## Efeitos da formação de classes de equivalência sobre a solução de problemas de adição e subtração

##### Abstract

Research has shown that some properties of the additive problem may cause difficulties to solve it. The most important are: the problems presentation form, the semantic structure and the position of the unknown. Two experiments were conducted with some methodological differences, but both with the aim of investigating if the formation of an equivalence class between different problems presentation forms could reduce difficulties in solving problems. Eight students from the 2nd to the 5th year of elementary school with difficulties in solving problems of the pretest analysis participated in Experiment 1. It was applied a procedure for teaching conditional discriminations between different forms of presentation of addition problems (numeralproblem, word-problem, collection-problem and balance-problem), followed by Posttest 1. There was an increase in the percentage of accuracy for all types of problems, but five participants had difficulties with the balance-problems. It was also assessed if an additional procedure for teaching algorithmic for problem solving additives with unknowns at positions a and b could produce even greater increase in the percentage of correct answers. There were two sessions for teaching the addition algorithm, followed by post-test 2, and two sessions for teaching subtraction algorithm, succeeded by Posttest 3 and generalization test. Four participants showed an increase in the percentage of correct responses at post-test 2 and six at Post-test 3. This result, although positive, suggests that procedural changes are necessary so that all may be benefited by learning algorithms. Participants achieved 100% correct responses at generalization test. In Experiment 2, three forms of presentation (numeral-problem, word-problem and balance-problem) were used. The goal was to produce the formation of two sets of equivalence classes, one of addition and one of subtraction, and evaluate its effect on problem solving performances. In order to reduce difficulties with balance, two sessions were designed to teach participants about their operation. Eight students in the 2nd to 5th year that presented difficulties in the pre-test problem solving participated. After the formation of classes, it was found in Post-test 1 that all participants increased the percentage of correct answers. It was then assessed whether a practice in solving balance-problems could further improve this performance, which was confirmed. In Generalization Test 1, all participants reached percentages above 75% accuracy. We evaluated whether it would be possible to improve the teaching of algorithms phase of Experiment 1. We performed a single session for teaching together addition and subtraction algorithms, which was followed by post-test 3, in which there was an increase in the percentage of correct responses. Then the generalization test was reapplied, in which all achieved 100% accuracy. In Experiment 2, at each post-test, there was improvement in performance. It was demonstrated that the teaching procedures adopted are important to reduce learning difficulties in problem solving.