Collaborative Work with Students

Collaboration with students is an area very important to me and to which I add significant value to a mathematics department. In fact, it is one of the main things I look for in an institution. In this section, I will discuss some specific examples of my work with students.

Research with an Engineering Major and Student Talks at ASPiRE

An engineering student, Nathan Hallmark, approached me in my office one day when he was a freshman in 2018 with some mathematical formulas that he had discovered while in high school. Nathan told me he was recommended by Ron Pepino, a professor in physics at Florida Southern, that he come talk to me about his work. I recognized that Nathan had some interesting work and decided to meet with Nathan weekly to discuss how he could improve his work to make it more rigorous. This meant Nathan needed to learn mathematical induction and how to type his results using a mathematical coding language called LateX. I continued working with Nathan through his four years at Florida Southern, aiming to eventually submit his work for publication into an undergraduate journal. 

Nathan had presented two formulas that required mathematical proof. After some initial work, we were able to complete the proof of the first of his formula. However, his second formula was considerably more complicated so that the technique used for the first formula was not as tractable. Over the next few years, we investigated alternative ways to work with the complicated summation formulas involved in his second formula. 

Unfortunately, Nathan and I were unable to complete the proof of his second formula before Nathan graduated in 2022. However, I am continuing to work with Nathan writing up our current results and working to complete the proof of his second formula. The current title is "On the difference of two numbers raised to the same exponent" and is planned to be published as joint work.

On February 1, 2020, I travelled with Roxanne Back (Math, Associate Professor) and about 15 students (including Nathan) to the ASPiRE (Advancing Student Participation in Research Experiences) conference at Florida Gulf Coast University. ASPiRE is an undergraduate conference where undergraduates present their research projects. Nathan gave a talk at ASPiRE about his work and how he has been able to overcome some of the challenges. Nathan presented his work a second time at the ASPiRE conference the following year on March 19, 2022. During this second talk, he was able to present updated work.

The First Florida Southern SCUDEM Team Publishes Their Model

As I discuss in more detail in the "Engagement Beyond the Classroom" section of this website, the SCUDEM challenge comprises undergraduate student teams of up to three students to create and justify a mathematical model for a given situation. The students are given three problems to choose from and have one week to create their model and conduct elementary analysis of their model and describe its limitations. After the week is over, each team presents their work in a conference style at a local site.

The first team of students that participated in the SCUDEM challenge (Zachary Fralish, Bernie Tyson III, and Anthony Stefan) did so well that they were able to continue their work after the challenge ended. One of my students received an email from Brian Winkel, the director of the SCUDEM organization who runs the modeling challenges, inviting the students to present their model and experience at the national Joint Math Meetings (an annual mathematics conference) in January 2019. He also suggested they publish their executive summary that was part of their original challenge submission. Brian also asked if our team could video record their talk so that he may include it on the SCUDEM website (along with other team submissions). Here is a link to their video submission.

I worked with the students as they expanded their model analysis for publication. The students each wrote a short testimony of their experience to include in their paper. The overall goal of their efforts was to showcase an exciting opportunity for STEM majors to get real-world experience in modeling and networking.  I also taught them how to code in LateX, so that the students could submit their article within the journal guidelines. In May 2019, Anthony Stefan, Zachary Fralish, and Bernie Tyson III submitted an article to the Rose Hulman Journal of Undergraduate Mathematics. On January 9, 2020, we got notification that their paper had been published:

“Using Differential Equations to Model Predator-Prey Relations as Part of SCUDEM Modeling Challenge” 

          Students: Anthony Stefan, Zachary Fralish, Bernard Tyson III

           Accepted for publication in Rose-Hulman Undergraduate Math Journal (August 2019)

                Link to the paper                   Link to journal publication 

In January 2019, I traveled with the students to the Joint Mathematics Meetings (JMM; an annual math conference) in Baltimore where they gave a talk as part of a special session held by Brian Winkel, the director of the SCUDEM event. For Zachary and Bernie, this was their first experience at a math conference. After the conference, all the students expressed a general excitement about the experience and a motivation to continue being involved in mathematics. Below is a picture of me with the students before their conference presentation in Baltimore as well as an email I received from Zachary after the conference.

The Second Florida Southern SCUDEM Team Publishes Their Model

In the following year, Bernie and Anthony had graduated. Zachary and I were able to recruit two new students, Jonathan Marshall and Nathan Hallmark, to form the second SCUDEM team at Florida Southern. Being successful at publishing their model the first time, Zachary was eager to publish again. After the students developed a new model, the students worked together to write an article summarizing their executive summary of their model results from the 2019-2020 SCUDEM challenge. I mentored the students during this process. Their article has been submitted to the International Electronic Journal of Mathematics Education. On January 9th 2021, we received notification that their article had been accepted for publication:

“Using Differential Equations to Model Phoretic Parasitism as Part of SCUDEM Challenge”

            Students: Nathan Hallmark, Zachary Fralish, Jonathan Marshal

            Accepted for publication in the Int. Electronic Journal of Math. Education (January 2021)

            Link to the journal publication                Link to the paper 

Future Collaboration with an Alumni

My former student Anthony Stefan is currently pursuing his P.hD. in mathematics at the Florida Institute of Technology (FIT). He has kept me up to date with the research he has been doing, and both him and his advisor at FIT, Aaron Welters, have invited me to collaborate with them on several open problems within the context of Anthony's research. I have started meeting with them virtually to discuss possible problems we could work on and steps to take that would eventually result in a publication. We have now recruited a fourth member, Ian Orzel, who recently graduated from FIT and had majored in both mathematics and computer science. Since our work involves incorporating several algorithmic processes, Ian will help us in the coding and analyzing of these algorithms from a computer science perspective. Below is a snapshot of me with Anthony and Aaron during one of our initial set of meetings.

Collaborations with Students Through Independent Studies

I have directed many independent studies and one senior seminar project while at Florida Southern. What I love about independent studies, other than when students get to learn a new area of mathematics, is that I often get to learn a new topic as well. Indeed, some of the independent studies I have led involved material that I have not studied before. I am always looking to learn new applications of math, not only for my own interests but also as possible applications to liven my course material. 

The following are the independent studies and senior project I have mentored while at Florida Southern:

• I directed an independent study on Monte Carlo simulation with one of the freshman math majors. This topic was something I had wanted to learn about and doing this independent study gave me the opportunity to learn some computer programming and the concepts behind the simulation process. 

• I have led two separate independent studies on linear algebra. One was for Zachary Fralish, a biochemistry major, and we focused the course on engineering applications. We essentially covered all the basics of linear algebra as well as some differential equations. The other student was an economics major, and we focused the course on economics applications (and used a text "for economists"). One of the main models we studied was the Leontief input-output model.  

• I was approached by two students asking me to advise an independent study of differential equations in spring 2020, based on the differential equations course within our curriculum.

• One student approached me to ask about learning finance topics. I had previously never studied finance, and always wanted to learn about it as it is so crucial to society. I decided to embark on a study of measure theory, which is a bit advanced for undergraduates but covers the underlying mathematical foundation in describing the price of financial assets. We also discussed stochastic calculus, which is used in the financial models. The main model we studied is the Black-Scholes option pricing model.

To dice deeper into this project, I reached out to a former colleague in finance at Florida Southern, James Farrell, to show me and my student how to use the Bloomberg terminals. Florida Southern Becker business building has a trading room with 8-10 Bloomberg terminals. Although the terminals sometimes do not connect with the Bloomberg server; there is no staff specifically for monitoring these terminals like there would be at a business relying on Bloomberg terminals. The student used the terminal to search for historic price data of various stocks and options to compare against the mathematical models.  

• I have mentored a senior seminar project in spring 2020. This project was on the Astros Baseball Scandal, and how statistics can be used to demonstrate that cheating was happening during certain Astros home games.

• One of my students in my linear algebra class in fall 2022 sent me a message about their wanting to learn abstract algebra and was asking if I would mind answering some of their questions as they try to read a particular textbook. I let the student know that abstract algebra is among my specialties, and I suggested doing an independent study in spring 2023. I also invited another student who I knew would benefit from an exposition on examples of groups, equivalence relations, cosets, and quotient groups. In spring 2023, I met routinely with the two students as they worked through materials that were inquiry-based.