If one were to peek into a mathematics lecture hall 5, 20, and 40 years ago, they would notice two counterintuitive facts: the equations remain the same, but the actual delivery looks completely different.
Chalk, analogue projectors, slides, markers, and digital presentations. Books, folded photocopies, moodles, and video tutorials. The acoustic backdrop, murmurs, clickety-clacks, and notifications' ping. It is easy to place each of these scenes in its proper era, yet it remains striking how the Pythagorean theorem survives as fixed Platonic ideals across any given format; Euclidean geometry has not shifted in millennia, and the fundamental rules of calculus remain as rigid as they were in the days of Leibniz. Even the most "modern" results I teach, Sylow and Burnside theorems, were proven around the end of the Victorian era.
There was, of course, a further time when proofs were spoken rather than written, when numbers were not yet digits, and when books were a synonym of power, but also marginalised, as seen when they were accused of corrupting the memory of Athenian learners... but teaching has never moved as fast as it does today.
For the past thirty years, things have changed so rapidly that it is no longer enough to rely on the methodologies of our own generation. Technology is now bypassing the human biological timescale. In many disciplines, the boundaries between pure thoughts and real applications are fading due to unprecedented technological advancement. Readaptation seems to happen almost every academic year. Evaluation methods change in parallel with the latest discoveries, and we have reached a Orwellian point where a student’s artificial intelligence is battling our own for detection.
But this is not our first sudden shift. The pandemic forced us to overhaul our approaches overnight. Flipped classrooms, audience response systems, and formative assessments (once distant concepts mentioned in passing by a colleague) suddenly had to be implemented by the following week. This was particularly challenging in mathematics, where the dialogue between student and lecturer inherently required (and requires) the physical support of a medium, whether digital or analogue.
Today, a single individual cannot possibly keep up with all the tools available, not even the educational ones. Thanks to the impressive impact of Large Language Models (LLMs), some of these tools are now ubiquitous among students. From personal calculators to the internet, and now to AI, technological adoption has always been a question of timing and opportunity cost. Now, with a single prompt, a student can receive a seemingly perfect answer, even lively, and that is happening in the problem classes of all my new modules. Before this, it took a few clicks on a search engine, and before that, a day trip to the local library in search of an understandable epigraph in a dusty book.
However, the problem is that genuine learning requires a necessary "waste" of time. In mathematics, the traditional deductive format (Definition-Theorem-Proof) actually shares a similar hidden structure with what these black-box chatbots offer: a definitive, written-in-stone discussion presented without additional aid, which is assumed to be true. It omits the wrong approaches, the incorrect hypotheses later proved wrong, or the stumbles in the middle of the discourse. If students accept these answers as absolute truth without grappling with the underlying concepts, their learning becomes fragile. The essential process of failing, trying again, and generalising with curiosity, the genuine inductive process, is lost, dramatically narrowing their own critical spirit and their appetite for novel, challenging problems.
However, there is another hidden catch but fundamental difference: behind a human mathematical proof, there is always an intention of a person offering a causal, logical solution. Deep learning models, conversely, offer just correlated ideas stitched together, and correlation does not imply causality.
How, then, do we solve this? This is where universities, especially well established ones, must take a central role. Rather than retreating to a defensive position with punishing policies against any use of AI, institutions and modules must adapt, not by changing the learners but by training their faculty to introduce these systems properly, establishing a culture from the bottom up where AI is viewed as a complement rather than an opponent. In this way, we will capitalise on the current weaknesses and we may adapt our evaluation systems towards more challenging approaches, as higher education is meant to be.
In this landscape, the "eternal student" in the lecture hall is no longer just the previous teenager in the front row trying to understand how groups and fields relate. Today, the eternal student is now the person at the front of the room, dropping some preserved truths while building the novel subjects and navigating a rapidly shifting educational paradigm.
In your own practice, what is the one eternal result of your subject that has been the hardest to adapt for the classroom of today?
Sinuhé joined HE as an Arena Fellow from UCL last year and is now serving as an Assistant Professor at the Universidad de Burgos in the newly implemented BSc program in Mathematics and Computation, as well as a guest researcher at the Max Planck Institute of Colloids and Interfaces.