Fall 2018 was our figure-out-how-things-work semester for the directed reading program at Wesleyan. If you've never heard of it, it's awesome! In it, grad students work one on one with undergraduates in a reading course setting. It's called the directed reading program for a reason; the focus is on reading math, not research. That means that we can focus on close reading, communicating math, and asking lots of questions as skills for the undergraduates we work with. As grad students, we get to work on being mentors.

Our first batch of grad students were Lisa Kaylor, Andre Oliveira, and myself, with Felipe Ramirez as our faculty mentor. We all met our students once a week for 1-2 hours, and near the end of the semester, the students each gave a 20 minute presentation to a group of undergrads, grad students, and math faculty. Of course, there was also pizza.

The student I worked, Winona, with was interested in learning more about hyperbolic geometry. However, she hadn't taken a real analysis class yet. So, instead of any of the book's I had originally thought about using, we used Marvin Greenberg's *Euclidean and Non-Euclidean Geometries: Development and History*. According to its preface, it can be used as a text for anything from high school geometry classes to undergraduate math majors. It also takes an axiomatic approach to things, which can be really fun! I also appreciated that there were exercises at the end of every chapter. We mainly used them as discussion tools , but there were a lot of really great questions.

Once I figured out a book to guide us, I had to figure out a plan for us to work with. I use zenkit for a variety of things, including keeping my thoughts together for research. I think of it as a bunch of digital post-its that I can rearrange onto different lists and make notes on. What I did for this, as to start with a vague idea of the chapters and ideas that I thought we'd get to each week, and the closer we got I would update with more details. After we met, I would go back and flesh out the details. Here's what a part of it looked like:

Another thing that was great at keeping both Winona and I organized was sending a weekly wrap-up email. A sort of detailed list of the things we talked about, the big questions she had that were answered, the questions we both still had, and the things to think about for our next meeting. I sent the first two, and then she sent one to me for each of the following weeks. They were great, and she was super observant. Her emails also gave me a better idea of what she felt like the main focus or most interesting parts of each of our meetings was. We also decided together what the 'big' or surprising things things were that we had talked about that week. I then kept those as a separate list to remind her of if she couldn't decide what to put in her presentation at the end of the semester.

I think for this spring semester, I'll keep the same organizational strategies, just maybe a little more fine tuned. I think something that will change for how we do things is to give the students in the directed reading program more opportunities to interact before the end of the semester, close to presentation time. Maybe a pizza lunch with mentors and students, or trying to set up a weekly study group where they can meet and talk about things. That was something that they all seemed interested in, so I hope to get something like that going.

**Main note on the first semester: **It was amazing and totally worthwhile, and I can't wait to do it again!

Bonus: Here are some fun pictures that Winona drew to show that under Hilbert's neutral geometry axioms (in Greenberg's book), if rectangles don't exist, then the negation of the parallel postulate holds. In that situation, that means we can choose "Rectangles do not exist" as an axiom to give us hyperbolic geometry instead of the hyperbolic axiom.

You start with a line l, and construct two unique lines, m and n, that are both parallel to l. (m is perpendicular to a line that l is also perpendicular to, so m and l are parallel. The same goes for n.

Because rectangles don't exist, m and n can't be the same line! If they were, there would be right angles at P and Q, meaning that m, l, and those two perpendiculars would form a rectangle.

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