Numbers and operations are to mathematics what the alphabet and syntax are to a language: an intimate connection linking us to the world around us. Our understanding of many phenomena is at stake, in physics, biology, history, geography and more. Initially, our sense of number begins to develop through our five senses and hands-on activities, before gradually becoming mental and then abstract. At first, the relationship is purely mental, without any formalism. Developing and strengthening it builds fluency with numbers and operations. This agility fosters a mental quickness that extends far beyond numbers and mathematics, while also building self-confidence—the confidence needed to keep exploring the world of mathematics. So where do mental methods stand in the teaching of arithmetic?
Mental arithmetic in step with technology ---------------------------------------------
Calculators, computers, smartphones and tablets have brought a succession of technological mini-revolutions over the past forty years. With screens everywhere, these devices are changing how we perceive the world, and therefore mathematics, raising legitimate questions about how numbers and arithmetic should be taught. What should we teach, and how should we teach arithmetic? The screen provides a new interface between our perceptions and our brain. It helps us form a mental relationship with numbers and operations by making those connections more immediate and bypassing written work. The growing place given to mental arithmetic in curricula over the past twenty years is therefore very much in step with this technological development! The picture is very different for written calculation methods. Of course, using these methods involves some mental activity, but this recedes behind the algorithmic nature of written calculations. This is a real paradox: although writing is supposed to stabilize meaning, in arithmetic it seems to take us further away from that quest! To understand why written calculation methods have played such a major role in teaching arithmetic, we must remember that fifty or more years ago, students had no technological aid to perform any of the four operations. Mastering these methods was vital. Their fundamental importance gave them an almost sacred place in education. Technology has advanced so quickly over the past fifty years that our educational habits have fallen out of step with our environment. Written methods still have a role, but a smaller one, positioned above all between mental methods and machines. The teaching of arithmetic needs a common-sense approach: mental calculation for simple operations, as in everyday life; written calculation when mental methods are not enough; and a machine when matters become technical and difficult. Within this framework, mental arithmetic becomes the common thread in learning arithmetic, with a constant focus on meaning and a variety of activities alternating between strategic calculation, automatic recall and games.
The key: making sense of it! -------------------------
There is no shortage of arguments supporting the increasingly prominent place of mental arithmetic in school curricula. The shift began with the 2002 curricula, which clearly distinguished automatic mental calculation from strategic mental calculation. Citizens can manage without a thorough command of written calculation methods—their mobile phones can come to the rescue—but a lack of fluency in mental arithmetic, whether in estimating orders of magnitude or using calculation techniques, can become a genuine handicap in everyday life (see the feature "The secrets of mental arithmetic" in Tangente 163). Moreover, the many national and international studies of our students' proficiency in arithmetic unfortunately all point to the same conclusion, regularly placing us near the bottom of international comparisons. This trend has continued to worsen since the first PISA studies, some twenty years ago. We can no longer take refuge in criticizing the methodology of one study or another.