Validation of a Scale to Assess the Reversibility of Thought in Verbal Arithmetic Problems

Abstract

Background : Reversibility is a key concept for the understanding and development of mathematical thinking. There is an agreement regarding problem-solving as a fundamental part of mathematical competence, and some authors regard reversible thinking as a requirement for it. Objectives : We want to validate an instrument that assesses the reversibility of thought when solving verbal arithmetic problems (word problems) involving various operations, semantic-mathematical structures and proximity of situational information. Design : A qualitative study was carried out from the data obtained by experts, and a quantitative study was carried out to determine the validity and reliability of the instrument. Setting and Participants : 318 students from different Spanish schools attending primary education (6 to 12 years) participated. Data collection and analysis: Participants performed 180 mathematical tasks distributed over three theoretical scales, two operations, and four semantic configurations. Results: To determine the consistency of the data, a reliability analysis was performed globally and on each of the scales, all values being greater than 0.90. Exploratory factor analysis resulted in three factors that explained more than 70%. To analyse the validity of the instrument, confirmatory factor analysis was performed, and its indices showed an adjustment of the models. Conclusions : We consider that the designed instrument is sufficiently robust to assess the reversibility of the basic addition and subtraction operations and, in addition, to analyse the discrimination of word problems according to the semantic-mathematical structure and their situational context.