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We present two empirical studies with 241 and 211 pre-service teachers that evaluate the explanatory power of word order matching and static comparison as models for the reversal error. We used tasks consisting of generating an algebraic equation representing a comparison given in a verbal statement. We introduce the types of magnitude involved in the statement as variables of analysis, something that was not previously tackled in previous works. Our results show that there are no statistical differences in the production of reversal errors depending on the information included in the name used to designate the variable, and that there are statistical differences depending on the syntactic configuration as well as the type of magnitude involved in the statement. The interpretation of these results indicates that both word order matching and static comparison have some potential as explanatory models for the reversal error, and that neither one of them, alone, is enough to completely explain the phenomenon. [ABSTRACT FROM AUTHOR]

An empirical study has been carried out to evaluate the potential of word order matching and static comparison as explanatory models of reversal error. Data was collected from 214 undergraduate students who translated a set of additive and multiplicative comparisons expressed in Spanish into algebraic language. In these multiplicative comparisons we used a format that can be translated from Spanish word-for-word as ' n times more than' (increasing comparison) and ' n times less than' (decreasing comparison) instead of ' n times as many', which is usual in other studies. Data analysis shows a significantly lower incidence of reversal error in the decreasing comparisons compared to the increasing ones. Additionally, no significant differences were found between additive and multiplicative comparisons. These results cannot be explained by the static comparison model. [ABSTRACT FROM AUTHOR]

Designers of interactive learning environments with a focus on word problem solving usually have to compromise between the amount of resolution paths that a user is allowed to follow and the quality of the feedback provided. We have built an intelligent tutoring system (ITS) that is able to both track the user's actions and provide adequate supervision during the resolution. This is done without imposing any restriction on the resolution paths that are allowed. Instead, the system attempts to enforce metacognitive learning by requiring an appropriate definition of quantities before they are used. The program (a) supports both the arithmetical and algebraic way of solving problems; (b) allows the recurrence to one or more equations when solving a problem in an algebraic way; (c) determines the validity of the expressions when they are introduced; and (d) admits the incorporation of new problems without the need of being reprogrammed. In this paper, we explain the design foundations, which are mainly based on (a) a reflection of the steps that a student should follow to solve a problem in an algebraic way, and (b) the use of a domain specific notation to represent both the problem structure and the current state of the resolution process. In particular, hypergraphs are introduced as an adequate way to support tracking in both the arithmetical and algebraic case. Moreover, we offer an extensive experimental evaluation which highlights the potential of the ITS as a learning tool. [Copyright &y& Elsevier]