Copyright © 1999 Massachusetts Institute of Technology
Numeracy is a term that has been used in a variety of ways. It encompasses formal and informal mathematics, cultural practices with mathematical content, and behavior mediated by mathematical properties even when these properties are not verbally accessible. The study of numeracy explores a broad range of mathematical competencies across species, cultures, and the human lifespan. Human numeracy has universal characteristics based on biological mechanisms and developmental trajectories as well as culturally variable representation systems, practices, and values.
Studies with diverse species show that animals are sensitive to number (see Gallistel 1990 for a review). This literature suggests that there are innate capabilities that have evolved to support numeracy in humans (Gallistel and Gelman 1992). A variety of sources of evidence point to early numerical abilities in human infants (see INFANT COGNITION). For example, infants as early as the first week of life have been shown to discriminate between different small numbers (Antell and Keating 1983). The possibility that this discrimination is carried out by a cognitive mechanism encompassing several modalities has been debated (Starkey, Spelke, and Gelman 1990). In addition, studies have shown that infants have some knowledge of the effects of numerical transformations such as addition and subtraction (Wynn 1992).
Numerical competencies evident in the human infant are strong candidates for universal aspects of human numeracy. However, this does not necessarily discount the role of culture in developing human numeracy. Within the framework of EVOLUTIONARY PSYCHOLOGY, it is argued that universal characteristics of numeracy provide innate starting points for numeracy development, which, in turn, is influenced by culturally specific systems of knowledge (Cosmides and Tooby 1994). Similarly, within the neo-Piagetian framework, children are seen not simply as passing through a universal set of stages, but also as setting out on a unique cognitive journey that is guided by cultural practices (Case and Okamoto 1996). In this sense, numeracy is viewed as a cultural practice that builds on innate mechanisms for understanding quantities. The result is a conceptual structure for numeracy that reflects both universal and culture-sensitive characteristics. These conceptual structures are relatively similar across cultures that provide similar problem-solving experiences in terms of schooling and everyday life. On the other hand, mastery levels of particular tasks or skills may differ from one culture to another depending on the degree to which they are valued in each culture (Okamoto et al. 1996). In addition, cultures influence mathematical practices through the belief systems associated with numeracy, as well as through tools and artifacts (e.g., symbol systems) that support numeracy. A broad array of human activities to which mathematical thinking is applied are interwoven with cultural artifacts, social conventions, and social interactions (Nunes, Schliemann, and Carraher 1993; Saxe 1991).
Cultures have developed systems of signs that provide ways of thinking about quantitative information (see LANGUAGE AND CULTURE). Different systems shed light on different aspects of knowing. That is, they provide a means to extend the ability to deal with numbers; at the same time, they constrain numerical activities. For example, the Oksapmin of Papua New Guinea have a counting system using body parts, with no base structure, that only goes up to 27 (Saxe 1982). This way of quantifying is fully adequate for the numerical tasks of traditional life. It does not, however, facilitate easy computation or the counting of objects beyond 27. In contrast, the perfectly regular base-10 system of many Asian languages appears to make the mastery of base-10 concepts easier for children beginning school than the less regular base-10 systems of many European languages, including English (Miura et al. 1993). These various representational systems are culture-specific tools to deal with counting and computing, and all cultures seem to have them. Other culture-specific representation systems have been identified for locating (geometry, navigation), measuring, designing (form, shape, pattern), playing (rules, strategies), and explaining (Bishop 1991).
Further cultural variations in mathematical behavior are manifest in the ways that people use mathematical representations in the context of everyday activities (see SITUATED COGNITION AND LEARNING). Although they use the same counting numbers for different activities, child street vendors in Brazil were observed to use different computational strategies when selling than when doing school-like problems (Nunes, Schliemann, and Carraher 1993). While selling, they chose to use oral computation and strategies such as decomposition and repeated groupings. On school-like problems, they chose to use paper and pencil with standard algorithms and showed a markedly higher rate of error. Research in other domains, for example measurement (Gay and Cole 1967) and proportional reasoning (Nunes, Schliemann, and Carraher 1993), further confirms that informal mathematics can be effective and does not depend upon schooling for its development.
One characteristic of ethnomathematics, as informal mathematics or NAIVE MATHEMATICS is commonly called, is that the mathematics is used in pursuit of other goals rather than solely for the sake of the mathematics as in school or among professional mathematicians. As new goals arise, representational systems and practices develop to address the emergent goals. For instance, as the Oksapmin became more involved with the money economy, their "body" counting system began to change toward a base system (Saxe 1982). Although the differences between informal and school mathematics are often stressed (Bishop 1991), skills developed in the informal domain can be used to address new goals and practices in the school setting. In Liberian schools it was found that the most successful elementary students were those who combined the strategies from their indigenous mathematics with the algorithms taught in school (Brenner 1985). Similarly, Oksapmin and Brazilian children benefit from using their informal mathematics to learn school mathematics (Saxe 1985, 1991). Because conflicts between informal and school mathematics frequently arise, a number of authors have argued for building bridges between these different cultures of mathematics (Bishop 1991; Gay and Cole 1967; Gerdes 1988).
In addition to the overt mathematical practices already described, Gerdes (1988) has described frozen mathematics as the mathematics embodied in the products of a culture such as baskets, toys, and houses. Although the history of these objects has typically been lost, the original designers of these cultural artifacts employed mathematical principles in their design, according to Gerdes. Mathematical traditions embodied in these artifacts can provide interesting mathematical investigations that help children understand their own cultural heritage as well as contemporary school mathematics.
The study of numeracy and culture draws from diverse disciplines within the cognitive sciences including psychology, linguistics, biology, and anthropology. The strengths of each discipline should be utilized to provide a more coherent view of what numeracy is and how it interacts with culture. Much future work remains to be done to better understand the universal and culture-specific aspects of numeracy.
Antell, S., and D. Keating. (1983). Perception of numerical invariance in neonates. Child Development 54:695-701.
Bishop, A. (1991). Mathematical Enculturation: A Cultural Perspective on Mathematics Education. Dordrecht: Kluwer.
Brenner, M. E. (1985). The practice of arithmetic in Liberian schools. Anthropology and Education Quarterly 16:177-186.
Case, R., and Y. Okamoto. (1996). The role of central conceptual structures in the development of children's thought. Monographs of the Society for Research in Child Development 61 (1-2, serial no. 246).
Cosmides, L., and J. Tooby. (1994). Origins of domain specificity: The evolution of functional organization. In L. A. Hirschfeld and S. A. Gelman, Eds., Mapping the Mind: Domain Specificity in Cognition and Culture. Cambridge: Cambridge University Press, pp. 85-116.
Gallistel, C. R. (1990). The Organization of Learning. Cambridge, MA: MIT Press.
Gallistel, C. R., and R. Gelman. (1992). Preverbal and verbal counting and computation. Cognition 44:43-74.
Gay, J., and M. Cole. (1967). The New Mathematics and an Old Culture. New York: Holt, Rinehart and Winston.
Gerdes, P. (1988). On culture, geometrical thinking and mathematics education. Educational Studies in Mathematics 19:137-162.
Miura, I. T., Y. Okamoto, C. C. Kim, M. Steere, and M. Fayol. (1993). First graders' cognitive representation of number and understanding of place value: Cross-national comparisons -- France, Japan, Korea, Sweden, and the United States. Journal of Educational Psychology 85:24-30.
Nunes, T., A. D. Schliemann, and D. W. Carraher. (1993). Street Mathematics and School Mathematics. New York: Cambridge University Press.
Okamoto, Y., R. Case, C. Bleiker, and B. Henderson. (1996). Cross cultural investigations. In R. Case and Y. Okamoto, Eds., The Role of Central Conceptual Structures in the Development of Children's Thought. Monographs of the Society for Research in Child Development 61 (1-2, serial no. 246), pp. 131 - 155.
Saxe, G. B. (1982). Developing forms of arithmetic operations among the Oksapmin of Papua New Guinea. Developmental Psychology 18:583-594.
Saxe, G. B. (1985). The effects of schooling on arithmetical understandings: Studies with Oksapmin children in Papua New Guinea. Journal of Educational Psychology 77:503-513.
Saxe, G. B. (1991). Culture and cognitive development: Studies in mathematical understanding. Hillsdale, NJ: Erlbaum.
Starkey, P., E. S. Spelke, and R. Gelman. (1990). Numerical abstraction by human infants. Cognition 36:97-128.
Wynn, K. (1992). Addition and subtraction by human infants. Nature 358:749-750.
Barkow, J. H., L. Cosmides, and J. Toob, Eds. (1992). The Adapted Mind: Evolutionary Psychology and the Generation of Culture. New York: Oxford University Press.
Crump, T. (1990). The Anthropology of Numbers. New York: Cambridge University Press.
Ginsburg, H. P., J. K. Posner, and R. L. Russell. (1981). The development of mental addition as a function of schooling and culture. Journal of Cross-cultural Psychology 12:163-178.
Hatano, G., S. Amaiwa, and K. Shimizu. (1987). Formation of a mental abacus for computation and its use as a memory device for digits: A developmental study. Developmental Psychology 23:832-838.
Lancy, D. F. (1983). Cross-Cultural Studies in Cognition and Mathematics. New York: Academic Press.
Miller, K. F., and J. W. Stigler. (1987). Counting in Chinese: Cultural variation in a basic cognitive skill. Cognitive Development 2:279-305.
Moore, D., J. Beneson, J. S. Reznick, P. Peterson, and J. Kagan. (1987). Effect of auditory numerical information on infants' looking behavior: Contradictory evidence. Developmental Psychology 23:665-670.
Nunes, T. (1992). Ethnomathematics and everyday cognition. In D. Grouws, Ed., Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan, pp. 557-574.
Reed, H. J., and J. Lave. (1981). Arithmetic as a tool for investigating relations between culture and cognition. In R. W. Casson, Ed., Language, Culture and Cognition: Anthropological Perspectives. New York: Macmillan, pp. 437-455.
Saxe, G. B., and J. K. Posner. (1983). The development of numerical cognition: Cross-cultural perspectives. In H. P. Ginsburg, Ed., The Development of Mathematical Thinking. Rochester, NY: Academic Press, pp. 291-317.
Song, M. J., and H. P. Ginsburg. (1987). The development of informal and formal mathematics thinking in Korean and U. S. children. Child Development 58:1286-1296.
Sophian, C., and N. Adams. (1987). Infants' understanding of numerical transformations. British Journal of Developmental Psychology 5:257-264.
Starkey, P., and R. G. Cooper, Jr. (1980). Perception of numbers by human infants. Science 210:1033-1035.
Stevenson, H. W., T. Parker, A. Wilkinson, B. Bonnevaux, and M. Gonzalez. (1978). Schooling, environment and cognitive development: A cross-cultural study. Monographs of the Society for Research in Child Development 43 (3, serial no. 175).
Strauss, M. S., and L. E. Curtis. (1981). Infant perception of numerosity. Child Development 52:1146-1152.
Copyright © 1999 Massachusetts Institute of Technology