Fictitious Pupils and Implicative Analysis: a Case Study
We present a case study, in the context of Didactics of Mathematics, in which we adopt the methodology of using fictitious data in the Statistical Implicative Analysis. On the one hand, unlike supplementary variables, the fact of adding fictitious data to
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Summary. We present a case study, in the context of Didactics of Mathematics, in which we adopt the methodology of using fictitious data in the Statistical Implicative Analysis. On the one hand, unlike supplementary variables, the fact of adding fictitious data to the sample does modify analyses results, so caution is needed. On the other hand, fictitious students are a tool for better understanding the data structure resulting from the analyses.
Key words: Contribution, entropic implication, fictitious subject, intensity of implication, quasi-implication, statistical implicative analysis, typicality.
1 Introduction The use of multivariate analysis in the field of Didactics of Mathematics (DM) has already got a long tradition in the frame of fundamental didactics. Important references can be found among the contributions of Journées de Caen (1995 & 2000), such as [3] and [10], in which new statistical tools are provided, motivated in the context of DM as in many other occasions, but fruitful for both the fields of DM and multivariate statistics. In the usual multivariate methods (generally factor analysis and principal component analysis), Brousseau [3] uses some supplementary individuals (fictitious individuals) in his data in order to be able to compare the a priori and a posteriori analysis of a questionnaire. The a priori analysis of the questionnaire leads to certain criteria of characterization of its questions (the variables). In this way, two matrices are obtained: one coming from the preexperimental analysis (the a priori matrix of the questionnaire: criteria × questions) and the empirical matrix, made of the collected data, where questions remain characterised by the present sample (answers × questions). Fictitious individuals allow for the simultaneous consideration of both the pre-experimental criteria and those provided by the sample in a single matrix. Fictitious individuals, as features of the variables involved in the P. Orús and P. Gregori: Fictitious Pupils and Implicative Analysis: a Case Study, Studies in Computational Intelligence (SCI) 127, 321–345 (2008) www.springerlink.com © Springer-Verlag Berlin Heidelberg 2008
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Pilar Orús and Pablo Gregori
experiment, contribute both to the improvement of the knowledge on the variables —enhancing the information management furnished by the sample and thoroughly analysed a priori —, and to the comparison of the a priori and a posteriori behavior of the same variables. The study of dependences between pieces of knowledge, approached in [9], highlighted the existence of non symmetric relations between variables —in a context of didactics— and motivated the search of new tools to analise such relations. Therefore the Statistical Implicative Analysis (SIA) is introduced [10–12] and mainly developed in research in DM [4,13,15,19], although it has also been used in other fields, such as in [7]. This theory is located among other procedures of data mining [1, 20, 21]. Our contribution intends to be located in the intersection of both contexts. On the one
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