Metacognitive training improves math problem solving

There is new evidence that training students to cooperatively evaluate their learning increases math achievement. A recent research study with 174 seventh-grade Israeli students suggests that metacognitive skills help students solve complex problems and improve their ability to use their knowledge on new, unfamiliar problems. Zemira R. Mevarech, Bar-Ilan University, Israel, reports that there is a growing consensus that cooperative learning has positive effects on student achievement, but there is less agreement about the kind of conditions most beneficial for group learning.

Some researchers emphasize the importance of group goals and individual accountability. Other studies suggest that students need to be taught specific learning strategies such as summarization and prediction to improve their reading comprehension and group discussion.

The purpose of the present study was to compare three cooperative-learning environments. The teachers were randomly assigned to classrooms, and intact classrooms were assigned randomly to one of the three treatment groups.

The metacognitive groups were taught to ask and answer questions about their learning. These questions focused on comprehending the problem, making connections and applying strategies. Metacognitive questions were designed to help students become aware of the problem-solving process and to monitor their progress. Each student took turns trying to solve a problem and explain his or her reasoning by answering metacognitive questions printed on a card.

Three types of metacognitive questions

There were three types of metacognitive questions: a comprehension question (What is the problem? Describe the problem in your own words without referring to the numbers); a comparison (How is this problem similar/different from the previous problem?); and a strategy question (What strategies are appropriate for solving the problem?).

Students in strategy groups were taught a diagramming strategy and the teacher discussed when and how to use it. Students practiced solving problems by using a visual model of unequal lines to represent the problem. They were trained to prompt one other to use the strategy with each problem. The teacher modeled specific strategies for individual teams.

The students in the control group practiced solving the same comparison problems in small groups without metacognitive or strategy training.

Study of Cooperative Problem-Solving

The study’s focus was on solving comparison problems that had both linguistic and nonlinguistic elements and required various problem-solving skills. Comparison problems can have either consistent or inconsistent language. Inconsistent problems have relational terms that imply an operation that is the opposite of what is needed to solve the problem accurately (for example, the word “less” is used but students need to add). Inconsistent problems are cognitively complex and are difficult for students to solve.

All students in the experiment took a 20-problem pretest to assess their ability to solve two-step comparison problems that had either consistent or inconsistent language. In addition, all students took a reading comprehension test. These tests uncovered no significant differences in reading or math between classrooms at the beginning of the study. All classes used the same textbook and studied the same set of problems. Math lessons were taught five times a week.

Students worked in heterogeneous groups of four students: one high achiever, two middle achievers and one low achiever. Each teacher introduced a comparison problem and discussed it with the whole class. She discussed how to transform the problem into an equation. The students then practiced the new material in their cooperative groups.


On the post-test, the metacognitive groups significantly outperformed the strategy groups, which, in turn outperformed the cooperative control groups. Analysis of the higher and lower achievers in each treatment group revealed important findings. High achievers in the metacognitive groups scored highest, followed by those in the strategy groups. However, the results were different for the lowest achievers. Low achievers performed best with metacognitive training, but worst with strategy training.

None of the low achievers in the strategy-trained group were able to apply their strategies to similar but unfamiliar problems, while 25 percent of low achievers in the metacognitive groups and 12 percent in the cooperative control groups successfully solved the unfamiliar problems.

When these results were analyzed by the type of problem — consistent-language versus inconsistent-language problems — the differences were apparent only for the more cognitively more complex problems with inconsistent language. Mevarech suggests that students in the strategy groups may be designing inaccurate visual models because they translate the key words into an equation without considering other information in the text and so do not detect inconsistent language.


Students involved in structured groups outperformed their peers who studied in cooperative settings with no problem-solving training. Strategy instruction proved less efficient than metacognitive training for enhancing mathematical reasoning and achievement in complex problem solving.

These findings show that students in heterogeneous classrooms can learn to solve complex cognitive tasks and that the progress of the lower-achieving students does not occur at the expense of the higher achievers or vice versa. Mevarech’s study indicates that structured group interactions that stress metacognitive processes foster mathematical reasoning. In this study, strategy instruction was not sufficient to increase achievement on complex problems in cooperative groups.

“Effects of Metacognitive Training Embedded in Cooperative Settings on Mathematical Problem Solving”, Journal of Educational Research, Volume 92, Number 4, April 1999, pp. 195-205, 1, 1999, pp. 53-72.

Published in ERN May/June 1999 Volume 12 Number 5

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