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73ENSEÑANZA DE LAS CIENCIAS, 2009, 28(1), 73–84

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74 ENSEÑANZA DE LAS CIENCIAS, 2010, 28(1)

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Q/9&4)"!Actuación de Nancy en el primer momento.



"""

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Q/9&4)"<Análisis del primer momento.

Q/9&4)"`Actuación de Nancy en el segundo momento.

Q/9&4)"cAnálisis del segundo momento.



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Q/9&4)"ZParte de la actuación de Nancy en el tercer momento.

Q/9&4)"aParte de la actuación de Nancy en el tercer momento.



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Q/9&4)"oAnálisis del tercer momento.

Q/9&4)"]Resumen de los tres momentos.



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Summary

Several research studies in mathematics education have reported the dif! culties that students have 4ith mathematical proof. Some of these dif! culties are related to the recognition of 4hat a mathematical proof is, the 4ay proofs are sho4n in textboo;s 4here the arguments are often accompanied by visual or intuitive <usti! cations, generic examples or naive inductions that invite the students to consider these as proofs =>reyfus, ?@@@A or sociomathematical norms =Bac;el C Cobb, ?@@E, pp. FE?A that establish the mathematical argumentation that is made in the classroom and create the studentsG beliefs about 4hen a mathematical explanation can be considered acceptable or 4hat is meant by a <usti! cation.

In addition, students have many dif! culties correctly handling the concepts and their de! nitions and properties. They do not understand the role of examples and counterexamples 4hen developing proofs 4ell enough. They sho4 a 4ea; linguistic and logical apparatus. They are not accustomed to verbally expressing ideas themselves in a context that is purely mathematical and have signi! cant shortcomings 4hen trying to 4rite a proof.

In this paper, the productions of Jancy, a student beginning undergraduate courses in mathematics, are analyzed. She formed part of an exploratory study analyzing the ans4ers to various problems for the purpose of approximating to her performance and advances. In order to describe and to analyze this studentsG ans4ers ToulminGs schemes have been used as an analysis tool. In this article, 4e focus only on one of the exercises. This 4as a logical implication problem that can be ans4ered 4ith a simple numerical argument but, in reality, the solutions that prevailed 4ere based on proving 4ith particular cases.

This study 4as conducted in three stagesL in the ! rst the initial problem 4as posed, secondly, the problem 4as slightly changed 4ith the aim that the student can provide further details about her ! rst solution to the problem and ! nally an intervie4 4as done so that she should explain their previous ans4ers and thus progressively she must deal 4ith her inconsistencies and this can allo4 us to understand ho4 she 4as solving the problem, and 4hy she committed mista;es. For the analysis, the

<usti! cations given by Jancy, her ob<ections, 4arrant and bac;ing =Toulmin, ?@NOA 4ere ta;en into account.

Some of her <usti! cations 4ere 4rong because the pattern used 4as verifying ! rst the consequent and then using the antecedentL !allac% o! a!! rmin, t.e conse1uent3 using particular cases only or not ta;ing into account the role of hypothesis. Qther <usti! cations 4ere correct but imprecise because she 4as una4are that a counterexample is enough to determine the falsity of an implication and she considered an implication to be true in some cases and not in others. Another imprecision 4as relative to the determination of the reciprocal truth value based on particular cases.

She also proved some statements in particular cases only, she 4as not a4are that statement proving only 4ith a ! nite number of cases is not enough to establish the general truth, she needed to base her ! ndings on her experience so she used speci! c examples to verify the properties. Initially she used apparently isolated examples, but then she 4as able to depict generic examples and ! nally she thought about ho4 to generate all possible cases than;s to the enrichment of her e4am5les s5ace and she could use the appropriate examples in the third stage. This trend contributes to a signi! cant advance in the use of language because she began to use formulas or the consideration of all possible cases in a structured 4ay. This development is perhaps attributable to the fact that during the intervie4 she should explain her ans4ers convincingly.

The 4arrant and bac;ing used by Jancy in her <usti! cations 4ere basically case studies, examples and truth tables. Although she managed truth tables correctly, it seems that they no longer ma;e sense 4hen she must ans4er some items and she felt the need to chec; 4ith particular cases. The ob<ections outlined in relation to the statements truth, 4ere examples that do not verify the antecedent and so they could not be used for proving the consequent, ie, she did not admit the conditional role of the hypothesis and this con! rms one of the dif! culties faced by students. Harel =TUUEA believes that this is mainly because students rush and focus on content that they do not distinguish from the state. Finally, logic and proof are sho4n to be conceived separately so that education should teach proof as a form of validation, and the usefulness of language as a necessity to develop and communicate a proof.

The Logical ,mplication in the Process of Mathematical Proof6 A Case 9tudy

Alvarado Monroy, Angelina4 5#González Astudillo, Mª Teresa6!"#$%&'()"*'"+),'-.,/%)$0"12/3'4$/*)*"5&.4'6"*'("#$,)*7"*'"8&4)2970"+:;/%7<"8'=)4,-'2,7"*'"8/*.%,/%)"*'"()"+),'-.,/%)">"8/*.%,/%)"*'"()$"?/'2%/)$"#;='4/-'2,)('$0"12/3'4$/*)*"*'"@)()-)2%)))(3)4)*7A&$)(0'$-)/,'A&$)(0'$

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