also will be covered as time permits: vector fields, line integrals, flux integrals, curl and divergence. Prerequisite: MATH 112 or a score of 4 or 5 on the BC Calculus AP exam or permission of instructor.

MATH 222

## Foundations

#### MATH 222

This course introduces students to mathematical reasoning and rigor in the context of set-theoretic questions. The course will cover basic logic and set theory, relations--including orderings, functions and equivalence relations--and the fundamental aspects of cardinality. The course will emphasize helping students read, write and understand mathematical reasoning. Students will be actively engaged in creative work in mathematics. Students interested in majoring in mathematics should take this course no later than the spring semester of their sophomore year. Advanced first-year students interested in mathematics are encouraged to consider taking this course in their first year. Students wanting to do so should contact a member of the mathematics faculty. Prerequisite: MATH 213 or permission of instructor. Offered every spring semester.

MATH 230

## Euclidean and Non-Euclidean Geometry

#### MATH 230

The Elements of Euclid, written over 2,000 ago, is a stunning achievement. The Elements and the non-Euclidean geometries discovered by Bolyai and Lobachevsky in the 19th century form the basis of modern geometry. From this start, our view of what constitutes geometry has grown considerably. This is due in part to many new theorems that have been proved in Euclidean and non-Euclidean geometry but also to the many ways in which geometry and other branches of mathematics have come to influence one another over time. Geometric ideas have widespread use in analysis, linear algebra, differential equations, topology, graph theory and computer science, to name just a few areas. These fields, in turn, affect the way that geometers think about their subject. Students in MATH 230 will consider Euclidean geometry from an advanced standpoint, but also will have the opportunity to learn about non-Euclidean geometries. Prerequisite: MATH 222 or permission of instructor.

MATH 324

## Linear Algebra II

#### MATH 324

This course builds on the concepts that arise in MATH 224. Topics will vary and will likely include some of the following: abstract vector spaces, inner product spaces, linear mappings and canonical forms, linear models, linear codes, the singular value decomposition, wavelets. Prerequisite: MATH 224. Offered every other year.

MATH 333

## Differential Equations

#### MATH 333

Differential equations arise naturally to model dynamical systems such as often occur in physics, biology, chemistry and economics, and have given major impetus to other fields in mathematics, such as topology and the theory of chaos. This course covers basic analytic, numerical and qualitative methods for the solution and understanding of ordinary differential equations. Computer-based technology will be used. Prerequisite: MATH 224 or PHYS 245 or permission of instructor. Offered every other year.

MATH 341

## Real Analysis I

#### MATH 341

This course is a first introduction to real analysis. "Real" refers to the real numbers. Much of our work will revolve around the real number system. We will start by carefully considering the axioms that describe it. "Analysis" is the branch of mathematics that deals with limiting processes. Thus the concept of distance also will be a major theme of the course. In the context of a general metric space (a space in which we can measure distances), we will consider open and closed sets, limits of sequences, limits of functions, continuity, completeness compactness, and connectedness. Other topics may be included, if time permits. Junior standing is recommended. Prerequisite: MATH 213 and 222. Offered every other fall.

MATH 441

## Real Analysis II

#### MATH 441

This course follows Real Analysis I. Topics will include a study differentiation and (Riemann) integration of functions of one variable, sequences and series of functions, power series and their properties, iteration and fixed points. Other topics may be included as time permits. For example: a discussion of Newton's method or other numerical techniques; differentiation and integration of functions of several variables; spaces of continuous functions; the implicit function theorem; and everywhere continuous, nowhere differentiable functions. Prerequisite: MATH 341. Offered every other spring.

MATH 493

## Individual Study

#### MATH 493

Individual study is a privilege reserved for students who want to pursue a course of reading or complete a research project on a topic not regularly offered in the curriculum. It is intended to supplement, not take the place of, coursework. Individual study cannot normally be used to fulfill requirements for the major. Typically, individual study will earn .5 unit or .25 unit of credit.\nTo qualify, a student must identify a member of the Mathematics Department willing to direct the project. The professor, in consultation with the student, will create a tentative syllabus (including a list of readings and/or problems, goals and tasks) and describe in some detail the methods of assessment (e.g., problem sets to be submitted for evaluation biweekly; a 20-page research paper submitted at the course's end, with rough drafts due at given intervals, and so on). Individual studies also require the approval of the department chair. The department expects the student to meet regularly with his or her instructor for at least one hour per week.\nStudents must begin discussion of their proposed individual study well in advance, preferably the semester before the course is to take place. Prerequisite: permission of instructor and department chair.

MATH 498

## Senior Honors

#### MATH 498

This course will consist largely of an independent project in which students read several sources to learn about a mathematical topic that complements material studied in other courses, usually an already completed depth sequence. This study will culminate in an expository paper and a public or semi-public presentation before an audience consisting of at least several members of the mathematics faculty as well as an outside examiner. Prerequisite: At least one "depth sequence" completed and permission of the department chair.