Advice on Undergraduate Research Projects in Mathematics


This is a brief guide to undergraduate research projects in Mathematics, along with some advice to help you get the most out of your experience.

Unique Aspects of Undergraduate Research Projects in Mathematics

It is important to realize that undergraduate research in mathematics is often different than research projects in other disciplines, even other sciences. In many other disciplines, it is often possible for a faculty member find an aspect of their own research to which an undergraduate can contribute. In these disciplines, an undergraduate research assistant works on one aspect of the professor's larger, existing research project (such as a potion of lab work), and in doing so the undergraduate student provides a service to the faculty member and actually helps the research be done more quickly.

In Mathematics, on the other hand, it is often very difficult for a faculty member to find an aspect of their own research to which an undergraduate can contribute. Much current research in mathematics requires a background in graduate-level mathematics, and it is often the case that even Ph.D. students must spend months or a year to get up to speed on a topic or just to get to the point where they understand what the relevant questions in the subject are asking. Consequently, it is often not possible for an undergraduate, in the short time period of one or two semesters, to obtain the background necessary to work on a faculty member's mathematics research and make (even small) contributions. Moreover, even when undergraduates can make contributions to a professor's mathematics research, the time and supervision required on the part of the professor often slow down the research rather than speed it up.

While some mathematics professors do occasionally include undergraduates in their own research projects, it is more common for a professor to create a project specifically for an undergraduate. These may be projects inspired by the professor's own research and in their area of interest, or they may be projects completely unrelated to the professor's research, but in a topic the professor is interested in or thinks would be a good learning experience for the students.

This situation has several advantages for you as the student:
  1. You know that your professor is involved in your project because they want to be and because they want to work with you. Everything they do in the project is for you and not for them. They are free to focus on mentoring and supervising you, rather than be concerned with the outcome of the research.
  2. Unlike other disciplines where you have to jump on to an existing research project, research projects in mathematics are often tailored with you in mind. Mathematics professors choose topics and projects because they think they will be good learning experiences for the students involved. The goal is not to "create new science", but rather to train you as a researcher.
  3. You do not have to worry about producing results or be under pressure to make your project "successful". If you learn something in the project and it contributes to your education as a mathematician, then it is a success.


Things to Keep in Mind about Undergraduate Mathematics Research Projects

When participating in an undergraduate research project in mathematics, you should keep the following things in mind:
  1. Focus on the "Research Experience" rather than the Product Produced. It is more important that you learn how to do mathematics research than it it is that you learn a particular topic in mathematics or discover something new. Mathematics research and applying mathematics in real world situations is often vastly different than classroom work. Unlike homework problems or proofs in your classes, which may require a few hours of work and during which you can ask for help, mathematics research usually requires you to work on problems that require weeks or months of study during which you spend much of your time being "stuck". Also, unlike classroom mathematics, the goal is not to learn a particular fact or master a particular technique by working exercises, but instead to look at an interesting situation or phenomenon and and explain it the best you can. Mathematics research involves exploration, investigation, and discovery. It is a much more creative activity than what you experience in the classroom, and it requires very different skills. Moreover, it doesn't matter if you discover "something new" in your research, as long as it is "something new to you". The act of discovery, and the skills you develop by figuring out a topic by yourself from the ground up, are exactly the same whether you are producing something that no one else has seen before or whether you are re-discovering a theory well-known to experts.

  2. Understand that Skills Learned in Research Projects are Transferrable. Regardless of what the particular topic of your research project is, understand that it is helping you to build and hone the skills that will make you a better researcher in all future projects. This includes not only mathematics research, but also applying mathematics to real world situations. The particular topic of your undergraduate research project is not so important, as long as it is something you are interested in. You are not choosing a research path for the rest of your life -- in fact, even if you go on to be a mathematician, it is incredibly likely that your future work will have anything to do with the topic you research as an undergraduate. What will matter is the research skills you develop. Research projects require you to repeated apply what you know in the context of motivating and meaningful problems. This leads to a greater understanding and retention of your current mathematics knowledge, and also helps you to make connections between ideas. It also develops new critical thinking skills, new problem-solving techniques, and helps you to develop a mathematical aesthetic as you practice making choices about which aspects of a problem to investigate.

  3. Write Up Your Results at the End of the Research Project and Give a Talk or Presentation on Your Work. Research consists of two parts: (1) Figuring Something Out, and (2) Communicating It to Others. If you learn something new, but don't share it, then you are only doing half of a researcher's job. Since the main goal of an undergraduate research project is to teach you how to be a researcher, it is important to incorporate training in mathematics communication as part of your project. You should write up the results of your research, with your professor's supervision and guidance, and in doing so learn the conventions and standards of mathematical writing. If you produce a particularly good write-up, you can even consider submitting it to a journal that publishes undergraduate research. In addition to writing up your results, it is also a good idea to give a talk (or poster presentation) on your research (possibly at a department seminar, an "undergraduate research day" at your institution, or at a conference that welcomes talks or poster presentations by undergraduates).

  4. Recognize Your Professor's Contributions. As stated above, most mathematics professors do not benefit their own research by supervising an undergraduate research project. When a mathematics professor supervises an undergraduate research project, they do it for altruistic reasons: they want to help students in their education and they want to contribute to the profession by training the next generation of mathematicians. Keep in mind that professors are incredibly busy and value their contributions. Any mathematics professor who does a research project with you is doing so voluntarily. They usually receive no compensation or recognition from the University for work supervising and undergraduate research project, and do it solely because they wish to help students.









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