Throughout the program, I asked students to consider the following overarching questions:

- How have historical events and cultural ideas shaped the history of mathematics?
- How have mathematical developments influenced history and culture?

To help answer these questions, I designed a set of readings, assignments, and site visits connected with the Florentine Renaissance, the Parisian Enlightenment and subsequent Romantic Age, the Scientific Revolution (broadly defined but with a particular focus on Galileo, Kepler, and Newton), and the 20th century.

I worked with the Education Abroad office at our University, and partnered with a logistics team in Europe to create a program that could accomplish my educational goals, while keeping the program fee at a competitive level (see Appendix 1: Fast Facts).

### Hub and Spokes

All of the students completed three weeks of online modules prior to departure to Europe. The assignments were based on the following materials. Students read or watched:

As well as selections from:

Another fine resource for biographical information on female mathematicians is the *Biographies of Women Mathematicians* website created by Larry Riddle of Agnes Scott College.

In addition to this central “hub” of assignments, each student selected a sub-topic “spoke” of their choice. My intent was that the spokes would give the program a slightly different flavor for each student. Then, because each student brought different background knowledge to Europe, they each played the role of co-instructor at some point during our travels. Examples of spokes included:

- Mathematics and War
- Codes and Ciphers
- Mathematics and Religion
- Mathematics and Architecture
- Contributions by Minority Scholars to Mathematics
- Calculus in the 19th Century
- “Impossible” Problems
- Mathematizing the Heavens
- Women in Mathematics

After students chose a spoke, I worked with them to find a small number of additional sources beyond the hub readings. For example, the students who chose Mathematics and Architecture watched the NOVA documentary “The Great Cathedral Mystery”, read “Are There Connections Between the Mathematical Thought and Architecture of Sir Christopher Wren?” by Maria Zack (2006), and also read “Architecture, Patterns, and Mathematics” by Nikos Salingaros (1999). When we visited St. Paul’s in London, those students were well-prepared to help our entire group better understand and appreciate that space.

### Assignments While Abroad

While abroad, I placed only modest expectations on the students to complete written academic work. Each student was responsible for researching one of our sites, and giving the group an introduction to that site. The students also completed a small set of “notebook assignments” associated with each site (see Appendix 2). These were questions meant to help students get more meaning from each site visit. Some notebook assignments asked students to look for certain objects—like a mathematical scavenger hunt. Others asked the students to reflect on buildings, spaces, people, or ideas. The questions were short, occasionally open-ended, and could be answered in less than 2 paragraphs. Finally, the students were responsible for updating and maintaining a daily blog of our journey (https://historyofmath.home.blog/).

### Final Papers

The major assignment of the course was a paper that addressed the two overarching questions mentioned earlier. All of the pre-departure readings and notebook activities were the raw material for the students’ final papers. Crucially, I asked the students to submit a rough draft of their paper *before *departing to Europe. Those drafts were based on their hub assignments and flavored by their spoke assignments. While abroad, I met with each student individually to discuss their draft and provide feedback. *After* our time abroad, students were required to re-write their papers and produce a final draft. I asked each student to write at least one full page of reflection on how their experiences abroad led to specific changes in their understanding of mathematics. All of the students (and myself) arrived in Europe with preconceived ideas about mathematics, history, culture, and the ways they interact. In the reflection piece, students described how their ideas changed. The reflection portions of their papers were the very best thing students have written for me in any course I’ve taught.

### Challenges

When we teach with historical sources at home, we make difficult decisions. I don’t give my students the original Latin text, so I have to choose a translation. At that point, they’re not reading Euler, they’re reading Blanton. How much cultural background information should I provide? How much do they need to know about Euler to appreciate his writing style? How can I assess if they’ve learned anything they wouldn’t have if I just taught a typical lesson on derivatives of trigonometric functions?

Teaching in historical places presents similar challenges: Cities evolve, buildings are torn down and new ones are erected. Artifacts are in museums, out of their original physical context, and are surrounded by other objects chosen to convey a particular message. When you visit a historical place, you’re surrounded by the ghosts of the past, but there are few tangible markers. In the Great Hall at Trinity College, today’s students eat their daily meals under the watchful eye of Henry VIII. There’s no sign to tell a visitor that this was where Hardy and Littlewood ate, and where Ramanujan could *not *eat, because of his diet and his faith. The casual visitor to Europe’s greatest sites is unaware of just how much mathematical history surrounds them. They can’t hear the whispers of the ghosts. My job was to help students imagine.

I am not an expert in Renaissance art, or architecture, or Enlightenment philosophy. My colleagues asked how I taught a lot of the content of the course, and I acknowledge that I’m still learning. What’s important is that I was not really the primary instructor of this program. I had a dozen co-instructors in my students, and in every city we visited I relied on local guides, professors, curators, and friends, to provide expertise. My role was to facilitate an atmosphere in which learning could occur. Some talk of being a “guide on the side” (Davis, 1994) in the classroom, and I literally was on the side (or in the back row, or even in a different part of the building) at many of the sites we visited.