Cognitive models underlying algebraic and nonalgebraic solutions to unequal partition problems

The structure of certain word-problems can be perceived in different ways, depending on the grammatical form of presentation of the problem and the student's expectation of how it will be solved. The results of our study involving 268 school students aged 14-16 show that, for a certain class of problems, different problem presentations promote the construction of different cognitive modeu, of the situation described. Our data provide support for the hypothesis of Nathan et al. (1992) that in the solution of algebra word-problems there are three components of interpretation and modelling: a propositional text base, a cOgnitive model of the situation, and a formal model of the mathematical relationships. However we show that, for certain problems, there are two equally valid cognitive models of the situation, only one of which can be linked to an algebraic representation of relationships. For problems of this type, the lack of correspondence between a cognitive model of the situation and an algebraic representation of relationships in the problem is a powerful obstacle to the use of algebraic methods. Students often have great difficulty in formulating algebraic equations fo