A3: Teaching Calculus III in the Age of AI

 

Ileana Vasu

Smith College

This is the third in a series of four blog posts in response to the call: For many reasons, including AI, we had to adapt our teaching strategies and learn as we go this year. So let’s share what we found that was useful, promising, and/or positive in some way!  Our audience is NE-COMMIT and anyone else curious about improving teaching mathematics.

The posts are ordered from those describing practices more resistant to AI to those describing practices more embracing of AI.

Here are links to the remaining posts: AI 1 Thinking Video Notes to my future forgetful self by Christine von Renesse, AI 2 Changing My Writing Assignments by Debbie Borkovitz, and AI 4 Experimenting with Potential Ways to Incorporate AI into my Teaching Practice by Rebecca A. Norton.

 This year, I made a deliberate choice to redesign parts of my Calculus III course around the reality that AI tools are now part of how students work. Rather than treat that as a problem, I built it into the course intentionally, leaning on two frameworks I keep coming back to and am genuinely grateful for: Tara Yosso's Community Cultural Wealth, which treats students' backgrounds and lived experience as assets rather than gaps to fix, and Francis Su's Mathematics for Human Flourishing, which asks what math is actually for beyond right answers. A colleague and I worked on two projects together this year, and I was genuinely glad to have someone to think this through with. I'll just be discussing one of them here. Here's what came out of that work this year: the parts that felt genuinely useful, promising, or just good, and the parts that reminded me why I love teaching this subject in the first place.

Math isn't neutral, so I do not teach it that way

I believe math has never been the culture-free, purely objective subject it's often presented as, and this year I tended to this belief in the Calculus projects. I built a surface from six parameters (two for each student in a group of three) drawn directly from each student's birthdate. Plug in your birthdays and out comes a genuinely strange equation, and when graphed, an equally strange bumpy little landscape, nothing like the tidy paraboloids in the textbook, and nothing any other group has. Students find critical points, analyze gradient fields, and then name their surface and write a short narrative describing it as a landscape. It's been lovely to watch students light up over a shape that's theirs. A surface built from your own birthday isn't something AI can meaningfully do for you. It doesn't know your surface exists until you build it. Of course, the math was not tidy, so students had to use Mathematica or Matlab, or Python to visualize their surface and its critical points, but they were free to use AI in learning how to use the software as long as their explanations were documented and their learning thorough.

Documenting AI use, instead of policing it

Every project includes a short, required section: what AI tool you used, what you asked it, what it gave you, what you had to fix, and what you learned from this interaction about mathematics, or coding. This one change did more for honesty than any detection software could. Students stopped hiding their AI use and started reflecting on their learning, often with real curiosity. The most common thing they discovered on their own: AI produces confident, plausible-looking code that is sometimes just wrong, and you only catch that if you understand the math well enough to check it. That's was the habit of mind I want them building, and I loved seeing them arrive at it themselves.

Multiple pathways, and room for real strategy

For the vector calculus portion, groups choose two of three options - gradient field analysis, directional derivatives, or flux integrals, based on their own strengths. This wasn't originally designed as an AI response, but it turned out to matter for the same reason: it shifted the emphasis from "did the AI produce the right output" to "did your group make a smart, thoughtful choice about how to approach this." That kind of judgment is much harder to outsource than code is, and it also felt like an honest nod to Yosso's idea of navigational capital, students strategically working with the tools and constraints in front of them, rather than following one prescribed path.

Creative narrative as a window into real understanding.

Every project ends with a short story describing the surface as a landscape: naming it, describing its features, weaving in the actual mathematics. One group named their surface "The Twin Comet Highlands," describing an origin story where  locals had a story that two comets had created the landscape and checked back with geological data to actually figure out whether the story matched reality or not.  Reading things like that has genuinely been one of the joys of my teaching year. It's also turned out to be one of the clearest windows into real understanding I've had in years: hard to fake, and just a pleasure to read like the one below:

Long ago, two comets are said to have grazed this land in the same season, one burying itself deep enough to raise Comet’s Crown, its ejecta pushing the ground up to 5.46 units above the plane, still by far the tallest point for miles. The other struck lighter, near the origin, and left only Whisker Ridge, a modest rise of 0.43 units that locals will still proudly call a mountain if you let them. Between the two impacts runs The Crossroads Pass, sitting at the exact center of the map,elevation zero. Every trade route through the highlands passes through here, because it is the only way across without a serious climb.  A visiting geologist, on hearing this legend, is said to have sighed and pointed out that a real comet strike leaves a crater with the peak rebounding up from inside it, not a lone summit rising out of open ground and that nothing in their surface is quite symmetric enough to have fallen from the sky in one piece anyway. The locals thanked her for her time and kept telling the story the old way….and the story continues

AI as an unexpected equity tool

I was not sure if AI might widen the gap between students who already knew how to code and those who didn't. In practice, it narrowed it somewhat, and that was a genuinely happy surprise. Students with no programming background could get a working starting point instead of staring at a blank file, then actually learn from modifying it. Not a level playing field, but a less tilted one, and a small, practical example of building on what students bring rather than what they lack.

The bigger takeaway

The adaptations that worked shared one thing in common: they made the mathematics personal enough, and the reasoning transparent enough, that AI became a tool students used rather than a shortcut around the point of the assignment. None of this happened by accident. It came from deliberately redesigning around the reality that these tools exist, and from a belief that mathematics was never neutral to begin with. Between the personalization, the storytelling, and paying closer attention to what students actually bring into the room, this was the year I felt calculus held a little more room for human flourishing, AI and all. I'm grateful to have gotten to teach it this way, and grateful, too, for a colleague willing to build alongside me.