Collins, A., Brown, J. S., & Newman, S. E. (1989). Cognitive apprenticeship: Teaching the crafts of reading, writing, and mathematics. In L. B. Resnick (Ed.), Knowing, learning, and instruction: Essays in honor of Robert Glaser (pp. 453-494). Hillsdale, NJ: Lawrence Erlbaum Associates.
Effective teachers "involve" students in learning as apprentices: they work alongside students and/or set up situations that will cause students to begin to work on problems even before fully understanding them.
A key aspect of an "apprenticeship" approach to teaching involves breaking the problem into parts so that students are challenged to master as much of a task as they are ready to handle.
In addition, teachers are encouraged to provide students with varying kinds of practice situations before moving on to more challenging tasks, allowing an understanding that surpasses the use of formulas.
Direct Quotes (and some comments):
"Only in the last century, and only in industrialized nations, has formal schooling emerged as a widespread method of educating the young. Before schools appeared, apprenticeship was the most common means of learning and was used to transmit the knowledge required for expert practice in fields from painting and sculpting to medicine and law. Even today, many complex and important skills, such as those required for language use and social interaction, are learned informally through apprenticeship-like methods--that is, methods not involving didactic teaching, but observation, coaching, and successive approximation" (p. 453).
"Cognitive apprenticeship, as we envision it, differs from traditional apprenticeship in that the tasks and problems are chosen to illustrate the power of certain techniques or methods, to give students practice in applying these methods in diverse settings, and to increase the complexity of tasks slowly, so that component skills and models can be integrated" (p. 459).
Regarding an exercise in which apprentices learned to put together a garment from pre-cut pieces: "This sequencing of activities provides learners with opportunities to build a conceptual model of how all the pieces of a garment fit together before attempting to produce the pieces. For cognitive domains this implies sequencing of lessons so students have a chance to apply a set of skills in constructing an interesting problem solution before they are required to generate or remember those skills. This requires some form of scaffolding" (p. 485). [In other words, once the student begins to understand what you are teaching, you can pull back and provide "only limited hints, refinements, and feedback" (p. 456). This is scaffolding.]
"Scaffolding can be applied to different aspects of a problem-solving process, for example, to management and control of the problem solving or to the subprocesses that are required to carry out the task. Global before local skills means that in the sequencing of lessons there is a bias toward supporting the lower level or composite skills that students must put together to carry out a complex task. In algebra, for example, students may be relieved of having to carry out low-level computations in which they lack skill to concentrate on the higher order reasoning and strategies required to solve an interesting problem" (p. 485).
"A critical element in fostering learning is to have students carry out tasks and solve problems in an environment that reflects the multiple uses to which their knowledge will be put in the future" (p. 487).
"The reason that Dewey, Papert, and others have advocated learning from projects rather than from isolated problems is, in part, so that students can face the task of formulating their own problems, guided on the one hand by the general goals they set, and on the other hand by the 'interesting' phenomena and difficulties they discover through their interaction with the environment" (p. 487).
"... problems emerge from interactions between the overall goals and the perceived structure of the environment. Thus, in projects students learn first to find a problem and then, ideally, to use the constraints of the embedding context to help solve it" (p. 488). [A fine example of this would be Lesh's (1985) suggestion that students wallpaper a room, a task which engages a substantial amount of mathematical knowledge and strategy, yet is neither fully nor rigidly defined from the start.]
"Drawing students into a culture of expert practice in cognitive domains involves teaching them how to 'think like experts.' The focus of much current cognitive research is to understand better what is really meant by such a goal and to find ways to communicate more effectively about the processes involved" (p. 488). [Schoenfeld (1987) also emphasizes the importance of creating a "microcosm of mathematical culture" in order to help students think like expert mathematicians. In his classroom, Schoenfeld solved problems alongside his students. "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" (p. 213).]
Lesh, R. (1985). Processes, skills, and abilities needed to use mathematics in everyday situations. Education and Urban Society, 17, 439-446.
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.