Math Capstone Course Examples: A Deep Dive into Senior Projects

Mathematics students often culminate their undergraduate studies with a capstone project during their senior year. This project serves as a demonstration of their accumulated knowledge and skills, fulfilling the Senior Comprehensive Requirement. For students in the honors program, this takes the form of an Honors Project. The capstone experience provides an opportunity for students to explore a mathematical topic in depth, guided by a faculty member, and to share their insights with the broader academic community. Many students choose topics that connect their mathematical understanding to their broader interests.

Capstone Project vs. Honors Project

The standard capstone project involves the student taking a one-credit course, MATH 492, during their final semester. In contrast, the Honors Project is a more extensive, year-long undertaking. Honors students engage in intensive work to define and solve a scholarly problem or develop a creative work. Their work is guided by an advisor and reviewed by two readers. Upon completion, the project is presented publicly to faculty, students, and guests. Honors students enroll in the three-credit course HON 498/499 during their senior year.

Examples of Capstone Projects

The diversity of topics explored in math capstone projects is impressive. Here are some examples of capstone projects and honors projects, illustrating the range of mathematical interests and applications:

1. Cauchy Residue Theorem and Real Definite Integrals:

One student, mentored by Dr. Keon, was inspired by an interest in problem-solving. This project demonstrated how the Cauchy Residue Theorem (CRT) could be used to evaluate certain real definite integrals that are challenging to compute using traditional methods.

2. Hamiltonian Cycles in Grids:

Dr. Julia mentored a project that sought to determine the minimum number of turns possible for a Hamiltonian cycle in an (m \times n \times \ell) grid. This project built upon existing research published on arXiv, with the student providing exact solutions for specific cases such as (2 \times 2 \times 2) and (3 \times 3 \times 3) cubes.

Read also: Navigating Math Courses

3. Lebesgue's Theorem and Riemann Integrability:

Under the guidance of Dr. Walter, a student proved a portion of Lebesgue's theorem. The theorem states that if a bounded function on a closed, bounded interval has a set of discontinuities of measure zero, then the function is Riemann integrable. The project also included an example of a function that is Lebesgue integrable but not Riemann integrable.

4. Mathematical Approaches to Gerrymandering:

Dr. Nathan mentored a project that explored promising mathematical approaches to detecting and reducing gerrymandering in the creation of congressional districts. The student discussed partisan symmetry and the efficiency gap as key concepts.

5. Sudoku Grids:

Dr. Kaylee supervised a project concerning Sudoku grids, which are solutions to Sudoku puzzles. The project touched upon the fact that the number of "essentially different" (9 \times 9) Sudoku grids has been calculated to be over five billion.

6. Bayesian Methods and Golf Handicaps:

Combining a love of mathematics and golf, Dr. Teddy mentored a project that explored how Bayesian methods could be applied to address the problem of "sandbagging" in golf handicaps. Sandbagging refers to the practice of a player intentionally inflating their handicap index to gain an unfair advantage.

7. Fermat's Last Theorem:

Dr. Brett guided a project related to Fermat's Last Theorem (FLT), one of the most famous problems in mathematics. The theorem, stated by Pierre de Fermat in 1637, asserts that the equation (x^n + y^n = z^n) has no positive integer solutions when (n > 2). The student followed the work of Carl F.

8. Survival Analysis:

Dr. Vinnie mentored a project focused on survival analysis, a statistical area concerned with time-to-event data, such as the deaths of living organisms or the failures of machines. The student provided a survival analysis for a COVID-19 study and a Primary Biliary Cholangitis (PBC) study, employing statistical methods like Kaplan-Meier curves and Cox Proportional hazards, and utilizing the R programming tool.

9. Data Analytics in Baseball:

Drawing on a student's experience as a catcher for the Bonnies baseball team, Dr. Teddy mentored a project that applied math to baseball. The project demonstrated how data analytics can be used to help pitchers refine their strategies against different types of batters, using the R program.

10. Elliptic Curve Cryptography:

Dr. Erica supervised a project that provided background on private and public key cryptography and elliptic curves. The student described the elliptic curve cryptosystem (ECC) in detail and explained its efficiency compared to the more commonly used RSA, concluding with applications of ECC to e-commerce.

11. Symmetries of Algebras:

Inspired by open questions in a paper by Chelsea Walton, Dr. Ben mentored a project that found the symmetries of a three-dimensional q-polynomial algebra and investigated the symmetries of the split-quaternion algebra.

12. Sudoku Grids and Latin Squares:

Dr. Gillian mentored a project that examined Sudoku grids as special cases of Latin squares.

Read also: Strategies for Adult Math Success

Real-World Applications and Interdisciplinary Connections

These examples demonstrate that math capstone projects are not confined to abstract theoretical concepts. They often involve real-world applications, interdisciplinary connections, and the use of computational tools. For instance, one project at Georgia College explored the impact of the COVID shutdown on student grades, another examined taxicab distances, and a third investigated mindsets about math abilities. These projects showcase the versatility of mathematics and its relevance to diverse fields.

COVID-19 and Academic Performance: One senior, Seth Rozelle, investigated whether the distribution of grades at Georgia College changed significantly due to the COVID-19 shutdown. His research revealed a notable increase in A's, a decrease in failing grades, and a slight increase in withdrawals during the spring of 2020.
Taxicab Geometry: Natalie Taylor explored the concept of "taxicab distance," which represents the distance traveled along a grid-like path, as opposed to the direct Euclidean distance.
Mindsets and Math Education: Morgan Grey studied the stigma surrounding math and found that while children initially have positive attitudes toward math, these attitudes tend to decline as grade levels increase. She also discovered a similar issue with the mindsets of teachers.

Project Options and Customization

Students may be given a choice of project options, allowing them to select a topic that aligns with their interests and skills. For example, pre-calculus, algebra 2, and geometry students might have several project options to choose from, with geometry students able to select options based on algebra content (e.g., parabolas) or topics like volume, area, and surface area. Some instructors may also allow students to design and create their own projects, fostering creativity and independent learning.

The Capstone Experience: Developing Essential Skills

The capstone project is more than just an academic exercise; it's a transformative experience that helps students develop essential skills for their future careers. As Dr. Blumenthal emphasizes, these projects foster critical thinking and the ability to make informed decisions based on evidence. Students gain experience in:

Research: Conducting literature reviews, gathering data, and analyzing information.
Problem-solving: Identifying and addressing complex mathematical problems.
Communication: Presenting their findings in a clear and concise manner, both orally and in writing.
Time management: Planning and executing a project over an extended period.
Independent work: Taking ownership of their learning and working autonomously.

Morgan Grey, a math education major, described her capstone experience as "nothing but a positive experience," highlighting the development of "proper research skills and time management techniques" and the enhancement of her professionalism.

tags: #math #capstone #course #examples