By Lew Ludwig

In Tuesday’s post, I talked about the centaur vs. cyborg analogy for how people approach AI. My student John had used the cyborg approach with my cheat-proof calculus exam, yielding a rare perfect score, and I wasn’t sure how to move forward from that with the final exam.

Playing to Space in Education

I decided to “play to space.” As a huge fan of the MLS Columbus Crew, I regularly attend their games with my family in the Nordecke fan section, proudly wearing our colors and waving the banners. One crucial concept in soccer is "playing to space." As players move the ball up the field, they pass it into open space ahead of their teammate, anticipating their path and maximizing momentum. If a pass is played directly to a teammate's feet, it often arrives behind them, disrupting the flow of play.

I could have given John the final, a pass to his feet, that would have disrupted our learning momentum. Instead, I asked him if he was interested in a challenge. Would he be willing to use ChatGPT to learn a totally new concept, then be verbally quizzed on the concept? He was game!

As he was leaving, I suggested he present the following Monday, the scheduled day of our exam. He said, “No, my dad is picking me up Thursday at noon to go home for the summer. Can we meet Thursday morning?” This was less than 48 hours away. He had other exams, had not packed, had no idea of this concept - the perfect recipe for disaster! I held my breath.

The cyborg reaches for mastery

On Thursday, John arrived at the appointed time. He had a slideshow to help explain his thoughts - okay, a good start. He quickly jumped into ideas new to him, “convergent” and “divergent,” and then explained what they meant. He then went on to do an example with 1/x2, explaining how he needed the antiderivative for the Fundamental Theorem of Calculus. Like most students at this level, he neglected the limit notation. Oh well.

He then turned the page and explained that you can’t do this directly as the function is not continuous at 0, so we need to use a limit! Luckily, I was looking over his shoulder so he couldn't see me tear up (as I am again right now). HE HAD CRUSHED IT! On his own, this business calc student, who had only taken Precalculus in high school, had used generative AI to teach him improper integrals! My emotional response was sparked not just by his mastery of the content but by the realization that he had embraced the challenge so fully, exemplifying the potential of AI to transform learning in ways that go beyond traditional boundaries.

Once again, John shared his chat discussion with me. This time, it was only 3800 characters, but the exchange was much more back-and-forth. The chatbot would use words like “converge” and “diverge,” and John would ask for clarification at a level he could understand. It computed things with limits; he asked why limits were needed. John was in the “cyborg state.”

Things were not all perfect. Through questioning, it was clear that John had a firm grasp of p-series for 0<p<1 and p>1, but p=1 was a bit problematic. He knew that he needed the ln x as the antiderivative, but he was struggling with the limit interpretation. This is where I, the human educator, stepped in. I sketched the graph of the exponential function and asked him to sketch its inverse. This helped him recall that ln x was the inverse. In a few more minutes, he used the graph to explain why the area under 1/x diverges - a concept and term that were entirely new to him.

It’s time to get our heads up and see the whole field

As we reflect on the journey, several lessons stand out. First, this was a unique opportunity. I was fortunate to have a student who trusted me enough to embark on this experiment. Moreover, he wanted to impress. He dedicated about three hours to learning and preparing the material—not a commitment every student is willing to make. This experience will remain a cherished memory for both of us.

Second, it reminded me to keep a vigilant eye on centaurs, those instances where AI might simply do too much of the work, risking the depth of students’ understanding and engagement. While my take-home test is still relatively AI-proof—you can't just copy and paste your way to completion—you can make significant inroads on some aspects with some simple centaur-like prompting.

Yet the most significant takeaway from our experience extends beyond the confines of any single test or academic term. It is about embracing a strategy that transcends traditional educational methodologies, a concept I’ve come to appreciate deeply through my love for soccer: playing to space. In the classroom, just as on the field, the goal isn't just to meet students where they are but to anticipate where they can be. It’s about challenging them to move towards areas they haven’t explored, leveraging tools like AI not just for incremental steps but for transformative leaps in understanding and capability.

By fostering an environment where students are encouraged to think ahead, to innovate, and to solve problems creatively, we equip them for a future where they are not just participants but pioneers. This means redesigning our educational approaches to not merely direct learning to where students stand, but to where they can thrive. If we’re to keep relevant and play a meaningful role in our students' learning journeys, we must adapt. We must rise to the challenge. We must play to space.

Lew Ludwig is a professor of mathematics and the Director of the Center for Learning and Teaching at Denison University. An active member of the MAA, he recently served on the project team for the MAA Instructional Practices Guide and was the creator and senior editor of the MAA’s former Teaching Tidbits blog.