In proof by reductio ad absurdum, the impossibility of a mathematical object is drawn from the deduction of a contradiction. The relationship between the statement and the contradiction is logical in nature and it is one of the main obstacles for students. An analysis of indirect argumentations produced by students in geometry enlightens how they sometimes by-pass this obstacle transforming the geometrical figure so that the (false) proposition becomes true and the link between the contradiction and the statement is reconstructed. This analysis reveals some interesting differences in the treatment of the contradiction in argumentations and in proofs, identifying important difficulties in understanding proof by contradiction.
Indirect argumentations in geometry and treatment of contradictions / Antonini Samuele. - In: PROCEEDINGS OF THE PME CONFERENCE. - ISSN 0771-100X. - STAMPA. - 2:(2008), pp. 73-80. (Intervento presentato al convegno Joint Meeting of PME 32 and PME-NA XXX tenutosi a Morelia, Mexico nel 17-21 luglio 2008).
Indirect argumentations in geometry and treatment of contradictions
Antonini Samuele
2008
Abstract
In proof by reductio ad absurdum, the impossibility of a mathematical object is drawn from the deduction of a contradiction. The relationship between the statement and the contradiction is logical in nature and it is one of the main obstacles for students. An analysis of indirect argumentations produced by students in geometry enlightens how they sometimes by-pass this obstacle transforming the geometrical figure so that the (false) proposition becomes true and the link between the contradiction and the statement is reconstructed. This analysis reveals some interesting differences in the treatment of the contradiction in argumentations and in proofs, identifying important difficulties in understanding proof by contradiction.
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