Metacognition is not a common word. In fact, every time I typed "metacognition" or "metacognitive" in this article it was automatically underlined in red. Microsoft doesn't consider it a word, a sad state of affairs for such an important term. Metacognitive strategies have been linked with successful and meaningful learning. Furthermore, there are a number of things teachers can do to help foster metacognition among their students.

**What is Metacognition?**

Metacognition is often described simplistically as "thinking about your thinking". According to the ERIC Digest on "Metacognition and Reading To Learn", metacognition involves "both the conscious awareness and the conscious control of one's learning." The term is used frequently in reading education circles, as a strategy to improve reading comprehension. But it is just as relevant for improving students' mathematical problem-solving.

NCREL, the North Central Regional Educational Laboratory, explains that metacognition consists of three basic elements: developing a plan of action, monitoring the plan, and evaluating the plan. Successful learners ask themselves metacognitive questions such as the following:

1) Developing a plan of action

- What in my prior knowledge will help me with this particular task?
- What should I do first?
- Do I know where I can go to get some information on this topic?
- How much time will I need to learn this?
- What are some strategies that I can use to learn this?

2) Maintaining/monitoring the plan

- Did I understand what I just heard, read or saw?
- Am I on the right track?
- How can I spot an error if I make one?
- How should I revise my plan if it is not working?
- Am I keeping good notes or records?

3) Evaluating the plan

- Did my particular strategy produce what I had expected?
- What could I have done differently?
- How might I apply this line of thinking to other problems?

Alan Schoenfeld, from the University of California at Berkeley, has been the predominant researcher to address metacognition in the mathematical arena. He looks at metacognition in terms of three areas:

- beliefs and intuitions
- your knowledge about your own thought processes
- self-awareness or self-regulation

The second and third areas seem to fit with others' descriptions of metacognition. What's different here is the first area: beliefs and intuitions. Schoenfeld argues that students' beliefs about mathematics shape the way they do mathematics. For example, if they believe (based on past experiences) that all mathematical problems can be solved in 5 minutes or less, then they will give up easily when they attempt to solve a more complex problem. Or if they believe math is formulaic and unrelated to the real world, then they will come up with an answer like "31 remainder 12" for the number of buses needed to move a group of people.

**
Why Is Metacognition Important?**

Many researchers believe that metacognition holds a great deal of promise for helping students do better. Metacognition has been linked to a wide variety of positive academic outcomes for students, such as better grades and performance on tests of intelligence. (e.g., Borkowski et al., 1987; Sternberg, 1984, 1986a, 1986b). There has been some suggestion that gifted students use more metacognitive strategies than non-gifted students. There has also been research suggesting that learning disabled students can benefit from using these strategies.

The ability to appropriately allocate cognitive resources, such as deciding how and when a given task should be accomplished, appears to be central to intelligence. Although most people of normal intelligence use metacognitive regulation when confronted with an effortful cognitive task, some are more metacognitive than others. Those with greater metacognitive abilities tend to be more successful in their cognitive endeavors. The good news is that people can learn how to better regulate their cognitive activities.

Metacognition is also important because it can increase the meaningfulness of students' classroom learning. (Schoenfeld, 1987) It can help students think of mathematics as a part of their everyday lives, help them make connections between mathematical concepts in different areas, and help them build a sense of a community of learners working together.

**
Opportunities for Developing Metacognition**

Schoenfeld asserts that creating a "mathematics culture" in a classroom is the best way to develop metacognition. Such a culture involves solving unfamiliar problems with your students, putting them on the board and working on them together. "Students participate with [the teacher], sometimes making mistakes and having to rethink where they have been. Such an approach exposes them to the process of thinking about the way a problem is being/could be solved; when they reflect on or talk about the process of problem solving, this is metacognition." (online summary of Schoenfeld, 1987)

Creating a mathematics culture and helping students develop metacognition are closely intertwined. Examples of teacher strategies to achieve both of these are:

- Have students monitor their peers' learning and thinking during group work
- Have students learn study strategies
- Have students relate ideas to what they already know
- Ask students which problems are hard and which are easy and why
- Have students develop and ask questions of themselves, about what's going on around them ("Have you asked a good question today?")
- Help students to know when to ask for help (when they do ask, require them to show how they have attempted to deal with the problem on their own)
- Show students how to transfer knowledge, strategies, and skills to other situations or tasks

The *Investigations* curriculum fosters metacognition in many ways. First and foremost, it encourages students to take responsibility for their own learning. Students learn about their own thinking partly just because they do so much of it, instead of doing a lot of imitating of methods demonstrated to them. They are continually asked to explain how they solved problems, step by step.

One useful metacognitive skill is asking yourself what you already know that might help you figure out what you don't know. As a curriculum with a constructivist philosophy, *Investigations* provides many opportunities for developing this skill. Examples include cluster problems, related problem sets, and learning addition and multiplication facts. In learning their number facts, students are also encouraged to develop and discuss cues that work for them.

An important metacognitive activity in *Investigations* is the classroom discussion. During discussions, students analyze their own strategies and those of others, thinking about why one method works and another doesn't. Teachers should encourage students to say things like, "at first I was thinking such-and-such, but then I noticed that . . . " or "I couldn't understand what so-and-so was saying until . . . " The goal is to reflect upon the problem-solving process, just as Schoenfeld advocates. This goal is apparent in many of the Dialogue Boxes in the units.

To develop metacognition through classroom discussions it is not enough to simply ask students to share their strategies and explain how they did it. Teachers need to have a clear mathematical agenda for each discussion, which often means choosing beforehand a certain strategy or two that they want to get discussed. During the discussion they might ask the class metacognitive questions such as:

- Which strategy would be more flexible with a wider range of problems?
- Which strategy is most clearly written, so that you can easily understand what the person was thinking?
- How could you make this strategy a little clearer (or more flexible? or more efficient?)
- Which strategies are both efficient and clearly understandable?

By asking these questions in discussions, teachers are helping students to see what is valuable about a strategy besides its accuracy: efficiency, a clear written explanation, flexibility, a demonstration of conceptual understanding, etc. Teachers should not shy away from comparing different students' strategies to make their points, though the classroom culture must support this first.

**
Conclusion**

Metacognition is a term that deserves a higher profile in the classroom and in professional development. Many influential documents in mathematics education, such as the NRC's report "Adding it Up", discuss the idea of metacognition without actually naming the term. The NCTM's Principles and Standards does mention metacognition in discussing problem-solving, citing research that indicates "students' problem-solving failures are often due not to a lack of mathematical knowledge but to the ineffective use of what they do know." (p. 53) Teachers need to recognize the importance of metacognition and make developing reflective habits an explicit goal in their classrooms.

Wendy Gulley, CESAME, Northeastern University

With thanks to Megan Murray, Susan Jo Russell and Cornelia Tierney, TERC

October 2001

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Bibliography and References:**

Borkowski, J., Carr, M., & Pressely, M. (1987). "Spontaneous" strategy use: Perspectives from metacognitive theory. *Intelligence*, 11, 61-75.

Brown, A. L. (1987). Metacognition, executive control, self-regulation, and other more mysterious mechanisms. In F. E. Weinert & R. H. Kluwe (Eds.), *Metacognition, motivation, and understanding* (pp. 65-116). Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Webpage on metacognition, *Educational Psychology Interactive*.

Schoenfeld, A. H. (1987). What's all the fuss about metacognition? In A. H.
Schoenfeld (Ed.), *Cognitive science and mathematics education *(pp. 189-215). Hillsdale, NJ: Lawrence Erlbaum Associates.
(For an online summary see: http://mathforum.org/~sarah/Discussion.Sessions/Schoenfeld.html)

Sternberg, R. J. (1984). What should intelligence tests test? Implications for a triarchic theory of intelligence for intelligence testing. *Educational Researcher*, 13 (1), 5-15.

Sternberg, R. J. (1986a). Inside intelligence. *American Scientist*, 74, 137-143.

Sternberg, R. J. (1986b).* Intelligence applied*. New York: Harcourt Brace Jovanovich, Publishers.

*This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.*