Éric Trouillot: Mental arithmetic, a path to jubilation | Tangente

Tangente: In what way can mental-arithmetic drills help with learning mathematics?**

Éric Trouillot: Fluency in mental arithmetic is the intimate relationship each of us builds with numbers and operations. We all need it every day. Intuition, orders of magnitude, connections between numbers… an individual's mental repertoire is something like the numerical toolbox available to solve everyday problems. From a teaching standpoint, numbers and operations are everywhere in problem-solving, both in primary school and in middle school, and it is only from high school onward that mathematics gradually breaks free of numbers. Recently, at the Paris Culture et jeux mathématiques fair, a student from the École normale supérieure told me that after spending his school and middle-school years passionate about mathematical games, working on puzzles from the Kangourou competition, or playing Mathador, he had the feeling that all of it had forged a particular neural network for his advanced studies. It was after being nourished on mathematical and numerical games that he came to do completely abstract mathematics, without a single number!

When calculators arrived in the seventies, we all thought — including within the institutions of the Éducation nationale — that they would free us from teaching calculation, and mental arithmetic in particular. That was also the era of the "modern maths" revolution, which aimed to reform teaching. In the spirit of the time, a good deal of geometry and mental arithmetic was seen as dusty. It thus dropped off the radar for several decades. For teachers, this put most primary school teachers and mathematics teachers in a very difficult position. For students, the result was a loss of meaning in the realm of numbers and operations, one we have not entirely recovered from.

We eventually understood how essential mental arithmetic is to learning, or more precisely, how important it is to internalize one's relationship with numbers and operations. This internalization is essential for laying the foundations of one's relationship with mathematics. How can a primary or middle school student hope to solve a problem without a minimum of mental fluency and ease with numbers? When it comes to fixing things, you won't get far with only a screwdriver and a 10 mm wrench!

Many students and adults have been conditioned by this way of working: you carefully write the numbers in a certain way, draw lines, align the carries, and in the end you get the result if you applied the method carefully. This old culture, which sees written calculation as the holy grail of calculation, does not encourage using one's brain to perform the operation in one's head.

After having long had great social usefulness, written calculation has now been replaced by the calculator, but the result for mental arithmetic is the same. By placing too much emphasis, too quickly, on written calculation techniques, we block many people's genuine entry into the world of numbers and operations, draining them of meaning through digit-by-digit tasks that make it impossible to grasp a number as a whole.

If I'm in a hurry and just want an order of magnitude, I do 60 × 20, which quickly gives me 1,200. If, on the other hand, I want an exact result, distributivity with its groups is often effective: I take ten groups of 59, which makes 590, I double it to reach twenty groups (result: 1,180), and I add another 59 to get the result: 1,239. This kind of practice builds a genuine relationship with numbers and generates confidence. With time, this ease can turn into jubilation.

What matters is always to foreground the reasoning, that is, the properties of numbers and operations that make it possible to carry the calculation through. It would be a mistake to systematically value only speed or the result: it is above all the reasoning process that matters. In this way, the automatic responses acquired through regular practice are not reflexes that come out of nowhere, but the natural extension of a genuine acquisition of knowledge.

The most important thing for the years ahead will be to start with this mental culture, placing written calculation as an extension of this internalization. We do need to know a few operational techniques, of course, but we must acknowledge that we no longer need them as much as we used to. We must also understand that these techniques are not an extension of mental arithmetic — the two worlds are even, in part, at odds with each other! In a sense, the very pairing of the words "arithmetic" and "mental" expresses this. Mental arithmetic indeed has its own didactics, which leads, for example, to considering numbers from left to right, thus respecting our natural relationship with them, rather than from right to left as is customary in written operational techniques.

The teaching triptych of mental/written/calculator-based calculation must give way to the societal diptych now in play: mental/calculator-based. This will happen through a didactics of mental arithmetic in teaching. The new 2025 curricula move in this direction by shifting the balance: much less written calculation (too time-consuming and of little use in building understanding), more thoughtful mental arithmetic.

That period reflected too hasty a push toward abstraction, taking a dim view of the whole tactile dimension, also associated with testing and trial and error. The manipulate/verbalize/abstract triptych, now in vogue, is far more in keeping with the more natural approach to mathematics, which consists in starting by observing, testing, manipulating, questioning oneself, in order to possibly model — abstraction being the final step. I think this approach applies in exactly the same way to numbers and operations. Automation should always be an extension of all these intermediate phases, where meaning is built and then takes hold. The neuroscientist Stanislas Dehaene puts it very well: the brain is an organ that constantly tests and gropes its way forward. The idea of an expert solution laid out a priori is foolish unless it is the considered extension of tests and comparisons between different approaches — exactly as in mental arithmetic.

The abacus, which is used in many countries to teach calculation, allows for a tactile relationship with numbers that certainly facilitates the building of automatic responses. It is fascinating, moreover, to watch the finals of mental-arithmetic competitions in these countries (see the article "La pensée du boulier" (Abacus Thinking)), where students move their fingers in the air as if their abacus were there: automation extends right to their fingertips! The speed at which they perform calculations is staggering, almost supernatural.