How to Be a Good Mathematician

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A mathematician is an expert in the field of mathematics, which involves both deciphering theoretical abstract concepts and applying these theories to solve problems in engineering, economics, business, and other technical disciplines. [1] What distinguishes a mathematician from other scientists and technical professionals is the passion for theoretical understanding, independent of how the theories apply to the real world: an engineer really wants to build a radio, a physicist really wants to understand what radio waves are, but a mathematician really wants to understand what a wave could be, how to define waves in general and work out their possible behavior. If you have a natural craving to understand shapes and numbers, and have the persistence to unravel hard problems every day, a mathematical career might be right for you. [2]

Part 1
Part 1 of 4:

Studying Mathematics

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  1. 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 2 + b 2 = c 2 , such as 3,4,5 with 3 2 = 9, 4 2 = 16, 5 2 = 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. [3]
  2. 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. [4] If you are easily discouraged by complex problems, mathematics may not be the right field for you. [5]
    • When learning math, don’t just memorize formulas. Think about the problem itself and try to figure out a way to solve it. At a higher level, you can't rely on memorizing question formats and regurgitating facts—rather, you need to understand why principles work and apply your skills to new and abstract scenarios. Think of math not as a set of isolated formulae but as a deeply interconnected discipline.
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  3. [6] 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.
  4. 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. [7]
    • Do problems from your textbook and find more problems online to master difficult concepts.
    EXPERT TIP

    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
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    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.

  5. 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. [8]
    • 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.
  6. 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.
  7. 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. [9] Make time to stop for conversation and discuss difficult problems with fellow mathematicians. [10]
  8. When you have encountered a particularly difficult concept, oftentimes, it is best to put it to the back of 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. [11]
    • 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.
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Part 2
Part 2 of 4:

Thinking Like a Mathematician

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  1. 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. [12]
  2. 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.
  3. 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. [13]
    • When writing a proof, always write in full sentences.
    • Study old proofs to get an idea of the proper format to use.
  4. 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. [14]
    • Always rewrite statements with the contrapositive to exercise that thought process.
  5. 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. [15]
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Part 3
Part 3 of 4:

Pursuing a Higher Education Degree

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  1. 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. [16] Choose a university that has a well-known reputation for their mathematics program: [17]
    • Harvard University
    • University of Cambridge
    • University of Oxford
    • Massachusetts Institute of Technology
    • University of Chicago
  2. 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 pushes the boundaries of what we know and comes up with new concepts to explain scientific and engineering questions.
  3. 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. [18]
    • 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. [19]
  4. Many aspects of mathematics require proficiency in computer programs such as R [20] , SAS [21] , and MATLAB [22] . 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.
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Part 4
Part 4 of 4:

Finding a Job

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  1. 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.
  2. 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: [23]
    • 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
  3. 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 [24]
    • European Mathematical Society
    • Society for Industrial and Applied Mathematics [25]
  4. A strong publication record is a great way to boost your resume and get noticed when applying for jobs. Journal 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.
  5. 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.
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  • Question
    How do I become good at math?
    Daron Cam
    Academic Tutor
    Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College.
    Daron Cam
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    Expert Answer
    Math all starts with fundamentals and working your way toward more complex equations. If you're struggling with a concept, go back to review and practice simpler equations so you understand it better.
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    Can I become a mathematician and a scientist at a same time?
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    Science of any form includes advanced maths, even chemistry. By studying science, you could also be gaining the knowledge of a mathematician. If you want a separate degree, you could always consider a second diploma on the area you want to follow, after maths or science.
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    What is a composite number?
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    A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself.
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      References

      1. ↑ http://learn.org/articles/Mathematics_Become_a_Mathematician_in_5_Steps.html
      2. ↑ http://www.howtobecome.com/how-to-become-a-mathematician
      3. ↑ https://www.imo-official.org/
      4. ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
      5. ↑ http://www.howtobecome.com/how-to-become-a-mathematician
      6. ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
      7. ↑ http://www.howtobecome.com/how-to-become-a-mathematician
      8. ↑ https://assets.press.princeton.edu/releases/gowers/gowers_VIII_6.pdf
      9. ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.
      1. ↑ https://assets.press.princeton.edu/releases/gowers/gowers_VIII_6.pdf
      2. ↑ https://assets.press.princeton.edu/releases/gowers/gowers_VIII_6.pdf
      3. ↑ http://www.kevinhouston.net/pdf/10ways.pdf
      4. ↑ http://www.kevinhouston.net/pdf/10ways.pdf
      5. ↑ http://www.kevinhouston.net/pdf/10ways.pdf
      6. ↑ http://www.kevinhouston.net/pdf/10ways.pdf
      7. ↑ http://learn.org/articles/Mathematics_Become_a_Mathematician_in_5_Steps.html
      8. ↑ http://www.topuniversities.com/university-rankings/university-subject-rankings/2015/mathematics#sorting=rank+region=+country=+faculty=+stars=false+search=
      9. ↑ http://learn.org/articles/Mathematics_Become_a_Mathematician_in_5_Steps.html
      10. ↑ https://assets.press.princeton.edu/releases/gowers/gowers_VIII_6.pdf
      11. ↑ https://www.r-project.org/
      12. ↑ http://www.sas.com/en_us/software/analytics/stat.html
      13. ↑ http://www.mathworks.com/products/matlab/
      14. ↑ http://learn.org/articles/Mathematics_Become_a_Mathematician_in_5_Steps.html
      15. ↑ http://www.ams.org/home/page
      16. ↑ http://www.siam.org/

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      To be a good mathematician, start by pursuing extracurriculars in high school that will let you hone your math skills, like quiz bowl or mathletes. Even if you’re studying something you don’t understand at first, keep studying it, since some math problems are very complex. In high school and college, seek out a small study group or a private tutor to help you understand the concepts behind the formulas. In your free time, read mathematical journals and books to learn about the history of mathematics as well as what’s happening today. Keep reading to learn how studying logic can improve your math skills!

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