Topics in Core Mathematics sounds like the blandest possible math class. The course title is meant to convey the one important aspect of the class for many students: this class fulfills the core math requirement for the College of Liberal Arts and Sciences. Box checked. However, the advantage of blandness is that it can be the base for anything. This spring, my colleague Katie Haymaker and I taught TCM at Graterford State Correctional Institution, a maximum-security prison in the Philadelphia area. I’ve written before about a talk we gave there and our math circle at Graterford, but this was our first chance to offer a course for college credit, through the Villanova Graterford program. Villanova is one of a large handful of US colleges and universities, including Bard College, Cornell University, and many community colleges, that bring professors into correctional institutions to offer classes for credit. This is one way of offering higher education to prisoners; other schools, like Adams State in Colorado, offer correspondence courses. Villanova’s program even gives faculty members full teaching credit for teaching at Graterford, which I have heard is fairly unusual. The goal of Villanova’s program is to be as similar to an on-campus degree as possible, with the same rigorous requirements in courses, and often the same professors teaching on-campus and off.
Some blog readers might have a chance to teach a course in this setting, though I realize that many will not. Some things about this process and course can give some wider early-career insights, though. For example, Katie and I would never have taught this course if Katie hadn’t been hanging out/networking at one of the monthly Faculty Friday happy hour events organized by the university. That’s how she first met Kate Meloney, who runs the Villanova Graterford program. Going to university-wide events like this has paid off for me in just meeting cool people, and sometimes these people connect me with something I’d never have known about otherwise. Also, co-teaching a course was a great experience for me. I had never attempted it before, and it turned out to be very rewarding. We were also very lucky that the dean of our college gave us both full teaching credit for the course. Before this, I didn’t know that I could apply for double-credit co-teaching with a colleague. I would definitely recommend it, especially if you can convince Katie to teach with you.
Our experiences might also be interesting if you are simply trying to teach a core math course on campus, since our course was attempting to be as much like an on-campus course as possible, though without much technology or access to office supplies. We designed our course around the text The Heart of Mathematics, by Ed Burger and Michael Starbird. We chose this book because I had used it with great joy in a Math for Liberal Arts course at Colorado State University as a graduate student. The idea of HoM is to “make mathematics fun and satisfying for everyone,” according to the back cover. I like the book because it starts with puzzles and works through some of the most fun big ideas of mathematics, without emphasizing computation or requiring any calculus or even much pre-calculus. Our plan was to basically avoid anything that looked like what the students had experienced before as “math,” and hopefully thereby avoid the twin issues of past algebra difficulty and math anxiety.
On the first day of class, we explained our plan to the students. As you might expect, someone asked if we would be teaching them anything they could use. Where are the practical applications? I felt that it was important to be clear from the start that applications of the sort you find in a calculus or algebra textbook were truly not the goal of the class. We would teach problem solving techniques that could be broadly applied. We would bring puzzles and weird ideas that would hopefully provide fodder for thought and conversation. But we would not be learning to compute the half-life of uranium. We would be learning about ideas, some of them with larger relevance. The goal was to take a new perspective on what mathematics could be. With that said, we started in on the puzzles, and all skepticism was set aside. Arguments, however, were not. We had SO MANY arguments about what the puzzles intended, what was fair in solving them, and what somebody really would have done in this ridiculous situation. But these puzzles and arguments were a great hook. This was definitely not what the students expected from math.
We eventually did sections on basic combinatorics, number theory, geometry, graphs, and probability from the textbook. Highlights were Euclid’s proof that there are infinitely many primes, Diffie-Hellman key exchange, Euler circuits, the Art Galley theorem, and Cantor’s diagonalization argument leading to different sizes of infinity. HoM is a really well-written book, and we used many of their ways of explaining things. We also came up with some of our own material, drawing on our own interests, and found other resources. For example, I wanted to teach the students about cryptography in my own way. So I talked about symmetric ciphers, including encryption using a Viginere square with a keyword, cracking these ciphers, then introduced the idea of a “key book” which would allow for no repetition of the keyword. The students then used Diffie-Hellman key exchange to agree on a page number in the key book to use for encryption. Here is the original assignment.
A student’s encrypted messages for his partners using Viginere square and key from a page of the book, agreed upon by Diffie-Hellman key agreement.
Another smaller activity that was very successful was our simulation of the Monty Hall problem. One of the big constraints of working in the prison is the lack of technology. We couldn’t just write a program to simulate keeping the door or switching a thousand times—we had to come up with some way to physically do it. We ended up splitting the students into pairs and using set cards to stand in for the doors. Two green set cards represented the donkey or can of soup, while a purple set card represented the new car. One player shuffled the three cards and gave the other person a chance to choose a door (by putting a paper clip on a card). The student with the cards would then show an unselected green card, and give the chooser a chance to switch. The groups recorded how many wins and losses they had by switching and staying. In some groups it seemed to make no difference, but when we added up everyone’s results we got very close to the predicted two thirds/one third proportion. This five-minute activity cemented the concept in a way that my lecture definitely had not.
Our syllabus included homework, two tests, and a final project/presentation. The students had presented puzzles in the first two weeks of class, and we wanted to end class with another presentation of some kind, where we could hear a bit from each person as an individual. We decided on a poster session, which had a whole extra set of challenges inside an institution. It wouldn’t be possible to assign the students to complete their posters outside of class, because the materials would be very expensive from the commissary and it wasn’t clear what they would have access to. In the end, we had students pair up and choose a section that we had not covered from the textbook. The students were to read their assigned section and meet with the partner (we had to make sure they were on the same cell block) to plan a poster. They would have an hour on the last day of class to actually construct the poster, followed by an hour and a half poster session. We also asked the students to prepare a sort of elevator speech, a 3-5 minute summary of their topic to accompany the poster. We actually weren’t sure that the markers, poster boards, and glue stick would be allowed into the institution, and we weren’t sure if the poster session would really work, but in the end everything came through and the session was one of the highlights of my semester.
Poster on voting paradoxes.
Poster on non-euclidean geometries.
Poster on Euler characteristic.
Poster and origami on platonic solids.
This was one of many places where having two professors was invaluable. Generally, our co-teaching took the form of alternating lectures. While one person would lecture, the other would sort of join the class, occasionally asking questions, adding some relevant thoughts, and helping to clarify questions from the class when possible. It took a lot of pressure off to share grading responsibilities, and enabled us to model the kind of interactions we wanted to see in group work and class discussion. This poster session would have been much more difficult with one teacher, just for the time involved with having meaningful one-on-one contact with each student. When it was time to start, we asked one member of each pair to stay with the poster while the other person walked around and learned about the other posters. Katie and I had developed a simple rubric for grading the posters and presentations: 5 points each for content depth, content correctness, poster clarity, poster creativity/visual appeal, and presentation. We assigned numbers as we went, to avoid a later session of agonizingly trying to remember our impressions. Between the two of us, it took about 45 minutes to see all the posters in the first round. We then had the students switch and we each went to the posters we hadn’t seen the first time around. The presentations varied in quality (as presentations do), but a few of them were really strikingly good. The posters also varied, in intent and execution, but again some were exceptionally well done. I’ve put a few here, but more posters are posted on my website.
I had more fun teaching this class than I have ever had teaching. The most striking thing about the course was the amount of energy in the classroom throughout the semester. The students were engaged and game, willing to dive in to any discussion, to speak up with questions, comments, and occasional complaints, and to try activities for themselves. Every day when I walked out of class, I felt that I had actually connected with the students. Along with this gameness, most of the students were fairly mature and serious about learning, while still being ready to make jokes and speak up in class. I wished I could have brought my on-campus students, as a demonstration of what a classroom can be like. I love working with my on-campus students, but I feel that self-consciousness and expectations of what a college classroom “should” be can really limit their experience. What could college be like if students really engaged every minute of class time and saw class as a dialogue? I have tried to create this classroom atmosphere in many classes, with varying degrees of success. At Graterford, this atmosphere just happened on its own.
Debates abound about the value of providing a college education, especially a liberal arts education, to incarcerated students. I believe strongly in the value of offering basic, vocational, and college education opportunities to incarcerated people. Educational opportunities are essential to opening viable paths for released individuals. I also believe that a college education can open doors in a person’s inner life, or at least provide access to some beautiful ideas and an intellectual joy that can be hard to find otherwise.
Two of the students in our class are graduating this year. Both were really cool students to have in class, and have been working on their degrees for many years. One was actually released in April—we worked with him by phone and mail to finish the course as he navigated the trials of re-entry. He will graduate on Villanova campus this Friday with his friends and family cheering him on. He is working two part-time jobs now, and just sent a text to let us know that he got a new job that he was really excited about. The other graduating student is not getting out right now. I have no idea when he will get out. He was one of the quickest students I have ever worked with, and brought so much to the class. Many of the Villanova graduates at Graterford become leaders in a very active alumni chapter at the institution, and help current and prospective students with classwork and navigating the program. I think this student will bring a lot to the next round of students at Graterford. I hope that our Topics in Core Mathematics course gave him something for himself, too.
In fact, I know that math infiltrated our students’ lives in unexpected ways. After our class on Cantor’s diagonalization proof, a student told me that he and his wife had a game on the phone–the I-love-you-more game, essentially. He said that where it used to stop at “I love you infinity,” the game now could proceed from “countable infinity” to “real number infinity.” Awesome.