Using errors as springboards for enhancing mathematical reasoning with three metacognitive approaches.

The authors examined effects of 3 metacognitive
approaches and 1 control group on mathematical reasoning,
conceptual errors, and metacognitive knowledge. The
metacognitive approaches were (a) diagnosing errors (DIA),
(b) improvement via self-questioning (IMP), and (c) a combined
approach (DIA+IMP). Controls (CONT) received no
metacognitive instruction. Participants were 115 9th-grade
students in an Israeli junior high school studying linear functions
and graph representations. Results indicated that students
using the combined approach outperformed their peers
in all measures of mathematical reasoning and metacognitive
knowledge. The IMP students outperformed the DIA and
CONT students in problem-solving skills and metacognitive
strategy use, and the DIA students outperformed their
peers in reducing conceptual errors. The authors discuss the
practical and theoretical implications of these metacognitive
approaches.