#### Tuesday, July 5th, 2011...4:04 pm

## A model for the community

Service-learning offers unique educational opportunities for students to apply their learning to issues of community interest. This post will discuss how I have integrated service-learning into a final project used in a discrete math modeling course. The course teaches such topics as Monte Carlo simulation, queuing theory, Markov Chains, and optimization. The capstone group project requires students to create a mathematical model for a problem posed by a local nonprofit organization. Topics included designing the layout of a parking lot for a local food pantry and assigning elementary students into practicum groups according to scheduling constraints and indicated interests for a local elementary school. The final model and recommendations from it were presented in a public poster session and in a paper.

Why use modeling in service-learning? The following quote from John von Neumann (a noted mathematician and computer scientist which is fitting given the ways this class integrates math and computing) motivates a reason:

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

Projects from the community that lend themselves to the discrete modeling topics of this course ideally contextualize learning math modeling. Why model? The world is often too complex to understand otherwise. Math modeling ideally can simplify a phenomenon and still capture its general behavior, offering analysis that is otherwise difficult if not impossible.

Seeing the role of modeling, I needed projects. At Davidson College, this is done with the aid of the Center for Civic Engagement. To aid that office in its call for proposals, I supplied examples of the types of problems my students study and solve. Such examples enable leaders in the non-profit organizations to think about their needs in relation to our skills. In the end, I need 6 projects for the class. Both years, I made personal contacts with organizations to create one or two projects that supplemented those submitted to the Center. Once the call is made, I receive the project proposals and in each case think about how I might tackle them. It should be noted that while I tell the students my ideas they are not bound to them.

After selecting the projects, I communicated with the community partners regarding follow up questions and the schedule (noting the weeks that students would want to meet and the time and date for the poster session). The email also thanked the partners for taking part in the educational process of the students. While the hope is that the students’ work would produce helpful and insightful results for the nonprofits, working on open-ended problems inherently involves unpredictable outcomes, which in itself is a unique and important learning environment for the students.

The students prepare for these final projects through the content learned through the course and a mid-semester project. The midterm project is similar although not set within a community-based problem. The earlier task requires working in a team of 3 on the Mathematical Content in Modeling (MCM) sponsored by COMAP. In 2011, three thousand teams participated from around the world. My students work on the discrete problem. The students’ success and fondness for these problems played an important role in my considering service-based problems as a final project.

In terms of content, the class introduces Monte Carlo simulation and queues. Simulation is often a tool employed on the MCM. They also learn Markov Chains which includes a discussion of PageRank (both as a simulation and as a Markov Chain). To see a way of introducing PageRank as a simulation and Markov Chain see my co-authored*Loci*article Google-opoly. Since Markov Chains can be solved using simulation, this technique, while important and useful, is rarely used on the MCM or final project. The last tool is linear programming. Students often use this on the final projects. A fun binary integer programming problem is applying such techniques to Sudoku. For a

*Loci*article on this, see my co-authored article Integer Programming Model for the Sudoku Problem.

To begin the projects, I select teams. While students can request team members, I make the final decisions. Teams are assigned such that each group has a programmer and a writer. The last person is selected with mathematical diversity in mind. Generally, the groups work quite well together and such composition is helpful for the projects.

The teams are announced on a Monday. At the same time, I present all 6 projects and give a summary of how I would approach the problem. Again, many teams end up developing their own approaches. Still, my ideas serve as a helpful springboard for their work. The next day, a representative from each group submits their team’s ranking for all 6 problems. I assign projects to teams in the order the teams submit their rankings, which begin at an announced time. The first team gets their first choice. All remaining teams get their highest ranked problem that has not yet been assigned. The submissions are made online using a Blackboard discussion list. This enables me to set a specific time, like 10 AM, as the *moment* submissions will be accepted. From here the class follows this schedule

- Week 1 – Teams assigned, topics presented and assigned
- Week 2 – Proposal due – These are returned with comments by the next class period.
- Week 3 – Revised proposal due – Each proposal is edited with my comments in mind.
- Weeks 3 & 5 – A day of class is set aside for project meetings. If helpful, students can meet elsewhere during this time, which sometimes allows for a visit with the community partner.
- Week 6 – Poster session that the community partners attend along with members of the public and college community.
- Week 7 – Final paper is due. The paper includes an executive summary that is suitable for the community partner to read and know the recommendations of the group. The paper 8-10 pages (a maximum of 20 pages) includes the details and is what I read to assess their modeling techniques.

Here are two projects that give a sense as to their variety.

**Community School of Davidson** (CSD) – Assign students into practicum groups (where each group is capped at a maximum size that may vary from group to group). In particular, students fill out an application giving their top three choices for assignment.

For this problem, students used linear programming. The problem contains a lot of decision variables. As such, it couldn’t be solved using the Solver tool of Excel. Students decided to automate the process and use the NEO Server for Optimization.

**Ada Jenkins Center** – Design layout for the food pantry’s new location that will include walk in refrigeration units, shelves and storage.

The biggest part of this problem involved comparing configurations of shelving. The team created by horizontal and vertical configurations and compared their efficiency using both linear programming (for how items were shelved) and Monte Carlo simulation (for how people moved through the pantry). Below you see two of the layouts created by the team.

The team compared configurations, balancing amount of people serviced by layout with the amount of food stored, that also did and did not allow passing of the shopping carts used to collect the food. This, like the CSD project, was well received by the community partner. The ideas were used. With a few tweaks to student work, the pantry “managed to get approximately 40% more space in the new pantry and it is now possible for carts to pass in the aisles.”

The service-learning projects are as applied, in many ways, as applied math can be! The students get to see their gifts and knowledge of mathematics benefit the community. They leave their intellectual imprint on the community in which they lived for 4 years of college. The students reflect these types of insights in their comments in the student evaluations. Here are two:

“I thought the MCM and final project were awesome. Not only did I learn a great deal about how to write a math modeling paper but I also grew very close to my teammates. By the end of the term, I felt like the class was a family.”

“Best class I have had at Davidson; very interesting, good blend of concepts vs. real life application, challenging but fair, and a community project where we can actually see the results of our work.”

Interested in service-based projects? They can take many, many forms. This is the one that fit this course. Look up such ideas online, look for books on the topic, and envision how your class can engage with the community with their content.