
Early Childhood Numeracy
Early number concept: Finding the springboard for further learningFrom birth children are exposed to fundamental informal mathematics through interaction with their surrounding social environment. As a natural consequence, it is not unusual for very young children to develop basic implicit notions of such concepts as number (i.e., more or less) and space or location (i.e., here or there). Regardless of culture, children's exposure to rich counting systems is virtually assured (Lave, 1988; Rogoff, 1990). Through interaction, most children will learn the counting words. Though these understandings are largely languageembedded, they nonetheless begin to expose children to certain principles that are essential precursors to further formal or operational understanding of mathematics concepts (Gelman & Gallistel, 1978). This exposure varies, however, as does the degree to which children progress. Study findings suggest wide variance in the level of natural development of intuitive number understanding children attain during their early preschool years (Case, 1985; Case, Griffin & Kelly, 1999; Griffin & Case, 1996, 1998; Griffin, Case, & Capodilupo, 1995; Griffin, Case, & Siegler, 1994; Hiebert, 1986; Siegler & Robinson, 1982). Still, analyses of findings reveal several general characteristics of preschool learning experiences that tend to produce the children who are most prepared for the elementary curriculum to come. In general, experiences that are most successful for helping preschool children form strong initial concepts of number and operations are:
The first two have to do with style—they are overarching considerations for how we put our plans into action for promoting children's learning. The third more closely relates to the learning plan itself—the instructional design we choose to employ. We'll first look at the fundamental principles that transcend understanding of any mathematical concept, and then address potential learning trajectories that are consistent with and lead to development of the mathematical understandings considered most essential for ensuring children's mathematical success upon entering math programs at the kindergarten through grade 2 level. Precursors to formal understandings of number concept and operationsThe elementary learning expectations set forth in Principles and Standards for School Mathematics (NCTM, 2000) place importance on the "Number and Operations" standard (along with "Geometry"). We follow by asking: 1. What can young children do that is sound both mathematically and developmentally? 2. What experiences can we provide to prepare them to meet future expectations? In exploring the research that informs these questions, we find that a solution to the first provides insight into the structure of a solution to the second. For example, in the context of early elementary objectives such as simple addition and subtraction, we of course do not often find a child less than four years old who can add or subtract numbers that are several digits removed from one another. However, it is not sufficient for us to say, as many would superficially interpret Piaget (Piaget & Szeminska, 1952), that prior to reaching an operational thinking stage of thinking—around age seven or eight—children cannot possibly begin to understand the operations of addition and subtraction. Rather more informative is that upon close analysis of conceptual shortcomings of school age children who do not understand such operations, we find an absence of certain preformal conceptions that are essential to formal understanding of addition and subtraction. Some of these conceptions—connecting one distinct number to each distinct object during counting, for instance—are easily within grasp of preschool children. Determining what conceptual understandings are appropriate for preschool children, when and in what order they are most effectively addressed, and what broader formal understandings they will eventually support, helps us to formulate a reasonable learning trajectory that will better prepare young children for the elementary mathematics standards and curriculum to come. As you review the following precursors, you will note that they are interrelated, and in some manner all apply to each trajectory discussed in the next section.
Sequencing children's experiences with number and operationsBy reviewing both the elementary expectations for learning and what research tells us about when children are developmentally able to understand certain concepts, we can construct a sequence—or trajectory—of number and operations expectations through the preschool years. Consider the following early elementary number concept expectations (NCTM, 2000) related to understanding of number quantity. Students will:
Consider also the early elementary expectations related to counting—that students will:
The trajectories in Figure 1 below each represent a potential preschool continuum of learning expectations. The first two illustrate children's progress toward the elementary expectations for understanding of number quantity and counting, repectively. The third represents an introductory preschool continuum of learning for addition and subtraction of whole numbers.
Though research generally indicates that most children are able to reach these levels of understanding at or near the ages indicated, it is noteworthy that individual differences in young children are as pronounced as for the rest of us—the most pronounced consistency is often the lack of consistency. It is also important to expect and to understand the contingencies that exist among continua—children typically must form certain understandings and abilities in one area before they are able to effectively progress to more advanced levels in other areas. For instance, recent research findings have indicated that a child's ability to count is instrumental in supporting a wide range of preschool number conceptions and the ability to perform singledigit addition operations. For more information and researchbased strategies for helping young children learn to count, read the article Count With Me! Likewise, the key to a child's progress in ability to perform whole number operations lies in the strategies that he employs. At very young ages, children should begin to progress through increasingly complex cycles for solving simple problems involving operations. Information is provided, along with a separate trajectory toward attainment of elementary standards and expectations, in our article Add With Me! Regardless of the differences that exist among young children, we nevertheless find a constant theme—through attentiveness to sequencing expectations, and by adjusting learning exercises accordingly, we can help promote children's continual growth in understanding by knowing where they should ultimately be heading and by staying at the edge of their level of understanding at any point in time. Case, R. (1985). Intellectual development: Birth to adulthood. New York: Academic Press. Case, R., Griffin, S., & Kelly, W. (1999). Socioeconomic gradients in mathematical ability and their responsiveness to intervention during early childhood. In D. Keating & C. Hertzman (Eds.), Developmental health and the wealth of nations: Social, biological, and educational dynamics (pp. 125149). New York: Guilford Press. Gelman, R., & Gallistel, C. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press. Griffin, S., & Case, R. (1996). Evaluating the breadth and depth of training effects when central conceptual structures are taught. Society for Research in Child Development Monographs 59, 90113. Griffin, S., & Case, R. (1998). Rethinking the primary school math curriculum: An approach based on cognitive science. Issues in Education, 4(1), 151. Griffin, S., Case, R., & Capodilupo, A. (1995). Teaching for understanding: The importance of central conceptual structures in the elementary mathematics curriculum. In A. McKeough, I. Lupert, & A. Marini (Eds.), Teaching for transfer: Fostering generalization in learning (pp. 121151). Hillsdale, NJ: Erlbaum. Griffin, S., Case, R., & Siegler, R. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students atrisk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 2449). Cambridge, MA: Bradford Books MIT Press. Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics. Hilldale, NJ: Erlbaum. Lave, J. (1988). Cognition in practice: Mind, mathematics, and culture in everyday life. Cambridge, MA: Cambridge University Press. Lay, M., & Dopyera, J. (1977). Becoming a teacher of young children. Lexington: D.C. Heath and Company. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. National Research Council. (1999a). How people learn: Brain, mind, experience, and school. In J. Bransford, A. Brown, & R. Cocking (Eds.). Washington, DC: National Academy Press. National Research Council. (1999b). How people learn: Bridging research and practice. In M. Donovan, J. Bransford, & J. Pellegrino (Eds.). Washington, DC: National Academy Press. Piaget, J., & Szeminska, A. (1952). Child's conception of number. London: Routledge & Kegan Paul. Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in social context. New York: Oxford University Press. Siegler, R.S., & Robinson, M. (1982). The development of numerical understandings. In H. Reese & L. Lipsitt (Eds.), Advances in child development and behavior (Vol. 16, pp. 242312). New York: Academic Press.
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