Actual cases of apprenticeship provide historically and culturally specific examples which seem especially helpful in exploring the implications of the concept of legitimate peripheral participation. As we have insisted, however, the concept should not be construed as a distillation of apprenticeship. Ethnographic studies of apprenticeship emphasize the indivisible character of learning and work practices. This, in turn, helps to make obvious the social nature of learning and knowing. As these studies partially illustrate, any complex system of work and learning has roots in and interdependencies across its history, technology, developing work activity, careers, and the relations between newcomers and old-timers and among co-workers and practitioners.

We have already outlined some reasons for turning away from schooling in our search for exemplary material, though schooling provides the empirical basis for much cognitive research on learning and also for much work based on the notion of the zone of proximal development. Such research is conceptually tied in various ways to school instruction and to the pedagogical intentions of teachers and other caregivers. In this context, schooling is usually assumed to be a more effective and advanced institution for educational transmission than (supposedly) previous forms such as apprenticeship. At the very least, schooling is given a privileged role in intellectual development. Because the theory and the institution have common historical roots (Lave 1988), these school-forged theories are inescapably specialized: They are unlikely to afford us the historical–cultural breadth to which we aspire.

Learning viewed as situated activity has as its central defining characteristic a process that we call legitimate peripheral participation. By this we mean to draw attention to the point that learners inevitably participate in communities of practitioners and that the mastery of knowledge and skill requires newcomers to move toward full participation in the sociocultural practices of a community. “Legitimate peripheral participation” provides a way to speak about the relations between newcomers and old-timers, and about activities, identities, artifacts, and communities of knowledge and practice. It concerns the process by which new comers become part of a community of practice. A person's intentions to learn are engaged and the meaning of learning is configured through the process of becoming a full participant in a sociocultural practice. This social process includes, indeed it subsumes, the learning of knowledgeable skills.

In order to explain our interest in the concept of legitimate peripheral participation, we will try to convey a sense of the perspectives that it opens and the kinds of questions that it raises. A good way to start is to outline the history of the concept as it has become increasingly central to our thinking about issues of learning. Our initial intention in writing what has gradually evolved into this book was to rescue the idea of apprenticeship. In 1988, notions about apprenticeship were flying around the halls of the Institute for Research on Learning, acting as a token of solidarity and as a focus for discussions on the nature of learning.

All theories of learning are based on fundamental assumptions about the person, the world, and their relations, and we have argued that this monograph formulates a theory of learning as a dimension of social practice. Indeed, the concept of legitimate peripheral participation provides a framework for bringing together theories of situated activity and theories about the production and reproduction of the social order. These have usually been treated separately, and within distinct theoretical traditions. But there is common ground for exploring their integral, constitutive relations, their entailments, and effects in a framework of social practice theory, in which the production, transformation, and change in the identities of persons, knowledgeable skill in practice, and communities of practice are realized in the lived-in world of engagement in everyday activity.

INTERNALIZATION OF THE CULTURAL GIVEN

Conventional explanations view learning as a process by which a learner internalizes knowledge, whether “discovered,” “transmitted” from others, or “experienced in interaction” with others. This focus on internalization does not just leave the nature of the learner, of the world, and of their relations unexplored; it can only reflect far-reaching assumptions concerning these issues. It establishes a sharp dichotomy between inside and outside, suggests that knowledge is largely cerebral, and takes the individual as the nonproblematic unit of analysis. Furthermore, learning as internalization is too easily construed as an unproblematic process of absorbing the given, as a matter of transmission and assimilation.

The most important chapter in this book is Chapter E: Exercises. I have left the interesting things for you to do. You can start now on the ‘EG’ exercises, but see ‘More about exercises’ later in this Preface.

The book, which is essentially the set of lecture notes for a third-year undergraduate course at Cambridge, is as lively an introduction as I can manage to the rigorous theory of probability. Since much of the book is devoted to martingales, it is bound to become very lively: look at those Exercises on Chapter 10! But, of course, there is that initial plod through the measure-theoretic foundations. It must be said however that measure theory, that most arid of subjects when done for its own sake, becomes amazingly more alive when used in probability, not only because it is then applied, but also because it is immensely enriched.

You cannot avoid measure theory: an event in probability is a measurable set, a random variable is a measurable function on the sample space, the expectation of a random variable is its integral with respect to the probability measure; and so on. To be sure, one can take some central results from measure theory as axiomatic in the main text, giving careful proofs in appendices; and indeed that is exactly what I have done.

Introduction – please read!

The purpose of this chapter is to give some indication of some of the ways in which the theory which we have developed can be applied to real-world problems. We consider only very simple examples, but at a lively pace!

In Sections 15.1-15.2, we discuss a trivial case of a celebrated result from mathematical economics, the Black-Scholes option-pricing formula. The formula was developed for a continuous-parameter (diffusion) model for stock prices; see, for example, Karatzas and Schreve (1988). We present an obvious discretization which also has many treatments in the literature. What needs to be emphasized is that in the discrete case, the result has nothing to do with probability, which is why the answer is completely independent of the underlying probability measure. The use of the ‘martingale measure’ ℙ in Section 15.2 is nothing other than a device for expressing some simple algebra/combinatorics. But in the diffusion setting, where the algebra and combinatorics are no longer meaningful, the martingale-representation theorem and Cameron-Martin-Girsanov change-of-measure theorem provide the essential language. I think that this justifies my giving a ‘martingale’ treatment of something which needs only junior-school algebra.

Sections 15.3-15.5 indicate the further development of the martingale formulation of optimality in stochastic control, at which Exercise E10.2 gave a first look. We consider just one ‘fun’ example, the ‘Mabinogion sheep problem’; but it is an example which illustrates rather well several techniques which may be effectively utilized in other contexts.