Newton's Grantham and the habit of working things out
King's School in Grantham educated Isaac Newton from around 1655 to 1659 — a fact the town rightly holds onto. Whatever the school's headmaster William Stokes saw in the boy who lodged with the local apothecary, it was not a finished genius awaiting discovery. Newton left Grantham and spent years making mistakes, correcting them, and making better ones. The apple story, appealing as it is, collapses a decade of painstaking work into a single afternoon. The real intellectual habit was slower and less cinematic: guess, observe the error, adjust, repeat.
That habit has a precise mathematical expression. Newton's method — the root-finding algorithm he helped develop — starts not with an answer but with a guess. From there it measures the gap between the guess and the truth, uses the slope of the function at that point to determine which direction to correct, and produces a new, better estimate. Run through enough iterations, and the approximations converge on the actual solution. Crucially, each step requires knowing where you currently are, how far off you are, and in which direction to move. Skip any of those three, and the method either stalls or spirals.
The question this raises for learners today is a simple one: what happens when someone hands you the root without showing you the function?
How the algorithm describes real learning
Think of a student revising an essay argument. They begin with a first version — imprecise, possibly wrong, but not nothing. A tutor marks not just that it is off-target but how: the claim is too broad, the evidence points the wrong way, here is where the logic breaks. The student adjusts, and the gap narrows. Repeat the cycle and a reasonable argument emerges. This pattern — initial position, error signal, directional correction, updated position — is precisely what Newton's method formalises, and it is offered here as a conceptual lens rather than an established pedagogical theory.
The mapping repays attention because each component carries independent weight. Prior knowledge is the starting guess: not a guarantee of accuracy, but necessary to begin. The gap between what the learner currently understands and what mastery requires corresponds to the function's value at the current point. Feedback that identifies which direction the error lies — and roughly by how much — corresponds to the derivative: it gives the correction its orientation. Without it, a learner can know they are wrong without knowing how to become less wrong.
The failure modes reveal what each component actually does. A starting guess too far from the truth does not simply slow convergence; it can send subsequent iterations in the wrong direction, compounding confusion rather than resolving it. Zero feedback is a flatline: the learner produces work, receives no directional signal, and has no basis for correction. Overcorrection — a tutor who rewrites the argument entirely — produces oscillation: the student adopts the new version without building any model that could generate a third.
The answer alone is not enough. Real learning requires exposure to all three: the function, the derivative, and the error signal that makes the next step legible.
What AI hands you instead
An AI tool asked to analyse a primary source will produce something that looks like analysis. It will have a thesis, evidence marshalled in support, and a conclusion. If a student submits that output without engaging with it, they have something more dangerous than a blank page: a plausible answer to a question they have not yet answered for themselves.
The structural problem is not that the output is wrong — it may not be. It is that the process generating it is entirely hidden. There is no starting estimate for the student to examine, no error signal they can locate, no directional correction they can trace back. All three components described in the previous section as essential to the learning cycle are absent simultaneously. What arrives is a root with no function visible behind it.
Without access to that process, the student has no internal model against which to test the output's validity. They cannot identify when the AI has missed context, collapsed a nuance, or — more subtly — answered a slightly different question from the one posed. The output, accepted uncritically, occupies the space where understanding should have grown.
This is not an argument against AI as a tool. It is a precise description of what AI, as it is typically used, removes from the learning loop.
The evidence for cognitive cost
Research on this pattern has accumulated quickly enough to take seriously, even if the evidence is not yet definitive.
Fan et al. (2024) gave the tendency a name: 'metacognitive laziness'. The term describes what happens when learners offload not just information retrieval but the regulation of their own thinking — the planning, the self-monitoring, the evaluation of whether their current understanding is sufficient — to a chatbot. The concern is not that AI provides answers but that it substitutes for the mental operations that consolidate learning.
Ahern (2025) sharpens this into a structural argument via an analogy most people find immediately legible: GPS navigation and hippocampal atrophy. Drivers who rely entirely on turn-by-turn instructions do not merely fail to memorise routes; some evidence suggests the spatial cognition that builds and maintains those maps degrades from disuse. Ahern proposes that AI offloading of problem-solving may carry comparable costs — not a habit problem but a capacity problem, written gradually into how the mind works.
The population figures add scale. A RAND Corporation report published in March 2026 found that around 67% of US students who used AI for homework believed frequent use was harming their critical thinking. More than 80% of educators expressed the same concern.
These findings are worth stating plainly for what they are: correlational data and self-reported beliefs. Neither proves permanent cognitive harm. What they do establish is that a large majority of the people inside the process share the intuition that something is being lost — and that the intuition has a plausible mechanism behind it.
Why the stakes are higher for adult and military learners
UA Grantham was founded in 1951 by a Second World War veteran who understood what education could mean for people navigating a life already crowded with obligation. Its student body still reflects that purpose: active-duty service members, veterans in career transition, and working adults fitting study around deployment cycles, shift rotas, and family commitments.
Those pressures make AI shortcuts feel rational in a way that is hard to dismiss. When a coursework deadline falls at midnight and a logistics estimate is due at 0600, accepting a plausible-sounding AI output rather than interrogating it has a clear short-term logic.
The problem is that the same temptation travels with these learners into professional life. A supply chain analyst who cannot identify why a cost model produced its figure, or a manager who cannot locate the assumption driving a risk assessment, is not merely underprepared academically. The inability to trace a conclusion back to what generated it becomes a liability when decisions carry real consequences.
No empirical study has directly measured first-principles outcomes in military or adult-online learner populations, so what follows is an argument from the logic of the situation rather than proved effect. That logic runs as follows: the people most incentivised to reach for a cognitive shortcut are also the people whose working lives most demand that they understand how an answer was reached.
Using AI as an interrogation partner
The evidence does not point only one way. Suriano et al. (2025), cited nearly 300 times in the two years since publication, found that active, interrogative engagement with ChatGPT — as opposed to passive copying — can promote complex critical thinking. The variable is posture, not presence.
The productive distinction is between receptive and interrogative use. Receptive use accepts the output; interrogative use puts it under pressure. Ask AI to argue against your own draft analysis. Ask it to name the weakest assumption in your position. Ask it what a well-informed sceptic would raise. These are not rhetorical exercises — they are the corrections that expose where a current model falls short and what revision would move it forward. Whether the approach works reliably depends on how consistently that posture is maintained; Suriano and colleagues demonstrate the possibility, not a guaranteed outcome under every condition. But it is teachable, and it does not require a technology policy: only a habit.
That habit mirrors what Newton's method actually demands of the solver. The algorithm converges because each iteration feeds new information about the gap between where you are and where the root lies — not because a better answer arrives from outside. For UA Grantham's learners, the practical version of that question is pointed: before accepting any output, ask what produced it. In a targeting solution or a cost model, the function that generated the answer is not a detail. It is the thing that tells you whether the answer is worth trusting.