# Undergraduate Study

The Department of Mathematics and Statistics offers the B.A. and B.S. degrees with a major in mathematics and the B.S. degree with a major in statistics. There is a program for a major in mathematics that qualifies a student for initial teaching licensure. A concentration in computational sciences is also available. Minors are offered in both mathematics and statistics. The Interdisciplinary Liberal Studies (IDLS) major is recommended for students interested in early or middle grades education and requires a collection of lower-division courses along with two upper-level concentrations. We also offer an algebra I add-on endorsement for IDLS majors.

The Department has four speciality areas, listed below. There are courses for the majors and minors that would be particularly relevant to each of these areas, and there are several faculty with active reseach programs in each area. If you are interested in learning more about the course offerings, career options, or potential research collaborators/advisors within these areas, please explore the links that follow.

Detailed information on the programs offered and the requirements for mathematics and statistics degrees can be found in the Undergraduate Course Catalog. Course information can be found in the Catalog's courses and descriptions.

## **Applied Mathematics**

### What is Applied Mathematics?

Applied mathematics uses mathematics to solve problems that come from the sciences, engineering, computer science, business, and industry. While the motivation is often to solve particular problems, the theory developed can be of wide-ranging interest.

Much classical applied mathematics involves developing mathematical models of a problem and solving them, often in the context of solving differential equations or optimizing some process. This dates back to the development of calculus, and continues to be important today. Here at JMU, our entire applied mathematics group uses differential equations in some way to solve problems in a wide range of disciplines.

Another important strand of applied mathematics is computational mathematics, the study of algorithms and techniques to approximate the solutions to problems that can’t be solved exactly. Many JMU faculty use computational techniques to aid them in their research.

### Jobs in Applied Mathematics

At one level, an applied mathematician is a problem solver in any area that requires mathematics. The website WeUseMath.org includes a long list of potential jobs for mathematicians. The Society for Industrial and Applied Mathematics (SIAM) includes a site on Careers in Applied Mathematics.

### What Do We Offer?

One of the main reasons for the calculus sequence is to represent situations where quantities vary continuously by differential equations, and solve them. As a result, the calculus sequence (MATH 235-7) followed by a course on differential equations (MATH 238 or MATH 336) is core to any applied mathematician. An introductory course on programming and computation for mathematics (MATH 248) also provides invaluable skills going forward.

At the intermediate level, there are a variety of courses with an applied bent. Methods of Applied Calculus (MATH 337) introduces a variety of techniques building on the calculus sequence for solving differential equations and modeling physical processes. Optimization (MATH 340) introduces a variety of techniques for finding the best possible solution, with applications in the sciences, economics and the social sciences. Nonlinear Dynamics and Chaos (MATH/PHYS 341) moves beyond classical (linear) modeling and looks at some of the strange (and chaotic) behavior of many problems. Mathematical Models in Biology (MATH 342) introduces and solves problems in biology using mathematics.

At the advanced level, there are two main sequences we offer. MATH 440-1 introduce partial differential equations, their applications and solution, and the qualitative theory of ordinary differential equations. MATH 448-9 introduce a variety of topics in numerical analysis, including numerical linear algebra and the numerical solution of systems of equations and differential equations.

Various faculty also teach special topic courses in areas including acoustics, solid mechanics, visualization and computer graphics, and computational applications in geometry and discrete mathematics.

### Faculty, Labs & Research Areas

Please visit the applied mathematics group's page here.

## **Mathematics Education**

### What is Mathematics Education?

Mathematics education is a discipline that is grounded in mathematics while simultaneously attending to the teaching and learning of mathematics. Students preparing to become teachers in the Commonwealth of Virginia must major in a content discipline (such as mathematics or Interdisciplinary Liberal Studies) and complete coursework in a teacher education program. So, although not a major, at James Madison University mathematics education is a crucial part of the preparation of teachers of mathematics at all grade levels PreK-12. In particular, one of our goals is to emphasize understanding how students learn mathematics and how this understanding can inform curriculum design, teaching practices, and assessment. Recommendations from the *Mathematical Education of Teachers* report of the Conference Board of the Mathematical Sciences inform the mathematics content courses for teachers offered through the Department of Mathematics and Statistics.

### What Do We Offer?

- JMU students who wish to become
**middle**and**high school teachers**major in mathematics and complete the secondary teacher education program through the College of Education. - JMU students who are primarily interested in teaching in a
**middle school**(grades 6-8) can major in Interdisciplinary Liberal Studies (IdLS) and complete the middle teacher education program through the College of Education. - JMU students who are interested in teaching
**elementary**(PreK-6) or**early childhood**(PreK-3) major in Interdisciplinary Liberal Studies (IdLS) and complete the elementary or early childhood teacher education program. - JMU students who wish to teach
**K-12 Special Education**major in Interdisciplinary Liberal Studies (IdLS) and complete the K-12 special education program. - JMU students who major in Interdisciplinary Liberal Studies (IdLS) can add an
**Algebra I endorsement**to their initial teaching license through additional coursework. - Current high school mathematics teachers who are interested in earning a
**masters degree**with a mixture of mathematics and education coursework can complete JMU’s M.Ed in mathematics.

### Faculty & Research Areas

Please visit the mathematics education group's page here.

## **Pure Mathematics**

**What Is Pure Mathematics?**

Mathematicians develop abstract structures to study complex realities. For example, it is often convenient to group like items together, as when the apples in the market are placed separately from the oranges and the pears, and mathematicians developed the notion of “set” to capture this. It sometimes happens that members of one set can be paired up with members of another set, and we use the concept of “number” to describe this tendency. In our daily lives we encounter numerous examples of related quantities, such as the relationship between temperature and the time of day, or between the distance we have travelled and the time during which we have been travelling. The notion of “function” is an abstract way of describing such relationships. Sometimes it is necessary to organize and manipulate large collections of data, and “matrices” are often useful for that purpose.

The pure mathematician takes the attitude that if abstractions like sets, numbers, functions and matrices are routinely useful for studying so many aspects of our daily lives, then they are also worth studying for their own sake. History records many instances of the usefulness of this approach. When Isaac Newton sought to understand the trajectories of projectiles, he found success not by studying projectiles, but by studying continuous functions. When Einstein was working out his theory of relativity, non-Euclidean geometry, a branch of mathematics developed for reasons having nothing to do with physics, proved to be indispensable. These are just two of many possible examples.

The usefulness of pure mathematics is only part of the story, however. There is also the tremendous beauty of the subject. It is hard to imagine an object more banal than the counting numbers, yet their structure is so complex that mathematicians routinely discover novel facts about them. Right triangles are all around us, but who would suspect that the square on the hypotenuse is equal to the sum of the squares on the other two sides?

It is this combination of beauty and usefulness that explains the importance and appeal of pure mathematics.

### Jobs in Pure Mathematics

Training in pure mathematics can lead to job opportunities in a variety of fields. In addition to teaching and research in mathematics, there are also fields like technical writing and investment banking to consider. Mathematics is also good preparation for those entering law and medicine. The Mathematical Association of America maintains a list of possibilities.

### What Do We Offer?

Students interested in studying pure mathematics should take MATH 245 as early as possible. This course provides an introduction to proofs and to topics in discrete mathematics. From there, our offerings can roughly be grouped into those covering discrete topics on the one hand, and those covering continuous topics on the other. For those interested in discrete topics, we offer elementary number theory (MATH 310), graph theory (MATH 353), abstract and linear algebra (MATH 430, 431, 434) and geometry (MATH 475). Continuous topics include real and complex analysis (MATH 410, 411, 360), and topology (MATH 435). We also offer a course in the history of mathematics (MATH 415).

### Faculty & Research Areas

Please visit the pure mathematics group's page here.

**Statistics**

### Why Statistics?

Statistics is used in almost any discipline including secondary education, biology, health sciences, engineering, computer science and pharmacology, in order to understand and account for randomness and make decisions accordingly. As a result the demand for statisticians is high and the choices available to statisticians are diverse.

### Jobs in Statistics

ThisIsStatistics.org is a great resource for students, parents, teachers and counselors who are wondering about the training, career options and career prospects for statisticians. The Bureau of Labor Statistics also gives employment projections on their website. The job titles given to statisticians vary so make sure to look for related occupations.

### What Do We Offer?

Our elementary statistics course (MATH 220) is, for most, the gateway course for the option 1 minor in statistics and a good course for those with an interest in the major, as it provides a survey of classical applied statistical methods. The upper level classes, generally speaking, each focus on one particular class of methods presented in MATH 220. Those with interests in applying statistics to their major discipline might consider a deeper foray into regression (MATH 322) - a tool for modeling relationships, the design and analysis of experiments (MATH 321), and/or methods for analyzing categorical data (MATH 327) such as you might get from surveys. Our other intermediate courses represent a broad array of topics so that you can choose those that best suit your interests e.g. those with an interest in applications to biology might be interested in biometrics (MATH 354) and multivariate methods (MATH 421).

The option 2 statistics minor is geared towards those majoring in a discipline that requires a calculus foundation. The required and elective courses in this minor reflect topics that would be useful for those going into industry or graduate school.

Calculus is at the heart of statistical methods; giving statisticians the means to model continuous phenomena. As such, those completing the statistics major will take single and multivariable calculus (MATH 236-7) and be introduced to calculus based statistics through our probability and statistics course (MATH 318). This course seeks to introduce you to a variety of classic statistical methods just as MATH 220 does, but from a more theoretical viewpoint e.g. you will see *why* the formulas are as they are. The other intermediate classes in the major overlap with those offered in the minor. Our upper level classes provide a solid foundation for graduate studies in statistics (any 400-level course), actuarial exams (MATH 426-7), and some graduate programs in engineering (MATH 423). Our statistics majors are required to take a course in statistical consulting (MATH 428) which provides practical experience with a real-world problem and training in the oral and written communicaton skills needed to be an effective statistical collaborator.

The use of statistics has only grown with the advancements and popularity of computers; computers being necessary for quick and reliable statistical calculations. The major requires some form of formal programming course, and such a course is optional for one of the minors. Regardless, our faculty use a variety of statistical programs so you would likely gain experience in at least two of the following: SPSS, R and/or SAS.

A variety of faculty also offer special topics courses each year related to their scholarly interests. These have included: Bayesian statistics, linear statistical models, statistical methods in clinical trials, and genomics.

### Faculty & Research Areas

Please visit the statistics group's page here.