Addition and Subtraction

A Cognitive Perspective

Thomas P. Carpenter, James M. Moser, Thomas A. Romberg, Thomas P. Carpenter, James M. Moser, Thomas A. Romberg

• Publisher: Taylor & Francis
• Year: 2020
• Print ISBN: 9780367495589
• eBook ISBN: 9781000089264
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

A hallmark of much of the research on children's thinking in the 1970s had been the focus on explicit content domains. Much of this research had been represented by an eclectic collection of studies sampled from a variety of disciplines and content areas. However, in the few years before this publication, research in several content domains has begun to coalesce into a coherent body of knowledge. Originally published in 1982, the chapters in this work represent one of the first attempts to bring together the perspectives of a variety of different researchers investigating a specific, well defined content domain.

This book presents theoretical views and research findings of a group of international scholars who are investigating the early acquisition of addition and subtraction skills by young children. Together, the contributors bring a blend of psychology, educational psychology, and mathematics education to this topic. Fields of interest such as information processing, artificial intelligence, early childhood, and classroom teaching and learning are included in this blend.

1	An Emerging Paradigm for Research on Addition and Subtraction Skills

Thomas A. Romberg

University of Wisconsin-Madison

For several centuries, being able to find “one’s sums and differences” has been considered one mark of a schooled person. Although today we may have expanded our expectations about what constitutes literacy, we still expect all children to efficiently carry out operations on whole numbers. Yet, in spite of these expectations about the skills of addition and subtraction, there has been little consensus about how such skills develop. Lack of consensus does not mean there has been little research. Recent reviews (Carpenter, Blume, Hiebert, Anick, & Pimm, 1981; Carpenter & Moser, in press; Riley, Greeno, & Heller, in press) have identified an extensive body of research on addition and subtraction. Some of these studies have been quite influential. For example, Thorndike’s instructional suggestions in his Psychology of Arithmetic (1922) became the model of how to teach arithmetic for decades,¹ and Brownell’s (1947) research on subtraction demonstrated the superiority of the “decompositions” subtraction algorithm over the “equal additions” algorithm when taught with rational explanation. This made the “fair trading” procedure central to contemporary instruction in subtraction. But, to a large extent, the many studies on addition and subtraction represent an eclectic morass. This copious literature has lacked an implicit body of intertwined theoretical and methodological beliefs that permit selection, evaluation, and criticism. However, today we believe a change is imminent. The research and theoretical positions set forth in this volume should be viewed as foreshadowing the emergence of a firm research consensus in this area.

To build this argument, I follow Thomas Kuhn’s description of the “route of normal science.” In his now classic treatise on the growth of science, The Structure of Scientific Revolutions (1979), Kuhn argues that a significant turning point in the history of science occurs when from the chaos of competing ideas about a problem area, a single paradigm emerges which implicitly defines for practitioners the legitimate problems and methods of research. A paradigm gains that status because it is more successful than others in solving a few problems a group of researchers have recognized as acute. In this sense Kuhn argues paradigms have two essential characteristics. First, the paradigm’s achievement in solving the acute problems is sufficiently unprecedented to attract a group of adherents. Simultaneously, the paradigm is open-ended, leaving all sorts of problems for the redefined group of practitioners to resolve. Kuhn calls the research carried out by this new group “normal science.” It consists of actualizing the promise of the paradigm, “extending the knowledge of those facts that the paradigm displays as particularly revealing, by increasing the extent of the match between those facts and the paradigm’s predictions and by further articulation of the paradigm itself” [p. 24].

It would be both presumptuous and incorrect to argue that a paradigm for research on the development of addition and subtraction skills has emerged and that the papers in this volume reflect work within a normal science. Rather, the papers reflect growing agreement around a constellation of ideas with the potential to become such a paradigm. Current work mirrors what Kuhn discusses as the “route of normal science.”

Historically, the road to a research consensus in any area is arduous. In the absence of a paradigm all facts that could possibly pertain to the development of a given science are likely to seem equally relevant. As a result, early fact-gathering is a nearly random activity. Furthermore, in the absence of a reason for seeking some particular form of information, early fact-gathering is usually restricted to the wealth of data that lie ready at hand. Thus facts accessible to casual observation and experiment are pooled together with data retrievable from reports of classroom teaching, curriculum development, or evaluation.

Although this sort of fact-collecting has been essential to the origin of many significant sciences, one somehow hesitates to call the resulting literature scientific. Similarly, it would be hard to describe early studies on addition and subtraction as scientific (Carpenter, et al., 1981). This is true because such studies juxtapose facts that will later prove revealing with others that will for some time remain too complex to be integrated with theory at all. In addition, since any descriptions must be partial, such a typical natural history often omits from its immensely circumstantial accounts just those details that will be sources of important illumination to later scientists. Because the casual fact-gatherer seldom possesses the time or the tools to be critical, natural histories often relate reasonable descriptions with others that we are now quite unable to confirm. This is the situation that creates the intellectual morass characterizing the early stages of a science’s development, and as Kuhn (1979) states:

No wonder, then, that in the early stages of the development of any science different men confronting the same range of phenomena, but not usually all the same particular phenomena, describe and interpret them in different ways [p. 16].

With regard to the development of addition and subtraction skills, a set of scholars is confronting the same range of phenomena from essentially similar perspectives, and is beginning to reach a consensus on the acute problems to be solved, and beginning to use the same language and research methods to attack these problems. The emerging general paradigm is to formulate precise models of the cognitive processes used by subjects when carrying out specific tasks and how those processes change over time.

Brown (1970) argues the origins of this paradigm stem from two primary sources-computer simulation of cognitive processes and the writings of Jean Piaget. The basic strategy for this simulation of human processing was sketched by Herbert Simon (1962).

If we can construct an information processing system with rules of behavior that lead it to behave like the dynamic system we are trying to describe, then this system is a theory of the child at one stage of the development. Having described a particular stage by a program, we would then face the task of discovering what additional information processing mechanisms are needed to simulate developmental change-the transition from one stage to the next. That is, we would need to discover how the system could modify its own structure. Thus, the theory would have two parts-a program to describe performance at a particular stage and a learning program governing the transitions from stage to stage [pp. 154–155].

In order to specify rules of behavior and modifications of behavior it is necessary to characterize the child as an organism functioning under the control of a developing set of central processes. Some of Piaget’s notions of child development, such as schema, assimilation, and accommodation, have gradually become the basis for creating dynamic models of children’s cognitive processes in solving specific problems. The rapprochement between these two quite different conceptualizations has not been easy, as Klahr and Wallace (1976) have argued.² Yet, today agreement on some aspects is emerging. The developing paradigm has four elements upon which there is some consensus:

1. detailed descriptions of the contexts within which specific tasks are embedded;

2. analyses of all the behaviors associated with the subjects’ responses to performing the task;

3. repeated assessment of performance behaviors over time; and

4. inferences about the cognitive processing mechanism which relates information about the task with performance, and about changes in this performance.

For several reasons, children’s processing of addition and subtraction information-the topic of this book-is one area where this emerging paradigm has proved revealing. Addition and subtraction are the first set of mathematical ideas typically taught in schools. Children bring to such problems well developed counting procedures, some knowledge of numbers, and some understanding of physical operations, such as “joining” and “separating,” on sets of objects. Thus, from this context researchers have a unique opportunity to examine variations in how children process information prior to, during, and after formal instruction. Identifying stages of development in strategies children use to solve such problems is the basic problem addressed in this book.

To solve a typical problem one first must understand its implied semantic meaning. Quantifying the elements of the problem comes next (e.g., choosing a unit and counting how many). Then, the implied semantics of the problem must be expressed in the syntax of addition and subtraction. Next the child must be able to carry out the procedural (algorithmic) steps of adding and subtracting. Finally, the results of these operations must be expressed.

As a group, the papers in this volume employ a variety of descriptions for the various cognitive processes or subprocesses children use on such problems. As yet, there is no agreement on terms for describing the problem contexts, the types of processes, or the processing mechanisms children use. Nevertheless, there is agreement that our aim is to formulate precise models that describe children’s addition and subtraction skills and how those skills change over time.

The importance of specifying task context is reflected in the chapters by Thomas Carpenter and James Moser, Pearla Nesher, and Gerard Vergnaud in this volume. Because addition and subtraction sentences can be used to represent a wide variety of problems with different semantic structures, it is important for these authors to classify different types of verbal problems, and to study whether children can solve such problems prior to formal instruction. If children can, investigation of whether they use different strategies with problems having different semantic structures, and investigation of the changes in choice of strategies, is appropriate. Thus, the notions of verbal comprehension and the strategies used to quantify, represent, and calculate are acute problems of interest.

In this volume J. Fred Weaver and Vasily Davydov present arguments about the conceptualization of problem context from a mathematical perspective. Weaver stresses an alternative “unary operation” notion about addition and subtraction whereas Davydov embeds addition and subtraction in a broader mathematical perspective which stresses quantification processes before operational processes.

With few exceptions the authors in this book go well beyond tallying the number of correct and incorrect responses when describing children’s behaviors in response to addition and subtraction problems. Identification of the actions and strategies children use on specific tasks is central to the papers by Carpenter and Moser; Vergnaud; Leslie Steffe, Patrick Thompson, and John Richards; Karen Fuson; and Prentice Starkey and Rochelle Gelman. Examining errors for prevalent patterns is a major emphasis in the investigation of both John Seely Brown and Kurt Van Lehn, and Lauren Resnick.

Inferences about cognitive processes used to produce responses and changes in responses over time are based on simulation models in both Brown and Van Lehn’s model and Resnick’s research; on notions of developmental stages in the work of Carpenter and Moser; Starkey and Gelman; Steffe et al.; and on instruction in Nesher’s research. Cultural background and its influence on performance is stressed in both Giyoo Hatano’s research and in Herbert Ginsburg’s studies. It should be noted that the latter two authors are on opposite sides of the issue. Hatano argues that cultural background is important and Ginsburg cites evidence that it is not. Finally, in three broader theoretical papers, Robbie Case, Kevin Collis, and Richard Skemp stress different considerations for future models of cognitive processing. Case emphasizes children’s developing memory capacity and how information is organized for storage, Collis agrees with Case but stresses learned outcomes, and Skemp argues for a theoretical formulation positing a “director system” at two levels.

All the papers build models to explain children’s behaviors. For example, because children bring to typical verbal problems well developed counting skills and use those skills to quantify and often solve such problems, the study of the development of counting skills themselves is of particular interest. Steffe et al. and Fuson examine this topic.

Carpenter and Moser, Vergnaud, Nesher, and Starkey and Gelman examine the way children represent or use representations of various problems. The use of physical manipulatives, pictorial illustrations, and symbolic statements is modeled by this group of researchers.

As previously argued, one feature of an emerging consensus on a paradigm is agreement on methods of inquiry into the questions of critical importance. In the research presented in this volume, clinical interviewing of students is the predominant methodology. Carefully designed tasks and probing questions presented to a small sample of children are accepted as appropriate procedures. The papers by Carpenter and Moser, Resnick, Steffe et al, and Fuson, in particular, reflect this strategy. Davydov, Steffe et al., and Resnick use the “teaching experiment” extension of this procedure.

In most chapters, the data are generally presented descriptively with little use of statistics to bolster the arguments. In fact, because the concern is on formulation of models, attention is drawn to questions which may not be answerable with usual methods of statistical inference.

Underlying the book is a belief that by using this paradigm, we will eventually derive information that can be used to improve instruction. In particular, Case and Davydov draw inferences for teachers based on current knowledge.

The strength of this emerging scholarly consensus on how addition and subtraction skills develop lies in the fact that new research can begin where the last left off. Such research can concentrate on subtle or even esoteric aspects of the phenomena, assured that findings will add new information to a conceptual whole.

The weakness of consensus on any paradigm rests primarily in the fact that adherence to a single perspective makes questions appear insignificant which are deemed critical from other perspectives. For example, the question, “Who decides what subtraction algorithm should be taught?” is critical to the curriculum theorist interested in the structure of the content to be covered. For a behavioral psychologist, answering the question “What extrinsic motivational procedures are effective in getting children to add or subtract with low error rates?” is critical. F...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright
• Contents
• Preface
• 1. An Emerging Paradigm for Research on Addition and Subtraction Skills
• 2. The Development of Addition and Subtraction Problem-Solving Skills
• 3. Levels of Description in the Analysis of Addition and Subtraction Word Problems
• 4. A Classification of Cognitive Tasks and Operations of Thought Involved in Addition and Subtraction Problems
• 5. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction
• 6. An Analysis of the Counting-On Solution Procedure in Addition
• 7. Children’s Counting in Arithmetical Problem Solving
• 8. The Development of Addition and Subtraction Abilities Prior to Formal Schooling in Arithmetic
• 9. Towards a Generative Theory of “Bugs”
• 10. Syntax and Semantics in Learning to Subtract
• 11. General Developmental Influences on the Acquisition of Elementary Concepts and Algorithms in Arithmetic
• 12. The Structure of Learned Outcomes: A Refocusing for Mathematics Learning
• 13. Type 1 Theories and Type 2 Theories in Relationship to Mathematical Learning
• 14. The Development of Addition in Contexts of Culture, Social Class, and Race
• 15. Learning to Add and Subtract: A Japanese Perspective
• 16. The Psychological Characteristics of the Formation of Elementary Mathematical Operations in Children
• Author Index
• Subject Index