Learning and Mathematics

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What's all the fuss about metacognition? - Schoenfeld (1987)

Schoenfeld wrote this chapter in response to a challenge from mathematicians (among them Joe Crosswhite, Henry Pollak, Anna Henderson, and Steve Maurer) to explain "what metacognition is, why it's important, and what to do about it -- all in clear language that we can understand." Schoenfeld's explanation describes metacognition, or reflecting on how we think, through a discussion of how a problem was solved and what it was about, or where and why a difficulty occurred in the process of problem solving. He also proposes some ways metacognition could be used in the classroom.
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Chapter:

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.

Overview:

Many students don't feel good about math, largely as a result of the way they have been taught. Because of the prevalent belief that classroom mathematics consists of mastering formulas, these students do not understand how mathematics can be meaningful.

Metacognition has the potential to increase the meaningfulness of students' classroom learning, and the creation of a "mathematics culture" in the class room best fosters metacognition. Schoenfeld believes that a "microcosm of mathematical culture" would encourage students to think of mathematics as an integral part of their everyday lives, promote the possibility of students making connections between mathematical concepts in different contexts, and build a sense of a community of learners working out the intricacies of mathematics together.

Direct Quotes (and some comments):

There are three ways to talk about metacognition in the learning of mathematics: beliefs and intuitions, knowledge, and self-awareness (self-regulation).

  1. Beliefs and intuitions. What ideas about mathematics do you bring to your work in mathematics, and how do they shape the way you do mathematics? Here, Schoenfeld takes the 'constructivist' viewpoint, seeing individual students as building their mathematical frameworks from their beliefs, intuitions, and past experiences in trying to understand and make sense of the world. One such belief which may result is that classroom mathematics is formulaic, non-negotiable, and not related to the outside world." (p. 190).

  2. Your knowledge about your own thought processes. How accurate are you in describing your own thinking? ... [G]ood problem solving calls for using efficiently what you know: if you don't have a good sense of what you know, you may find it difficult to be an efficient problem solver (p. 190)." In other words, your approach to a task and your understanding of how to solve that task are affected by the extent to which you can realistically assess what you are capable of learning. For example, research shows that young children have little idea how well they can memorize; they may say that they can memorize a hundred unrelated words when in fact they can memorize only four or five. As children grow older, however, their estimates of their memory skills become more and more accurate.

  3. Self-awareness, or self-regulation. "How well do you keep track of what you're doing when, for example, you're solving problems, and how well (if at all) do you use the input from such observations to guide your problem solving actions?" (p. 190). Another way to think about this is as "awareness of your thinking and your progress as you're problem solving." Schoenfeld suggests that you can think of this using a management approach: "Aspects of management include (a) making sure that you understand what a problem is all about before you hastily attempt a solution; (b) planning; (c) monitoring, or keeping track of how well things are going during a solution; and (d) allocating resources, or deciding what to do, and for how long, as you work on the problem (pp. 190-191)."

Schoenfeld describes the process of solving unfamiliar problems with his students. He puts such problems on the board, and everyone works on them together. Students participate with him, 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.

"I argue that, largely as a result of their instruction, many students develop some beliefs about 'what mathematics is all about' that are just plain wrong -- and that those beliefs have a very strong negative effect on the students' mathematical behavior" (p. 195).

"One of the problems on the National Assessment of Education Progress secondary mathematics exam, administered to a stratified sample of 45,000 students nationwide, was the following: An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed? Seventy percent of the students who took the exam set up the correct long division and performed it correctly. However, the following are the answers those students gave to the question how many buses are needed: 29% (1 in 3) said the number of buses is '31 remainder 12'; 18% said the number of buses needed is "31"; 23% said the number of buses needed is '32,' which is correct; (30% did not do the computation correctly)... One out of three students said '31 remainder 12'- without checking to see if the result made sense! In essence, they treated the problem as calling for a formal computation. Despite the 'cover story' about the buses, the computation had little or nothing to do with the real world" (p. 196).

"Many students come to believe that school mathematics consists of mastering formal procedures that are completely divorced from real life, from discovery, and from problem solving... If mathematics problems were meaningful to students, could students possibly report that the number of buses they need is 31-remainder-12? (A colleague has pointed out that a group of students in the schoolyard, deciding on how many cars they needed to go someplace, would never make the same mistake)" (p. 197).

"Mathematical thinking as a culture cannot be put into an essentially alien culture, the traditional classroom." [It is necessary to rethink what the traditional classroom should consist of, to rethink, for example, who should be doing the talking in the classroom and when. It is also necessary to think about what students know already and what they are ready to work on next.] (p. 214).

"I would argue that ... interactions, and the sense of community -- a culture of mathematics, if you will -- are part of what sustains mathematics. And participation in that culture is how one comes to understand what mathematics is" (p. 212).

"I realize that what I succeeded in doing in the most recent versions of my problem-solving course was to create a microcosm of mathematical culture. Mathematics was the medium of exchange. We talked about mathematics, explained it to each other, shared the false starts, enjoyed the interaction of personalities. In short, we became mathematical people. It was fun, but it was also natural and felt right. It wasn't a separate 'school experience' for a few hours a week. By virtue of this cultural immersion, the students experienced mathematics in a way that made sense... For that reason, the course has a much greater chance of having a lasting effect" (p. 213).

"I focused on elaborating aspects of mathematical thinking... [of] what it is to live in a culture of mathematics. But the flip side of the coin, understanding the development of mathematical thinking in our culture at large, is equally crucial" (p. 214).

"[W]hat we need is a program of 'cultural design' for schooling. Understanding enough about the social contexts that promote the need to develop and understand mathematical ideas, and about the environments that support the growth and development of those ideas, may allow us to create classroom cultures in which students do mathematics naturally. When that happens, the 'transfer problem' [i.e., not making connections between mathematical concepts in different contexts] will no longer be a problem" (p. 214).

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