ARITHMETIC/ALGEBRAIC PROBLEM-SOLVING AND THE REPRESENTATION OF TWO UNKNOWN QUANTITIES

We deal with the study of the senses and the meanings generated in the representation of the unknowns in the resolution of word problems involving two unknown quantities. The discussed cases show the difficulties that the students beginning the algebra learning have to deal with when using the equality between "unknown things". For them, applying the equality transitivity property between different (but equivalent) algebraic expressions, or replacing an unknown quantity with its representation in terms of another one is not derived from an extension of the transitivity between numerical equalities or from the numerical substitution. This may have important implications in the algebra problem-solving teaching domain, in which it is usual to take for granted that students spontaneously transfer these numerical issues to the algebraic realm. Previous research has been undertaken to probe cognitive processes that take place in solving word problems in the transition from arithmetic to algebraic thinking. Bernardz, Radford, Janvier, & Leparge (Bernardz, Radford, Janvier, & Leparge 1992; Bernardz, 2001) have substantially contributed to this research area. Puig and Cerdan (1990) have formulated criteria to determine when a word problem can be considered as algebraic. From a different perspective, A. Bell (1996) has approached this matter by showing through examples how generic problems can provide algebraic experiences that develop manipulative algebraic abilities. Rojano and Sutherland (2001) have studied how technological environment can help students to represent and solve word problems without having to take on board with the algebra symbolic code, from the very beginning. The present paper addresses the theme of arithmetic/algebraic problem-solving from a different point of view. The results presented are part of the research program “The Acquisition of Algebraic Language ” (Filloy and Rojano, 1989; Rojano, 1994), which intends to throw light on the uses of Mathematical Sign Systems (MSSs) (Filloy, 1990) which will culminate in the competent use of the System of Signs of Symbolic Algebra. In previous reports (Filloy, Rojano and Solares; 2003), we have dealt with the problems of the significations and senses generated in the acquisition of the syntactic abilities needed for the manipulation of what is “unknown” in solving equation processes. In the study we report here , we approach the same theme in the context of the arithmetic/algebraic world problem resolution. We emphasize the role of teaching interventions that promote the use of Signs Systems in which suitable strategies for the solution process may be developed.