Mathematics college curriculum

Summary: Collegiate mathematics education is determined by the student’s choices within the constraints of graduation and department requirements.

For thousands of years, mathematics has been considered an important part of a liberal arts education. Examples of this idea abound, including schools and scholars in ancient Greece, China, and the medieval Islamic world, as well as in the rise of North American colleges in the seventeenth century.

Debate has existed for decades about which topics should be a part of the college curriculum and how best to teach them. Common curricula, such as geometry, or educational tools, like the abacus, have been replaced by other focuses as societies’ needs have changed and technology has advanced.

New discoveries in mathematics and emerging disciplines also result in curriculum changes. In the twenty-first century, the mathematics curriculum at the university level varies depending on the educational goals of the student. In the United States, the types and the number of mathematics courses required in the curriculum are typically based on a student’s major subject of concentration. In this regard, there tend to be three broad categories into which a typical college student may be classified: a student who needs to fulfill a general education requirement in the mathematical sciences; a student majoring in a partner discipline, such as the physical sciences, the life sciences, computer science, engineering, economics, business, education, and the social sciences; and a student whose major is in the mathematical sciences, including pure (theoretical) or applied mathematics, statistics, actuarial sciences, and mathematics education. At most colleges and universities, curriculum is approved by both internal governing bodies, such as curriculum committees, and external accrediting agencies. Local, national, and specialized accrediting agencies may approve programs at the department or college level.

History

There is a rich history of mathematics in higher education contexts. From the schools of ancient Greece to the universities of the Renaissance, mathematics was an important component of the seven liberal arts, and mathematics was seen as a way to understand reality. Three of the liberal arts, the Trivium, consisted of grammar, rhetoric, and logic. In the Quadrivium, the other four liberal arts, arithmetic was the study of numbers, geometry was the study of numbers in space, music was the study of numbers in time, and astronomy or cosmology was the study of numbers in space and time. The first college in the United States was Harvard University, founded in 1636. Harvard and other institutions of higher learning included mathematics in their curriculum. Around the time of the Revolutionary War, advanced mathematics included topics in surveying, algebra, geometry, trigonometry, and calculus. In 1776, Congress advised that disabled veterans, “[w]hen off duty, shall be obliged to attend a mathematical school, appointed for the purpose, to learn geometry, arithmetic, vulgar and decimal fractions, and the extractions of roots.” This obligation led to the official founding of the United States Military Academy in 1802. After World War II and the beginnings of the Cold War, the growing emphasis on computer technology greatly impacted the mathematics curriculum in the United States.

Teachers have long explored different methods to help students succeed in mathematics. The philosopher Socrates is known for the Socratic Method, and in the early part of the twentieth century, topologist Robert Lee Moore developed a Socratic style of teaching that became widely known as the Moore Method. Versions of the Moore Method, or a modified Moore Method, continue to be used in twenty-first-century undergraduate and graduate mathematics classrooms. In some implementations, students work on problems and present proofs or solutions they develop on their own, with the class being responsible for corrections and the teacher acting as a guide. In the 1980s, a calculus reform movement that is often referred to as the “calculus wars” spurred debates among mathematicians regarding various aspects of teaching, including the use and balance of lectures, technology, and rigor in calculus classrooms.

Calculus education had already undergone many changes in the twentieth century, such as a shift to calculus being taken earlier in the college curriculum. Following the ethos of “calculus should be a pump, not a filter,” educators explored many different approaches, often based on empirical studies. Some campuses embraced new approaches, while others soundly rejected them. In the early twenty-first century, mathematicians continue to discuss and refine the calculus course as well as other mathematics courses. There are also discussions at both the college and federal level of the possibility of standardized college mathematics assessments.

General Education Mathematics Requirement

For the college student majoring in a subject area that does not require specific mathematics courses, the extent of the mathematics curriculum may consist of mathematics courses that satisfy general education core requirements. At most colleges and universities, these courses enroll almost twice as many students as all other mathematics courses combined. These students represent a broad variety of majors, including students from the humanities, fine arts, elementary education, and several branches of the social sciences.

Courses that fall into this category may be termed or described as one of the following: quantitative literacy; liberal arts mathematics; finite mathematics; college algebra with modeling; or introductory statistics. These courses are designed to have students learn to think effectively, quantitatively, and logically, and may actually also be requirements for a student’s major. Such courses often serve as students’ final experience of college mathematics. While these courses may be terminal, such courses could also entice students to study mathematics further, and therefore, such course offerings may act as a springboard or gateway through which a student chooses to continue the study of the mathematical sciences.

There is a wide variety of topic options in these courses. Some professors incorporate topics directly from daily life, like financial mathematics, while others focus on algebraic or statistical techniques that might be important in future coursework. General education courses are also seen as the final place to impact students’ perceptions about mathematics and its role in society. In the same way that a survey course on important literature might include works by William Shakespeare, some mathematicians select course topics from the masterpieces of mathematics, which might include great theorems, like Euler’s theorem, named for Leonard Euler; interesting applications, like Chvátal’s art gallery theorem, named for Václav Chvátal; interdisciplinary topics, like fractals, perspective drawing, or the philosophy of mathematics; or beautiful mathematical topics, like the golden mean. Some classes focus on the breadth of mathematics, while others try to cover a few topics in depth. There is also a wide variety of teaching methodologies and pedagogy. In some classrooms, the focus is on lectures, while in others it is on discussion or presentations. Technology may be a fundamental part of the class, or the class might focus on pencil-and-paper methods.

Mathematics and Partner Disciplines

During the second half of the twentieth century and into the twenty-first, there has been an enormous growth and development of scientific and technological disciplines, and, consequently, the role of mathematics is increasing in an expanding array of subject areas and professional programs. Students may be required to take specific mathematics courses that complement their major field of study. These partner or client disciplines include physics, chemistry, biology, computer science, engineering, business, finance, economics, nursing, psychology, and education. Partner and client discipline courses may impact mathematics as well as the respective discipline.

Some of these courses are taught in mathematics departments; others are taught as a quantitative course in the major, as in some psychology departments. This system provides numerous opportunities for faculty and students in mathematics departments to collaborate with their counterparts in other academic departments on campus. It is not uncommon for students who major in these partner disciplines to also study advanced mathematics, often resulting in dual majors or a minor in mathematics. One emerging area in the twenty-first century has been calculus for the life sciences. Cutting-edge pedagogies may come from mathematics or a client discipline. Faculty in either mathematics or a client discipline may lead efforts in interdisciplinary curricular development, or departments may resist changes because of staffing or philosophical considerations, sometimes leading to friction between departments.

Such courses need not be limited to calculus-based courses. For example, students in the sciences often benefit from the skills and techniques used in introductory statistics and discrete mathematics courses, which may not have a calculus prerequisite. The ability to visualize in three dimensions is also valued by partner disciplines, and courses that emphasize geometric and graphical reasoning, linear systems, and vector analysis may also be required.

Precise, logical thinking is an essential part of mathematics. While it remains a component of the mathematics courses taken by students who study the aforementioned partner disciplines, additional needs specific to such fields of study are also imbedded in the courses taken by these students. Logical and deductive reasoning skills may need to be developed in a specific context, and certain disciplines may or may not have a need for the use of formal proof found in the mathematics courses. Also, the level and type of logical reasoning may vary depending on discipline. For example, business majors may require more quantitative or statistical analysis, while engineering students need to engage in more formal analysis in a course like multivariable calculus. Students studying the natural sciences benefit from heuristic arguments and data analysis, while computer scientists and software engineering students need the ability to use logic to write simple proofs. The courses that bridge various other subject areas with mathematics attempt to balance the rigorous proof and deductive reasoning inherent to mathematics with the skills these partner disciplines require of their students.

Students who are preparing to teach elementary or middle school mathematics also fall into this category. The curriculum designed for future primary mathematics teachers varies state by state, with many requirements set by schools or teacher program accrediting agencies. They often aim to provide these students with a firm foundation in various mathematical topics, such as number and operation, algebra and functions, geometry and measurement, and data analysis, probability, and statistics. These topics are studied at a level above and beyond that which they will eventually teach. Courses are designed to provide students with an understanding of these broad areas as well as an ability to make connections among various mathematics topics and with other subjects taught in the elementary and middle school curriculum. The intent is that the future teachers will be able to guide their students in ways that instill mathematical breadth and depth and “plant the seeds” of ideas that will come later. From 2003 to 2009, the Mathematical Association of America ran a program that was funded by the National Science Foundation called Preparing Mathematicians to Educate Teachers (PMET). PMET strove to improve the mathematics education of teachers by targeting the development of faculty awareness and teaching as well as instructional materials.

Concentration in the Mathematical Sciences

For students who choose to major in the mathematical sciences, their college curriculum is centered on this goal of study. Major programs in the mathematical sciences include courses that focus on pure (theoretical) mathematics, applied mathematics, statistics, actuarial sciences, or secondary mathematics education. Depending on the college or university, the programs and faculty for statistics or applied mathematics, as well as actuarial sciences and mathematics education, may be housed in a department distinct from the traditional mathematics department.

The actual course of study for mathematics majors will differ depending on the specific college or university. In general, students in their first years of study will take a sequence of courses in calculus consisting of single- and multivariable calculus, which include the topics of differentiation and integration, sequences and series, vector analysis, and differential equations. Beyond calculus, mathematics students often take a transition course that includes an introduction to proof-writing techniques demonstrated by a study of various foundational topics in mathematics, such as logic, set theory, functions and relations, and cardinality.

Other commonly required courses for the mathematics major include linear algebra, abstract algebra, and real analysis (or advanced calculus). Several other advanced courses in mathematics that make up the major include ordinary differential equations, partial differential equations, discrete mathematics, probability and statistics, modern geometry (Euclidean and non-Euclidean geometry), complex analysis, topology, combinatorics, and number theory. Students who are interested in learning more applied mathematics may take courses in dynamical systems, numerical analysis, cryptanalysis, and operations research. A course in the history of mathematics may also be offered, especially for those students preparing to teach mathematics.

Because there are numerous topics that connect mathematics with other disciplines, various interdisciplinary courses may also be offered by mathematics departments in conjunction with other academic departments on campus. Some schools use common syllabi or exams for certain courses, and other schools allow more flexibility in what is taught and how it is taught. Regardless, there are at least some common expectations because mathematical definitions, ideas, and proofs build upon one another across courses, and so earlier courses in the major impact later ones. For example, a single-variable calculus class impacts multivariable calculus, and an analysis course impacts courses in complex analysis and topology.

The curriculum for students majoring in mathematics is designed so that there is a progression from the study and practice of computational methods and procedures toward an extensive understanding of the subject, which may include logical reasoning, generalization, abstraction, sophisticated applications, and formal proof. Students majoring in mathematics are also encouraged to demonstrate their mathematical knowledge in both written and oral formats. Students should also gain experience in the analysis of data, gaining the ability to move between context and abstraction—an especially important ability for students whose course of study focuses on applied areas of mathematics as well as for those becoming mathematics teachers. While mathematics students may prefer one area of mathematics over another, they are encouraged to gain a broad view of the subject, recognizing the complementary nature of the following concepts: theory versus application; discrete versus continuous; algebraic versus geometric; and deterministic versus probabilistic.

In addition to specific mathematics courses, students majoring in mathematics may take courses in computer science. On some college and university campuses, mathematics and computer science are classified in the same department or division. The natural affinity between the skills used by mathematicians and computer scientists makes this partnering possible, since the application of logical reasoning to the task of programming enhances the learning of both disciplines. For mathematics majors who are preparing to enter the nonacademic workforce, experience with teamwork, creativity, and problem synthesis skills is enhanced by computer programming coursework.

Undergraduate Research in Mathematics

Many mathematical science departments require their mathematics majors to engage in some form of research at the undergraduate level. This research can take many forms, such as a capstone course, a thesis, or some other form of a project during the senior year of college. The area of study for such research may connect knowledge of previous courses in an advanced manner. Such research often culminates in both a written paper and an oral presentation. This presentation provides the opportunity for mathematics students to not only study the mathematics, but write and speak about their results in the fashion conventional to the discipline.

Separate from major program requirements, research in mathematics at the undergraduate level can also be performed at National Science Foundation (NSF) programs, such as Research Experiences for Undergraduates (REUs) held at various schools across the country, often during the summer months. These opportunities allow students to become actively involved in current mathematical research projects under the guidance of faculty, and thus demonstrate how mathematical research is done and how it differs from research done in other fields. Programs such as REUs demonstrate how the activities of a professional mathematician are performed, including the various stages: formulating and solving a problem, writing a mathematics paper, communicating the results in a talk or poster (perhaps at a local or national mathematics conference), and possibly publishing a research article. The topics of study in REUs go beyond the standard undergraduate curriculum and also draw upon previous coursework and experience. By conducting research before they graduate from college, students get a taste of what happens in graduate school programs in mathematics, specifically the research component of the dissertation requirement.

Two-Year Colleges

A significant percentage of students who receive a bachelor’s degree in the mathematical sciences have taken some of their mathematics courses at two-year colleges. While many college students may fulfill their general education requirement in mathematics by taking such courses at a two-year college prior to attending a four-year college or university, many potential mathematics majors complete a variety of mathematics courses that satisfy requirements in the major program. Such courses include developmental mathematics, precalculus, introductory calculus, multivariable calculus, linear algebra, differential equations, discrete mathematics, and statistics. While an associate’s degree in mathematics may not be obtainable from a two-year college, it is becoming more common that future mathematics majors are beginning their mathematics career at these schools, including students who are also preparing to become mathematics teachers at the various school levels.

Technology

With the advances made in science and technology during the latter half of the twentieth century, many new instructional techniques are being designed and utilized in the mathematics classroom at all levels, including the collegiate. With an emphasis on critical thinking and deductive reasoning and a movement away from rote memorization of mathematical theories and algorithms, there has been an increase in the use of technology for teaching and learning advanced mathematics. Accurate visualization of graphs and geometric objects and easy manipulation of algebraic constructs are some of the benefits of current technology available for mathematics education.

Computational technology changed rapidly during the latter part of the twentieth century. At the beginning of the twenty-first century, computer algebra systems (CAS), such as Mathematica, MATLAB, and Maple, are often helpful tools for both in-class demonstrations and independent student assignments. These software packages are commonly implemented in a variety of courses, such as calculus, linear algebra, differential equations, statistics, real analysis, and complex analysis.

Other software packages, such as Geometer’s Sketchpad and Exploring Small Groups (ESG), are more course-specific to geometry and group theory, respectively. In addition to desktop or laptop computer technology, the development of handheld graphing calculators, such as the various models produced by Texas Instruments (TI-83+, TI-84, TI-86, and TI-89), has also influenced the use of this technological tool in the classroom. Computer programs and graphing calculators are also being used at the secondary school level, and the transition to using such technology in the mathematics classroom at the collegiate level is often a smooth experience for the mathematics student.

Bibliography

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American Mathematical Society (AMS) and The Mathematicians and Education Reform (MER) Forum. “Excellence in Undergraduate Mathematics: Confronting Diverse Student Interests.” http://www.math.uic.edu/~mer/pages/Excellencepage/index.html.

Cajori, Florian. The Teaching and History of Mathematics in the United States. Washington, DC: Government Printing Office, 1890. http://www.archive.org/details/teachingandhist03cajogoog.

Committee on the Undergraduate Program in Mathematics (CUPM) of The Mathematical Association of America (MAA). Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide 2004. Washington, DC: MAA, 2004. http://www.maa.org/cupm/cupm2004.pdf.

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