An Inside-Out Course on Number Theory (Pt 8)

Link to previous posts: Pt1 Pt2 Pt3 Pt4 Pt5 Pt6 Pt7

Last Friday was the final meeting of this class. The outside students from Claremont were not present as it was finals week.

The inside students wanted to do a bit of math and just chit chat. We spent about an hour thinking about Fermat’s theorem on the sum of two squares (the only odd primes that can be written as the sum of squares of two natural numbers are the ones that are one more than a multiple of 4). We wrote out a whole bunch of prime numbers on the board and then we all attempted to write them as the sum of two squares. Then, we looked at our work and tried to figure out which ones could be written as the sum of two squares and which couldn’t. It was pretty satisfying!

After we concluded our work on that problem, we chatted a bit about mathematics and how the students envisioned mathematics being a part of their lives. One student is leaving prison in just a few days. He said that he will be taking more classes to try to finish up his bachelor’s degree. Another student said that he intends to enroll in college when he gets out and work toward a degree in mechanical engineering. Another will parole in a few months and he wants to be a mathematics teacher. The inside students also had a lot of questions for me about what graduate studies in mathematics was like and what kind of research I do, we so chatted about that.

It was intensely sad to say goodbye. I really enjoyed spending time with each and every person in this course and we got to know each other in a way that rarely happens in my usual math classes.

In retrospect, I think the thing that I will take away the most is the way that this particular pedagogical approach (working in groups with very little direct instruction, letting students’ interests drive the course) felt so right for this setting (a course with both traditional college students and incarcerated students in a prison). It felt re-humanizing, joyful, and fun. My thoughts now turn to why I’m not also using similar strategies in my usual Harvey Mudd College classes.

What’s next for Inside Out at the Claremont Colleges? One unfortunate issue about our Inside Out program at the Claremont Colleges is that we’re limited right now by who is trained to teach Inside Out and who is available in any given semester. (For me, teaching Inside Out was something that I did on top of my regular teaching load, but others are able to get their Inside Out course counted as part of their regular teaching load.) The consequence is that the courses form a strange patchwork of unrelated courses.

One exciting development is that the Claremont Colleges are working toward the ability to offer inside students a bachelor’s degree in organizational studies starting next fall. The reason for the choice of organizational studies is that a community college that also is working in this facility is offering several associates degrees in psychology, politics, and business. So, this organizational studies BA would build on all three of those pathways nicely.

It would be great to be able to offer more advanced mathematics courses, but both sides of the supply and demand for such courses is not robust enough. So, for now, I will look forward to the next time that I get to offer this class again inside.

An Inside-Out Course on Number Theory (Pt 7)

Link to previous posts: Pt1 Pt2 Pt3 Pt4 Pt5 Pt6

This blog post consists of guest posts by two of my Claremont Colleges students in this course. I asked all of the students to write a reflection at the end of the class. I gave them the option of sharing their reflection on this blog if they chose to. (I edited slightly to preserve anonymity and to improve clarity for audiences outside of the Claremont Colleges.)

This was an incredible experience! When I first heard of the Inside Out program, I thought that it sounded like an exercise in some kind of savior-dynamics that took advantage of the inside students to create some sort of cultural experience for the outside students. We feel privileged to be able to walk in (and out) of the prison, for the incarcerated students, it is a cursed life. The whole idea of Inside Out left a bad taste in my mouth.

I signed up because (a) the math sounded interesting (b) this was the only course at the Claremont Colleges that related to mathematics education that I had heard of. The Inside Out experience was an added–though sort of undesired–bonus.

After my first week, my perspective had changed. It was clear from day one that all the students in the class were there together to practice math–to grow as students and people. For the first half of the semester certain topics were unheard of. We talked about education: what classes we were in, what our aspirations were. We saw each other as humans, students. We chatted and looked for patterns in numbers, but seldom did we discuss the elephantous divide between us.

At some point in the semester that started to shift. Inside students would drop “I have 6 more years here, so I should be able to get a bachelor’s degree before I get out.” It would be a blip in the morale-o-meter, but we would keep working, ignoring what our minds and hearts were actually telling us.

Around that same time, inside students would ask what we were doing on the weekends and instead of our usual vague “nothing” or “studying” we would let slip that we were going to the beach or to visit our parents or something.

It was around Thanksgiving that I felt like I knew the inside students. I could tell their interest in the math, knew what they were likely to rant about, knew where some of them had in-prison jobs, what their personalities were like. They knew that we would go home at Thanksgiving to family and travel.

By the end of the semester, though we didn’t pry for information, inside students talked about their court dates and hassles. I could almost feel bits of my heart breaking hearing these stories. Leaving the prison, I would shake off the melancholy on long runs on a different type of tax-funded land: national parks. I don’t know if the inside students were fairly tried, if their sentence reflects some guilty crime. That is not what Inside Out asks you to consider (though it crosses my mind). Inside Out asks you to consider seeing people you might not normally encounter and without needing to hear their stories to sit and learn together. To share in our collective humanity by succumbing to the unknown and to a pursuit of thought.

In this case that pursuit was working together to identify patterns, the malleability of numbers, the interesting feats of special cases of numbers (primes / square numbers), and to block new concepts together. What I had thought would be the crux of my Inside Out classroom experience was tangential to learning about beautiful, human relationships and an oppressive, not uplifting system.

As you look back over the problem sets and work that we have done, what are some things that you have enjoyed learning about the most? The way everything we had been learning about in this course came together on the last day of class was close to mind-blowing. It was amazing to feel like I understood encryption from the ground up–what had felt like basic math in previous classes suddenly became important building blocks for a complicated encryption system. This was one of the first times that I felt like I truly understood a complicated concept in math beyond just plugging things into equations.

What were some of the most challenging things that you encountered in this course and how did you face those challenges? I think the most challenging part of this class was the power dynamic between the inside and outside students. Every week the guys I’d work with would assume that because I’m attending the Claremont Colleges that I’d be better at understanding the problem set for the day than they’d be, when in reality, when we got down to it, I was often one of the last people to have it click. Also, I had several conversations where my inside classmates would be surprised that we got class credit for the course because to them it seemed that we were just volunteering. I forget who said this on the last day of class, but someone joked that this class might actually be an anthropology course for the outside students. I think what these guys were picking up on was very real–it feels uncomfortable that I can just choose to take a class in prison to see what prisons and the people inside them are like, and then just leave. A large part of what drew me to this class was the fact that I could learn in an environment I’d never been before and have classmates I normally wouldn’t have, and I think that the guys on the inside could sense that motive. I don’t know how I or anyone else in the Inside Out program can avoid this dynamic.

What have you learned about yourself in relation to mathematics as a result of this course? This is going to sound so corny, but I learned it doesn’t really matter if I’m good at math, it matters that I’m having fun. And I was having fun!! Perhaps another way of looking at it is, if I’m having fun while doing math, it doesn’t mean I’m not learning. This class was the first math class I’ve taken that didn’t make me feel “good” or “bad” at math. I was just doing it and not taking away any personal value judgments from that. This class inspired me to take another related math class next semester.

An Inside-Out Course on Number Theory (Pt 6)

Link to previous posts: Pt1 Pt2 Pt3 Pt4 Pt5

A lot of feelings today, which was the day of our last class in which both outside (Claremont) students and inside (incarcerated) students were together. Next week is finals week for our students, so our outside students don’t attend class.

We were able to successfully get through all of the mathematical machinery that we needed to understand the RSA cryptosystem, and we spent the day going through some examples, encrypting and decrypting messages (for very small numbers, since we only had four-function calculators). Fun times! It felt so satisfying that RSA was such a nice way to tie together everything that we had been doing. I had no idea that this was going to be the “goal” of the class when we set out. It was an off-comment that I made one day about how number theory can be used in cryptography that seemed to be interesting to students, and so I plotted a course for us to end with RSA.

But the most meaningful moment today was this: One of our inside students has not been able to attend class for the last few days because he was reassigned to some other kind of vocational class (against his will) that met at the same time as our class. Nevertheless, we have been exchanging work so he has been keeping up with the class. Today, in his packet of work he included a written reflection and a note that he wanted me to share it with the whole class.

In this reflection, he explained that he originally signed up for a lot of classes but didn’t get into any of them except this number theory course. He was not excited about it. He said that he avoids social contact with anyone and considers it a good thing if he can get through an entire day without talking to anyone. Coming to this class and being forced to work with other people was initially a big challenge for him. But, over time, he found the interactions with other students really enjoyable and he grew to love coming to class and working with others. He concluded his reflection by thanking all of us in the class for helping him grow mathematically and also socially. Was hard to read his reflection today without crying.

The outside students said their goodbyes to the inside students and vice versa. They shared their appreciations for each other. One guy on the inside thanked us for helping to restore his faith in people “on the outside” because we interacted with him as fellow human beings.

So many feels today. Next week will be my last week with the guys inside.

An Inside-Out Course on Number Theory (Pt 5)

Link to previous posts: Pt1 Pt2 Pt3 Pt4

Class is going really well, at least from a mathematical perspective. Next week is our last class meeting and we are on track to be able to wrap up the whole class in which we use all of the tools that we’ve been building up over the entire semester to explain how the RSA cryptosystem works. (This book by my colleague Mohamed Omar has been super helpful.)

Today, we took care of the final two mathematical tools that we’ll need to understand the RSA cryptosystem.

(1) We learned about how to use the Euclidean Algorithm to find the greatest common divisors between any two numbers. The students were particularly enamored with the “square-cutting” visual (see below) that I learned from Bowen Kerins as we have co-taught the math course for the Teacher Leadership Program at IAS/Park City Mathematics Institute.

(See for an animated version.)

(2) We also learned how to solve problems of the form “ax=1 in mod n”. We learned the conditions under which such problems will have a solution.

And, a few students got to the fun part, which is that while (1) and (2) seem unrelated, it turns out that you can use (1) to help you find the answer to (2).

Finally, I just got a copy of Mathematical Outreach: Explorations in Social Justice Around the Globe, edited by Hector Rosario. In it, Robert Scott makes some great observations about teaching mathematics in prisons. One quote has been resonating in my head since I read it:

“A math pedagogy premised upon following the rules, accepting that there is only one right answer, and relying on practice/repetition in order to habituate oneself to pre-determined axioms would seem to reprise the culture of incarceration itself.”

Robert Scott, “Math Instructors’ Critical Reflections on Teaching in Prison”, page 213 of Mathematical Outreach: Explorations in Social Justice Around the Globe, edited by Hector Rosario, 2020