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 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.
Note: You will need Adobe Acrobat Reader to read the CUPM Curriculum Guide.If Acrobat Reader is not installed on your computer, you can download it for free from Adobe.
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.
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.
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.
In addition to the foundational knowledge described in the General Foundation section, concentrations in applied mathematics should include the following.
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.
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.
Mathematical Association of America’s CUPM Curriculum Guide 2004