The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise

The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise

The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise

The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise

Synopsis

This volume focuses on two related questions that are central to both the psychology of mathematical thinking and learning and to the improvement of mathematics education: What is the nature of arithmetic expertise? How can instruction best promote it? Contributors from a variety of specialities, including cognitive, developmental, educational, and neurological psychology; mathematics education; and special education offer theoretical perspectives and much needed empirical evidence about these issues. As reported in this volume, both theory and research indicate that the nature of arithmetic expertise and how to best promote it are far more complex than conventional wisdom and many scholars, past and present, have suggested. The results of psychological, educational, and clinical studies using a wide range of arithmetic tasks and populations (including "normally" and atypically developing children, non-injured and brain-injured adults, and savants) all point to the same conclusion: The heart of arithmetic fluency, in general, and the flexible and creative use of strategies, in particular, is what is termed "adaptive expertise" (meaningful or conceptually based knowledge). The construction of adaptive expertise in mathematics is, for the first time, examined across various arithmetic topics and age groups. This book will be an invaluable resource for researchers and graduate students interested in mathematical cognition and learning (including mathematics educators, developmental and educational psychologists, and neuropsychologists), educators (including teachers, curriculum supervisors, and school administrators), and others interested in improving arithmetic instruction (including officials in national and local education departments, the media, and parents).

Excerpt

Many educational researchers are interested in how students can be taught subjects so that they develop adaptive expertise—the ability to apply meaningfully learned procedures flexibly and creatively. From this perspective, school instruction is successful when students are able to use what they have learned to invent effective procedures for solving new problems. Simply being able to complete school exercises quickly and accurately without understanding—what we call routine expertise—is not particularly valuable. This is because such competence is extremely limited in that it can be applied effectively only to familiar tasks. Unfortunately, little is known about the process of cultivating adaptive expertise.

Researchers who have studied transfer have almost unanimously concluded that, when solving real-world problems, students seldom aptly apply schooltaught procedures that were learned in a short period of time. Similarly, people who have had years of experience solving problems in a given domain may be unable to solve problems outside their experience. Such “experts” can solve familiar types of “problems” quickly and accurately but may not understand why their procedures work. As a result, when these experts are faced with a changed condition or a new problem, they are unable to modify known procedures or invent new ones. This is particularly true for experts in knowledgelean domains, such as abacus operation, which I studied in detail in the early 1970s. However, even in knowledge-rich domains, some experts' knowledge may consist of narrow, unconnected problem-solving schemas. As a result, these experts may simply classify problems and apply the single-routine solution associated with a particular problem type.

The notion of adaptive experts, which I introduced in Hatano (1982), was a theoretical ideal rather than a model derived from a series of empirical studies. I began by considering the following two questions: What kind of knowledge do flexible and inventive experts construct? And how do they construct it? In other words, I speculated about both the product of adaptive expertise and its acquisition process. Regarding the former, my attention focused on the conceptual knowledge underlying procedures. “Flexibility and adaptability seem to be possible only when there is some corresponding conceptual knowledge to give meaning to each step of the skill and provide criteria for selection among alternatives possibilities for each step within the procedures” (p. 15). I assumed (and still assume) that conceptual knowledge enables experts to construct mental models of the major entities in a content domain, models that can be used in mental simulations. I was very pleased to see that, in their formulation of adaptive expertise in the Preface and chapter 1, the book's editors emphasize the importance of connections among pieces of knowledge, especially between procedural and conceptual knowledge.

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