Early Childhood Education

Mathematics

 

The turn of the century has seen a dramatic increase in attention to the mathematics education of young children, for at least five reasons. First, increasing numbers of children attend early care and education programs. Second, there is an increased recognition of the importance of mathematics for individuals and society. Third, there is a substantial knowledge gap in the mathematics performance of U.S. children living in economically deprived communities (Griffin, Case, and Siegler, 1994; Saxe, Guberman, and Gearhart, 1987). Fourth, researchers have changed from a position that young children have little or no knowledge of or capacity to learn, mathematics to one that acknowledges competencies that are either innate or develop by or before the pre-K years. Fifth and finally, research indicates that knowledge gaps appear in large part due to the lack of connection between children’s informal and intuitive knowledge and school mathematics, and especially due to the poor development of this informal knowledge in some children (Baroody and Wilkins, 1999; Ginsburg and Russell, 1981; Hiebert, 1986). As these reasons suggest, positions regarding young children and mathematics have changed considerably over the years. Early childhood professionals increasingly acknowledge that better mathematics education can and should begin early. Even preschoolers show a spontaneous interest in mathematics. Caring for them well, in any setting, involves nurturing and meeting their intellectual needs, which includes needs for mathematical activity.

Research on mathematics has played a central role in contributing to these changing attitudes and understandings. For example, very young children are sensitive to mathematical situations and thus all have the potential to become mathematically literate. Research has demonstrated that babies in the first six months of life can discriminate one object from two, and two objects from three (Antell and Keating, 1983). This was determined via a habituation paradigm in which infants “lose interest” in a series of displays that differ in some ways, but have the same number of objects. For example, say that infants are shown a sequence of pictures that contain a small set of objects, such as two circular regions. The collections differ in attributes such as size, density, brightness, or color, but there are always two objects. The differences between successive pictures initially keep infants’ attention—they continue to look at each picture in turn. Eventually, however, they habituate to the displays; for example, they begin to look at the screen less, and their eyes wander. Then they are shown a collection of three circular regions that are similar in attributes to those they had previously seen and their eyes focus intently on this new collection. Thus, the researchers know that they are sensitive in some way to number. This empirically established insight has convinced many that young children can engage with substantive mathematical ideas.

These changing understandings of young children and mathematics have not been without controversy, however. Over the course of the twentieth century, research on mathematics moved from a cautious assessment of the number competencies of children entering school, to a Piagetian position that young children were not capable of true numeric thinking, to the discovery of infant sensitivity to mathematical phenomenon, to the present debate about the meaning of these contradictions and an attempt to synthesize apparently opposing positions. The last phase includes a paradox: studies contradicted Piagetian positions on children’s lack of ability, but supported the basic constructivist Piagetian framework. That framework has been so influential that even substantive new theories were borne in reaction to it. For example, significant experiences are often those produced by the child’s own actions, including mental actions. Further, children can and do invent concepts and strategies, and, even when incorrect, exhibit intelligence. Their search for patterns is fundamentally mathematical in nature. As a specific example of the paradox, Jean Piaget was incorrect in claiming that early number knowledge was meaningless. Piaget was, however, accurate in describing children’s construction of logicomathematical knowledge that is increasingly general and that eventually compels children to make warranted generalizations resistant to confounding by distracting perceptual cues.

Recent debates are among three theoretical frameworks for understanding young children’s mathematical thinking: empiricism, (neo) nativism, and interactionalism. In traditional empiricism, the child is seen as a “blank slate,” truth lies in correspondences between children’s knowledge and reality, and knowledge is received by the learner via social transmission or abstracted from repeated experience with a separate ontological reality. An extension, traditional information processing theory, uses the computer as a metaphor for the mind and moves slightly toward an interactionalist perspective. In contrast, nativist theories, in the traditional of philosophical rationalism (e.g., Plato and Kant), emphasize the inborn, or early developing, capabilities of the child. For example, quantitative or spatial cognitive structures present in infancy support the development of later mathematics, and thus innate structures are fundamental to mathematical development. In this view, a small number of innate and/or early-developing mathematical competencies are privileged and easy to learn. These are hypothesized to have evolutionary significance and be acquired or displayed by children in diverse cultures at approximately the same age. Neither the empiricist nor nativist position fully explains children’s learning and development. An intermediate position appears warranted, such as interactionalist theories that recognize the interacting roles of nature and nurture. In interactionalist, constructivist theories, children actively and recursively create knowledge. Structure and content of this knowledge are intertwined and each structure constitutes the organization and components from which the child builds the next, more sophisticated, structure (Clements and Sarama, in press-a).

Research from these positions reveals a picture of young children who possess an informal knowledge of mathematics that is surprisingly broad, complex, and sophisticated (Kilpatrick, Swafford, and Findell, 2001). In both play and instructional situations, even preschoolers can engage in a significant level of mathematical activity. In free play, they explore patterns and shapes, compare magnitudes, and count objects. Less frequently, they explore dynamic changes, classify, and explore spatial relations. Importantly, this is true for children regardless of income level and gender (Seo and Ginsburg, 2004). In a similar vein, most entering kindergartners, and even entering preschoolers, show a surprising high entry level of mathematical skills. For example, most entering kindergartners can count past ten, compare or relate quantities, read numerals, recognize shapes, make patterns, and use nonstandard units of length to compare objects. As mentioned, these capabilities are well established by most entering kindergartners from middle- and high-income, but by a smaller proportion of children from low-income communities. Research has shown, however, that high-quality mathematics curricula (see Curriculum, Mathematics) can help children from low- resource communities develop mathematical concepts and skills (see Interagency Education Research Initiative [IERI], as well as Clements and Sarama, in press-b). Without intervention, many of these children later have trouble in mathematics and then school in general. With support, most primary grade children can construct surprisingly sophisticated and abstract concepts and strategies in each of these topical areas.

Number and operations is arguably the most important of the main concepts that should be developed in the early childhood years because (1) numbers can be used to tell us how many, describe order, and measure; they involve numerous relations, and can be represented in various ways; and (2) operations with numbers can be used to model a variety of real-world situations and to solve problems; they can be carried out in various ways.

Early numerical knowledge associated with these concepts has four interrelated aspects: instantly recognizing and naming how many items of a small configuration (“subitizing”; e.g., “That’s two crackers.”), learning the list of number words to at least ten, enumerating objects (i.e., saying number word in correspondence with objects), and understanding that the last number word said when counting refers to how many items have been counted. Children learn these four aspects initially by different kinds of experiences, but they gradually become more connected. Indeed, having children represent their quantitative concepts in different ways, such as with objects, spoken words, and numerals, and connecting those representations, are important aspects of all mathematics. Each of the four aspects begins with the smallest numbers and gradually includes larger numbers. Seeing how many, or subitizing, ends at three to five items and moves into decomposing/composing where small numbers are put together to see larger numbers as patterns. Like all mathematical knowledge, knowledge of number develops qualitatively. For example, as children’s ability to subitize grows from perceptual, to imagined, to numerical patterns, so too does their ability to count and operate on collections grows from perceptual (counting concrete objects), to imagined (with six hidden objects and two shown, saying, “Six ... seven, eight! Eight in all!”), to numerical (counting number words, as in “8 + 3? 9 is 1, 10 is 2, 11 is 3 ... 11!”).

Regarding operations, even toddlers notice the effects of increasing or decreasing small collections by one item. Children can solve problems such as six and two more as soon as they can accurately count. Children who cannot yet count-on often follow three steps: counting objects for the initial collection of six items, counting two more items, and then counting the items of the two collections together. Children develop, and eventually abbreviate, these solution methods. For example, when items are hidden from view, children may put up fingers sequentially while saying, “1, 2, 3, 4, 5, 6” and then continue on, putting up two more fingers, “7, 8. Eight.” Children who can count-on simply say, “S-i-x—7, 8. Eight.” At this point, children in many parts of the world learn to count up to the total to solve a subtraction situation because they realize that it is much easier. For example, the story “Eight apples on the table. The children ate five. How many now?” could be solved by thinking, “I took away 5 from those 8, so 6, 7, 8 (raising a finger with each count), that’s 3 more left in the 8.”

After they have developed these strategies, children can be encouraged to use strategic reasoning. For example, some children go on to invent recomposing and decomposing methods using doubles (6 + 7 is 6 + 6 = 12. 12 + 1 more = 13). Primary-grade children can extend such strategies to their work with large numbers and place-value concepts. For example, they might learn first to count by tens and ones to find the sum of 38 and 47, and later learn to decompose 38 into its tens and ones and 47 into its tens and ones. This encourages the children to reason with ten as a unit like the unit of one and compose the tens together into 7 tens, or 70. After composing the ones together into 15 ones, they have transformed the sum into the sum of 70 and 15. To find this sum, the children take a 10 from the 15 and give it to the 70, so the sum is 80 and 5 more, or 85. Strategies like this are modifications of counting strategies involving ten and one just like strategies for finding the sum of 8 and 7 are modifications of counting strategies involving only one (e.g., children who know that 8 and 2 are 10 take 2 from 7 and give it to 8. So, 10 and 5. 15). We know from studies of cognition in everyday life, including adults and children selling candy in the streets of Brazil, that such strategies can be invented in supportive cultures.

To develop computational methods that they understand, children benefit from experiences in kindergarten (or earlier), including hearing the pattern of repeating tens in the numbers words and relating these words to quantities grouped in ten. First graders can use quantities grouped in tens or make drawings of tens and ones to do two-digit addition with regrouping and discuss how, recording numerically their new ten: for example, 48 + 26 makes 6 tens (from 40 and 20) and 1 ten and 4 (from 8 + 6), so there is a total of 7 tens and 4 for 74. Children invent and learn from each other many effective methods for adding such numbers and many ways to record their methods. Second graders can go on to add 3-digit numbers by thinking of the groups of hundreds, tens, and ones involved. They can subtract (e.g., 82 — 59) by thinking of breaking apart 82 into 59 and another number. Computers can help provide linked representations of objects and numerals that are uniquely helpful in supporting this learning (see Curriculum, Technology). Although some teachers and critics worry that calculators will interfere with such learning, research results consistently reveal that—used wisely, to further problem-solving efforts and in combination with other methods—calculator use is not harmful and can be beneficial (Groves and Stacey, 1998).

Geometry, measurement, and spatial reasoning are also important, inherently, because they involve understanding the space in which children live. Two major concepts in geometry are that geometry can be used to understand and to represent the objects, directions, locations in our world, and the relationships between them; and that geometric shapes can be described, analyzed, transformed, and composed and decomposed into other shapes. Initial knowledge of these concepts is not beyond the cognitive capabilities of young children. Very young children know and use the shape of their environment in navigation activities. With guidance, they can learn to mathematize this knowledge. They can learn about direction, perspective, distance, symbolization, location, and coordinates. Some studies have identified the primary grades as a good time to introduce learning of simple maps, such as maps of objects in the classroom or routes around the school or playground, but informal experiences in prekindergarten and kindergarten are also beneficial, especially those that emphasize building imagery from physical movement. Again, computers can help “mathematize” these experiences.

Children can learn richer concepts about shape if their educational environment includes four features: varied examples and nonexamples, discussions about shapes and their characteristics, a wider variety of shape classes, and interesting tasks. All are important, because concepts of two-dimensional shapes begin forming in the prekindergarten years and stabilize as early as age 6. Therefore, children need rich opportunities to learn about geometric figures between 3 and 6 years of age. Curricula should develop these early concepts aggressively, so that by the end of grade three children can identify examples and nonexamples of a wide range of geometric figures; classify, describe, draw, and visualize shapes; and describe and compare shapes based on their attributes. Young children move through levels in the composition and decomposition of 2-D figures. From lack of competence in composing geometric shapes, children who are given appropriate experiences can gain abilities to combine shapes into pictures, then synthesize combinations of shapes into new shapes (composite shapes), eventually operating on and iterating those composite shapes. Helpful experiences include making pictures and solving puzzles with geometric shapes such as pattern blocks and tangram sets.

Measurement is one of the main real-world applications of mathematics. Measurement of continuous quantities involves assigning a number to attributes such as length, area, and weight. Together, number and measurement are components of quantitative reasoning. In this vein, measurement helps connect the two realms of number and geometry, each providing conceptual support to the other. Two main concepts in measurement are that comparing and measuring can be used to specify “how much” of an attribute (e.g., length) objects possess and that repeating a unit or using a tool can determine measures.

Prekindergarten children know that properties such as mass (amount), length, and weight exist, but they do not initially know how to reason about these attributes or to measure them accurately. At age 4-5 years, however, many children can, with opportunities to learn, become less dependent on perceptual cues and thus make progress in reasoning about or measuring quantities. This involves learning many concepts, including the following: the need for equal-size units; that a line segment made by joining two line segments has a length equal to the sum of the lengths of the joined segments; that a number can be assigned to a length; and that you may need to repeat, or iterate, a unit, and subdivide that unit, to find that number (to a given precision). By the end of the primary grades, children can learn relationship between units and the need for standard units, the relationship between the size and number of units, and the need for standardization of units.

Two other areas can be woven into the main three areas of number, geometry, and measurement: algebra and data analyses. Algebra begins with a search for patterns. Identifying patterns helps bring order, cohesion, and predictability to seemingly unorganized situations and allows one to recognize relationships and make generalizations beyond the information directly available. Although prekindergarten children engage in pattern-related activities and recognize patterns in their everyday environment, an abstract understanding of patterns develops gradually during the early childhood years. Children eventually learn to recognize the relationship between patterns with nonidentical objects or between different representations of the same pattern (e.g., between visual and motoric, or movement, patterns), identify the core unit (e.g., AB) that either repeats (ABABAB) or “grows” (ABAABAAAB), and then use it to generate both these types of patterns. In the primary grades, children can learn to think algebraically about arithmetic, for example, generalizing that when you add zero to a number the sum is always that number or when you add three numbers it does not matter which two you add first.

The beginning of data analysis, also accessible to young children, contains one main concept: Classifying, organizing, representing, and using information to ask and answer questions. Children can learn to classify and count to order data, then organize and display that data through both simple numerical summaries such as counts, tables, and tallies, and graphical displays, including picture graphs, line plots, and bar graphs. They can compare parts of the data, make statements about the data as a whole, and generally determine whether the graphs answer the questions posed initially. These sorts of activities can be generated in a variety of mathematics experiences as well as more integrated curriculum strategies such as the Project Approach or those based on the Reggio Emilia approach to long-term projects.

In summary, young children have the interests and ability to engage in significant mathematical thinking and learning, more so than is typically introduced in most educational or curriculum programs. Mathematical processes, such as reasoning, problem solving, and communicating, are also critical. Children, especially considering their minimal experience, are impressive mathematical problem solvers. They are learning to learn, and learning the rules of the “reasoning game.” Research on problem solving and reasoning also reveals surprising early abilities. Although the processes definitely improve, recent research claims appear valid: domain-specific knowledge is essential. However, what is then often neglected is the recognition that usually the reasoning from domain-specific knowledge simultaneously builds, and builds on, the basis of mindful general problem solving and reasoning abilities that are evident from the earliest years. See also Classroom Environment.

Further Readings: Antell, S. E., and D. P. Keating (1983). Perception of numerical invariance in neonates. Child Development 54, 695-701; Baroody, A. J., and J. L. M. Wilkins (1999). The development of informal counting, number, and arithmetic skills and concepts. In J. V. Copley, ed., Mathematics in the early years. Reston, VA: National Council of Teachers of Mathematics, pp. 48-65; Clements, D. H., and J. Sarama (in press-a). Early childhood mathematics learning. In F. K. Lester, Jr., ed., Second handbook of research on mathematics teaching and learning. New York: Information Age Publishing; Clements, D. H., and Sarama, J. (in press-b). Effects of a preschool mathematics curriculum: Summary research on the Building Blocks project. Journal for Research in Mathematics Education; Ginsburg, H. P., and R. L. Russell (1981). Social class and racial influences on early mathematical thinking. Monographs of the Society for Research in Child Development 46 (6, Serial No. 193); Griffin, S., R. Case and R. S. Siegler (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly, ed., Classroom lessons: Integrating cognitive theory and classroom practice. Cambridge, MA: MIT Press, pp. 25-49; Groves, S., and K. Stacey (1998). Calculators in primary mathematics: Exploring numbers before teaching algorithms. In L. J. Morrow and M. J. Kenney, eds., The teaching and learning of algorithms in school mathematics. Reston, VA: National Council of Teachers of Mathematics, pp. 120-129; Hiebert, J. C. (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum; Kilpatrick, J., J. Swafford and B. Findell (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press; Saxe, G. B., S. R. Guberman, and M. Gearhart (1987). Social processes in early number development. Monographs of the Society for Research in Child Development 52(2, Serial No. 216); Seo, K.-H., and H. P. Ginsburg (2004). What is developmental^ appropriate in early childhood mathematics education? In D. H. Clements, J. Sarama, and A.-M. DiBiase, eds., Engaging young children in mathematics: Standards for early childhood mathematics education. Mahwah, NJ: Lawrence Erlbaum Associates, pp. 91-104.

Douglas H. Clements and Julie Sarama