In Mathematics literature some records highlight the difficulties encountered in theteaching-learning process of integers. In the past, and for a long time, manymathematicians have experienced and overcome such difficulties, which becomeepistemological obstacles imposed on the students and teachers nowadays. The presentwork comprises the results of a research conducted in the city of Natal, Brazil, in thefirst half of 2010, at a state school and at a federal university. It involved a total of 45students: 20 middle high, 9 high school and 16 university students. The central aim ofthis study was to identify, on the one hand, which approach used for the justification ofthe multiplication between integers is better understood by the students and, on theother hand, the elements present in the justifications which contribute to surmount theepistemological obstacles in the processes of teaching and learning of integers. To thatend, we tried to detect to which extent the epistemological obstacles faced by thestudents in the learning of integers get closer to the difficulties experienced bymathematicians throughout human history. Given the nature of our object of study, wehave based the theoretical foundation of our research on works related to the daily lifeof Mathematics teaching, as well as on theorists who analyze the process of knowledgebuilding. We conceived two research tools with the purpose of apprehending thefollowing information about our subjects: school life; the diagnosis on the knowledge ofintegers and their operations, particularly the multiplication of two negative integers;the understanding of four different justifications, as elaborated by mathematicians, forthe rule of signs in multiplication. Regarding the types of approach used to explain therule of signs arithmetic, geometric, algebraic and axiomatic , we have identified inthe fieldwork that, when multiplying two negative numbers, the students could betterunderstand the arithmetic approach. Our findings indicate that the approach of the ruleof signs which is considered by the majority of students to be the easiest one can beused to help understand the notion of unification of the number line, an obstacle widelyknown nowadays in the process of teaching-learning
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