Culture and Cognitive Development

Studies in Mathematical Understanding

Geoffrey B. Saxe

• Publisher: Psychology Press
• Year: 2015
• Print ISBN: 9780805802733
• eBook ISBN: 9781317728085
• Topic: Psychology
• Subtopic: Developmental Psychology
• Index: Psychology

Researchers examining children's mathematics acquisition are now questioning the belief that children learn mathematics principally through formalized, in-school mathematics education. There is increasing evidence that children gain mathematical understanding through their participation in out-of-school cultural practices and that their mathematics only occasionally resembles what they learn in the classroom.

Culture and Cognitive Development presents the latest research by Dr. Geoffrey Saxe on this issue. In examinations of the mathematical understandings of child candy sellers in an urban center in northeastern Brazil, Dr. Saxe finds sharp contrasts between mathematics as practiced in school and in real-world settings. In this unique research project he presents a penetrating conceptual treatment of the interplay between culture and cognitive development, filling a void in current research literature.

Subjects examined include:

the interplay between sociocultural and cognitive developmental processes

the differences between math knowledge learned in and out of the classroom

the ways math learning in the classroom is modified by children's out-of-school mathematics and, correspondingly, how practical out-of-school mathematics use is modified by formal education

I	CULTURE AND COGNITION: A METHOD OF STUDY

Some years ago, I visited a Papua New Guinean highlands group, the Oksapmin, for the purpose of studying the development of mathematical understandings in a non-Western culture. As a student of cognitive development, I was struck by the differences between the Oksapmins’ indigenous mathematical practices and my own. In house building, arrowhead making, string bag weaving, and counting, the Oksapmins’ approach to solving mathematical problems of measurement and numeration involved very different ways of thinking and very different procedures for accomplishing everyday problems (Saxe, 1982). For instance, Oksapmin often conceptualize numerical and measurement problems in terms of an indigenous, 27-body-part number system with no base structure. A number is expressed by pointing to a particular body part (like the neck) and saying the body-part name.

I had two initial reactions to the mathematical practices of the Oksapmin which were linked to my graduate training in developmental psychology, training in the structural-developmental tradition of Piaget and focusing on mathematical cognition. The first reaction was an intellectual excitement: The same mathematical operations of correspondence and measurement captured in Piagetian psychogenetic analyses that I had studied in Western children were apparent in the activities of a people from an extraordinarily foreign culture. The second was an intellectual frustration: The aspects of cognition and the texture of everyday life in Oksapmin that seemed so marvelously different from that of the West—like the Oksapmins’ use of their numeration system in everyday practices—were not captured by Piagetian core constructs like conservation. While Piaget’s epigenetic constructivism—the thesis that the individual generates novel intellectual structures by reorganizing prior knowledge to resolve contradictions—appeared critical to me for conceptualizing developing cognitive processes, reducing thought to a small set of Piagetian categories seemed weak as a method for the study of culture-cognition relations.

It has been 10 years since my first visit to Oksapmin. The volume that follows illustrates my efforts to synthesize a research program that provides insight into the distinctiveness of children’s cognitive development across cultures but at the same time reveals universal regulative processes which transcend cultural boundaries.

In the first part of this volume, I introduce a general analytic model that targets cultural practices as important contexts for study. In subsequent parts, I apply the model to a single cultural practice—candy selling—as it has emerged in the lives of children living in northeastern Brazil. In candy selling, the relations between culture and cognitive development stand out in particularly clear relief and are particularly amenable to study.

1	Culture and Cognitive Development

Treatments of cognitive development can be understood as rooted in one of three fundamental views on the character of knowledge. Each view carries with it both problems and advantages for an adequate account.

The empiricist view is that the environment is the source of knowledge, and through experience children’s knowledge comes to reflect the environment with increasing exactness. The position has its roots in philosophy (e.g., Locke) and is also formulated in modern treatments of learning (Bijou & Baer, 1961; Gagné, 1985). The promise of these treatments is to explain the way children’s knowledge is shaped and organized directly by experience. But to date, the promise is far from being realized and many have argued that, in principle, rational thought structures like logical deduction cannot be accounted for solely by reference to the environment (e.g., Chomsky, 1957; Piaget, 1970).

The nativist position acknowledges the need for fundamental knowledge structures to organize experience into categories and logical systems, and its claim is that these structures are a hereditary endowment. Again, the position has ancient philosophical roots (Plato) and finds its current articulation in treatments of language (Chomsky, 1972) and cognition (Fodor, 1983). While such models explain the independence of rational cognitive structures and experience, they at the same time do not offer compelling treatments of development nor the variability of cognitive forms across cultures.

The constructivist position—in which the treatment I develop in this volume has its roots—is that fundamental aspects of knowledge neither come preformed in the genes nor in the environment, but are actively constructed by the developing individual. The aim of constructivist accounts is to show how novel cognitive structures emerge as a function of the developing subjects’ commerce with a social and physical environment, and the focus is on explaining cognitive development with reference to principles of self-regulated change and interaction.

The problem which is the focus of the volume is to understand the interplay between sociohistorical and cognitive developmental processes. Most fundamentally, my concern is to understand how artifacts and forms of social organization—products which have emerged over the course of social history—come to be interwoven with and are intrinsically related to the nature of children’s intellectual constructions. These historical products may be conceptual as in the case of scientific concepts entailed in Newtonian mechanics, symbolic forms as in the case of numeration or writing systems, or material as in the case of tools like the lever or the computer.

In the following discussion, I consider the problems and prospects in the ways researchers in the constructivist tradition have conceptualized and studied the interplay between sociohistorical processes and cognitive developmental ones. While constructivist treatments share core assumptions, they often differ in their analytic units and in their levels of analysis. All have had difficulty in producing rich and systematic conceptual frameworks for the analysis of intrinsic relations between cognitive developmental and sociohistorical processes. After reviewing key features of these treatments in this chapter, I then turn, in Chapter 2, to my effort to work toward a more complete framework for the analysis of the interplay between cognitive developmental and sociohistorical processes.

CONSTRUCTIVIST APPROACHES

A basic assumption of constructivist treatments is that individuals create new knowledge in their goal-directed activities; in turn, new knowledge leads subjects to identify new goals. Cassirer (1957), in his philosophical treatment, expresses the constructivist dialectic between conceptual advances and goal-directed activities extraordinarily well.

[E]ach newly acquired concept is an attempt, a beginning, a problem; its value lies not in its copying of definite objects, but in its opening up of new logical perspectives … one of its essential tasks is not to let the problems of knowledge come prematurely to rest, but to keep them in a steady flux, by guiding them toward new goals…. Here again we find that the concept is far less abstractive than prospective; it not only fixes what is already known, establishing its general outlines, but also maintains a persistent outlook for new and unknown connections. (Cassirer, 1957, p. 306)

Thus, from the constructivist perspective, goals themselves are rooted in individuals’ understandings.

Empirical research on culture and cognitive development has been influenced by two constructivist treatments which differ in their description of the role of social processes in the development of the individual’s self-regulative autonomous reasoning. One treatment is associated with the structural developmental approach of Piaget, and the other is associated with the sociohistorical approach of Vygotsky.

Piaget and the Structural-developmental Approach

Piaget’s treatment of cognitive development is rooted in a neo-Kantian epistemology in which a principal assumption is that the world is not known directly but is assimilated by intellectual structures. Kant and Piaget shared a concern to understand how the subject comes to know the necessity of propositions about logico-mathematical and physical phenomena. The solution for both was in the properties of these cognitive structures. Piaget, however, did not share Kant’s monism and took a developmental perspective. Using empirical research, he produced evidence supporting a sequence of four stages of cognitive development that extended from infancy through adolescence.

In Piaget’s analysis, each successive stage constituted a new cognitive equilibrium, and for each stage Piaget’s central analytic concern was to present an analysis of its structural or formal properties. In Piaget’s scheme, reality for the infant at birth is no more than the extension of its hereditary reflexes, like sucking and grasping (Piaget, 1954, 1963). In the course of progressively more complex interactions with its environment, the infant transforms these hereditary reflexes into cognitive structures which make it possible, at about 18 months, for the infant to “re-present” experience, and semiotic systems (imitation, imagery, language) begin to emerge (Piaget, 1962). The representational capacity carries with it, however, new problems of coordinating representations, problems which are not solved until the next stage when concrete operational classificatory and relational structures emerge in middle childhood (Inhelder & Piaget, 1969; Piaget, 1952). Concrete operational structures provide the basis for a wide range of novel and stable concepts (e.g., quantity conservations, Euclidean and projective spatial understandings). It is not until early adolescence, however, that the individual constructs an integration of concrete operational class and relational operations into a system of formal operations which is the basis for hypothetico-deductive reasoning (Inhelder & Piaget, 1958).

To explain progress through his stages, Piaget, like Cassirer, argued that the subject is purposeful, constructing goals based on prior knowledge and creating coherent solutions to these goals in the form of novel cognitive developments. In Piaget’s analysis, development proceeds by a self-regulated construction, or “equilibration,” a dialectical process in which the subject resolves perturbations in the coherence of his or her structuring activities by coordinating and constructing new, more adequate cognitive structures. For Piaget, it is equilibration that guides the direction and organization of cognitive development (Piaget, 1977).

In Piaget’s treatment of equilibration, the interplay between social life and cognitive developmental processes was not a core concern. Indeed for Piaget, the focus was on the formal properties of action without regard for the situatedness of actions in a sociohistorically articulated web of meanings. Nonetheless, questions of social influences on cognitive development have emerged in the work of various researchers who have attempted to extend Piaget’s analyses to social processes, and Piaget himself has noted the effect of social processes on the rate of progress through his stages (Piaget, 1966, 1972). However, we find in these extensions that social life is related to cognitive development as an external process, and the way sociocultural life may be deeply interwoven with the character of intellectual functioning is unanalyzed.

Empirical Research on Culture and Cognitive Development Related to the Piagetian Framework

To study sociocultural influences on cognitive development from within the Piagetian framework, researchers have contrasted individuals’ performances from different cultural groups on Piagetian tasks (for a review, see Dasen, 1972; Dasen & Heron, 1981). Typically, researchers have focused on the transition to concrete operations, using Piaget’s original tasks or slight adaptations of these in varying content areas (e.g., conservation (Laurendeau–Bendavid, 1977; Opper, 1977), classification (de Lacey, 1970), motion (Za’rour & Khuri, 1977), space (de Lemos, 1974). This literature has revealed both cultural similarities and differences. Age norms from many groups suggest that children progress up through Piaget’s stage of concrete operations, though there is little documentation of the emergence of formal operational structures across groups (see Neimark, 1975; Piaget, 1972). Further, the age norms for passing concrete operational tasks vary widely across cultures. Whether these cross-cultural differences reflect merely lack of appropriate accommodations of method to the different cultural contexts or actual differences in conceptual development is not entirely clear and has been the subject of considerable discussion (see Hallpike, 1979; Jahoda, 1980; Kamara & Easley, 1977; Nyiti, 1982; Piaget, 1972). Regardless, this literature sets the stage for more focused investigations of the ways that dimensions of sociocultural life may be associated with progress through Piaget’s stage sequence. I will briefly consider three such dimensions.

Cultural Practices. One sociocultural dimension isolated for study is participation in cultural practices thought to favor equilibrations. Researchers have analyzed relations between participation in such practices as pottery making and children’s understanding of mass conservation in rural Mexican groups (Price–Williams, Gordon, & Ramirez, 1967; Steinberg & Dunn, 1976), practices of economic exchange and children’s understanding of number conservation in West African groups (Posner & Baroody, 1979), practices of hunting and nomadic life style on spatial concepts in Canadian Eskimos (as opposed to agricultural and sedentary groups; Dasen, 1975), and schooling and various concrete operational concepts (Goodnow & Bethon, 1966; Laurendeau–Bendavid, 1977; Mermelstein & Shulman, 1967). The results of studies on practice participation and schooling are varied. Some indicate effects on concrete operational concepts; others do not. Rarely is there a detailed analysis of which aspects of practice-linked experience that were related to cognitive developmental change, and herein lies a problem with this focus. Practices are “packaged variables” (Whiting, 1976), and efforts to unpack a practice requires an analytic model that links the structure of practices with the structure of cognitive developmental processes, an analytic model that has yet to emerge in any sophisticated form within the Piagetian framework.

Social Interactions. A second sociocultural dimension isolated for analysis is social interaction (Doise & Mugny, 1984). Unlike the practice-based research, the social interaction studies are typically laboratory-based. Investigators typically draw activities from Piagetian assessment tasks and instruct dyads to work collaboratively to reach agreement on a solution. Such studies often make use of an experimental design in which children are randomly assigned to experimental (social interaction) and comparison (no social interaction) groups, and interactions in the experimental groups are often videotaped for later analysis. These studies sacrifice the naturalistic features of the cultural practice research. However, they typically provide a more detailed account of social interactional proccesses hypothesized to be implicated in stage change, most notably social conflict which, it is argued, sets in motion the process of equilibration. Using this kind of paradigm, researchers have studied spatial perspective taking (Bearison, Magzamen, & Filardo, 1986; Doise & Mugny, 1984; Doise, Mugny, & Perret–Clermont, 1975, 1976), proportionality in the...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• PART I. CULTURE AND COGNITION: A METHOD OF STUDY
• PART II. COMPONENT 1: EMERGENT MATHEMATICAL GOALS
• PART III. COMPONENT 2: FORM-FUNCTION SHIFTS IN CANDY SELLERS’ MATHEMATICS
• PART IV. COMPONENT 3: THE INTERPLAY BETWEEN LEARNING IN AND OUT OF SCHOOL
• PART V. CULTURE AND COGNITIVE DEVELOPMENT
• Appendix A: Recruitment of Sellers
• Appendix B: Background Information on Sellers and Their Practice
• Appendix C: Extended Description of Tasks
• References
• Author Index
• Subject Index