The Second Derivative of Knowledge: What AI Changes About the Skills We Need

The article introduces the metaphor of the "second derivative of knowledge" — the meta-competence to recognize that I have a problem, to formulate the right query, and to critically assess what the AI returned — and argues that mastering it is precisely what will decide who thrives in the age of artificial intelligence and who remains a mere "button for firing off prompts." The author grounds the thesis in a synthesis of research from 2023–2026 (Bloom's taxonomy, cognitive load theory, randomized studies from Harvard and Wharton, policies of the OECD, UNESCO, and WEF, and examples of schools from San Francisco through Singapore and Beijing to the Czech Republic), showing that AI's effect on learning is determined by design, not the technology itself. At the same time, it warns that meta-competence cannot be built without solid foundations — without automated knowledge in memory, both the "newcomer's dilemma" and the very ability to recognize that an AI's answer makes no sense run aground.

Under Austria-Hungary, pupils learned the extended multiplication tables — the whole thing, all the way up to twenty times twenty. They could multiply three-digit numbers in their heads. Then came logarithmic tables, then calculators, and they lost that ability. When pocket calculators appeared in Czech schools in the 1980s, teachers warned in a panic: children will never learn to do arithmetic. They were partly right — they were just wrong about what it meant.

The generation that grew up with the calculator can't do the extended multiplication tables. But they can program. The generation growing up with AI may not program — but they will be able to formulate problems that no previous generation could even name. The same thing happens every time: one ability recedes, and the brain redirects the freed-up capacity elsewhere. From mechanical calculation to the ability to direct a computation. From writing loops to the ability to design an architecture. From applying knowledge to the ability to know that I have a problem, which discipline solves it, and how to tell that the AI has handed me nonsense.

I call this shift the second derivative of knowledge. And the data from 2023–2026 suggest that mastering it is precisely what will determine who prospers in the age of AI — and who becomes a mere button for launching prompts.

Three derivatives: from facts to the ability to think about thinking

The metaphor from mathematical analysis is no accident. Imagine knowledge as a curve. The zeroth derivative — the value of the function — is the facts themselves: the capital of Bolivia, the formula for calculating resistance, the year of the Battle of Austerlitz. For centuries these formed the core of education. Today AI retrieves them in a fraction of a second.

The first derivative — the slope, the rate of change — is the ability to combine those facts, apply them, see the relationships between them. The engineer who knows not only Ohm's law but can design a circuit. The doctor who not only knows the symptoms but makes a diagnosis. This was the domain of experts — and this is exactly where generative AI is now penetrating.

The second derivative — the curvature, the change in the rate of change — is the meta-competence: to recognize that I have a problem. To identify which field solves it. To formulate the right query. To critically evaluate what the AI has returned to me. To know what I don't know — and to know how to find it out.

This metaphor is not just an elegant shorthand. It finds surprisingly strong support in the academic literature.

Bloom's taxonomy under pressure from AI

Bloom's taxonomy of cognitive objectives — six tiers from remembering through understanding, applying, analyzing, evaluating, all the way to creating — has served since 1956 as a foundational framework for education around the world. AI is turning it upside down.

Oregon State University, in its analysis, recommends fundamentally rethinking the "Remember" level of Bloom's taxonomy, because AI tools can produce accurate answers to queries for basic information. Hmoud and Ali (2024) go further and propose an entirely new taxonomy for the age of AI — six levels: Collect, Adapt, Simulate, Process, Evaluate, Innovate — that replace the traditional pyramid.

The key change, however, lies not just in the lower tiers being "automated." A white paper by the publisher Anthology (2025) argues that AI does not transform only the lower levels but changes the meaning of every level of the taxonomy: it shifts the emphasis from the question "what do you know?" to the questions "how do you know it?" and "how do you know it is trustworthy?"

Gonsalves (2024), in the pages of SAGE Journals, identified a disruption of the traditional hierarchy: students working with generative AI move in non-linear cycles among the cognitive, affective, and metacognitive domains. In other words: Bloom's pyramid disintegrates into a network.

And it is at exactly this point that the second derivative comes into play. Tankelevitch et al. (2024), in work honored at the CHI conference (ACM), arrived at a key insight: the metacognitive demands of working with generative AI resemble the demands on a manager who delegates tasks to a team. The manager must understand and formulate their goals, break them down into communicable tasks, assess the quality of the output, and adjust the plans.

This analogy describes the second derivative in practice exactly. You are not the one who computes. You are the one who says what to compute, why, and whether the result makes sense.

Harvard Project Zero names this shift as a transition from "reckoning" — calculation and prediction, which is the domain of AI — to "judgment": decision-making under uncertainty, ethical considerations, and practical wisdom, which must be provided by a human.

But there is a warning that cannot be ignored. The Online Learning Consortium (2025) points out that students cannot meaningfully engage AI for higher-order critical evaluation without first independently mastering the lower-order analytical and synthetic skills. I will return to this — it is a fundamental complication.

The theories that predicted it

Three learning theories formulated the core of the second derivative long before the arrival of ChatGPT.

George Siemens's connectivism (2005) introduced a principle that today sounds almost prophetic: the ability to know more is more important than what a person currently knows. To the traditional know-how and know-that, Siemens added the concept of know-where — as a key competence for the digital age. In an era when AI functions as a universal knowledge node, navigating networks of knowledge becomes more important than memorizing it.

Heutagogy — the theory of self-determined learning by Hase and Kenyon (2000) — brings the concept of double-loop learning. Single-loop is solving a problem (the first derivative). Double-loop is reflecting on one's own assumptions about how I learn and what I need to learn (the second derivative). Hase wrote: the primary concern of heutagogy is the questions that the learning experience raises, not the provision of answers.

A comparative study (2024) of connectivism and rhizomatic learning confirmed that both theories emphasize technological interconnectedness, distributed knowledge, and adaptability — precisely the competences that the second derivative requires.

International institutions are beginning to put these theories into practice. The OECD Learning Compass 2030 introduces the Anticipation–Action–Reflection (AAR) cycle, where the ability to anticipate the impacts of one's own decisions and to reflect on one's learning forms the core of the competences for 2030. In September 2024 UNESCO issued two competency frameworks — 12 competences in 4 domains for students and 15 competences in 5 domains for teachers — with a progression from understanding through application to creation.

The WEF Future of Jobs Report 2025, based on a survey of more than 1,000 global employers representing over 14 million workers, ranks analytical thinking first among key competences — seven out of ten companies consider it essential. Among the fastest-growing skills through 2030, alongside AI and data analysis, are precisely creative thinking, resilience, and curiosity — by and large the meta-competences of the second derivative.

Schools that teach differently: from San Francisco to Beijing

The theory is nice. What does it look like in practice?

Minerva University in San Francisco is the most radical example. Its philosophy is explicit: information today is abundant and easily accessible, content has become a commodity. The curriculum is not organized by disciplines but around learning outcomes in four competences: critical thinking, creative thinking, effective communication, and effective interaction. No lectures, no final exams — only active seminars of at most twenty students.

The results are impressive: first-year retention of around 96%, a graduation rate of approximately 89% — nearly 26 points above the US national average. An admission rate of around 3%, however, suggests a strong selection bias — the success may be explained more by the students admitted than by the pedagogy.

Finland builds on decades of media-literacy tradition. In 2025 the Finnish National Agency for Education (EDUFI) issued recommendations for AI in education, integrating AI literacy from preschool age through vocational education. Phenomenon-based learning, introduced in the 2016 curriculum, teaches students to tackle interdisciplinary projects instead of isolated subjects. The key challenge: the guidelines have not yet been incorporated directly into the curriculum, whose revision takes place roughly once a decade. Only about 20% of Finnish teachers actively use AI in their teaching.

Singapore structured its approach to AI in education into four layers: Learn about AI, Learn to use AI, Learn with AI, Learn beyond AI — with an emphasis on strengthening domain knowledge and human values that AI will not replace. The Student Learning Space platform deploys the LEA (Learning Assistant) tool, which poses guiding questions instead of direct answers. Data from TALIS 2024 show that 75% of Singaporean teachers use AI — the most in the world.

In 2025 China took a historic step: the Ministry of Education issued guidelines under which, from September 2025, AI is a compulsory subject for all primary and secondary school pupils, with a minimum of eight hours per year. The structure is progressive: at primary school, exposure to voice recognition and image classification; at lower secondary school, technical principles and critical thinking for identifying disinformation in AI outputs; at upper secondary, applied innovation and interdisciplinary systems thinking.

Estonia followed up on its legendary Tiger Leap program of the 1990s with the AI Leap program (2025): students and teachers gained access to AI tools through partnerships with OpenAI and Anthropic, with planned expansion to tens of thousands more students.

And what about the Czech Republic?

In July 2024 the Czech government approved an updated National Artificial Intelligence Strategy of the Czech Republic 2030. In 2025 the NPI ČR (National Pedagogical Institute) issued a methodological document, "How to Teach and Assess in the Age of AI," focused on critical thinking, responsibility for AI outputs, and AI literacy. The "Major Revision" of the RVP (Framework Educational Programme) (2025) integrates AI as a cross-cutting competence — not a separate subject — with mandatory implementation from September 2027.

The numbers, however, show a chasm between strategy and reality. According to an NPI survey (2024, 950 head teachers and 1,449 teachers), only about 20% of Czech teachers actively use AI in their teaching. For comparison: in Singapore it is 75%. The Czech Republic has a strategy, but it does not have the army to carry it out.

Global policies are converging, but practice lags behind

The direction is clear and surprisingly consensual. The OECD and the European Commission are jointly preparing an AI Literacy Framework for primary and secondary schools — a working version from May 2025 defines 22 competences in 4 domains: Engage with AI, Create with AI, Manage AI, Design AI. At the same time, the OECD is preparing an entirely new assessment domain for PISA 2029 — Media and AI Literacy (MAIL) — which will be the first international comparative test of AI literacy.

The EU AI Act (in effect since February 2025) introduces, in Article 4, a requirement for adequate AI literacy among employees. In September 2025 the Commission published guidance for teachers on the key priorities of digital education. In November 2025 South Korea announced a plan to develop AI talent worth USD 960 million.

Implementation, however, remains the main problem. Data from TALIS 2024 (OECD) show that only 41% of teachers globally use AI in their teaching. Three quarters of non-using teachers cite a lack of knowledge and skills as the main barrier. On average, only 38% of teachers have completed training — but the variance is enormous: 76% in Singapore versus 9% in France.

Inequalities add a further dimension. The Brookings Institution (2023) warns of a "third digital divide": wealthy children will get both AI and human teachers, while poor children will get only AI. The Stanford Center for Racial Justice (2024) found that 72% of white American teenagers had heard of ChatGPT compared with 56% of Black teenagers.

Where the truth is more complicated: why you cannot skip the basics

It would be convenient to say: forget facts, teach meta-competences. Cognitive science says: it is not that simple.

John Sweller's cognitive load theory (1988) shows that working memory can hold a maximum of four to seven items at once. Knowledge stored in long-term memory functions as "schemas" — compressed wholes that occupy the space of a single item in working memory. Without them, working memory is overwhelmed and higher-order thinking becomes literally impossible. You cannot manage a team (the second derivative) when you do not understand what the team is doing (the zeroth and first).

Oakley et al. (2025, arXiv), in their work "The Memory Paradox," argue that effective human–AI interaction depends on strong internal models — biological schemas — that enable users to evaluate, refine, and direct AI outputs. In other words: to be a good "AI manager," you need precisely the knowledge you are about to delegate.

A six-year randomized study (Grissmer et al., 2023, University of Virginia, n = 2,310) found that pupils in schools with a content-rich curriculum (Core Knowledge) scored 16 percentile points higher on state reading tests. A curriculum rich in art, history, and science led to a marked improvement in reading skills — not a curriculum focused on general "skills."

Robert Bjork's concept of "desirable difficulties" adds another layer. Four demonstrably effective practices — spacing learning over time, interleaving topics, retrieval from memory, and active generation — produce markedly better long-term retention, even though in the short term they appear less efficient. AI systematically removes precisely these difficulties by providing immediate answers. It optimizes short-term performance at the cost of damaging long-term learning.

Punya Mishra (2025) analyzed a Microsoft study (319 knowledge workers) and identified the "novice's dilemma": learners who have neither domain expertise nor expertise in working with AI are doubly vulnerable — unable to assess the accuracy of AI outputs and unaware of when and how AI may lead them astray.

This is the strongest counterargument: the second-derivative thesis presupposes that the basics exist. For a novice who knows nothing, the second derivative is undefined — you have nothing to differentiate.

It is best illustrated by an analogy with language. Active knowledge — the ability to speak, write, argue — is conditioned by passive knowledge: you must understand words before you can use them. No one learns fluent Czech by skipping the phase in which they understand but do not yet speak. Likewise in mathematics: higher-order logical reasoning rests on mastered arithmetic, even if you then never do it by hand again. A child who has never done arithmetic has no intuition for numbers — does not know whether a result "makes sense." And it is precisely this intuition that you need when evaluating an AI's output. The question, then, is not "whether to teach the basics" but how much, how, and when to move higher.

But history shows: the human brain never lost the capacity — it always redirected it. The pupils who could multiply large numbers in their heads did not have more neurons than today's programmers. They simply had differently allocated cognitive resources. The question is not whether the brain will lose an ability — that one will move elsewhere. The question is whether we direct where it moves, or whether we leave it to chance and the algorithm. This is precisely why the second derivative is so important: it is not just a new competence, it is the competence to direct one's own cognitive reorganization.

What the hard data say: design decides everything

Two studies from 2025 together paint a surprisingly clear picture — and that picture is not black and white.

A Harvard randomized study (Kestin et al., 2025, Scientific Reports, n = 194) found that students with a carefully designed AI tutor — based on GPT-4, pedagogically structured to ask short questions, guide step by step, and support the student's own thinking — learned more than twice as much in less time compared with active learning in the classroom.

On the other hand: Bastani et al. (2025, PNAS, Wharton/UPenn, approximately 1,000 high-school students at a Turkish school) found that the group with unrestricted access to ChatGPT solved 48% more problems during practice but scored 17% worse on a subsequent test without AI.

Together, these two studies say something fundamental: the effect of AI on learning depends entirely on design, not on the technology itself. A well-designed AI tutor with deliberate "desirable difficulties" dramatically improves learning. Unmanaged access to a chatbot damages it. And it is precisely the ability to distinguish these two modes — that is the meta-competence of the second derivative.

Gerlich (2025, MDPI, 666 respondents) found a strong negative correlation between the frequency of AI use and critical-thinking scores. A report by the University of Technology Sydney (2026) introduces the term "performance paradox": AI increases a student's immediate performance while simultaneously weakening durable learning. Students, moreover, demonstrably offload critical thinking and other cognitively demanding operations to the AI.

Three meta-analyses from 2025 confirm an overall positive effect of AI on academic performance — Dong et al. (29 studies, 2,657 participants) report an effect size of 0.924, Ma et al. (34 studies) an effect size of 0.68 — but with high heterogeneity. The effects depend dramatically on context and design.

Encouraging, by contrast, are the data on the teachability of meta-competences. A meta-analysis of metacognitive interventions in mathematics (2025, Cogent Education, 43 studies, 13,924 participants) found an effect size of 1.11 for mathematical performance and 1.18 for metacognitive skills themselves. Eberhart et al. (2024, Metacognition and Learning, 67 studies) confirmed the effectiveness of metacognitive interventions in children, with a persistent effect at follow-up measurement. A meta-analysis of long-term effects in university students (49 studies, 5,786 participants) found that programs based on metacognitive theory had higher effects on academic performance than programs based on purely cognitive theory.

The second derivative, then, can be taught — and its effects are measurable and lasting. But only if there is a foundation on which it stands.

Employers confirm it: they want meta-competences

As we showed above, the WEF Future of Jobs Report 2025 identifies analytical thinking as the number one key competence. The PwC Global AI Jobs Barometer (2025, analysis of approximately 1 billion job advertisements) found that skills in AI-exposed professions change 66% faster than in other professions, and that workers with AI skills command a 56% wage premium.

The ZipRecruiter Annual Employer Survey (2025, over 1,500 professionals) paradoxically found that the three most sought-after skills are human ones: collaboration, customer service, and communication. The most lacking skill is critical thinking.

Meanwhile, AI adoption among students is exploding: according to the College Board, most American high-school students use generative AI for schoolwork, with up to 92% of students overall reporting that they use AI. But according to the RAND Corporation (2025), more than 80% of students report that their teachers never explicitly taught them how to use AI.

A dangerous situation is arising: mass adoption is outpacing pedagogical preparation. Students have the tool of the second derivative in their hands, but no one is teaching them how to use it — and no one is teaching them that they need to know how to use it.

The three derivatives are not a choice, but a hierarchy

Let us return to the cascade from the introduction. The calculator. The programming language. AI. Each time the same thing happened: people lost one ability — and their brain redirected the freed-up capacity elsewhere. Programmers stopped writing assembler but learned to design systems. No one lost a piece of their brain. The brain reorganized itself.

With AI it is the same, just an order of magnitude higher.

The research from 2023–2026 does not support the interpretation that the zeroth derivative — factual knowledge — is useless and can be skipped. Cognitive science clearly shows that without automated schemas in long-term memory, working memory is overwhelmed and higher-order thinking fails. You cannot manage a team whose work you do not understand. But at the same time it is clear that traditional education invested disproportionately much time in the zeroth derivative at the expense of the first and second. Students spent thousands of hours memorizing information they can now find in a second — and yet they were never systematically taught how to recognize a problem, formulate a question, or evaluate the quality of an answer.

Three insights that are not obvious from the individual studies but emerge from their synthesis:

Design decides more than technology. The Harvard AI tutor doubled learning. Unmanaged ChatGPT worsened it by 17%. Meta-competences — the ability to distinguish when AI helps and when it harms — are therefore existentially important.

Immediate performance ≠ durable learning. Without metacognitive awareness — the second derivative — a student does not even realize that they are at that moment offloading hard cognitive work to the machine. As the so-called performance paradox showed, AI raises results here and now but may weaken the building of durable knowledge.

A new type of inequality: The meta-competences of the second derivative are already unevenly distributed. The "novice's dilemma" afflicts precisely those who need education most.

The solution is neither "back to rote learning" nor "leave everything to AI." It is a shift in proportions: less time on memorizing facts, more on applying them — and substantially more on the ability to recognize what I don't know, where I will find it out, and how I will tell that an answer makes sense.

Every time the dominant tool has changed in history, people lost one ability — and the brain redirected the freed-up capacity elsewhere. Now it is happening again. The question for the next ten years is not whether the brain will redirect — that is certain. The question is whether we will direct that shift, or whether we will leave it to chance.

Methodological note: This article synthesizes research from 2023–2026, including meta-analyses, RCT studies, OECD surveys (TALIS 2024), and policy documents from UNESCO, the WEF, and the EU. Key limitations: most studies on the effects of AI on learning are short-term (weeks to months); long-term longitudinal studies are lacking. Both the Wharton study (Bastani et al.) and the Harvard study (Kestin et al.) have relatively small samples. The meta-analyses exhibit high heterogeneity, which suggests that the effects depend strongly on context. Data on Czech education are limited to the NPI surveys from 2024. Additional sources: OECD Learning Compass 2030, EU AI Act, UNESCO AI Competency Frameworks (2024), WEF Future of Jobs Report 2025, PwC Global AI Jobs Barometer 2025, RAND Corporation 2025.

The conception, structure, and editorial line of the article are the work of the author, who prepared the content outline, established the key theses, and directed the entire creation process. Generative AI (Claude, Anthropic) was used as a tool for research, fact-checking, and fleshing out the author's draft.

The author edited the outputs throughout, verified the key findings, and approved the final wording. No part of the text was published without human oversight. All factual data were verified against the publicly available sources cited in the text.

The procedure complies with the requirements of Art. 50 of EU Regulation 2024/1689 (AI Act) on the transparency of AI-generated content. #poweredByAI