Students' systematic errors: Tools for promoting conceptual understanding and knowledge transfer

This study compares the effects of students' participation in whole-class discussions of correct, versus both correct and incorrect (mixed), solutions to a type of equivalence problem. The study examines the effects of: (a) students' solution types on the content of their class discussions and (b) both solution types and discussion content on students' procedural and conceptual understanding of mathematical equivalence.;Pairs of fifth-grade classes from several elementary schools participated in the study. One class from each school participated in a teacher-led discussion of correct solutions to three identically formatted equivalence problems; the second class participated in a teacher-led discussion of one correct and two incorrect methods for solving a single equivalence problem of the same type as discussed in the correct-solutions classes. All teachers were provided with the same set of open-ended questions to ask their students during the course of their discussions. A variety of pretest and posttest measures were used to determine what students learned from their discussions.;Participation in discussions of correct solutions allowed significantly more students to learn to solve posttest problems than did participation in discussions of mixed solutions. Comparisons of the subgroups of participants from each condition who learned to solve the posttest problems suggest, however, that students who discussed mixed solutions acquired a deeper understanding of the concept of equivalence than their counterparts in the correct-solutions classes. The former students solved more transfer problems and demonstrated a better understanding of the relational nature of the concept of equivalence in their written explanations than did the latter students.;The class discussions held in the two conditions appeared to be more similar than different. Discussions of correct solutions, however, seemed to focus students' attention on simple solution procedures that could be applied without conceptual understanding. The limited posttest success of participants in mixed-solutions discussions appears to have been due to the failure of several students per class to understand that their task was to decide which one of the three alternative solutions they discussed was correct, rather than to learn three equally acceptable solution methods.