Varieties: The Spice of Life

Every year, I look forward to the summer. But not for the reason that most people would think of with me being a teacher.

For me, it's because I'm excited to go teach at Georgia's Governor's Honor Program (GHP).

If you've never heard of it, GHP is a four-week program at Berry College in Rome, GA, where enthusiastic high school students from all around Georgia interview and audition to intensively study a major subject of their choosing. It's the kind of environment every teacher dreams of — students who are there because they want to soak in as much knowledge as possible and be around equally-motivated students. (It's also, as far as I'm aware, the only such governor's school that operates free of charge to its students, which is an important contribution to the diversity of the student body.)

I also see GHP as an opportunity to push myself as a teacher — it takes a lot of work to keep up with these students! We guide them as they explore mathematics that they normally wouldn't see until undergraduate or even graduate courses. I like to try a lot of new ideas with my teaching at GHP, ideas that I can then bring back to my own classroom. (And the fact that there are no grades to worry about certainly helps eliminate some of the usual confounding variables!)

This year, I felt like taking a bit more of a risk, and decided to teach a course in computational algebraic geometry.

...yes, to high schoolers.

If you're not familiar with algebraic geometry, at a bird's-eye level, it studies geometric shapes called varieties, defined as the zero sets of one or more polynomials in multiple variables, using techniques from abstract algebra. (For example, the unit circle in \(\mathbb{R}^2\) can be thought of as all the points where \(x^2+y^2-1\) equals zero.) The "computational" part comes in when you start looking at algorithms to manipulate those polynomials (for example finding a convenient basis of polynomials to work with).

I actually had just taken a course in computational algebraic geometry last year with Dr. Daniel Miller at Emporia State University, which I absolutely loved. The entire time, I kept thinking to myself, "You know ... I bet GHP students could handle this." It touches on so many things that high school students already see in their curriculum:

• Coordinate geometry
• Systems of equations
• Real and complex number systems
• Polynomial zeros, factoring, and division
• The Fundamental Theorem of Algebra
• Rational functions
• Conic sections
• Parametric curves

What's even better is that it beautifully ties together all these concepts — something that unfortunately can't be said for most of the high school mathematics curriculum.

I also realized this would be a great opportunity to work in one of my all-time favorite topics: projective geometry and division by zero.

All of this led me to conclude that this would be a perfect course to offer at GHP, so I went ahead with it. I decided to call the course Varieties: The Spice of Life. (Thanks to @notamoon1 on Twitter for that suggestion!)

My main source was Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea. I also referenced Elliptic Tales by Ash and Gross for some of the projective geometry material toward the end.

Once I started teaching the course, there were certainly some challenges.

To begin with, I initially underestimated how quickly my students would pick things up. For the first day, I wrote up an activity that would have students begin familiarizing themselves with SageMath (a Python-based computer algebra system) on the CoCalc website, which we'd be using throughout the summer. I planned for the activity to take until the end of class, but some students blasted through it because of prior programming experience. I encouraged those students to explore a little further to see what else they could get CoCalc to do, but even once everyone else caught up, I still found myself with 15 minutes to spare at the end of class (which I used to briefly introduce students to the idea of rings and fields.)

I realized that day that I was going to need to differentiate my instruction to make sure that all my students could stay engaged no matter what pace they were working at individually.

Also, I wanted to make sure the class was as student-centered as possible — the last thing I wanted to do was lecture at students for an hour straight in the summer, when they get enough of that during the school year. I'm a firm believer in the principle that to learn mathematics, you must do mathematics.

After some thought, I came up with a structure that worked for the rest of the summer.

• During the first 15-20 minutes of the day, I would introduce the main topic for that day, in sort of an interactive mini-lecture. I'd pose some questions, have students notice-and-wonder, and use that to motivate and define some key terms. I'd also demonstrate how to use CoCalc to do some of the heavy lifting and visualize what we were talking about.
• Then, for the bulk of class, the students would work on problem sets that I wrote up. I made the problem sets way longer than anyone could complete in a single class period, but I specified that everyone should work one or two particular chosen problems in class. If they finished those questions, they could work on whichever other problems piqued their interest. (I also told them they should investigate some of these other problems when they got home from GHP — who says that when school stops, the learning has to stop as well?)
• Finally, at the end of class, we'd go over those chosen problems, answer student questions, tie things together, and pose some new food-for-thought cliffhangers that would be resolved in a future class.

Making the problem sets was definitely a challenge! I carefully rewrote many of the problems to strike a balance between guiding them and giving them room to explore. Often I'd write up some description of a particular concept, only to realize I could instead just have students play with particular examples meant to elicit noticing-and-wondering so they could connect the dots themselves.

On the last day, I asked students to give me feedback on how the course went.

• For the most part, the students really enjoyed the course! In particular, many students mentioned they enjoyed tackling familiar concepts at a deeper level and seeing how they tied together in new ways. A few students had even taken linear algebra already (yes, in high school!), and they noticed a lot of parallels there to topics like row reduction and bases.
• Some of the students did say that the material was sometimes difficult, and they had to really stay on top of it to keep up (since again we only had four weeks, and students took three courses total). One student actually said this was the first time they'd been confused in math in a long time — but they then said that it was a good feeling to step up their game to overcome that challenge!
• Students also mentioned they appreciated the student-centered approach, not just being told a bunch of facts but getting to work them out for themselves. A number of them mentioned that they left the course better understanding the "why" behind the things they'd previous been told to just accept.
• The division-by-zero topic was of course a hit! To be fair, I mentioned that we'd be learning to "break the rules" and divide by zero in my initial pitch for the class, and I did dangle it in front of students for a few weeks before we finally did it to prepare for projective varieties at the end. There's something alluring about "forbidden knowledge!"
• CoCalc was also something students really enjoyed, both for novice and expert programmers. One student mentioned that they loved how for just about everything we explored, it was soon followed by "Guess what? CoCalc can do that too!"
• One of the students actually had a $2 bet with another student about whether I was going to cover Hilbert's Nullstellensatz on the last day. I wasn't planning to initially, but I did briefly discuss it with those students while everyone was playing with algebraic surfaces in Surfer. (I still don't know what came of the bet.)

So looking back, despite the above difficulties, I'd say the course went pretty well overall.

As a final note, here's my biggest takeaway from having taught the course:

We're teaching math completely out of order.

Let me explain what I mean.

When doing mathematics "rigorously," we develop things in a very meticulous way. We state our axioms, define our terms carefully, prove our theorems logically, prove more theorems on top of those, and so on. You're not allowed to use something unless you've proven it, lest the foundation of your arguments be put at risk. This is the norm for writing mathematical papers, as it should be, since that level of rigor is crucial for advancing the field.

But that's not how we actually do mathematics.

We tinker with examples, notice interesting relationships, and fiddle around until things become clear. Only then do we decide on the best order to elegantly define our terms and prove our results.

So why is it that in textbooks and in the classroom, we start with the cleaned-up end result, rather than letting students partake in the journey that gets us there?

We know the terrain well — which paths are full of beautiful scenery and fruitful discovery, and which paths lead to dead ends. What we're doing instead is essentially laying down a sanitized concrete walkway that gets students from point A to point B in an ostensibly straightforward way. But that leaves students wondering why anyone would bother to take that walkway in the first place.

And while we're at it, we really need to dispense of this idea that just because concept X is needed for a mathematically "rigorous" foundation for concept Y, students cannot explore concept Y before they've thoroughly learned concept X.

None of my students had a course in abstract algebra before. But they didn't need it to get their hands dirty and start playing around. They especially didn't need the formal set-with-two-binary-operations-satisfying-these-axioms definition of a ring — we just thought of them as any collection of things you could add, subtract, and multiply. That was enough for them to start seeing that there are a lot of parallels in how we think about polynomials and how we think about integers.

Yes, you could call it hand-waving. But I think it's justified. It's not about covering up holes in a shoddy argument. Rather, it's about temporarily hiding some of the nitty-gritty details so students can build an intuition for the concept. Those details can always be filled in at a later time.

In short, we need to stop conflating logical foundation with pedagogical foundation.

So, what do you think?

Could students better learn mathematics if we rethink our notions of what order things should be done in?

Please leave a comment below — I'd love to hear your ideas!

■