Early Childhood Education

Curriculum, Mathematics

 

Mathematics curricula for early childhood is an area of substantial recent research and development activity. For example, re-stimulated by research demonstrating that achievement gaps between children from low- and higher-resource communities begin in the earliest years, developers recently have produced a wide variety of innovative preschool curricula. Such flurries of activity may mislead some people to believe that early childhood mathematics is a new phenomenon. However, history shows that mathematics, as well as conflicts about the type of mathematical experiences that should be provided, have long histories in early education.

Conflicts stemmed from different opinions of the appropriateness of mathematics for young children. Negative opinions usually were based on broad social theories or trends, not observation or study of children. Those who actually worked with young children historically provided a rich mathematics curriculum. For example, mathematics was pervasive in the work of Friedrich Froebel, the founder of kindergarten (which originally included children from three to seven years of age). Froebel’s fundamental gifts were largely mathematical manipulatives and his occupations were mathematical explorations and constructions. As early childhood education was institutionalized, these deep mathematical ideas were largely forgotten or diluted. For example, in the first half of the twentieth century, U.S. psychologist Edward Thorndike wrote about learning as associations that were strengthened or weakened by consequences such as rewards. His implications for education was to do things directly. To emphasize health, he suggested replacing the first Froebelian gift (small spheres) with a toothbrush and the first occupation with “sleep.” The mathematical foundation of the gifts was thus ignored.

Froebel was a crystallographer. Almost every aspect of his kindergarten crystallized into mathematical forms—the “universal, perfect, alternative language of geometric form.” Its ultimate aim was to instill in children an understanding of what an earlier generation would have called “the music of the spheres”—the mathematically generated logic underlying the ebb and flow of creation. Froebel used “gifts” to teach children the geometric language of the universe, moving from solids (spheres, cylinders, cubes) to surfaces, lines, and points, then the moving back again. Cylinders, spheres, cubes, and other materials were arranged and moved to show these geometric relationships. His mathematically oriented occupations with such materials included explorations (e.g., spinning the solids in different orientations, showing how, for example, the spun cube can appear as a cylinder), puzzles, paper folding, and constructions. Structured activities would follow that provided exercises in basic number, arithmetic, and geometry, as well as the beginning of reading. For example, the cubes that children had made into the chairs and stoves would be made into a geometric design on the grid etched into every kindergarten table, and later laid into two rows of four each and expressed as “4 + 4.” In this way, connections were key: the “chair” became an aesthetic geometric design, which became part of a number sentence.

Consider several other examples. Children covered the faces of cubes with square tiles, and peeled them away to show parts, properties, and congruence. Many blocks and tiles were in carefully planned shapes that fit in the grid in different ways. “All the blocks and sticks and rings and slats were used in plain view on the ever-present grid of the kindergarten table, arranged and rearranged into shifting, kaleidoscopic patterns or decorative, geometric borders” (Brosterman, 1997, p. 38). Using these materials, Froebel developed skills that had been—and usually remain to this day—reserved for students in higher grades.

Another example of an historical curriculum material that emphasizes mathematics is the building block set. Children create forms and structures that are based on mathematical relationships. For example, children may struggle with length relationships in finishing a wall. Length and equivalence are involved in substituting two shorter blocks for one long block. Children also consider height, area, and volume. The inventor of today’s unit blocks, Caroline Pratt, tells of children making enough room for a toy horse to fit inside a stable. In Pratt’s example, the teacher told preschooler Diana that she could have the horse when she had made a stable for it. Diana and Elizabeth began to build a small construction, but the horse did not fit. Diana had made a large stable with a low roof. After several unsuccessful attempts to get the horse in, she removed the roof, added blocks to the walls to make the roof higher, and replaced the roof. She then tried to put into words what she had done: “Roof too small.” The teacher gave her new words, “high” and “low” and she gave a new explanation to the other children. Just building with blocks, children form important ideas. Teachers such as Diana’s who discuss these ideas with children, giving words to their actions, can foster such intuitive ideas through constructive play. With such materials and through teacher guidance, children can be helped to distinguish between different quantities such as height, area, and volume.

As part of K-12 schools, mathematics education in the primary grades has its own historical path. In the colonial times, counting and simple arithmetic was taught, but not usually to girls. Between 1815 and 1820, U.S. educators revised the teaching of arithmetic in response to the Swiss reformer Joseph Pestalozzi. Warren Colburn’s text, for example, started with practical examples, used objects (manipulatives) for solutions, and asked students to explain how they solved problems. However, many teachers failed to understand the reform efforts, and routine pedagogy remained common. At this time, “stimulus-response” theory, built on Thorndike, dominated psychology, and its effects were seen in the emphasis on drill procedures in arithmetic textbooks (reflecting a limited understand of even that limited theory).

From the 1920s, social utility theory influenced curricula to focus on those skills needed in everyday life. Emphasis was on practical use, but not on mathematics as a discipline or students’ understanding. In the 1930s, Gestalt theory, which focused on insight and relationships, led to recommendations by mathematics educators such as William Brownell that understanding of mathematics principles was a key foundation for learning.

Jean Piaget’s research led to a renewed focus on children’s thinking about mathematics. Mathematics curricula based on his theories took many forms. Some consisted almost solely on attempting to teach children to correctly respond to Piagetian tasks, such as number conservation, seriation, and classification. Others emphasized the constructivist philosophy of Piaget, and emphasized child- centered exploration. In a more recent extension of that approach, Kamii offers everyday experiences and games that encourage children to construct notions of number, and physical knowledge experiences such as bowling, balancing cubes, and pick-up sticks, for low-achieving young children before they experience any specific mathematical content. Evaluations of these approaches have been positive.

The constructivist theories of Piaget and Bruner motivated developers in the mid-twentieth century to incorporate “discovery learning,” emphasizing process goals and students’ exploration and invention of solution methods. The “space race” led to several different curriculum modifications, from those that emphasized the structure of mathematics itself (e.g., set theory in the “new math”) to those that build upon new psychological insights and reform movements to build new types of manipulatives (e.g., the “geoboard” and base ten blocks) and tasks for mathematics education. Thus, there were a variety of curricula through the 1960s and 1970s, although many shared at least some characteristics, such as increased emphasis on mathematical structures and precision, guided discovery approaches, and moving content to lower grade levels. “Laboratory” curriculum materials continued to be developed up to 1980, but excesses of some of these approaches led to some curricula following a “back to basics” approach.

Since that time, two main types of primary-grade curricula have been developed. The first type includes commercially published, traditional text books, which still dominate mathematics curriculum materials in U.S. classrooms and to a great extent determine teaching practices. Ginsberg and others claim that the most influential publishers are a few large conglomerates that often have profit as their main goal, leading them to follow state curriculum frameworks, attempting to meet every objective of every state—especially those that mandate adherence to their framework. They also tend to be eclectic in their teaching approaches. The second type of curricula includes those developed by researchers and innovators, often with external funding and frequently attempting to follow the reform-oriented positions of the National Council of Teachers of Mathematics (see www.nctm.org for this and recent recommendations for Curriculum Focal Points). The resulting innovative primary-grade curricula often provide educational experiences that are simultaneously more child-centered and more challenging. Building on children’s mathematical intuitions and problem-solving ability, these curricula ask children to develop their own ideas and strategies, and guide that development toward increasing levels of mathematical sophistication. The curricula develop skills in conjunction with learning the corresponding concepts, because research indicates that learning skills before developing understanding can lead to learning difficulties. Successful innovative curricula and teaching build directly on students’ thinking (the understandings and skills they possess), provide opportunities for both invention and practice, and ask children to explain their various strategies. Such programs facilitate conceptual growth and higher-order thinking without sacrificing the learning of skills. They also pose a broader and deeper range of problems in arithmetic and geometry than traditional curricula. However, they also require a more knowledgeable teacher, and are, perhaps, more vulnerable to misconceptions and therefore misuse, such as believing that accuracy is unimportant. Traditional curricula, which still are used in a majority of schools, have been offering more problem-solving opportunities for students in recent years, but often do not reflect all that is known about teaching and learning early mathematics.

Not traditionally part of the elementary school curriculum, preschool mathematics curricula have followed a different, but related course. Originally based on traditions from Froebel, traditional early childhood practices, and then Piaget, curriculum development has recently been influenced by newer theories that put number in a foundational role. Such curricula have shown substantial positive effects. For example, in one study, four-year-old children were randomly assigned to one of three educational conditions for eight weeks: Piagetian logical foundations (classification and seriation), number (counting), and control. The logical foundations group significantly outperformed the control group both on measures of conservation and on number concepts and skills. However, inconsistent with Piagetian theory, the number group also performed significantly better than the control group on classification, multiple classification, and seriation tasks as well as on a wide variety of number tasks. Further, there was no significant difference between the experimental groups on the logical operations test and the number group significantly outperformed the logical foundations group on the number test. Thus, the transfer effect from number to classification and seriation was stronger than the reverse. The areas of classes, series, and number appear to be interdependent but experiences in number have priority.

Recent curriculum development and research in preschool mathematics education has built on these beginnings, as well as the wealth of research on young children’s learning of mathematics. For example, contemporary curricula emphasize number, geometry, and to a lesser extent, measurement and patterning, because research shows that young children are endowed with intuitive and informal capabilities in these areas and because these areas form the foundation of later mathematical learning. These curricula have helped children make strong, significant gains in each of these various areas of mathematics in their preschool year. Thus, most recently developed research-based preschool curriculum is based on the notion that children have more capability and interest in mathematical activities than often assumed. They consider children to be active builders of mathematics rather than passive receivers of facts and procedures (see Constructivism). They ask children to solve mathematical problems, albeit beginning problems, with understanding and talk about what they have done.

Equity has often been a driving force in creating and studying preschool mathematics curricula (see also Technology Curriculum). Research indicates that children from low-resources communities who experience a high-quality mathematics curriculum can learn basic mathematical ideas and skills. For example, they learn the number skills, including number recognition, counting, comparison, and simple arithmetic. This closes the gap between children from low-resource and those from higher-resource communities. The development of geometry and spatial sense are also important. Research on the Agam and building blocks curricula show that rich geometric and spatial activities have multiple benefits. Such activities include finding shapes in the environment, from more obvious examples to embedded shapes; reproducing designs with shapes; composing shapes to make pictures, designs, and other shapes; and forming mental images of shapes. Such curricula increase children’s knowledge of geometry and spatial skills, including foundations of the visual arts. In addition, they increase children’s arithmetic and writing readiness capabilities.

Some of these curricula are more structured than others. Some use whole- group instruction, others small-group instruction, often with games. Most have been successful, if performed in high-quality settings. All approaches have a shared core of concern for children’s interest and engagement and content matched to children’s cognitive level. Young children benefit from a range of mathematical experiences, from the incidental and informal to the systematic and planned. However, a core of intentional, systematic activities appears to hold particular promise, making unique contributions to children’s development.

The ecological perspective suggests that many aspects of the child’s environment affect the success of a curriculum. The ecological factor that has most often been identified as influential involves the role of the teacher. Professional development on early mathematics curricula is consistently identified as the main criterion of a high-quality implementation, along with other support for the teacher (see Interagency Education Research Initiative [IERI]). Early childhood teachers often lack experiences that develop deep knowledge of the mathematics taught, knowledge of the specific developmental paths of children’s learning of that mathematics, and innovative ways of helping children learn mathematics. Indeed, especially for a mainly female group, mathematics is often avoided and viewed as difficult and distasteful. Such knowledge is enhanced when curricula are built around understanding children’s development of mathematical ideas and strategies. Family involvement and a classroom environment filled with potential for mathematic explorations are also components of most early mathematics curricula.

In summary, there is a long history of worthwhile mathematics curricula for early childhood, from the preschool years through the primary grades. Achievement gaps between children from low- and higher-resource communities, which begin in the earliest years, lend urgency to building on historical and recent development and research efforts to provide high-quality implementation of innovative curricula to all children.

Further Readings: Balfanz, R. (1999). Why do we teach young children so little mathematics? Some historical considerations. InJ. V. Copley, ed. Mathematics in the early years. Reston, VA: National Council of Teachers of Mathematics, pp. 3-10; Brosterman, N. (1997). Inventing kindergarten. New York: Harry N. Abrams; Carpenter, T. P., E. H. Fennema, M. L. Franke, L. Levi, and S. B. Empson (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann; Clements, D. H., and J. Sarama (2007). Effects of a preschool mathematics curriculum: Summary research on the Building Blocks project. Journal for Research in Mathematics Education. Clements, D. H., J. Sarama, and A.-M. DiBiase (2004). Engaging young children in mathematics: Standards for early childhood mathematics education. Mahwah, NJ: Erlbaum; Cobb, P., T. Wood, E., Yackel, J. Nicholls, G. Wheatley, B. Trigatti, et al. (1991). Assessment of a problem- centered second-grade mathematics project. Journal for Research in Mathematics Education 22(1), 3-29; Ginsburg, H. P., A. Klein, and P. Starkey (1998). The development of children’s mathematical thinking: Connecting research with practice. In W. Damon, I. E. Sigel, and K. A. Renninger, eds. Handbook of child psychology, Vol. 4: Child psychology in practice. New York: Wiley, pp. 401-476; Gravemeijer, K. P. E. (1994). Developing realistic mathematics instruction. Utrecht, The Netherlands: Freudenthal Institute; Griffin, S., and R. Case (1997). Re-thinking the primary school math curriculum: An approach based on cognitive science. Issues in Education 3(1), 1-49; Hiebert, J. C. (1999). Relationships between research and the NCTM Standards. Journal for Research in Mathematics Education 30, 3-19; Kamii, C. K., J. Rummelsburg, and A. R. Kari (2005). Teaching arithmetic to low-performing, low-SES first graders. Journal of Mathematical Behavior 24, 39-50; Mokros, J. R. (2003). Learning to reason numerically: The impact of Investigations. In S. Senk, ed. Standards-based school mathematics curricula. What are they? What do students learn? Mahwah, NJ: Erlbaum, pp. 109-132; Piaget, J., and A. Szeminska (1952). The child’s conception of number. London: Routledge and Kegan Paul.

Douglas H. Clements and Julie Sarama