# Mathematics Concentration for Students Matriculated After July 1, 2012

Mathematics is a continually-evolving field characterized by quantitative, deductive and analytical reasoning. Some mathematicians see mathematics as the hidden language of the universe and appreciate mathematics for its logical system of thought and the beauty of the unexpected connections discovered among different ideas. Many mathematical journeys have been followed because they are interesting, and only later was it recognized that these journeys could be used to explain parts of physical reality. However, other mathematicians take a more practical approach, focusing on mathematics as a tool for solving complex problems through modeling.

College-level mathematics builds upon the kinds of elementary mathematical objects, concept, and structures with which we are all familiar, such as number systems and arithmetic operations. However, college-level mathematics primarily involves manipulating, applying and generally reasoning about more complex/sophisticated abstract mathematical structures, objects and ideas.

Mathematical fields are distinguished from nonmathematical fields that use mathematics extensively, yet in which the reasoning is primarily conducted in the language of the other field. For example, in accounting, the reasoning is primarily in terms of business concepts, rather than the concepts of mathematics.

Concentrations in mathematics include a range of approaches and titles. Students seeking to think and invent in the language of mathematics as an endeavor for its own sake would most likely be working toward a concentration simply titled "mathematics." Students interested in using mathematical reasoning in order to solve practical problems might consider building a concentration in "applied mathematics." The applied mathematics concentration may focus on business-oriented applications of mathematical reasoning, the study of science and engineering-related problems that arise in research and industry, or one of many other topics determined by the student’s academic interests. Beyond these, there are other, typically interdisciplinary, specializations that have a strong emphasis on using mathematical reasoning as a tool, but which are not generally considered to be subfields of mathematics itself. Examples of these vary widely, and include such fields as actuarial science, quantitative psychology and theoretical physics.

Note: Empire State College cannot facilitate teacher certification directly, but can provide the mathematical content needed to prepare a student to enter a master’s program leading to teacher certification. Students who are seeking to teach must consult certification requirements in the state/region in which they intend to obtain certification, and should review the Mathematical Association of America’s CUPM Curriculum Guide 2004; for K-8, see the discussion on pages 38-42, and for secondary school, see the discussion beginning on page 52 and the recommendations on pages 54-56.

## ‌Concentrations

### General Foundation for All Concentrations

Concentrations in mathematics and applied mathematics should include both theoretical and applied studies, rather than focus exclusively on only one perspective in the field. Likewise, concentrations in mathematics and applied mathematics should include studies in both the continuous and the discrete, although a weighted preference may be given to one branch or the other. Similarly, concentrations in mathematics and applied mathematics should include studies in both the stochastic and deterministic, as well as in both the algebraic and the geometric.

There is a common core of foundational knowledge areas that concentrations in mathematics and applied mathematics are expected to include.

• Calculus: Calculus, often described as the study of continuous motion or change, forms the computational basis for classical (Newtonian) physics, and is an essential foundation for further studies in all areas of mathematics that concern continuous variables. The study of calculus is built upon the notion of a limit, which is a precise mathematical construction used to describe closeness among “infinitesimal” quantities. Calculus topics are typically studied in a sequence of either three or four terms, depending on curricular design. To address the calculus expectation, students are expected to learn differential calculus, integral calculus, multivariate calculus and vector analysis.
• Linear Algebra: Linear algebra is the detailed study of linear systems of equations and their properties, and forms an essential foundation to further studies in mathematics. The concepts introduced in linear algebra appear in virtually every area of advanced-level undergraduate mathematics, including the continuous and the discrete, the theoretical and the applied. Linear algebra is also known by such titles as "matrix algebra," "matrix theory" and "finite dimensional vector spaces."
• Proofs: Logic and the construction of proofs are essential for success in any advanced-level undergraduate study in mathematics. These core skills are best developed in a study dedicated to formal logic and the construction of proofs that can serve as a transition to advanced-level studies.
• Tools: Proficiency in the use of tools employed by mathematical professionals is vital. Students are expected to employ such tools in problem solving, analysis and the communication and understanding of mathematical ideas; students should be exposed to these tools throughout their concentration studies. Tools might include, but are not limited to, computer algebra systems (such as Maple, Mathematica, or MATLAB), statistical packages and an algorithmic programming language. Students should select those tools that are relevant to their academic interests.

### Core for Concentration in Mathematics

In addition to the foundational knowledge described above in the General Foundation section, concentrations in mathematics should include advanced-level, proof-based studies in the core areas described below. At least one study in each of these areas is essential. A second term of study in each of these areas is recommended, and is critical for students interested in attending graduate school in mathematics.

• Analysis: Students in mathematics are expected to learn the theoretical underpinnings of calculus. To address this expectation, students are expected to develop the ability to understand and to create proofs about the theory of calculus, to revisit familiar concepts in the context of mathematical proof, and to analyze mathematical details and their theoretical implications. Studies that usually contain the content that would meet this expectation include real analysis, applied analysis, advanced calculus, real variables and theory of calculus.
• Abstract Algebra: Students in mathematics are expected to learn the theory of algebraic structures in which the sets and operations in question do not exclusively consist of the familiar numbers and arithmetic operations. To address this expectation, students are expected to develop the ability to understand and create proofs about the abstract algebraic structures known as groups, rings and fields. Studies that would usually contain content that would meet this expectation include abstract algebra, modern algebra, introduction to group theory and Galois theory.

Students are expected to build on the core in a variety of ways, depending upon their academic interests and goals. Indeed, beyond the traditional areas mentioned above, interested students also may build on the core mathematics knowledge areas to create an individualized (possibly interdisciplinary) degree program that explores connections of mathematics to other areas of their interest, such as art, music, history or philosophy.

### Core for Concentration in Applied Mathematics

In addition to the foundational knowledge described in the General Foundation section, concentrations in applied mathematics should include the following.

• Modeling: Students in applied mathematics are expected to learn how to formulate and analyze mathematical models. To address this expectation, students will develop the ability to investigate not only simple models, but also to analyze more complicated models. The analysis of complex models typically requires the use of technology tools.
• Statistics: Students in applied mathematics are expected to develop proficiency with the use of statistics and probability on data sets. This includes the identification of appropriate tools, the application of these tools and the analysis of the results. The theoretical underpinnings of probability and statistics are usually explored within this study of statistics. Studies that usually contain the content that would meet this expectation include: mathematical statistics, probability and statistics and theory of statistics.
• Numerical Methods: Students in applied mathematics are expected to learn classical mathematical approaches to solving real-world, large-scale problems for which it is not possible to obtain closed-form solutions using elementary functions.
• Applied Analysis: Students in applied mathematics are expected to learn the theoretical underpinnings of some of the mathematical methods that are commonly applied to the analysis of problems posed in the sciences. Students will develop the ability to understand and to create proofs about these methods, and to analyze mathematical details and their theoretical implications. Studies that usually contain the content that would meet this expectation include real analysis, applied analysis, advanced calculus, real variables, complex variables, partial differential equations and special functions.

### Concentrations in Other Areas

A student might be interested in specializations related to the mathematical sciences, which include, but are not limited to, statistics, actuarial science and mathematical programming. Students interested in any of these specializations should investigate traditional curricula that are aligned with their area of interest. For each specialization, a variety of studies outside of mathematics will be important. Since many of these specializations are highly interdisciplinary in nature, the interested student might consider building such a degree program in the interdisciplinary studies area of study.

## Rationale

Students should discuss explicitly in their rationale essay how each of the above topics is incorporated in their degree program and how the program is designed to meet their goals.

### Resources

Mathematical Association of America’s CUPM Curriculum Guide 2004