Mathematics in the age of artificial intelligence

Salvatore Mosaico · 2026

Author’s note

The classroom scenes are not transcripts: they are simulations designed to show how a teaching idea becomes dialogue, discovery, error and argument.

Contents

1. 1. The question artificial intelligence poses to school
2. 2. Knowing is not merely obtaining an answer
3. 3. Mathematical skills that become central
4. 4. Simulated lesson 1: chickens and rabbits
5. 5. Simulated lesson 2: x² − y² = 15
6. 6. Simulated lesson 3: the twelve coins
7. 7. Simulated lesson 4: conjectures and counterexamples
8. 8. Simulated lesson 5: questioning and checking AI
9. 9. How exercises, assignments and assessments change
10. 10. The teacher’s new role
11. 11. A possible five-session workshop
12. 12. Conclusion: not less mathematics, but more authentic mathematics

1. The question artificial intelligence poses to school

I have watched artificial intelligence solve equations, geometry problems, sequences, logic questions and even puzzles whose difficulty lies not in calculation but in finding the idea. This raises an unavoidable and deliberately provocative question: if a machine can do all this, what must we still teach students so that, without it, they do not become incapable of thinking?

Neither rejecting technology nor surrendering to it is an answer. Banning AI as if it were a passing accident means educating for a world that no longer exists. Handing every activity over to it means abandoning precisely what mathematics can develop: attention, autonomy, abstraction, appreciation of proof and the habit of distrusting the first plausible answer.

Calculators did not make numbers useless. They shifted the boundary between what was worth doing by hand and what was worth understanding. AI creates a deeper shift: it automates not just calculation but part of the language used to explain calculation. School therefore cannot simply add a new tool to the old lesson; it must redefine what counts as mathematical performance.

This essay’s thesis is simple: school mathematics must continue to provide basic techniques and fluency, but should assess above all the ability to choose, represent, check, argue and generalise. The final answer remains necessary, but ceases to be the exclusive centre of learning.

In the twentieth century, school often rewarded those who could find answers. In the AI age, it must also teach students to formulate good questions, judge the answers they receive and construct arguments they can defend before others.

2. Knowing is not merely obtaining an answer

Suppose a student photographs an equation and receives the correct answer within seconds. What do they actually possess? An output, not necessarily knowledge. Knowledge appears when they can explain which transformations are legitimate, recognise existence conditions, anticipate order of magnitude, notice contradictions and transfer the method to another problem.

At least five levels should be distinguished. First, execution: applying a known procedure. Second, understanding: explaining why it works. Third, choice: deciding which procedure fits. Fourth, control: checking assumptions and results. Fifth, creation: modifying or generalising the problem, or inventing a counterexample.

AI is often strong at the first level and can assist with the others; but epistemic responsibility—the responsibility for deciding why an answer deserves trust—must remain with the person. This avoids two opposite mistakes: thinking every automatic answer is false, or believing that a well-written answer must be true.

Manual calculation also retains a purpose. It should not survive as a punitive ritual but as targeted practice. Multiplication tables, fractions, conversions, fundamental algebraic transformations and mental agility reveal structures that would otherwise remain opaque. Without an internal mathematical vocabulary, students cannot even check the external tool.

The criterion is therefore not “manual or automatic”. It is: which part of the activity develops the thinking we want to cultivate today?

3. Mathematical skills that become central

A contemporary approach can be organised around eight practices already present in the best traditions of mathematics education, now made more urgent.

1. Start with problems. A formula introduced after a need emerges has meaning; a formula presented as an instruction to imitate is soon delegated to a machine.

2. Build representations. Drawings, diagrams, words, case tables, graphs and equations are not decorations: they are different ways of seeing the same object. Translating between representations is a robust skill.

3. Compare strategies. Two correct solutions need not be educationally equivalent. One may be more general and the other more intuitive; one faster and the other more transparent.

4. Estimate before calculating. Estimation creates an expectation against which to check the exact result. It is one of the best antidotes to blind trust.

5. Argue. “Why?” should be an ordinary question. Formal proof is not always required, but a reason appropriate to the learner’s age always is.

6. Seek counterexamples. Many statements look true because they work in the first cases. Looking for the case that breaks them teaches the value of hypotheses and the difference between evidence and proof.

7. Model. Moving from reality to mathematics involves choices: what to count, what to ignore and which quantities to relate. The model must then be brought back to reality and criticised.

8. Use AI openly and verifiably. Students should be able to say what they asked, received, checked and changed. The aim is not to catch tool use but to make the intellectual work following its use visible.

4. Simulated lesson 1: chickens and rabbits

Problem: a yard contains 35 animals, chickens and rabbits, with 94 legs in total. How many chickens and rabbits are there?

The standard school method introduces two unknowns and a system of equations. It is correct, but for upper-primary or early-secondary pupils it may hide the idea beneath notation. A thought experiment is a better starting point.

CLASSROOM SCENE — FROM STORY TO MODEL

Teacher: Pretend for a moment that all 35 animals are chickens. How many legs would we have?

Pupil: Seventy, because we give each animal two legs.

Teacher: But we observe 94. How many legs remain unexplained?

Class: Twenty-four.

Teacher: When we turn an imaginary chicken into a rabbit, how many legs must we add?

Pupil: Two. So 24 ÷ 2 = 12 rabbits.

Teacher: And the chickens?

Class: 35 − 12 = 23 chickens.

The decisive step is not division. It is the temporary “all chickens” assumption, which creates a reference situation. The 24 extra legs become information: every rabbit contributes two more legs than a chicken.

The teacher can now let algebra emerge rather than impose it. If p denotes chickens and c rabbits, the statements become p+c=35 and 2p+4c=94. Subtract twice the first relation from the second to obtain 2c=24. The system is no longer a mysterious ritual: it is the compact notation of the same reasoning.

The lesson can continue with variations: what if there were 40 animals? What if there were 95 legs? The latter is impossible because a sum of even numbers is even. A property lets us judge the statement before calculating.

What is assessed? Not just 12 and 23, but the ability to explain why we start from 70, divide by 2, and check that 12×4+23×2=94.

5. Simulated lesson 2: x² − y² = 15

Finding all integer pairs (x,y) satisfying x²−y²=15 illustrates the difference between manipulating symbols and seeing a structure.

CLASSROOM SCENE — A TRANSFORMATION THAT OPENS THE PROBLEM

Teacher: We could try many values. But do you recognise a familiar form?

Student: A difference of squares: x²−y²=(x−y)(x+y).

Teacher: What do we then know about the two factors?

Student: They must be integer divisors of 15 whose product is 15.

Class: We start with 1 and 15, and 3 and 5, then reverse their order and also consider the negative pairs.

Setting x−y=1 and x+y=15 gives 2x=16, hence x=8; subtracting gives 2y=14, hence y=7. The factors 3 and 5 give x=4, y=1. Reversing the factors changes the sign of y; negating both changes the signs of x and y. The eight solutions are (8,7), (8,−7), (−8,7), (−8,−7), (4,1), (4,−1), (−4,1), (−4,−1).

Why do x−y and x+y have the same parity? Their sum is 2x and their difference is 2y. Because 15 is odd, its factor pairs are both odd and yield integer x and y. Writing u=x−y, v=x+y gives x=(u+v)/2 and y=(v−u)/2. The ordered factor pairs listed exhaust the possibilities, so the solution list is complete.

Changing the domain changes the problem. Over the integers there are finitely many solutions; over the reals the equation describes a hyperbola and has infinitely many. “Which numbers are allowed?” is not a formality: it determines the nature of the answer.

AI can quickly list pairs. Students must explain why the list is complete. The word all requires a completeness argument, not a collection of examples.

6. Simulated lesson 3: the twelve coins

Twelve coins look identical; one has a different weight, but we do not know whether it is heavier or lighter. Using a balance scale, we must identify it in three weighings.

The complete procedure is ingenious, but before its decision diagram comes a more powerful question: how much information can one weighing provide? There are three outcomes: left heavier, balance, right heavier.

Three weighings distinguish at most 3³=27 outcome sequences. There are 24 cases: each of the 12 coins may be heavier or lighter. The inequality 27≥24 means the information count does not rule out success; it does not by itself prove that a physically realisable strategy exists. Two weighings yield only 3²=9 sequences and cannot suffice.

CLASSROOM SCENE — STRATEGY BEFORE MOVES

Teacher: If I weigh six against six and the scale tilts, what happens?

Pupil: The unusual coin could still be any of the twelve; we have not identified it or whether it is heavy or light.

Teacher: Have we divided the cases well?

Class: No. Four against four leaves more manageable groups, and an imbalance also identifies normal coins among those left out.

This introduces an idea anticipating information theory: a good question divides possibilities evenly. Instead of memorising a trick, students understand why some first moves are unproductive and others useful.

AI can generate or check the case tree. Students formulate the invariant: after each weighing, the remaining cases must be few enough to distinguish with the remaining weighings. More than nine cases after the first weighing cannot all be separated by two more. That numerical bound is necessary, but each branch must also be achievable with legal weighings.

An interesting assessment need not demand the complete algorithm. It could offer three first weighings and ask which preserves the possibility of success, with an information-based justification. For six against six, each imbalance leaves twelve possible explanations, exceeding nine; four against four leaves eight per possible outcome.

7. Simulated lesson 4: conjectures and counterexamples

A formula produces many primes in its first trials. It is natural to conjecture that it always does. But “it worked a hundred times” does not mean “it works for every natural number”. AI makes calculating thousands of cases easy, which makes teaching the different roles of numerical checking and proof especially important.

CLASSROOM SCENE — BREAKING AN ATTRACTIVE RULE

Teacher: Consider n²+n+41. For n=0,1,2,… we obtain many primes. Can we proclaim a law?

Student: We can say it seems true in the tested cases.

Teacher: Try a value related to 41.

Student: For n=41, we get 41²+41+41=41×43, so it is not prime.

Teacher: One counterexample defeats a universal statement.

The activity can be arranged into three conceptual columns, even without a table: calculating produces cases; conjecturing formulates a rule suggested by cases; proving establishes the rule for every permitted case. A counterexample, by contrast, suffices to show a universal rule is false.

Error becomes teaching material, not a stain to erase quickly. Locate it: what is the first unjustified step? Which hypothesis was missing? Does the result remain true for a smaller class of cases?

This habit matters beyond mathematics. Many observations may support a prediction without proving logical necessity. Learning the difference prepares students to read statistics, graphs and algorithmic claims more clearly.

8. Simulated lesson 5: questioning and checking AI

The poorest educational use of AI is to ask for a solution and copy it. The richest is to make the automatic answer an object of analysis.

CLASSROOM SCENE — AI UNDER EXAMINATION

Task: solve 2ˣ+x=5, then evaluate a simulated AI answer claiming x=2 because 2²+2=6.

First observation: substituting x=2 immediately shows that 2 is not a solution.

Second observation: f(x)=2ˣ+x−5 is strictly increasing, because both 2ˣ and x increase. There can therefore be at most one real solution.

Search: f(1)=−2 and f(2)=1. Continuity gives a solution between 1 and 2; strict increase makes it unique. It can be approximated numerically.

Final question: which part has been proved exactly, and which part is only approximated?

The class can assign roles: one group formulates the request, another checks every step, a third seeks an alternative solution and a fourth changes the data to test the method. The final submission must clearly separate the tool’s contribution from students’ decisions.

Even a correct answer can be criticised: does it use a theorem that is too advanced, conceal an excluded case, or introduce unnecessary notation? Confusing correctness and quality would be a mistake. A good solution is correct, complete, understandable and appropriate to the problem and audience.

Teachers can also ask students to improve their requests. Instead of “solve”, ask “suggest two methods”, “do not omit conditions”, “include a plausible error to detect”, “give only a hint”, or “generate a similar but different problem”. Formulating the request becomes metacognition: students must know what help they need.

9. How exercises, assignments and assessments change

If an assignment remains a sequence of identical standard exercises, AI makes it difficult to know who has learned. The answer is not more surveillance, but designing performances in which the process is visible.

A good assignment can have four moments: an initial prediction without tools; a solution, possibly assisted; an independent check; and reflection on choices. Questions become: which method did you choose and why? Which step was weakest? How do you know no solutions are missing? What changes when one datum changes?

A mixed assessment can retain a short tool-free part for fundamentals alongside a part using declared tools. The latter assesses quality of judgement rather than execution speed.

EXAMPLE OF AN AUTHENTIC ASSESSMENT

Part A — Fundamentals: carry out and explain two essential transformations without tools.

Part B — Choice: solve a problem using an arithmetic, algebraic or graphical method and justify your choice.

Part C — Critique: examine an automatic solution containing an error. Identify the first error and correct only what depends on it.

Part D — Transfer: change a datum so the problem has no solution, or more than one. Justify.

Part E — Transparency: declare the tools used and the checks performed personally.

Assessment itself must change. A possible rubric weights five dimensions: understanding the statement and assumptions; strategy; correctness; argument and checking; ability to revise. A calculation error then does not erase valid reasoning, while an unjustified correct number does not automatically receive full marks.

A short oral discussion gains value. Targeted questions suffice: “why this step?”, “what do you predict if I double the datum?”, “give an example that does not work”. Students who understand can adapt; those who merely copied tend to lose their way when the problem changes.

10. The teacher’s new role

A teacher does not become less important because a machine can explain. They become less central as the exclusive distributor of procedures and more important as a designer of cognitive experiences. They choose fruitful problems, listen to incomplete strategies, decide when not to intervene, connect ideas and establish criteria of validity.

A good teacher’s question is often worth more than a long explanation. “What do you already know?”, “can you represent it?”, “which simple case would you try?”, “how could you disprove your idea?”, “where do you use this assumption?” These questions do not take the problem away from students: they give them footholds.

AI can lighten preparation: creating variations, adapting language levels, suggesting examples and producing deliberately flawed solutions. But choosing what is educationally meaningful remains situated: it depends on the class, observed errors, available time and students’ actual knowledge.

Equity also needs protection. Not everyone has the same tools, connection or ability to question them. Essential activities must be accessible at school; usage rules must be explicit; personal data must not become the price of learning.

The international guidance cited below stresses a human, responsible approach: AI requires attention to privacy, age, inclusion, transparency and the irreplaceable role of educational judgement. It is neither an oracle nor simply an enemy, but a new environment in which to educate.

11. A possible five-session workshop

Transformation does not require rebuilding the curriculum from scratch. A small workshop integrated into ordinary lessons is a starting point.

Session 1 — The plausible answer. Present three solutions: correct, containing a calculation error, and containing an illegitimate assumption. Students identify the first point where each argument earns or loses trust.

Session 2 — Many methods, one problem. Solve chickens and rabbits through drawing, extra-leg reasoning, a trial table and equations. Compare accessibility, speed and generality.

Session 3 — From example to proof. Test a formula with a spreadsheet or AI, then seek a counterexample and explicitly distinguish conjecture from proof.

Session 4 — Information and strategy. With coins, a simulated balance or three-outcome questions, design decisions that divide the space of cases well.

Session 5 — AI as reviewer and defendant. Each group proposes a problem, obtains an automatic answer, annotates and corrects it, and presents a reasoned verdict. Assessment focuses on the quality of checks.

Finally, each student completes a short “thinking diary”: what did I predict? Where did I change my mind? Which check convinced me? What could I do again without help? This makes learning observable without turning it into bureaucracy.

12. Conclusion: not less mathematics, but more authentic mathematics

The fear that AI makes us less intelligent contains a conditional truth: it will if we use it to avoid every cognitive effort. But this is not destiny. A technology producing answers can finally force us to value questions, proofs, checks and discussion of methods more highly.

Students must still learn to calculate. Without foundations, checking becomes fragile. Above all, however, they must learn when to calculate, what a calculation represents, which conditions make it valid and how to recognise an absurd result. They must move from concrete problems to models and back to reality. They must distinguish examples from proofs and convincing phrases from correct arguments.

School does not compete with AI in exercises solved per minute. Its task is to form people who can navigate abundant answers. Faced with an elegant solution, they should still ask: “Why does it work? Always? Under which conditions? Can I check it another way?”

AI does not mark the end of mathematics education. It may mark the end of an overly mechanical version. If we seize the opportunity, we return to the discipline’s heart: not calculation as obedience, but reasoning as a form of freedom.

The decisive question is not whether students can obtain an answer without us. It is whether they can recognise a good answer, defend it with reasons and have the courage to reject it when it is wrong.

Essential references

• G. Pólya, How to Solve It, Princeton University Press — Understand, plan, execute and review.
• H. Freudenthal, Mathematics as an Educational Task — Mathematics as human activity and guided reinvention.
• A. H. Schoenfeld, Mathematical Problem Solving — Strategies, metacognitive control and belief systems.
• UNESCO, Guidance for Generative AI in Education and Research (2023) — Human-centred, safe, equitable and meaningful AI use.
• European Commission, Ethical Guidelines on the Use of Artificial Intelligence and Data in Teaching and Learning for Educators (2022) — Transparency, responsibility, inclusion and data protection.
• National Council of Teachers of Mathematics: problem solving, reasoning, representation and communication — Problem solving, reasoning, representation and mathematical communication.

Written for online publication. Short excerpts may be quoted with attribution to the author and title.

Read also

The programmer in the age of artificial intelligence