At its most general mathematics studies patterns, structure and change. As well as being the language of (and indeed underlying) the sciences and engineering, it is abstraction beyond the concrete and understanding how abstract structures lead to more complex realities. Being able to go from the specific to the general, recognize commonalities, and prove the correctness of overarching structures is common to most of mathematics.

While there are an enormous number of sub-specialities within mathematics, these days we somewhat arbitrarily divide the field into three major areas. Applied mathematics is primarily motivated by solving 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 and go well beyond the original motivation. Pure mathematics is primarily motived by the study of the abstract structures, and following where sometimes very simple assumptions lead. And yet, an enormous amount of classically pure mathematics has become amazingly practival over the years, making the boundary between "pure" and "applied" quite fuzzy. Finally, statistics deals with the collection, analysis, interpretation, and presentation of data. While the theory of statistics is usually developed to understand and solve problems with data, there is significant overlap of statistics with pure and applied mathematics in terms of developing new techniques to solve new problems. Data science is a recently emerging field that calls on expertise overlapping all these areas.

A degree in mathematics or statistics will not make you an expert in science or engineering, but will give you the underlying language and skills to work with and support those in a multitude of areas. The critical thinking and logical skills that are a core outcome of a mathematics or statistics degree are the building blocks sought after by most professional employers.

Double Majors at JMU

Due to its underlying significance in so many other fields, a motivated student will be well served by completing a double major (or at least a minor) with mathematics and some other related area. The advantage, of course, is to have a much richer understanding of the underlying structure that describes a discipline. Examples that are very common are mathematics with quantitative science, physics, chemistry, biology and computer science. 

Applied Mathematics at JMU

At its core, applied mathematics is the mathematics initally developed to solve practical problems, and usually develops the theory well beyond that required to simply solve problems. Much of classical applied mathematics concerns developing "mathematical models" that describe a physical situation in such a way that the tools of mathematics can be used to solve for what is of interest. Many of these problems involve understanding how quantities change continuously and their relationships to each other. This is the basis of calculus, the most useful mathematical tool in the history of science and engineering, from its origins to today. A derivative, the slope of the tangent to a line, describes how something changes in time. Relations between derivatives and other fuctions lead to differential equations, the main way most scientific problems are formulated. Integration, measuring accumulated change, is used to undo a derivative, and is used to solve differential equations. Most of our 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.

As well as the calculus sequence leading to solution methods for elementary differential equations, JMU offers courses on elementary programming and computation for mathematics, optimization (finding the best solution for some problem), nonlinear dynamics and chaos (moving from easy linear problems to more realistic scenarios), mathematical models in biology, the qualitative theory of ordinary differential equations, and an introduction to applications and solutions of partial 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.

Pure Mathematics at JMU

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.

A math major includes core courses in discrete mathematics and proof techniques, linear algebra, analysis (a more detailed look at the underlying theory of calculus) and abstract algebra. JMU also offers elective courses on number theory, graph theory and combinatorics, advanced analysis, linear algebra and abstract algebra, topology, history of mathematics, and geometry. Special topics courses are regularly offered in areas of particular interest to faculty. 

Computational Mathematics at JMU

Another important strand of modern mathematics is computational mathematics: the study of algorithms and techniques to approximate the solutions to problems that either can't be solved exactly, or would take far too long to do by hand. Many JMU faculty use computational techniques to aid them in their research across the whole spectrum of mathematics, including numerically solving differential equations, investigating problems in systems of polynomial equations, combinatorics, abstract algebra, and vizualizing and analyzing large data sets. JMU includes electives on numerical analysis (solving nonlinear equations, numerical linear algebra, interpolation and approximation, approximating integrals etc.) and the numerical solution of ordinary and partial differential equations. However, computational tools are used in many of our courses in all disciplines.

Should I consider a degree in Statistics?

Statistics at JMU

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

All statistics majors (and indeed a vast swath of JMU students) take an elementary introduction to statistics that provides a survey of classical applied statistical methods. Upper level classes, generally speaking, each focus on one particular class of methods presented an elementary class. These include a deeper foray into regression (a tool for modeling relationships), the design and analysis of experiments, and methods for analyzing categorical data such as you might get from surveys. JMU also offers courses for those with an interest in applications to biology and multivariate methods. 

Another collection of statistics courses justify various statistical methods introduced in an elementary course using  the theory of calculus to understand random process that vary continuously. JMU offers a core calculus based probability and statistic course, explainign the why behind various methods, not just how. JMU offers advanced courses on the theory and application of probability and statistics that are core to the major, as well as courses like an introduction to biometrics, an introduction to machine learning, stochastic processes, decision theory and genomics. Our statistics majors are required to take a course in statistical consulting 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. Our faculty use a variety of statistical programs so you would likely gain experience in at least two of SPSS, R, 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.

Should I consider a degree in Math Education?

Mathematics Education at JMU

Mathematics education is a discipline grounded in mathematics while simultaneously attending to the teaching and learning of mathematics.  At James Madison University, mathematics education is a crucial part of the preparation of future mathematics teachers at all grade levels, PreK-12, led by a team of expert faculty.

Our goal is to help students unpack mathematical ideas so that they will be able to develop a deeper understanding of those ideas and so that they will be able to understand the way that students think about those ideas.  It is critical that our students not only know how to carry out mathematical procedures, but also why those procedures work (as well as why other procedures do not work).  This kind of knowledge is crucial for the depth and flexibility of understanding that future teachers need in order to teach mathematics for understanding and proficiency.

Recommendations from the Association of Mathematics Teacher Educator’s Standards for Preparing Teachers of Mathematics and the Mathematical Education of Teachers report of the Conference Board of the Mathematical Sciences inform the mathematical content courses for future teachers offered at JMU.

Our department supports the mathematical training of future and practicing teachers:

Elementary Education (PreK-6): JMU’s four-year elementary education program is rooted in foundational content knowledge and content-based pedagogical knowledge that helps majors in this program build the knowledge, skills, and empathic understanding required of successful elementary teachers.  Students in this program are required to take a sequence of lower-level mathematics content courses and have the option of taking additional upper-level mathematics content courses that lead to the Algebra I Add-On Endorsement (AAO).  Previous elementary education majors have remarked that the AAO Endorsement gave them the opportunity to 1) be hired in a middle school in addition to PreK-6 classrooms and 2) demonstrated their mathematical confidence and competence. Both qualities made them more marketable as teachers and practically guaranteed them a job after graduation.

Middle Education (6-8): JMU’s four-year middle education program is designed to help future teachers be equipped to walk alongside middle grades students as they develop their content knowledge, social and emotional skills, and learning strategies. Students in this program major in Middle Grades Education with at least one content area concentration.  Those concentrating in mathematics take a sequence of lower-level and upper-level mathematics content courses that lead to the Algebra I Add-On Endorsement (AAO). The AAO Endorsement licenses middle education majors to teach algebra, which can include teaching algebra in a high school, thus broadening their job prospects.

Secondary Education (9-12): JMU’s secondary education program is content focused, giving future teachers a firm foundation in the content they will teach as well as attending to and developing their pedagogical skills.  Those interested in teaching secondary mathematics have two options that lead to licensure:

• Double majoring in Secondary Education and Mathematics. This option leads to licensure with a bachelor’s degree after four years.
• Majoring in Mathematics and minoring in Secondary Education with an additional year spent earning a Master of Arts in Teaching. This option leads to licensure with a bachelor’s and a master’s degree after five years.

Master of Education in Mathematics: The Master of Education (M.Ed.) in Mathematics is designed for practicing secondary teachers of mathematics to deepen their understanding of mathematics and explore current educational theories and practices. This low-cost, completely online program offers practicing secondary teachers of mathematics opportunities for exciting career advances (e.g. teaching Dual Enrollment or community college courses) and substantial salary increases.