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The purpose of this paper is to suggest how to overcome the difficulties which arise when secondary school students have to deal with too formal and abstract proofs. In fact, such difficulties emerge very often in math classes. As a result, students do not achieve a real and deep comprehension of the proofs, but limit themselves to learn them by heart. On the other hand, the introduction of proofs based on concrete and visual elements greatly helps them to gain a clear understanding of the underlying mathematical concepts. An alternative way to prove the irrationality of the square root of two will exemplify how to make proofs less formal and more semantic. [ABSTRACT FROM AUTHOR]