Mathematics learning with multiple solution methods: effects of types of solutions and learners’ activity

Abstract

It is commonly suggested to mathematics teachers to present learners different methods in order to solve one problem. This so-called “learning with multiple solution methods” is also recommended from a psychological point of view. However, existing research leaves many questions unanswered, particularly concerning the effects of different types of solution methods and different degrees of learner’s activity. In this context, two experiments were conducted. In Experiment 1, a 2 × 3-factorial design was implemented, with the first factor concerning multiple versus uniform solutions and the second factor addressing different combinations of formal and informal solution methods. No “multiple solutions effect” was found. An integration of informal methods did not affect learning outcomes; however, it significantly reduced the subjective difficulty of the problems. Then, in Experiment 2, the effectiveness of multiple versus uniform solutions and of measures to foster an active processing was examined using a 2 × 3-factorial design (“number of solutions”: multiple versus uniform; “activity”: complete examples versus incomplete examples versus example-problem pairs). The “multiple” conditions significantly outperformed the “uniform” conditions, and complete examples and example-problem pairs significantly outperformed incomplete examples. Based on the results of Experiment 1 and 2, preconditions under which multiple solutions can improve learning outcomes are discussed.