{"slug": "math-education-and-llm", "title": "Math Education, and LLM", "summary": "A new analysis argues that regardless of advances in large language models, learning mathematics requires a non-trivial minimum of human effort because math is fundamentally about building intuition and abstraction, not just calculation or theorem proving.", "body_md": "# Math Education, and LLM\n\n2026-06-16\n\n*Abstract: This article defines math and math education, and argues for a lower bound of\nhuman effort required to learn mathematics (or any other abstraction-heavy subject) regardless\nof LLM capability.*\n\nLLMs are evolving rapidly; within a year AIs are able to tackle math problems - we had\nthought of them as the hardest for AI to automate - but they are getting there. Let it be OpenAI's\n[marketing bluff](https://openai.com/index/model-disproves-discrete-geometry-conjecture/)\nor not (of course humans helped, how much? we can never know...), we can at least say that\nfrontier models are useful for assisting with math research. The natural question to ask here is\nthat, is AI helpful at math education? To what extent?\n\nMathematics is a truly unique subject from all other sciences, in that it is not\nnatural at all, despite the fact that universities like to put math in a physical science\nbuilding of some sort. While it originates in counting and measuring the physical world, it has\nevolved out of its physics context into a discipline purely focused on a-priori reasoning and\nabstraction during the [formalism](https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics))\nmovement. My preferred way to define math is \"the study of a-priori constructions\". I'd also\nlike to think of all math knowledge as an infinitely large graph of theorems where one node\npoints to another if one can be deduced from the other by the axioms chosen. In this interesting\nperspective, math is much like art, poetry, or music, where every theorem already [exists somewhere](https://en.wikipedia.org/wiki/G%C3%B6del_numbering)\nand we are just discovering them. An implication of this view is that, humans have to occupy a\nposition in math research, since we are the ultimate judge to say whether an abstraction or\ntheorem is interesting and worth developing or not. Math is tightly connected to personal and\ncollective taste and intellect: *\"The product of mathematics is clarity and understanding.\nNot theorems, by themselves.\"* [[1]](#cite-budneyWhatsMathematician2010)\n\nAs a result, calculation or theorem proving is only a small part of doing math, and the goal is\nrather to cultivate good instinct [[2]](#cite-taoThereMoreMathematics2026) - the ability to fluently\nnavigate and manipulate some levels of abstraction, and thus \"sense\" how to get from one node to\nanother or which nodes are worth exploring. People used to develop abstraction out of physical\nproperties, such as the invention of calculus which was used to describe continuous physical\nphenomena. But now, the abstraction is so far removed from the physical world that it is often\nthe case that mathematical abstractions are invented before any application is found: Riemannian\ngeometry, a 19th-century invention, is now the language of general relativity (1915).\n\nMath education gets hard here, since the properties of good math education, in contrast to math\nitself, the most rigorous of subjects, are interestingly ill-defined, heavily depending on human\ncreativity and interpretation. I can only name properties of good math education: I learn math\nbest when I am in the middle of the material, and suddenly I \"click\" and can predict what comes\nnext. The \"moment of insight\" reminds me of Grant Sanderson's repeatedly emphasized \"want you to\nfeel like you could have reinvented <math topic> yourself\" in his [channel](https://www.youtube.com/@3Blue1Brown).\nIt also aligns with the \"generation effect\" [[3]](#cite-slameckaGenerationEffect1978) in cognitive\npsychology, which states that people remember better if they generate the answer themselves\ninstead of just reading it. In my experience, it is non-trivial to write a prompt\nas it is non-trivial to write a textbook which is good enough to guide students to have the\n\"moment of insight\".\n\nRegardless of LLM capability, it still requires a non-trivial minimum human effort to learn math;\nsince math is all about building intuition about abstractions, the old, usual, and perhaps the\nonly way is to see and practice a lot of concrete examples, after which the motivation for\nbuilding some abstraction can be understood, and after which the abstraction itself can be fully\ngrasped. For example, the \"group\" abstraction requires one to see a lot of integers, reals,\npolynomials, modular arithmetic, matrices, and so on before knowing why we want such a thing.\nIt's unskippable.\n\nI was motivated to write this after reading [the Daily Californian's report on UCB](https://www.dailycal.org/news/campus/academics/failing-grades-soar-as-professors-see-greater-ai-usage-dwindling-math-skills-in-uc-berkeley/article_16fad0bf-02cb-4b8c-8d88-888ffd9f8608.html)\nthat soaring failing grades *correlates* with increasing AI usage. It is consistent with my\nabove point that one always needs to grind through to build math skills, and also reveals the\nproblematic side, not on LLM itself but on the problem of laziness in human. It does not imply\nstudents have gotten more lazy because of AI though, but rather that AI removes a lot of\nfriction for laziness: people used to copy each other's homework, google an answer key, and now\nthey can simply ask AI to solve arbitrary math problems. Since there do exist people who are\ngenuinely willing to throw their entire lives into math, laziness may not be a human nature but\nrather a product of a flawed education system. The solution is beyond the scope of this essay,\nbut it certainly won't be found by simply trying to \"ban AI\".## References\n\n- W. Thurston, “What’s a mathematician to do?,” MathOverflow. Accessed: May 11, 2026. [Online]. Available: https://mathoverflow.net/q/43690\n- T. Tao, “There’s more to mathematics than rigour and proofs.” Accessed: Jun. 17, 2026. [Online]. Available: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/\n- N. J. Slamecka and P. Graf, “The generation effect: Delineation of a phenomenon,” Journal of Experimental Psychology: Human Learning and Memory, vol. 4, no. 6, pp. 592–604, 1978, doi: 10.1037/0278-7393.4.6.592.", "url": "https://wpnews.pro/news/math-education-and-llm", "canonical_source": "https://ycao.net/posts/math-education-llm", "published_at": "2026-06-17 11:40:24+00:00", "updated_at": "2026-06-17 11:53:00.701666+00:00", "lang": "en", "topics": ["large-language-models", "ai-research", "ai-ethics"], "entities": ["OpenAI", "Grant Sanderson", "3Blue1Brown"], "alternates": {"html": "https://wpnews.pro/news/math-education-and-llm", "markdown": "https://wpnews.pro/news/math-education-and-llm.md", "text": "https://wpnews.pro/news/math-education-and-llm.txt", "jsonld": "https://wpnews.pro/news/math-education-and-llm.jsonld"}}