How to Be a Good Mathematician

Studying Mathematics

Develop your interest in high school. Maybe you have always been fascinated by numbers and how formulas explain the world in which we live. If you think mathematics might be something you want to pursue further, focus on your math courses in high school. Ask your teachers for extracurricular activities you can participate in to hone your math skills, such as quiz bowl or mathletes. Try to solve this classical problem: A Pythagorean triple is three whole numbers a,b,c with a + b = c, such as 3,4,5 with 3 = 9, 4 = 16, 5 = 25, and 9 + 16 = 25. Find other examples; find a formula to generate more and more of them; finally find all possible such a,b,c. Make the problem not something you learn about, but something you understand for yourself (along with friends). Compete in mathematics competitions such as the International Mathematical Olympiad.

Be persistent. Math is not a subject that you can just sit down and memorize. Many of the concepts take time to fully understand. Do not be discouraged if you don’t comprehend right away. If you are easily discouraged by complex problems, mathematics may not be the right field for you. When learning math, don’t just memorize formulas. Think about the problem itself and try to figure out a way to solve it.

Seek out a small study group or private tutor. There are some very easy concepts in math and there are some very difficult ones. While self-study and online tutorials are helpful, you cannot become a mathematician all on your own. Find a like-minded group of peers that you can study with or study one-on-one with a professional math tutor. If you are still in high school, ask one of your teachers if they know of any groups that you can join. If you are at the college level, find a math club or society to join.

Do math. This step may seem obvious, but it is one thing to study math and quite another to actually do math. Explore mathematical problems and concepts that go above and beyond what you may have learned in school. Do problems from your textbook and find more problems online to master difficult concepts. EXPERT TIP Joseph Meyer Joseph Meyer Math Teacher Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. Joseph Meyer Joseph Meyer Math Teacher Master math basics before exploring online tools. Traditional math learning methods like using manipulatives and working on paper build a strong foundation. Once you're proficient in those, online tools can help you tackle complex problems, visualize concepts in new ways, and solidify your understanding.

Read mathematical journals and books. While it is important to stay current on what is happening today, it is equally as important to read about the history of mathematics and how it developed. Study the original papers and proofs that were developed hundreds of years ago to obtain a solid foundation in the subject. Many mathematical proofs and concepts have evolved and been simplified over time. Reading the original paper will help you understand how it has been streamlined to the form in which it exists today.

Be passionate about math. In order to stay properly motivated, you must have a passion for what you are doing. If you are not motivated by the idea of discovery or working through a problem, you will be easily discouraged in your studies. Don’t try to force your passion. If it’s not something you truly love, it is not likely to magically become something you love. On the flip side, you may be passionate, but lack inherent skill. Math may not come easily to you, but if you love it and are willing to put in the necessary time and effort, you can become a good mathematician.

Collaborate with your peers. Gaining a fundamental understanding of mathematics mostly happens with intense and consistent studying on your own; however, asking questions and conversing with your peers can lead to personal insights or a new perspective on an old problem. Make time to stop for conversation and discuss difficult problems with fellow mathematicians.

Take a walk. When you have encountered a particularly difficult concept, oftentimes, it is best to put it to the back your mind for a bit. Let your brain chew on the problem without pencil and paper and see if you can come up with an interesting solution. Even if you don’t come up with a groundbreaking new theory, taking some time out of your day to enjoy nature is a productive use of your time.

Thinking Like a Mathematician

Question everything. When looking at something that is said to be true, see if you can prove it. If someone tells you something is true, ask them to prove it to you. Trying to prove something to yourself will help you exercise your math brain and start thinking about all math problems in this way.

Study logic. Logic is the foundation of a mathematical argument and necessary to developing a proper proof. There are self-study guides online or textbooks you can purchase to study logic. You can also sign up to take a logic course at the local college. Spend time working through logic problems and cultivating that type of thought process. Logic will help you write proofs in an ordered way that people won’t be able to find error with.

Write in sentences. Although it may seem strange to use sentences to explain equations and numbers, a proper mathematical proof is written out in a grammatically correct sentence. If you can’t write your argument in a sentence, you may not fully understand what you are trying to say and need to think about it more. When writing a proof, always write in full sentences. Study old proofs to get an idea of the proper format to use.

Rewrite statements with the contrapositive. The contrapositive of a statement “If A, then B” is “If not B, then not A”. As an example, the contrapositive of the statement “If I’m from New York, I am American” is “If I am not American, I am not from New York”. The contrapositive is always true which can sometimes be difficult to wrap your head around. Always rewrite statements with the contrapositive to exercise that thought process.

Ask yourself about the converse of a statement. The converse of a statement such as “If A, then B” is “If B, then A”. The converse is not necessarily a true statement. When studying a problem, always ask yourself if the converse is true and then try to prove it. Even if it’s not true, it’s a good exercise to keep in mind when thinking about problems.

Pursuing a Higher Education Degree

Obtain a bachelor’s in mathematics. It’s necessary to have at least a bachelor’s degree to start a career in mathematics. A master’s degree is recommended, but a bachelor’s is where to start. Choose a university that has a well-known reputation for their mathematics program: Harvard University University of Cambridge University of Oxford Massachusetts Institute of Technology University of Chicago

Choose between applied and theoretical mathematics. When pursuing an advanced degree in mathematics, it is necessary to make a preliminary choice between studying applied and theoretical mathematics. While there is overlap between the two, advanced research will usually focus on one or the other. Applied mathematics uses established theories to solve problems in the real-world with mathematical modeling. Theoretical mathematics push the boundaries of what we know and come up with new concepts to explain scientific and engineering questions.

Complete a master’s degree and/or doctorate. Advanced degrees are often considered to be necessary to be a mathematician. While you can start your career with a bachelor’s, pursuing a master’s or doctorate will increase the job opportunities open to you. When choosing a university, again consider the reputation of the program and the professors who teach there. Persistence is important when pursuing a doctorate. The first year or two can be very difficult because all of that you must learn to develop a research project.

Learn computer modeling software. Many aspects of mathematics require proficiency in computer programs such as R, SAS, and MATLAB. These programs are not intuitive so taking a course or getting private tutoring is recommended. There are many online tutorials that are useful for learning these programs, as well.

Finding a Job

Explore potential careers. Options for a career in mathematics vary widely. There are many careers available both in industry and in academia. Jobs in industry generally involve mathematical modeling for scientific/biomedical research, economics, encryption, business analytics, actuary analytics, and more. Careers in academia can involve teaching at any level and advanced research similar to that done during a doctorate. While pursuing advanced degrees, speak with alumni from your program and your professors to find what is a good fit for you.

Apply for internships. Internships are a great way to test out a career and gain some necessary experience. You can apply for internships at any stage of your career or education. Many universities have programs that will help place you into an internship that aligns with your career goals. Math societies also promote internship opportunities. Relevant internship opportunities might include: A research position with a professor Mathematical modeling with a biomedical or pharmaceutical company Statistics and analytics in economics Deciphering encryptions for military or law enforcement

Attend mathematics conferences. There are hundreds of mathematical societies that put on conferences every year. Some of them are more general mathematics, but there are many that are more concentrated on specific branches such as geometry or calculus. Conferences are a great venue to learn about the current research and network with peers in your field. Regular conferences are hosted by societies such as: American Mathematical Society European Mathematical Society Society for Industrial and Applied Mathematics

Publish in mathematical journals. A strong publication record is a great way to boost your resume and get noticed when applying for jobs. Journals articles show that not only have you produced research that has advanced the field, but that you know how to communicate that research to the public.

Network with other mathematicians. The best way to find a job is to meet people within the field. The more people you know, the more job opportunities you will hear about. These individuals might also be able to put in a good word for you when you are applying for the job to make your resume stand out from the crowd. Conferences are a great place to meet new people in your field of study. Being part of mathematical societies is also a great way to expand your network. Join the alumni board of your university after you graduate to get in touch with older students. If speakers come to give talks at your university, ask them to lunch or out to coffee.