Usually teachers explain the results that they set out; for example, they give the steps leading to a proof. Explanation is essential if one wants that the students accept a result, and use it. However, even when it has received an explanation, the student is not fully satisfied: he believes that the result is correct, but he cannot see how he could have found it.

In order to clarify this point, I will take a mathematical example, but this can be applied for any kind of problem. Let us show that there is an infinity of prime numbers. The classic proof is a reductio ad absurdum: we assume that the proposition is false, there is only a finite number of primes, and we show that that leads to a contradiction. If there is a finite number of primes, one of them is larger than all the other primes, let us call it N. Now, we consider the number N!+1, N! representing factorial N, that is the product of all positive integers less or equal to N. Either this number is prime, or it is not. If it is prime, as it is larger than N, there is a contradiction. If it is not a prime, it has at least one prime divisor, let us call it P. As N! is divisible by any positive integer less or equal to N, N!+1 is not divisible by any of these integers; therefore, as it is divisible by P, P is greater than N and we also have a contradiction. As both possibilities lead to a contradiction, our hypothesis is false: there is no prime number greater than all the other prime numbers, there is an infinity of prime numbers.

I was convinced and impressed by this proof. However, I was not satisfied: how could I have had the idea to consider this weird number, N!+1? My teacher had given an explanation that justified the result, he had not given a meta-explanation that indicates how a mathematician had found it, many years ago. This absence is serious: the student considers that mathematics is a science that can only be developed by super-humans, which have a special gift for mathematics. This gift enables them not only to understand proofs, but also to find them. Meta-explanation is essential for both human and artificial beings so that they could learn to find proofs.

Giving meta-explanations is not an easy task for teachers, because usually they do not know them: unconscious mechanisms give the idea to consider the most difficult steps, and we cannot observe them. This absence is not due to a lack of goodwill, but to the limitations on our consciousness.

We have already seen that artificial beings such as CAIA can receive knowledge in a declarative form. A particular form of knowledge, called meta-knowledge, indicates how to use knowledge. Therefore, as artificial beings can have access to declarative knowledge, they can build a trace, which contains all the actions necessary for finding a solution, and a meta-trace, which contains all the reasons for choosing the preceding actions.  The explanation is built from the trace that we can create; on the contrary, as we can find only snatches of the meta-trace, we cannot build meta-explanations.

I had the impression that I could have easily found all the steps of the preceding proof, except the key: how somebody could have the idea to consider N!+1? However, it is possible that nobody had ever the idea to consider this number when he was trying to prove this result. It could have happened differently: one day, someone interested in the factorial function, considered N!+1, and decided to play with this number. It is evident that it cannot be divided by any prime number less or equal to N. Therefore, to any number N one can associate a number P that has at least a prime divisor greater than N. It is sufficient to apply this result to prime numbers: one has proven the theorem without wanting to prove it! Naturally, this meta-explanation is questionable, this is the case of most of them. Nevertheless, I am satisfied: I believe that I could have found this proof in that way.

CAIA can explain all its results, and it also can meta-explain them: it indicates the conditions enabling it to choose all the actions that it has considered, including the unsuccessful ones. Using meta-explanations, one can know why one has made excellent or poor decisions: they could be very useful for learning to solve problems efficiently. However, I still have to modify CAIA so that it can improve the meta-knowledge used for choosing its attempts.