Mathematics, or just plain math, is one of the primary pillars of human inquiry, knowledge, and intellectual pursuit. It lies at the very foundation of our rational understanding of the physical world. And yet, it is woven so deeply into the fabric of our culture and consciousness that it largely eludes easy definition. In a quip that is often attributed to Charles Darwin, mathematicians are likened to blind men in a dark room looking for a black cat, which isn’t there. When the great 20th-century philosopher and mathematician Bertrand Russell set out to define the discipline in his 1903 book *The Principles of Mathematics*, he may have illuminated that room slightly by proposing that mathematics is a system of symbolic logic. But that insight only further complicated the quest for a straightforward definition of mathematics.

Math does begin with numbers, and with the assignation of numerical values to objects in a manner that both describes the physical world and allows for further investigation and elaboration in the form of calculations. In that sense, it is a language of quantification, with its own grammar, syntax, and semantics. It is this concept of mathematics that informs the notion of “quantitative literacy,” which is the ability to use numerical data and the language of mathematics to solve everyday problems. But, at higher levels, mathematics bleeds into metaphysical abstraction; into realities that are as far from daily life as the distant galaxies astronomers gaze toward, and as ephemeral as the dark matter atomic scientists theorize about. So, math ends up in a place that can be as straightforwardly mundane as making change for a dollar, and as inscrutably complex as the interior of a black hole. To bridge that gap, it can be useful to consider the following components of mathematics as it has evolved.

**Arithmetic**: encompasses the basic syntax of mathematics, including addition and subtraction; multiplication and division; and fractions, percentages, exponents, and square roots.

**Geometry**: moves into the practical application of arithmetic to spatial relations and shapes in two and three dimensions in order to compute area, determine volume, measure angles, and quantify the size of forms in relation to other objects in space. In its more advanced forms, geometry moves into trigonometry.

**Algebra**: involves the introduction of abstract values into mathematics in order to solve theoretical problems using linear and quadratic equations and more complex systems that may include both positive and negative values.

**Calculus**: is an incrementally complex system and language for measuring non-static values by incorporating time and change over time to algebraic equations.

The following resources offer more information about the field of mathematics and its components:

The University of Oregon hosts a blog about mathematics by Dr. David Moursund, a former chair of the Department of Computer Science and current professor at the College of Education.

The Mathematics Department at Fordham University includes an overview of the discipline by Dr. Robert H. Lewis, a Professor of Mathematics at the school.

The Robert F. Wagner Graduate School of Public Service at New York University has a page devoted to algebra and algebraic terminology.

The Association of American Colleges & Universities has developed a Quantitative Literacy VALUE Rubric, and has a page devoted to defining quantitative literacy across various academic disciplines.

A statistic is a numerical value derived from the mathematical analysis of a dataset that either describes features of that set or makes inferences and predictions based on patterns in that data. Statistical science is a discipline that uses mathematics as one of its primary and fundamental tools, and mathematical statistics refers specifically to the rigorous application of mathematical principles to the collection, interpretation, and presentation of data. Statistics encompasses probability theory, a branch of mathematics concerned with quantifying randomness and uncertainty. The application of probability theory and other forms of logical reasoning, including theories of decision making and Bayesian analysis, allows statisticians to create scientific models that can forecast the weather, find predictive patterns in consumer behavior, set betting margins, and calculate the likelihood of various outcomes in an election. Advances in the information collection, storage, and processing capabilities of computers have facilitated the application of statistical methodologies and other related forms of data analytics to a growing number of problems and questions, with a greater degree of efficiency and accuracy.

For more information on statistics, consult the following organizations and resources:

The American Statistical Association is a professional association devoted to the practice and profession of statistical science. The ASA also maintains STATtr@k, a website for professionals in the fields of data and statistics, and This Is Statistics, a resource for students, parents, educators, and counselors.

STATS is a statistical literacy project run by Sense About Science USA, a non-profit organization devoted to promoting the study and public understanding of science.

The Department of Statistics at Florida State University maintains a web page that addresses the question, “What is Statistics?”

The School of Mathematics and Statistics and the University of Melbourne has a Statistical Consulting Centre that provides an overview of statistics.

In the simplest terms, an algorithm is a set of step-by-step instructions for completing a particular task or a discrete series of operations. When used in conjunction with quantitative data and statistical probability modeling, an algorithm becomes a powerful tool for complex analyses, decision-making, automated reasoning, and machine learning. Much of what we think of as computer programming is, in fact, algorithmic in nature. A common type of algorithmic coding involves an if/then instruction, such as, if the numerical value is greater than ten, then subtract three; if the numerical value is less than five, then add three.

A computer algorithm will often use mathematical functions and statistical modeling techniques to quickly mine, sort, and/or analyze complex and/or very large datasets. These algorithms can be thought of as mathematical equations that may contain layers of instructions, or routines and subroutines, and handle multiple variables, or inputs. A common way to illustrate algorithmic logic is to compare a set of instructions to a recipe. For example, an algorithm for preparing a peanut butter and jelly sandwich might read: spread peanut butter on one side of a piece of bread; spread jelly on one side of another piece of bread; press the two pieces of bread together. It does not contain numerical values for the amount of peanut butter and jelly: those would be inputs, or parameters. A more advanced computer algorithm might then use statistical data mined from the shopping habits of thousands of families to determine the optimal amount of peanut butter and jelly per sandwich, or to calculate the proportion of sandwiches that should contain grape or strawberry jelly when feeding a population of 100 five-year-olds.

To explore algorithms further, consult the following resources:

Harvard University computer scientist David J. Malan narrates a TED-Ed animated video on “What’s an Algorithm?”

Software developer and former University of Iowa mathematics professor Alexander Bogomolny maintains “Interactive Mathematics Miscellany and Puzzles,” a website dedicated to mathematics that has a section on “What is Algorithm?”

The BBC has a multi-media webpage that offers an overview of several approaches to understanding “What is an Algorithm?“

Mathematics and statistics are taught in various forms at all levels in higher education, from undergraduate Bachelor of Arts (BA) and Bachelor of Science (BS) programs, to master’s and doctoral degrees that often offer the options of specializing in particular areas of research in the field of mathematics and/or statistics. Even at the undergraduate level, it is not uncommon for math and statistics majors to pursue a complementary area of study by minoring or double majoring in a related discipline. Mathematics, for example, has obvious applications in computer science, physics, and engineering, while statistics is itself an application of mathematics that is integral to all of the social sciences (psychology, sociology, economics, political science), and also to business and management. Because statistics is applied mathematics, both disciplines are often housed within the same academic department.

While there are undergraduate degrees in statistics, at the bachelor’s degree level it is more common to find majors in mathematics and/or applied mathematics that incorporate training in statistics. A BA in mathematics and/or statistics is a broader, liberal arts degree that includes general education coursework in the humanities and arts and sciences. In contrast, a BS in mathematics, or in mathematics with a concentration in statistics, will have fewer general education requirements, but may require a secondary concentration in an area like computer science, engineering, education, or business and economics. Some colleges and universities are also beginning to offer BS degrees in Data Science, which is an area of applied mathematics and statistics we’ll discuss below.

A bachelor’s degree in mathematics and/or statistics can offer sufficient training for entry-level positions in a range of fields, but it is often a way of preparing for further training in a master’s degree program. At the graduate level, this training gets more specialized, and includes degrees in areas like: Actuarial Science; Applied Mathematics; Applied Statistics; Biostatistics; Computer Science and Information Technology; Data Science and Analytics; Math Education; Mathematical Engineering; and Pure Mathematics or Mathematical Theory. These may be Master of Arts (MA) or Master of Science (MS) degrees, and there are also Master of Education (MEd) degrees for those aiming to teach math at the primary, secondary, high school, and/or community college level.

Pursuing a PhD in mathematics or statistics is generally reserved for those aiming for an upper level research position in academia, industry, or government, and for those who want to teach in these disciplines at the college level. These are research-intensive degree programs that can require six or more years to complete. In some cases, students may receive a master’s degree in the process of working toward a PhD, and master’s programs are often associated with PhD programs in mathematics, applied mathematics, and statistics.

For more information on degrees in mathematics and statistics, consult the following resources:

“Why Major in Mathematics, Applied Mathematics, and Statistics,” by Professor Daniela Calvetti, Department of Mathematics, Applied Mathematics, and Statistics at Case Western Reserve University.

Math with Bad Drawings is a website by Ben Orlin, a mathematics teacher who has written on the subject for Vox, The Atlantic, Slate, The Los Angeles Times, and The Journal of Humanistic Mathematics.

The American Mathematical Society has an online tool for exploring graduate degrees in mathematics and statistics.

The American Statistical Association has a page devoted to resources for statistics students.

“What is the major difference between statistics and math major students?” is a thread on Quora.com that includes insights from Harvard University Professor of the Practice of Statistics Joeseph K. Blitzstein.

The convergence of computer technology, mathematical theory, and statistical modeling has created what many have termed the Data Revolution, which encompasses everything from public policy and healthcare, to business and marketing strategy, to professional sports and the entertainment industry. Simply put, there are fewer and fewer areas of human endeavor in which quantitative literacy, applied mathematics, and the use of statistics are not highly valued. There may be no better example of this than FiveThirtyEight.com, the website and media company founded in 2008 by data journalist and blogger Nate Silver, and which now covers politics, sports, science, health, economics, and culture from a quantitative perspective. The list below covers a broad but by no means complete breakdown of some of the more prominent fields that are open to those with training in mathematics and statistics:

Mining, storing, sorting, and analyzing big data sets, and designing and maintaining the IT systems that facilitate these processes requires programming knowledge and training in mathematics and statistics. There are increasingly openings for data scientists and analytics professionals in all sectors of the economy, from government to private industry, in manufacturing and in marketing. Professional data science and analytics resources include the:

Optimization and efficiency are two of the big buzzwords in today’s data-driven business world, and business intelligence (BI) is where theories of business management are tested and improved through quantitative analysis. Whether it’s in the area of executive decision-making, supply-chain dynamics, employee hiring, or marketing campaigns, BI and the mathematical and IT proficiencies it requires, has become integral to all types of business operations. Strategic and Competitive Intelligence Professionals (SCIP) is an international organization that serves professionals in the realm of BI.

Mathematics and statistics have long been an integral component of IT and computer science, and these are areas that mathematicians and statisticians have traditional found employment in. Career opportunities in this sector have been broadening, particularly as concerns about cybersecurity have intensified. Professional cybersecurity and IT resources include:

- Association of Information Technology Professionals
- International Association of Computer Science and Information Technology
- Information Systems Security Association

Another area that has traditionally been a magnet for those with backgrounds in mathematics and statistics is engineering, which itself is a broad category that includes professionals working in aerospace, biotech, civil, industrial, materials, mechanical, and structural engineering. Professional engineering organizations include:

Actuarial scientists are the statistical and probability specialists who make the insurance industry possible, and who play crucial roles in financial investing and management. There are graduate programs in Actuarial Science, and the Society of Actuaries dates back to 1889 and is the world’s largest actuarial professional organization.

Statistical analysis and knowledge of statistical methodologies are integral components in psychology, sociology, economics, and political polling research. This can lead to careers in data journalism, public policy, or even business and marketing. Professional behavioral science resources include:

- American Psychological Association
- American Sociological Association
- American Political Science Association
- American Economic Association
- Behavioral Science & Policy Association

Physics, chemistry, biology, geology, astronomy, and all of the other physical sciences are another traditional home for those inclined toward mathematics and statistics. The National Science Foundation has a group devoted to Mathematical and Physical Sciences.

A degree in mathematics and an ability to work with large datasets is one of the more obvious qualifications for careers in accounting and finance, and many mathematics programs offer courses that target specific proficiencies in these fields. Professional accounting and finance resources include:

- National Society of Accountants
- American Accounting Association
- American Institute of CPAs
- Association for Financial Professionals

The healthcare industry, medical research, and health services policy are three areas that have become more and more data driven in the digital age. There are career opportunities for those with training in statistics and applied mathematics throughout this sector of the economy. Two groups that serve this growing field are:

- American Association of Health Data Organizations
- American Health Information Management Association

STEM, or Science, Technology, Engineering, and Mathematics research and education have become a national priority.

- The US Department of Education has devoted significant resources to STEM initiatives.
- The National Education Association has a section on STEM education.
- The STEM Education Coalition is a professional organization that promotes development in this area.