Salvatore Mosaico · 2026
Contents
- 1. The first challenge: does knowing things no longer matter?
- 2. First example: a valid answer can be incomplete
- 3. Second example: an answer can create a question we did not have
- 4. Third example: it works, but is it well designed?
- 5. The hardest objection: are we wasting time?
- 6. Returning to the original saying
The teacher projects a statement for the class to discuss:
“Intelligence is not the ability to store information, but knowing where to find it.”
An AI assistant is available in the classroom. Its answers are projected for everyone to discuss. All the dialogue below, including the AI’s answers, is part of a teaching simulation.
1. The first challenge: does knowing things no longer matter?
Teacher: This statement distinguishes intelligence from remembering lots of information. How would you change it now that we have artificial intelligence?
Anna: I would say: “Intelligence is knowing what to ask AI.”
Luca: Does that mean someone who knows nothing becomes intelligent simply by asking questions?
Anna: That is not what I said. But if I can get an explanation when I need it, why must I learn everything beforehand?
Marta: I think you are confusing remembering an answer with understanding it.
Davide: Here is a practical question. If I need to solve a problem and AI solves it, the problem is solved. What difference does my understanding make?
Teacher: That depends on our goal. Do we only want this particular result, or do we also want to become capable of tackling other problems?
Davide: In real life, I often just want the result.
Teacher: That is a reasonable objection. We do not need to become experts in everything we use tools for. But we must understand when we can rely on a result and when we need to check it. Let us try an example.
2. First example: a valid answer can be incomplete
The teacher writes:
x(x − 1) = 2(x − 1)
Teacher: Let us ask AI to solve it.
AI: Dividing both sides by x − 1 gives x = 2.
Anna: There is our answer.
Luca: But you always tell us to check.
Teacher: Let us check it.
Marta: Substituting x = 2 gives 2(2 − 1) = 2 on both sides. It works.
Davide: So we used AI, checked the answer and finished.
Luca: Wait. In the original equation, x = 1 works too: both sides are zero.
Anna: Is the AI’s answer wrong, then?
Marta: Two really is a solution. The problem is that one is missing.
Teacher: Exactly. We checked that the proposed solution works, not that every solution has been found.
Davide: So checking the result is not enough?
Teacher: Not here. We must also check the reasoning. What happened when we divided by x − 1?
Luca: For x = 1, we divided by zero.
Teacher: That step is only allowed when x ≠ 1. We must examine x = 1 separately. Alternatively, we can avoid division:
x(x − 1) − 2(x − 1) = 0
(x − 1)(x − 2) = 0
A product is zero when at least one factor is zero. Therefore x = 1 or x = 2.
Anna: To spot the problem, I needed to know about division by zero already.
Teacher: Yes. Mathematical knowledge helped us evaluate the answer.
Anna: What if I did not know it?
Teacher: You could learn it through this very discussion. We do not need to know everything before using AI. But getting an answer does not make learning unnecessary.
Marta: I did not spot it myself. I noticed when Luca tried one.
Teacher: That matters: checking need not be a lone student’s struggle against a machine. It can grow out of discussion with classmates, the teacher and AI.
Davide: You could miss a solution too.
Teacher: Certainly. Our conclusion should not be “trust the teacher instead of AI.” It should be: ask for reasons we can examine, whoever proposes the answer.
3. Second example: an answer can create a question we did not have
Teacher: So far we have discussed errors. Now let us consider a useful answer that takes us beyond our original question.
√(128 · 98)
Teacher: Ask AI to calculate this square root and show its method.
AI: We can write the two numbers as follows:
128 = 8² · 2
98 = 7² · 2
√(128 · 98) = √(8² · 7² · 4)
= 8 · 7 · 2 = 112
Davide: Good. This time the result is complete.
Marta: Something interests me: neither 128 nor 98 is a perfect square, but their product is.
Luca: Is that because they are both even?
Anna: We could ask AI.
Teacher: We can. But first let us test that idea ourselves. Which numbers would you choose?
Davide: Eight and ten. They are both even, and their product is eighty.
Marta: Eighty is not a perfect square: it lies between sixty-four and eighty-one.
Luca: So being even is not enough.
Teacher: Good. We have found a counterexample. Looking again at the factorisation, what do 128 and 98 have in common?
Anna: Each is a square multiplied by two.
Marta: And the two factors of two make four, another square.
Teacher: Can we use that observation to construct another example?
Marta: Let us try:
18 = 3² · 2
50 = 5² · 2
√(18 · 50) = 3 · 5 · 2 = 30
Davide: Indeed, eighteen times fifty is nine hundred.
Luca: We have not just done the calculation: we have found a way to invent more examples.
Teacher: Exactly. Now ask AI to express your observation in general terms.
AI: Choose positive integers a, b and k. Construct ka² and kb². Their product is:
(ka²)(kb²) = k²a²b² = (kab)²
√[(ka²)(kb²)] = kab
This gives a way to construct two numbers whose product is a perfect square.
Anna: I would now change what I said at the beginning.
Teacher: How?
Anna: I said intelligence means knowing what to ask AI. But we did not know this question beforehand. It arose from looking at the answer.
Marta: I only wanted a calculation. The factorisation helped me notice something I was not looking for.
Teacher: So the relationship is not just:
Ask a good question → get a good answer.
It can also be:
Ask a simple question → encounter an interesting idea → form a new question → try to test it.
Davide: And the new question can come from a classmate, not just AI.
Teacher: Exactly as it did here.
4. Third example: it works, but is it well designed?
Teacher: Let us turn to computing. We have already built
RandomNumero(N). Its contract is to return a random integer from 1 to N, inclusive, with equal probability for each integer, for positive integer N.
Now we want RandomRange(a, b), returning a random integer between a and b, inclusive. We assume a and b are integers and a ≤ b.
Anna: Let us ask AI.
The AI proposes an independent function built directly on the underlying random generator. The class examines it and checks that the method is correct.
Davide: We checked the endpoints this time too. It works.
Teacher: Yes. But have we used what we had already built?
Marta: No. We already had
RandomNumero(N).
Luca: Must we use it? If the new function works, why call it wrong?
Teacher: It is not automatically wrong. We need to decide whether our project is better served by reusing a function that already expresses part of the work, rather than duplicating its logic.
Davide: So the question is about organising the solution, not its result.
Teacher: Exactly. Let us build
RandomRangeusing onlyRandomNumero. Start with an integer between 10 and 20.
Anna: I would use
RandomNumero(10)and add ten.
Marta: Wait. That makes the smallest value eleven, not ten.
Luca: There are eleven integers from ten to twenty, including both endpoints.
Teacher: So?
Marta: Use
RandomNumero(11)and add nine. One becomes ten, and eleven becomes twenty.
Davide: Every intermediate value shifts in the same way.
Teacher: Now generalise. How many integers lie between a and b, inclusive?
Luca: b − a + 1.
Teacher: What must we add to turn 1 into a?
Anna: a − 1.
Marta: Therefore:
RandomRange(a, b) =
RandomNumero(b − a + 1) + a − 1
Teacher: Check the endpoints:
1 + a − 1 = a
(b − a + 1) + a − 1 = b
Davide: The new function builds on the previous one.
Luca: I still think an independent version might be fine in such a small program.
Teacher: You are right not to treat reuse as an absolute rule. We must justify it. In this exercise, building on an existing function was part of our learning objective. Reuse was not an incidental detail.
Anna: Then even a technically correct AI answer might miss what we want to learn.
Teacher: Yes. We need to make that objective explicit rather than expect it to be guessed.
Marta: We could ask: “Do not start from scratch. Use
RandomNumero(N)and explain how to transform its interval.”
Teacher: That is a better request. Notice that you formulated it after discussing an initial solution, not before.
5. The hardest objection: are we wasting time?
Davide: Let me play devil’s advocate. We spent ten minutes discussing a formula AI could give us in ten seconds.
Teacher: True. What did those ten minutes give us beyond possessing the formula?
Marta: We understood why it contains the plus one.
Luca: We learned that a correct function may not be our preferred design for a particular project.
Anna: And we learned to build one function from another.
Davide: But I could ask AI for all those explanations too.
Teacher: Certainly. That would be a good use of it. The question is what you do with the explanations: skim them, repeat them, or actually use them?
Davide: How can you tell?
Teacher: Change the problem slightly. Suppose we want a random integer from −3 to 2. How would you use the existing function?
Davide: There are six possible numbers. Use
RandomNumero(6)and subtract four. One becomes minus three, and six becomes two.
Teacher: The calculation is right. But we have already built the higher-level function. To use it, we simply write:
RandomRange(-3, 2)
Marta: So negative numbers do not require a new function?
Teacher: Exactly. Your reasoning explains what the function does internally. Let us count the values, including both endpoints:
b − a + 1 = 2 − (−3) + 1 = 6
−3, −2, −1, 0, 1, 2
Anna: There are six. To turn the first value, 1, into −3, subtract four: a − 1 = −3 − 1 = −4.
Teacher: Substitute a = −3 and b = 2 into the general formula. The following lines show an equivalent calculation, not a new function declaration:
RandomRange(-3, 2)
≡ RandomNumero(2 - (-3) + 1) + (-3) - 1
≡ RandomNumero(6) - 4
Davide: RandomNumero(6) produces one of the integers from 1 to 6. Subtracting four shifts each to the right position:
1, 2, 3, 4, 5, 6
↓ −4
−3, −2, −1, 0, 1, 2
Teacher: Each starting value corresponds to exactly one final value. If the six starting values are equally likely, the final ones remain equally likely. Negative numbers do not change the algorithm.
Marta: To understand the mechanism I can reason about
RandomNumero(6) - 4; to use the existing service I callRandomRange(-3, 2).
Teacher: Exactly. Here are the two reusable functions and the call in JavaScript, for small integer intervals such as this example:
function RandomNumero(N) {
return Math.floor(Math.random() * N) + 1;
}
function RandomRange(a, b) {
return RandomNumero(b - a + 1) + a - 1;
}
const numero = RandomRange(-3, 2);
console.log(numero);The formula b − a + 1 counts the values, while a − 1 shifts the interval into position. Understanding the mechanism does not mean rewriting it every time.
Teacher: You have now applied the reasoning to a different case, rather than merely recognised a formula on the page.
Anna: So the criterion is not how long I worked without AI.
Teacher: No. Nor is it how many questions you asked. What matters here is the capability you developed through using it.
Luca: Someone could just copy the answer.
Marta: They could copy a textbook or a classmate too.
Teacher: We must address the risk of replacing work with a ready-made answer. Choosing a tool is not enough: we must design the activity. Today I asked you not just for results, but to find an omission, propose a counterexample and adapt a solution.
6. Returning to the original saying
The teacher projects the statement again.
“Intelligence is not the ability to store information, but knowing where to find it.”
Teacher: After our discussion, what would you change?
Anna: Add “and knowing what to do with it.”
Luca: I would add “and knowing when not to trust it.”
Marta: Not just distrust. In the square-root example, the AI’s answer helped us see something new.
Davide: Then: “knowing how to use answers to go further.”
Teacher: What about the knowledge we carry in our own heads?
Luca: We need it. Without knowing about division by zero, we would not have understood the first mistake.
Anna: But we can also develop knowledge through dialogue. We do not need to possess it all beforehand.
Marta: So knowing and asking are not alternatives. We can ask in order to understand better.
Davide: And understanding better can change our question.
The teacher writes a proposal that has emerged from the discussion:
“With AI, intelligence is not just knowing where to find answers, but understanding them, testing them and using them to build new knowledge and new questions.”
Luca: I like it, but how do we know we truly understand, rather than merely follow an explanation that sounds convincing?
Teacher: A slogan cannot settle that. Start with a test: choose one of today’s examples, change a condition, and explain to your classmates what changes and why.
Marta: May we use AI?
Teacher: Yes. But distinguish what it suggested, what you checked and what you still find unconvincing.