Connections among the Disciplines

Mathematics has many natural connections to other fields in science disciplines, as well as areas in economics and finance.

Helping students connect mathematics to other disciplines is very important to me. In lower-level courses, I enjoy exposing non-majors to mathematical ideas that are useful in the real-world. In upper-level courses, I aim to connect mathematical content to applications in other fields (see examples below).

I have found that using application problems to practice course material can showcase the importance and relevance of otherwise seemingly unimportant and mundane material. It is my aim to include numerous applications that bring the material to life. I also try to incorporate those applications relative to the students' majors.  

I find that many of my students do not have a background in basic physics, and if I cover a physics application too deeply that some students do not enjoy it. I have worked to include any prerequisite knowledge when covering application topics, and limit the amount of physics related applications. The exception is in calculus 3, where I describe the ideas of electric charge, electric potential, and electric field, all in order to apply vector calculus to describe Maxwell's equations in electrodynamics by the end of the course. I enjoy inserting some history of math into my courses and Maxwell's accomplishments in the late 1800's was a major mathematical achievement which today is simplified through our modern use of calculus notation. I use this as a reoccurring application that builds to an understanding of Maxwell's equations in relation to certain line, surface, and volume integrals. The following comment is from a section of calculus 3 I taught the semester after the previous comment.

Connecting Calculus 1 to other disciplines

Most of the students who sign up for my Calculus 1 courses are biology majors and very few are math majors. I therefore created many activities and applications based on biological concepts. These applications include the following topics: Mechanics, Fluid Dynamics, Species Richness, Niche Overlap, Pathogenesis, Disease Virulence, and Blood Flow (venous & cerebral). Below are several other applications from my lecture notes that I provide to my students.

Connecting Linear Algebra to other disciplines

I often see non-math majors in linear algebra, such as economics, chemistry, and computer science, and I prefer choose applications based on the interests of the students that sign up. As I get to know my students, they reveal their interests and I can plan my applications accordingly. For example, when I taught linear algebra in fall 2022, a student showed me something they knew about (so-called vampire matrices) that was related to the material in the course and was something they found interesting. I was able to look up the topic more, learn about it, and incorporate that topic as an application later in the course. Examples of other applications I have included are problems in: 

• finding chemical reactions among geological phases

• using Google’s page rank algorithm

• decoding linear codes using nearest neighbor decoding

• solving a system of linear differential equations of order one using eigenvectors

• cryptography—using matrix inverse to encode/decode a message

I provide two excerpts below from my linear algebra class notes that showcase an application I incorporated into the course. The first one is about balancing chemical equations using a system of linear equations. The second one is about the Leontief Input-Output model that describes economic dependence using a system of linear equations.

I have incorporated several short essays throughout the course as part of the final grade (complete 1 for a D, 2 for a C, 3 for a B, 4 for an A). The essay prompts I used were:

 In 7 - 10 sentences,

•       describe the different ways a system of linear equations may or may not have a solution. 

•       describe how a system of linear equations is used to balance a chemical equation. 

•       describe in your own words how nearest neighbor decoding works. 

•       describe in your own words the correspondence between a matrix and a linear transformation.

•       describe in your own words what the determinant of a matrix represents graphically.

These essay prompts were meant to have students looking up associated history, paraphrasing concepts in their own words, and using a worked-out example to aid in their discussion. Some students excelled at composing the essays, but I was surprised at how much other students struggled to compose an essay. It is apparent that our students will greatly benefit from having more practice with writing and I aim to include such assignments in my courses. 

Connecting Differential Equations to other disciplines

Differential equations comprise the mathematics of modeling physical phenomenon. Thus, this course has natural connections to physics and chemistry in articular. Some of the applications I have used in this course involve the following:

• Growth/decay models

• Newton’s law of cooling

• Circuit analysis with batteries, resistors, and capacitors

• Logistic growth and other population models

• nth-order Chemical reactions

• Mixing solutions

• Problems involving Newton’s second law

Integrating mathematics and finance

During the summer and fall in 2021, I worked with James Farrell (in the Finance Department) to create a finance-related course for math majors with little to no prior exposure to financial topics. My interest in finance began the previous year when a student in the math program asked me to mentor an honor's project based on the mathematics in finance. My idea for the course was to introduce the concepts and math behind the time value of money and common financial instruments such as bonds, stocks, and stock options. I wanted to also describe how these instruments can be priced, leading up to the Black and Scholes options pricing model and some of "The Greeks" by the end of the course.

Not being an expert in finance, I asked James Farrell if he was willing to teach the course with me. He agreed to it as an overload. We decided to call the course "Math Behind Finance" with a double prefix MAT/FIN. 

Based on the success of teaching the course and the feedback we got from the students, I plan to break this course into a two-semester sequence that will could become part of a finance and accounting concentration within an applied math major. The first semester will comprise the background information of stocks, bonds, forwards, and options, as well as using Excel and a financial calculator, all at a slower pace. The second course will investigate pricing models for options based on the instruments studied in the first semester.

I end this section with some comments I have received in my evaluations that mention the applications of my course material.