*By Victor Piercey, Ferris State University*

As a graduate student working in algebraic geometry, I was often star struck at the impressive speakers who attended the local seminars I frequented. While many of these memories are faded and vague, one instance stuck with me. About three minutes into a talk, one famous algebraic geometer in the audience stopped the speaker and asked “Why do we care about this problem?” Watching such an exchange, it occurred to me that everyone needs motivation, even top mathematicians involved in abstract research. We all need purpose. Why should our students expect any less?

I have since gained a great deal of respect for the question “When are we ever going to use this?” when asked by students. These students recognize that learning mathematics takes a nontrivial amount of effort, and they are looking for purpose. The mathematician at the seminar was no different: knowing that the speaker was going to embark on a journey that took effort to follow, they wanted purpose too.

Many of our students, whether they are majors or non-majors find meaningful purpose in realistic applications. The emphasis should be on the word realistic – students will (and should) roll their eyes if a person is buying 68 cantaloupes at a grocery store in a problem!

This is where interdisciplinary collaboration comes in. It can be challenging to find realistic applications for mathematics. What’s more, you have to figure out how much to teach about the application and how much that obscures the mathematics. When working with collaborators from outside mathematics, not only do you find great applications, you get to experience being a student again. This helps you determine how much a student might need to know or learn about your applications and contexts, as well as how much a particular context makes the mathematics harder to learn.

Over the last three years, several institutions across the country have been part of a NSF-funded grant to support collaboration between mathematics and the partner disciplines to improve the teaching of mathematics in the first two years. The project is called SUMMIT-P [1] (see http://www.summit-p.com). The work of the consortium rests on research conducted by a committee of the Mathematical Association of America culminating in a series of reports called the Curriculum Foundations. [4][5][6]. There is a variety of institutions, including a community college, small liberal arts colleges, comprehensive public universities, and large research-oriented universities. There is also a variety of partner disciplines, from engineering to biology to psychology. The mathematics courses addressed include quantitative reasoning, college algebra, introductory statistics, calculus 1, 2, 3, and differential equations. Our goal is to establish collaborative, interdisciplinary communities at our institutions that facilitate the inclusion of realistic partner discipline contexts into mathematics while incorporating mathematics into partner discipline courses.

## Interdisciplinary Collaboration at Ferris State University

At my institution, Ferris State University, we are working with a faculty member from social work (Mischelle Stone) and another from nursing (Rhonda Bishop) on a 2-semester hybrid quantitative reasoning/algebra course (for the connections between quantitative reasoning and algebra, see [7]). The course sequence originated out of collaboration with business faculty (see [8]). Almost every lesson in the class is couched in some application that comes from the partner disciplines. However, the strongest and most meaningful applications come in case studies that students work on at the end of each chapter. So far, we have created case studies addressing human trafficking, genocide, a disease outbreak, and construction and management of the death star. Each case study requires a brief writing assignment framed as recommendations to a supervisor or board of directors. As an example, some of the tasks involved in the human trafficking case study are:

- Examine human trafficking data to prioritize resource allocation,
- Prepare a budget for the medical needs of human trafficking victims in a location,
- Forecast fundraising needs for a program to combat human trafficking in hotels,
- Prepare an annual budget broken down by months for a shelter for human trafficking victims (based on assumptions about how the number of guests per month changes), and
- Determine how much food to order for a human trafficking victims’ shelter from two different suppliers while minimizing the environmental impact.

For more detail, let’s look a little closer at the last item: determining how to determine which supplier to select to purchase food for a human trafficking victims’ shelter. In discussions with Mischelle about human trafficking, Mischelle shared the challenges associated with ordering supplies, especially at scale, while balancing a concern for the environmental impact of shipping the supplies. This leads to a desire to buy local and solicit donations. For smaller shelters, this is reasonable. For larger shelters, there are nontrivial logistical problems.

Our discussion led to a linear programming problem where human trafficking shelter managers have to make a constrained decision about how much food to order from different suppliers while minimizing environmental impacts measured by the total mileage involved in shipping. After a couple of preliminary questions in which students determine that they need at least 2700 meals per month (assuming a 30-day month) and want to spend less than $7500 per month, they are confronted with the following problem:

We need to decide how many trucks of food to order from each supplier while minimizing the total number of miles driven by all of the trucks from the two suppliers.

Our data is as follows:

- Our first supplier is 250 miles away. The new supplier is 400 miles away.
- Trucks from the first supplier carry 300 meals each. The trucks from the new supplier can carry 900 meals each.
- Each truck of food costs $1,500 (from either supplier).

We determined the minimum number of meals and maximum costs earlier. Within these constraints, how many trucks should we ask for from each supplier in order to minimize our environmental impact?

The problem could show up in a standard textbook as:

Minimize subject to:

The problem could also show up with some context, such as determining an optimal bundle of CDs and DVDs to purchase. But students find the human trafficking context much more compelling. They *care* about human trafficking, and they might also care about the environment. The problem feels like a legitimate professional decision they could run into. They don’t care about optimal bundles of CDs and DVDs for many reasons. First of all, such a problem is woefully out of date. Commercially published textbooks adapt slowly. But even with a more up-to-date context, students would dismiss the problem as artificial since they have been making these kinds of decisions for much of the lives without resorting to mathematical techniques such as linear programming. In addition, one may also object that such a problem promotes consumerist values, but I recognize that this is not a universal concern.

One unexpected byproduct of this problem that Mischelle and I came up with is that it can be adapted to other contexts. I had brief conversations with one of Ferris’ history professors (Barry Mehler) who studies genocide about the Shoah Visual History Archive (https://sfi.usc.edu/vha). This archive contains recorded testimonials of genocide survivors from all over the world. Ferris has recently obtained access. In my discussions with Barry, I learned that providing food for refugees fleeing genocide raises a similar problem (with different parameters). In particular, this problem could be applied to current refugee camps in Bangladesh for Rohingya fleeing genocide in Myanmar.

For both the human trafficking and genocide problems, students are asked to watch a video prior to working. For human trafficking, Mischelle provided us with a video about management challenges at a human trafficking victims’ shelter in Tampa Bay. For genocide, students watch a testimonial in the Shoah archive from a refugee discussing food distribution at a camp. While neither of these videos is mathematical, they enrich and humanize the context. This allows us to tap into the “caring” and “human dimension” components of Fink’s taxonomy of significant learning [3, pg. 2], each of which are easy to miss in a math class.

## Generating Problems and Scenarios

The point of this is that I would never have come up with this problem without collaborating with Mischelle. I probably wouldn’t have even thought of using human trafficking in a math class. And I would not have been able to extend the problem to genocide refugees.

What’s more, once you have one problem, you can generate more by asking students to go back and reconsider the original parameters. For example, in the human trafficking problem, the solution is to order all of the food from the second supplier. One could ask whether the first supplier could lower their price sufficiently to change the outcome in their favor. This leads to deeper mathematical reasoning beyond just solving a linear programming problem. In addition, it asks students to put themselves in a different role, allowing them to see further complexity in human society.

In addition to the case studies, Mischelle, Rhonda, and I have designed role-playing simulations that open and close the second course in the sequence. The first is based on a fictional budget crisis at a rural health clinic and has few mathematical prerequisites (see http://bit.ly/RuralHealthClinic) while the second is based on the Flint Water Crisis and uses most of the content learned in the class. One unexpected consequence we have noticed is that students see more than the connections between mathematics and the other disciplines, they see connections among the partner disciplines as well!

To carry our collaboration further, we are facilitating a faculty learning community (see [2]) with three mathematics faculty, two business faculty, two nursing faculty, and two social work faculty. The members of the faculty learning community are split into three teams. Each team has one mathematician and two faculty from different partner disciplines. The teams are currently developing scenarios that will be translated into course materials for both the quantitative reasoning sequence and in the partner discipline courses. The scenarios they have developed are:

- Managing a hurricane shelter for low-income families that includes several individuals with chronic illnesses.
- Managing a 50
^{th}wedding anniversary banquet, following which contaminated food leads to an outbreak of food-born illnesses. - Examining local police-stop data for racial profiling and preparing a budget to implement recommendations to the police department.

While rich and realistic applications appeal to students’ practical desires, they may strike you as too utilitarian. There is much more to mathematics than how it is used. There is the thrill of problem solving, and there is beauty (see e.g. https://www.artofmathematics.org/). However you frame it, though, you are addressing a purpose to mathematics, even if that purpose is more intrinsic than extrinsic. These purposes can also be served by interdisciplinary collaboration, whether with those in the fine arts or those in game-design.

## Tips for Successful Collaboration

Collaborating effectively requires a great deal of listening. Find out what your colleagues teach in their courses. Find out what they know about what is taught in your mathematics courses. You will be quite surprised! Be patient with one another, and avoid disciplinary microagressions. One of the activities that the SUMMIT-P institutions engaged in is a fishbowl conversation: partner disciplines sit in the middle of the room and discuss questions from a protocol while the mathematicians sit along the perimeter of the room and don’t speak.

You will find language and conventions are very important when collaborating across disciplines. Create a dictionary of terms used in the partner disciplines and their mathematical equivalents. For example, economists refer to the derivative of a function as a marginal quality (marginal cost, for example). Sharing that dictionary with students will help them to see the connections between the mathematics and the application in economics.

To be clear, the kind of collaboration I am talking about happens behind the scenes, in the design of a course or course materials. There are other forms of collaboration in teaching and learning, such as team-teaching or teaching a student learning community. However you approach it, interdisciplinary collaboration can help you to define mathematical purpose for your audience, whether it is the student who wants to know why they have to learn implicit differentiation or the star professor listening to your talk who wants to know why your problem is interesting.

## References

[1] Collaborative Research: A National Consortium for Synergistic Undergraduate Mathematics via Multi-institutional Interdisciplinary Teaching Partnerships (SUMMIT-P); proposal funded by the National Science Foundation (NSF-IUSE Lead Awards 1625771 and 1822451).

[2] Cox, M. D. (2004). Introduction to faculty learning communities. In Cox, M.D. & Richlin, L. (Eds.),* Building faculty learning communities *(pp. 5-23). *New directions for teaching and learning*: No. 97, San Francisco: Jossey-Bass.

[3] Fink, D.L. (2005). Integrated Course Design. IDEA Paper 42. Available at https://www.ideaedu.org/Portals/0/Uploads/Documents/IDEA%20Papers/IDEA%20Papers/Idea_Paper_42.pdf

[4] Ganter, S.L. and Barker, W. (Eds.) (2004). Curriculum Foundations Project: Voices of the partner disciplines. MAA Reports, Mathematical Association of America, Washington, DC.

[5] Ganter, S.L. (2009). The Curriculum Foundations Project: phase II. MAA Focus, February/March, Mathematical Association of America, Washington, DC.

[6] Ganter, S.L. and Haver, W.E. (Eds.) (2011). Partner discipline recommendations for introductory college mathematics and the implications for college algebra. MAA Reports, Mathematical Association of America, Washington, DC.

[7] Piercey, V (2017). A quantitative reasoning approach to algebra using inquiry-based learning. *Numeracy*, Vol. 10, Issue 2, Article 4.

[8] Piercey, V and Militzer, E (2017). An inquiry-based quantitative reasoning course for business students. *Problems, Resources, and Issues in Mathematics Undergraduate Studies*, Vol. 27, Issue 7, pgs. 693 – 706.

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This is really cool! I love how the mathematical modeling was applied to an idea the students brought to the table. I’ve also found that students are much more engaged when they have some say in what direction to go with a topic.