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14/09/2023•27 minute read

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The Importance of Mathematics

Pure vs Applied Mathematics

Specialized Majors & Careers

Academic Tracks for Math

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*Bank statements. Excel spreadsheets. A receipt attesting to the damage incurred from a shopping spree... Numbers and mathematics are part and parcel of our daily lives. They also serve as the language of science — providing ways to calculate and model all kinds of factors and dynamics that make up the natural, physical, organic, and man-made realities we live in, from life expectancy to aerodynamics and black holes...*

*But what are the academic pathways that aspiring math majors need to consider? And what kinds of math concepts and applications are unique to each pathway? This post provides a full exploration of math majors and degree tracks in college and highlights which math majors are best suited to specific career goals and interests, so you can approach your college decision making with more certainty, clarity, and confidence!*

**Since time immemorial, mathematics has played a key role in shaping human civilization.** There are traces of mathematical notations and formulas going back to important ancient civilizations in Asia and the Mediterranean. And, even in ancient contexts, there’s evidence it was used for record keeping, for commercial transactions, and also for geometry and physics.

In the Renaissance and Enlightenment periods, during the 16th and 17th centuries, new pioneers of thought used experimentation and tremendous conceptual powers to make **stunning** **advances in mathematical understanding**.

During this same period, Isaac Newton and Gottfried Leibniz introduced **an elegant and versatile system of math called calculus — one of the most powerful tools of modern day mathematics and physics.**

In the more modern era, math has remained intimately intertwined with explosive growth in scientific advances and top career opportunities requiring strong and specialized mathematical foundations — for locomotion, energy generation and transfer, telecommunications, aeronautics and space exploration, astrophysics, and computer science…

Given the need for math in modeling, exploring, and verifying physical realities too small to observe with the human eye or as large as the infinite reaches of the universe…

And…in light of the ever-evolving technologies and technological solutions that are the engines of progress and economic expansion in industrialized nations today… **Mathematics has a very bright future, and so do today’s math majors!**

In addition, new frontiers in economic theory and modeling, financial modeling, quantum physics, and machine learning, among others, are forged by and require skilled mathematicians and ongoing advances in mathematical and STEM research.

This means **aspiring math majors have fantastic career prospects, but as you start your college journey you need to choose from among many possible mathematics specializations and pathways.**

From here, let’s explore the most foundational branches of math learning and how they differ. Indeed, charting a course toward either *pure mathematics* or *applied mathematics* is one way to begin to home in on the kind of math courses and specializations you want to pursue in college and beyond.

Pure mathematics is a branch of mathematics that focuses on abstract concepts and principles.

**Studying pure mathematics is kind of like studying the very nature and boundaries of math itself and imagining new possibilities as well!**

Math majors who choose a pure mathematics track study the broad principles, concepts, and techniques of math. Pure math courses are heavily proof-based and deal with theoretical possibilities of math.

As a result, a pure math pathway incorporates lots of advanced mathematics curriculum, with a specific focus on abstract concepts, theorems, formulas, and algorithms.

The study of pure mathematics in a bachelor degree program will typically entail a variety of rigorous courses, including:

**Calculus**– students learn everything from single-variable calculus to multivariable functions, theorems, and vector analysis.**Complex Variables**– students do specialized learning about variable-driven aspects of mathematics, including geometry, analytic functions, and residue theory.**Data Structures**– in this area of math study students are introduced to data structures and algorithms. Topics covered range from stacks and queues to analyzing and plotting graphs.

Because math itself is the object of study in pure mathematics, **study and work in this branch may be disconnected from present or real-world applications.** In practice, and over time, however, pure math can and does contribute to the critical role that mathematics plays in practical scientific applications and advances.

Applied mathematics focuses on using mathematical tools and methods to solve real world problems in various fields. In other words, applied mathematics stays connected to practical applications. It involves the application of mathematical models, algorithms, and statistical analysis to understand, explain, and predict phenomena across different domains, such as physics, logic, or financial modeling and computation.

**Applied mathematics majors will learn how to use equations, proofs, and algorithms in ways that apply to specific jobs and tasks.**

**Courses required for an applied mathematics major include:**

**Calculus**— Calculus offers a foundation for using math in physics and for applications related to change processes, such as motion, change, or transfer processes. In the world of knowledge and computing it can help with modeling processes that occur on a continuum.**Linear Algebra**— Linear algebra explores the properties and operations of vectors and matrices, which are essential for solving systems of linear equations and a wide range of mathematical and scientific applications. Linear algebra provides tools to analyze and manipulate data, model real-world phenomena, and solve problems in various fields, including physics, computer science, engineering, and economics.**Numerical & Complex Analysis**—**Numerical Analysis**focuses on developing algorithms and methods to approximate and solve mathematical problems using numerical techniques. Numerical analysts work to ensure the accuracy and efficiency of computations — crucial for applications in engineering, physics, computer science, and more...**Complex analysis**is a field of mathematics that explores functions of complex numbers — both real and imaginary. Complex analysis has applications in physics, engineering, and various areas of mathematics.

As computer technology and innovation become increasingly dominant fields of research and development, it makes sense for applied mathematics majors to get exposure to this specialized area of applied math.

Applied Math | Computer & Data Science Applications |
---|---|

Binary Mathematics | Binary mathematics is an essential math field for computer programming. The binary number system uses only two digits, 0 and 1. It simplifies the coding process and is essential for low-level instructions used in hardware programming. From pictures to games and even videos, computers store data using the binary system. |

Linear Algebra | This branch of mathematics provides concepts crucial to many areas of computer science, including graphics, image processing, cryptography, machine learning, information retrieval, web search, and more. Linear algebra is what makes your video games look so breathtaking, and answers whatever questions you may be asking! |

Calculus | Calculus is used in an array of computer science areas, including creating graphs or visuals, simulations, problem-solving applications, specialized coding, and algorithms. Differential equations are particularly important in making software tools that can predict future outputs. |

Discrete Mathematics | Discrete mathematics studies mathematical structures and constitutes a common language in the universe of computer science engineers and software technicians. It is used in many computer science applications, including algorithms, computer architecture, databases, machine learning, computer networking and security, and the design of operating systems. |

Statistics | Statistics help computers process massive amounts of information in the blink of an eye. Statistics is used for data mining, data compression, image analysis, and artificial intelligence. |

Source: Mathnasium

As machine learning evolves the practical applications of data science are becoming more powerful and more widely used and sought after.

As with computer science in general, **advanced mathematics is fully intertwined with data science computation.** Linear algebra is used for machine learning, data compression, and image processing.

Probability theory is also used in machine learning, to build models that use patterns in data to make predictions about new data.

Statistics helps computer programs turn raw data into useful information or meaningful predictions about future events. **Whether it's hypothesis testing, machine learning, or data visualization, statistics is an important application in the data science universe.**

Calculus is essential for working with large datasets and complex models. In particular, **calculus is used in data science to perform optimization,** for complex problem-solving functions within many constraints and parameters.

Now you know that *pure mathematics* and *applied mathematics* are the two key pathways that many colleges and universities use to guide curriculum for math majors.

But, because advanced mathematics skills and knowledge are so foundational for most STEM fields, some ‘math majors’ may actually want to consider majoring in a STEM field instead — which at most schools would include an option to minor in math as well.

Let’s put it this way: if you get a degree in mathematics you might naturally find your way to a career in a STEM field where your math knowledge will be highly valuable! In fact, the right advanced math skills should help you get into leading STEM programs in college.

Therefore, another way to develop strong math knowledge and ability is to **study advanced math concepts as you pursue a STEM major.**

Questions about women in STEM fields and professions? Check out:

At its core, statistics entails the collection, analysis, and interpretation of data to acquire insights and make informed decisions. It is indispensable in various fields like scientific research, economics, and finance.

Some subfields within statistics include descriptive statistics, inferential statistics, probability theory and regression analysis. These sub-fields provide methods for data collection and hypothesis testing, allowing for sound data-based predictions to be made.

Statisticians are high in demand across many industries like healthcare, finance, and government. The core of a statistician’s work often entails helping organizations make informed decisions, and future projections.

Actuarial science is a niche field that integrates mathematics, statistics, and business to assess and manage risk in finance and insurance sectors. **Actuaries use mathematical models and statistical methods for financial forecasting** used by insurance underwriters, financial analysts, and economists for risk management and other financial modeling.

Actuaries often work for insurance, consulting, or financial companies.

Source: Bureau of Labor Statistics

**Operations Research**

Operations research applies mathematical and analytical methods to optimize complex systems and decision making processes in management and business. It entails the use of mathematical models and data analysis, as well as optimization techniques.

**Operations research mathematics is particularly relevant in logistics, supply chain management, and inventory control, to name just a few examples.** With a strong background in mathematical models and algorithms, operations researchers are able to optimize processes to help companies bolster productivity and thereby reduce costs of production.

Operations researchers often work in industries like manufacturing, logistics, and consulting, where process optimization is in high demand.

**Mathematical biology plays a significant role in the understanding of diseases, ecology, and complex biological processes.** It allows researchers to acquire insights, make predictions, and devise strategies for disease management and conservation efforts.

Career opportunities include roles in research institutes, healthcare, and government agencies…

Computational mathematics focuses on developing algorithms and computational techniques to solve mathematical problems. **Computers can be programmed to use the computational algorithms to perform calculations and modeling.**

Computational mathematicians may pursue areas of technical specialization, such as numerical analysis, optimization, and computer-aided design.

Majoring in computational mathematics opens doors to a career in technology and research institutions, and is also a great pathway for careers in software development.

Want to explore the professional world of computational mathematics in applied mathematics? Check out:

SIAM (Society for Industrial and Applied Mathematics)

Source: Bureau of Labor Statistics

As the name suggests, this major focuses on mathematical pedagogy and learning. **Mathematics education is a field devoted to the research, development, and effective implementation of teaching methods and curriculum for math instruction.**

The ubiquity of mathematics in our daily lives necessitates the need for mathematical education, especially with regard to basic numerical skills, logical reasoning, and problem-solving abilities.

Knowing advanced math is one important step to a career in math education, but the ability to teach well — and importantly, to teach *math* well — is a craft that can only be honed through practice, and not the mere internalization of facts.

To learn more about professional pathways in math education, check out:

AMTE (Association of Mathematics Teacher Educators)

Even in grades before high school, some extra focus on mathematical literacy can help younger minds build a stronger foundation for success in more accelerated and rigorous math programs in the later grades.

**In high school, seeking opportunities to get into advanced college prep math courses — and stay that course for all four years — is a great way to jumpstart your transition to an undergraduate math program in college.**

Also, with achievements in advanced math courses, including Honors or AP math classes, and involvement in extracurriculars that emphasize math exploration, you should improve your chances of getting into leading colleges and math programs.

**Ways to amplify your math learning and accomplishments before applying to college:**

- Talking to school academic counselors, as early as middle school, about qualifying for advanced math courses, as some advanced high school math classes may require completion of advanced math courses prior to high school.
- Pursuing opportunities to enroll in accelerated, Honors, or AP math courses in high school
- Participating in math-oriented extracurriculars: math circles, math clubs, pre-college academic programs, or STEM-related extracurriculars with math components.
- Working as a math tutor, or participating in internships or part-time jobs that involve applied mathematics.

Wondering about math enrichment opportunities where you live? Some organizations offer local programs, so be sure to research opportunities unique to your community. To research more opportunities you can also try:

Math Circles provides support and resources for after-school programs for students, professional learning communities for teachers and mathematicians…There are nearly 300 Math Circles around the U.S., including approximately 150 Math Teacher Circles and another 150 Math Student Circles. Find a math circle near you!

Khan Academy provides online resources — practice exercises, instructional videos, and a personalized learning dashboard. Subjects include math, science, computing, history, art history, economics, and more.

SummerApply! offers tools and resources to help parents and students find, research, and join a wide range of pre-college summer programs, including academic enrichment, residential and online college bridge programs, internships, study abroad, and more.

Undergraduate degree pathways for math majors include:

**AA or AS degree in math**

**BA or BS degree in pure math or applied math**

Students who want a math-rich curriculum could also opt to pursue an undergraduate degree in other STEM fields, in lieu of mathematics:

- Physics
- Chemistry
- Various branches of computer and data sciences
- Various branches of engineering

**Other math-rich disciplines exist outside of STEM fields too, including:**

- Economics
- Math Education
- Business, Finance, & Insurance
- Accounting

Questions about undergraduate degrees? Check out:

Exploring the Spectrum: Understanding Different Types of College Degrees

A Comprehensive Overview of the Associate Degree

Charting Your Academic Path: Bachelor of Arts vs Bachelor of Science

How Much Do Engineers Make? The Financial Future of a Career in Engineering

Students who complete a BA can pursue job opportunities aligned with their degree program and qualifications. And, **college graduates interested in taking their math expertise to the next level, can opt to pursue an MA or MS in mathematics, or go even further and seek a PhD.**

- With
**a masters or PhD in pure mathematics,**you would take important steps in qualifying to work in research centers or higher education roles. - With
**a masters or PhD in a specialized branch of applied mathematics,**you might seek roles in higher education or, depending on your area of specialization, in any number of STEM fields, or in economics, finance, or business.

Advanced degrees in fields such as these can open doors to a world of career opportunities in the business sector, nonprofit and research sectors, in education, and in government.

Questions about postgraduate degrees? Check out:

Exploring the Spectrum: Understanding Different Types of College Degrees

Where Should I Pursue a Graduate Degree? US, UK, or Elsewhere?

Today’s technologically driven economy has meant that the demand for individuals equipped with mathematical skills is on the rise. **A dual or interdisciplinary study path can provide you with the upper hand in entering the relevant industry and securing competitive jobs.**

One popular dual major is mathematics and economics. In this kind of program you would develop specialized expertise in applying math concepts to economics, including tasks related to financial forecasting, analysis of economic policies, statistical research, and more.

A dual major like this is highly beneficial for careers in finance, consulting, and economic research.

Obviously, a dual major in mathematics and computer science would also open doors to many opportunities. With both advanced math skills and a strong foundation in computer science, you should be well prepared to pursue challenging entry-level jobs and have an outstanding foundation for pursuing further CS certifications or technical specializations, including postgraduate degree tracks as well.

A mathematics and computer science dual major is particularly relevant for careers in software development, data science, and cryptography.

Finally, a dual major approach that combines math with a branch of engineering is another excellent way to equip yourself for a bright future with many potential career prospects.

With numerous math majors available, picking the right major can be a daunting proposition to many. Here’s a few next steps to do right now…

- As you explore pathways, check out some of the links on this page that will provide you with more information.
- Explore your current passions and interests with a supportive parent, guardian or mentor, and with input from an experienced school counselor or professional admissions advisor.

Congratulations! You now have a truly comprehensive understanding of the world of mathematics, the academic pathways to choose from, insights into larger career prospects, and tips for the next steps on your college journey!

And while math can be a hard subject in college for some, what is certain is that math is not *just* challenging. It is an amazingly devised language and an elegant, adaptable code. It’s workings span various types of concepts and systems that make it possible to quite literally unlock the mysteries of physical reality — from atoms to galaxies — and the complexities of mental ones as well, across sectors stretching from aerospace engineering to economics, business, and computer science…

And now that you’ve read this far, I hereby nudge you to continue exploring the fascinating world of mathematics — particularly if math is right up your alley!

And, if you’re feeling stuck on your college journey…

Don’t let the opportunities of a lifetime pass you by.

While the costs of college have gone up, **there’s good evidence that the return on educational investments are usually well worth it!**

What Makes Crimson Different

Want greater clarity, a better strategy, or simply want help figuring out how high you should aim in terms of college choices?... A Crimson Advisor is ready to answer your questions.

Our Advisor and Tutoring networks, and many other targeted services, have helped over 6,000 students get into leading schools around the globe. And, our unique college pathways technology can help you manage your college journey with ease, even on the runway to the school of your dreams!