In my first-year calculus exam, my lecturer forbade the use of calculators. For a course that made extensive use of common trigonometric ratios, it made for a tense few weeks beforehand, working out how to recall them. There were a couple of ways to do this. The first was to memorise the pattern in the table of values, and trust yourself to reproduce it under exam conditions. The second was to know two special triangles, an isosceles triangle with two sides of length 1 and an equilateral triangle with side length 1 cut in half, and derive the ratios from those. I couldn’t tell you today which one I did, but I’m fairly sure it would have been the first.
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Third-year, an optimisation assignment I couldn’t make head nor tail of. In quiet desperation, I worked my way along the library’s optimisation shelf, looking for anything that might help me understand what we were actually being asked. Then, to my surprise and, I’ll admit, some delight, I found a textbook with exactly the same questions in it as our assignment. What do you think I did next?
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This week I chaired a teaching and learning committee in my college. We were given a presentation on software that has students write their assignments straight into it: a process-capture environment, recording how much text is typed and when, flagging apparently copied content, and monitoring students. Quietly, I was horrified. It’s a surveillance tool, built on a culture of suspicion. At best, the benefit to students seemed to be that they could use it to prove their innocence. I thought our legal system ran the other way.
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If AI can do mathematics better than we ever can, why would anyone put in the hard work to learn it? Alex Kontorovich, as part of the ‘What Is Math For in the Age of AI?‘ panel discussion at the International Congress of Mathematicians six weeks ago, gives a helpful analogy.
Kontorovich asks his students to picture three scenes. In the first, you’ve got pallets that need to go onto a truck, so you get a forklift, drive it over, and load them. In the second, you’re learning to drive a forklift, so you move some pallets around for no reason except that you’re learning to operate the machine. In the third, you fancy a workout, so you drive the forklift to the gym and do your reps with it instead of your own arms. One of these, he says, is the wrong use of technology.
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Artificial Intelligence has intruded into every domain of our lives. Setting aside that it’s a knowledge-hungry, water-guzzling machine trained without permission on other people’s work, I’m intrigued by its promise, alarmed by its threats, and tired by its constant mention.
AI is the latest in a dilemma that’s plagued us in education for more than a century, framed by Dewey as two questions: should we emphasise the doing or the result, and is it valued for its own sake or as a means to something else? AI just makes it easier to land in the wrong cell of that grid without noticing. And it’s by no means the first technology accused of doing that.
There’s an inclination to protect the process by restricting or monitoring tools, but that doesn’t always follow. Banning the calculator moved my effort from understanding trigonometric ratios to memorising a table. A process-capture environment moves a student’s effort from writing to proving they wrote it. Long before there was a chatbot to ask, there was a library shelf with a textbook that could answer my questions for me.
And not every tool costs you the process. When I got to algebraic fractions in later years, I would use my calculator on simple examples to see if I could work out the right “rule”. The tool did the checking, I did the understanding. On the exam-writing committee for Year 12 mathematics, I was pleasantly surprised by how graphics calculators helped us ask questions that dug into real conceptual understanding rather than skirt around it.
Ravi Vakil, on the same ICM panel as Kontorovich, put it plainly: the purpose of doing mathematics was never to get the answer on a test, it was to train your mind. Kontorovich described the same split showing up in students’ test scores: those using AI to accelerate their understanding pull ahead, those using it to do their homework for them fall behind.
This all leaves me with an uncomfortable thought. Have we caused this ourselves, with our focus on evaluating the product rather than process? As much as we can talk to students about the value of working things out for themselves, if we only ever assess the product, we’re driving the very behaviour we say we want to prevent. Perhaps the real question isn’t whether students should have the keys to the forklift, but what it is about the gym we’ve built that keeps pointing them toward it.
Photo by Filip Szalbot on Unsplash
Wonder in Mathematics 