Math Education, and LLMMath Education, and LLM<br>2026-06-16Abstract: This article defines math and math education, and argues for a lower bound of<br>human effort required to learn mathematics (or any other abstraction-heavy subject) regardless<br>of LLM capability.LLMs are evolving rapidly; within a year AIs are able to tackle math problems - we had<br>thought of them as the hardest for AI to automate - but they are getting there. Let it be OpenAI's<br>marketing bluff<br>or not (of course humans helped, how much? we can never know...), we can at least say that<br>frontier models are useful for assisting with math research. The natural question to ask here is<br>that, is AI helpful at math education? To what extent?Mathematics is a truly unique subject from all other sciences, in that it is not<br>natural at all, despite the fact that universities like to put math in a physical science<br>building of some sort. While it originates in counting and measuring the physical world, it has<br>evolved out of its physics context into a discipline purely focused on a-priori reasoning and<br>abstraction during the formalism<br>movement. My preferred way to define math is "the study of a-priori constructions". I'd also<br>like to think of all math knowledge as an infinitely large graph of theorems where one node<br>points to another if one can be deduced from the other by the axioms chosen. In this interesting<br>perspective, math is much like art, poetry, or music, where every theorem already exists somewhere<br>and we are just discovering them. An implication of this view is that, humans have to occupy a<br>position in math research, since we are the ultimate judge to say whether an abstraction or<br>theorem is interesting and worth developing or not. Math is tightly connected to personal and<br>collective taste and intellect: "The product of mathematics is clarity and understanding.<br>Not theorems, by themselves." [1]As a result, calculation or theorem proving is only a small part of doing math, and the goal is<br>rather to cultivate good instinct [2] - the ability to fluently<br>navigate and manipulate some levels of abstraction, and thus "sense" how to get from one node to<br>another or which nodes are worth exploring. People used to develop abstraction out of physical<br>properties, such as the invention of calculus which was used to describe continuous physical<br>phenomena. But now, the abstraction is so far removed from the physical world that it is often<br>the case that mathematical abstractions are invented before any application is found: Riemannian<br>geometry, a 19th-century invention, is now the language of general relativity (1915).Math education gets hard here, since the properties of good math education, in contrast to math<br>itself, the most rigorous of subjects, are interestingly ill-defined, heavily depending on human<br>creativity and interpretation. I can only name properties of good math education: I learn math<br>best when I am in the middle of the material, and suddenly I "click" and can predict what comes<br>next. The "moment of insight" reminds me of Grant Sanderson's repeatedly emphasized "want you to<br>feel like you could have reinvented yourself" in his channel.<br>It also aligns with the "generation effect" [3] in cognitive<br>psychology, which states that people remember better if they generate the answer themselves<br>instead of just reading it. In my experience, it is non-trivial to write a prompt<br>as it is non-trivial to write a textbook which is good enough to guide students to have the<br>"moment of insight".Regardless of LLM capability, it still requires a non-trivial minimum human effort to learn math;<br>since math is all about building intuition about abstractions, the old, usual, and perhaps the<br>only way is to see and practice a lot of concrete examples, after which the motivation for<br>building some abstraction can be understood, and after which the abstraction itself can be fully<br>grasped. For example, the "group" abstraction requires one to see a lot of integers, reals,<br>polynomials, modular arithmetic, matrices, and so on before knowing why we want such a thing.<br>It's unskippable.I was motivated to write this after reading the Daily Californian's report on UCB<br>that soaring failing grades correlates with increasing AI usage. It is consistent with my<br>above point that one always needs to grind through to build math skills, and also reveals the<br>problematic side, not on LLM itself but on the problem of laziness in human. It does not imply<br>students have gotten more lazy because of AI though, but rather that AI removes a lot of<br>friction for laziness: people used to copy each other's homework, google an answer key, and now<br>they can simply ask AI to solve arbitrary math problems. Since there do exist people who are<br>genuinely willing to throw their entire lives into math, laziness may not be a human nature but<br>rather a product of a flawed education system. The solution is beyond the scope of this essay,<br>but it certainly won't be found by simply trying to "ban AI".References<br>W. Thurston, “What’s...