Abstract:

Mental calculation (MC) skills are important because they accelerate arithmetic counting without the aid of a calculator. It is useful both in mathematical activities and in everyday life. There are many positive aspects in the development of this ability that may be considered a real domain of experience in Boer's sense (Boer, 1989). For example, a solver with good mental arithmetic skills does not interrupt the reasoning when solving a problem because he doesn't need to concentrate on using the calculator rather than problem solving (Vergnaud, 2007). The skilful use of this domain frees students from the “chains” of the algorithms of the written procedure and stimulates the creation of new solution strategies. In this way, they increase their level of confidence with the properties of numbers, including some not sufficiently practiced in standardized written algorithms.

The use of MC, however, is not convenient on every context. Experience, in fact, teaches that favourable conditions must be presented for it to be more advantageous than written one. Many students think that learning this technique is boring and this is an obstacle to its learning. The aim of this research is to test an innovative tool that is an incentive for the acquisition of arithmetic counting skills.

The research idea is inspired by Butlen's thought: "The practice of mental calculation increases students' adaptability and initiative. (...) As students improve their knowledge of numbers and operations, they allow themselves more trial and error (...) by adapting their strategies to the numerical data involved in the calculation” (Butlen, 2007).

The learning dynamics of MC are very similar to those of strategic games and this observation inspired our experimentation. For the purposes of our research, an educational game has been developed for the reinforcement of this technique. The experimentation has been conducted on a group of Italian students in the first year of secondary school and has been structured in five phases.

In the first phase the researchers submit a questionnaire to investigate what students think about mental counting and what skills they already have with this methodology. Then, students are guided in 'discovering' some of the most common techniques and stimulate them to create new personal computation strategies. In the third phase students invent new questions that can be solved with the didactic approach they have learned or with the ones they have invented. After, students enhance their skills using a MC game, an educational artefact. In this game, in order to win, it is necessary to apply what learned in the previous phases and to do many mathematical operations without using a calculator (i.e. addition, differences, multiplication, division, powers, factoring into prime numbers, multiples and submultiples of given numbers, least common multiple, greatest common factor). In the last phase the students have to answer a questionnaire with the aim of quantifying their progress compared to the initial levels.

The analysis of the results of the experiment showed a widening implementation in MC compared to the starting levels in terms of speed, accuracy and expansion of knowledge of calculation strategies.

Keywords:

Mental calculation, Game-Based Learning, Education, Ludodidactic.