The effects of metacognitive training versus worked-out examples on students' mathematical reasoning.

Background: The present study is rooted in a cognitive-metacognitive approach. The study examines two ways to structure group interaction: one is based on worked-out examples (WE) and the other on metacognitive training (MT). Both methods were implemented in cooperative settings, and both guided students to focus on the problem's essential parts and on appropriate problem-solving strategies.

Aims: The aim of the present study is twofold: (a) to investigate the effects of metacognitive training versus worked-out examples on students' mathematical reasoning and mathematical communication; and (b) to compare the long-term effects of the two methods on students' mathematical achievement.

Sample: The study was conducted in two academic years. Participants for the first year of the study were 122 eighth-grade Israeli students who studied algebra in five heterogeneous classrooms with no tracking. In addition, problem-solving behaviours of eight groups (N = 32) were videotaped and analysed. A year later, when these participants were ninth graders, they were re-examined using the same test as the one administered in eighth grade.

Method: Three measures were used to assess students' mathematical achievement: a pretest, an immediate post-test, and a delayed post-test. ANOVA was carried out on the post-test scores with respect to the following criteria: verbal explanations, algebraic representations and algebraic solution. In addition, chi-square and Mann-Whitney procedures were used to analyse cooperative, cognitive, and metacognitive behaviours.

Results: Within cooperative settings, students who were exposed to metacognitive training outperformed students who were exposed to worked-out examples on both the immediate and delayed post-tests. In particular, the differences between the two conditions were observed on students' ability to explain their mathematical reasoning during the discourse and in writing. Lower achievers gained more under the MT than under WE condition.

Conclusions: The findings indicate that the kind of task and the way group interaction is structured are two important variables in implementing cooperative learning, each of which is likely to have different effects on mathematical communication and achievement outcomes.