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## Interdisciplinaria

*versión On-line* ISSN 1668-7027

#### Resumen

ATTORRESI, Horacio Félix et al. Application of the Fischer LLTM model to the analysis of the sources of difficulty of deductive reasoning items.* Interdisciplinaria* [online].
2009,
vol.26, n.1, pp. 77-93.
ISSN 1668-7027.

The processes involved in deductive reasoning have been studied by Cognitive Psychology since the seventies. Many hypotheses have been put forward to explain the difficulties in solving simple reasoning problems when considering their logical connectives, content and context of the tasks in which they are presented. These hypotheses have led to the development of different theories of reasoning like those based on the formal inference rules approach (Braine, 1978; Braine & O'Brien, 1991; Braine & Rumain, 1983; Rips, 1994), the Pragmatic Schemas Theory (Cheng & Holyoak, 1985) and the theory of semantic mental models (Johnson-Laird, 1983, Johnson-Laird & Byrne, 1991). The componential models of the Item Response Theory have allowed Psychometry to explain said these processes (Embretson, 1994). Thus, for instance, the Linear Logistic Latent Trait Model (LLTM) (Fischer, 1973, 1997), an extension of the Rasch model, expresses item difficulty as the sum of the effects due to the sources of difficulty predicted by the mentioned cognitive theories, which enables us to decide whether these effects are significant and estimate them. In other words, the Rasch item parameters β_{1} are linearly decomposed in the form where p is the number of components considered, α_{l} -the basic parameters of the model, expresses the difficulty of each component l, w_{il} is the weight of α_{l} with respect to the difficulty of the item i and c is an arbitrary normalization constant. Formula (1) implies that the application of the LLTM model makes sense only when the Rasch model fits the data. On the other hand, if the proposed components were sufficiently exhaustive to explain the differences between the items, formula (1) would allow us, once the basic parameters α_{l} have been estimated, to recover estimates similar to those obtained directly by the application of the Rasch model, which would imply a high correlation between the parameters estimated under both models. The identification of the difficulty components and the estimate of their effects may be useful to generate items with preset difficulty parameters. This paper describes the process to find a subset of deductive reasoning items to which the LLTM model fits well. A set of 24 deductive reasoning items were designed and created considering the sources of difficulty predicted by cognitive theories and educational practice. The objective is to verify the suitability of such sources and to guide the construction of new items. Each item may consist of one, two or three premises and one conclusion. The individual must decide whether the conclusion is true or false. Nine items are made of concrete content, neutral to avoid any bias due beliefs or opinions, and the remaining ones have abstract or symbolic content. They were administered to a sample of 251 students of Psychology (Universidad de Buenos Aires - Argentina), composed of 24% males and 76% females, whose average age is 22.68 (DS = 6.35). Good fit for the Rasch model (p = .89) and for the LLTM model (p = .11) were obtained for 12 of them. The Wald z-values were not significant for the 12 items mentioned before. The linear correlation between the parameters estimated under both models was r = .99. Five components that turned out to be significant were considered. These components are listed in a decreasing level of difficulty: (a) affirmation of the consequent and negation of antecedent fallacies, (b) negation when affecting disjunction / conjunction, (c) abstract or symbolic content, (d) quantifiers and (e) conditionals. The two assumptions that refer to both, the item and subject parameter invariance, were checked. The results of this exploratory step encourage us to go on constructing new items taking into account the sources of difficulty that were found.

**Palabras llave
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**Fischer LLTM model; Rasch Model; Sources of difficulty; Deductive reasoning; Cognitive theories; Educational practice.