Helping Early Childhood Educators to Teach Mathematics

Excerpted from Chapter 7 of Critical Issues in Early Childhood Professional Development, edited by Martha Zaslow, Ph.D., & Ivelisse Martinez-Beck, Ph.D.

Copyright © 2006 by Paul H. Brookes Publishing Co. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

This chapter is about helping prospective and current child care providers, teachers, and other early childhood education professionals teach mathematics effectively to young children at the preschool and kindergarten levels. Many education agencies across the country are faced with a mandate to implement programs of early childhood mathematics education (ECME). Teaching mathematics to 3-, 4-, and 5-year-olds is a challenge requiring considerable knowledge and skill. Yet few early childhood professionals have had the benefit of sound preparation in ECME. The United States therefore faces a pressing need to develop programs that train prospective and practicing early childhood professionals to teach mathematics effectively. This chapter describes two such programs: one involving a set of higher education courses in ECME and the other using in-service workshops that revolve around a specific mathematics curriculum. Drawing on observations of how an early mathematics program was implemented in inner-city settings, the chapter concludes by describing some ways in which the educational system can be structured to support teachers' professional development.

THE NEED FOR PREPARATION IN EARLY MATHEMATICS EDUCATION

American children's mathematics performance is weaker than it should be. Children from China, Japan, and Korea outperform their American counterparts in mathematics achievement, perhaps as early as kindergarten (Stevenson, Lee, & Stigler, 1986), and certainly during the early school years (Mullis et al., 1997) and beyond (Mullis et al., 2000). Within the United States, children from low income backgrounds — a group comprised of a disproportionate number of African Americans and Latinos (National Center for Children in Poverty, 1996) — show lower average levels of academic achievement than do their peers from middle- and upper-income backgrounds (Denton & West, 2002).

Although there is no single remedy for this unfortunate situation, a strong foundation in preschool education, including mathematics, can help to promote learning in the later years (Bowman, Donovan, & Burns, 2001; Peisner-Feinberg et al., 2001). However, ECME is not only preparation for schooling. Learning mathematics is developmentally appropriate and can be enjoyable for 3-, 4-, and 5-year-olds: They already possess a surprisingly competent informal mathematics (Ginsburg, Klein, & Starkey, 1998), they spontaneously engage in everyday mathematical activities (Seo & Ginsburg, 2004), and they are ready to learn complex mathematical ideas (Greenes, 1999).

In response to the educational need and opportunity, many states and education agencies have decided to introduce programs of early childhood education. By the beginning of the new century, Texas and Illinois began to expand preschool programs, particularly for children considered at risk. Georgia, New York, and Oklahoma adopted a policy of universal preschool education. Furthermore, educators have come to recognize that mathematics needs to occupy a central place in early childhood education. Thus, in New Jersey, many preschools and child care centers serving children from low income backgrounds are faced with a mandate to teach mathematics as well as literacy.

This explicit emphasis on education places a heavy burden on early childhood professionals. In addition to fulfilling all of their other responsibilities, they must now become proficient in teaching new programs, both in literacy as well as mathematics, and all for very low pay. Yet through no fault of their own, few early childhood professionals have had the benefit of sound preparation for ECME. One reason is that child care providers typically receive minimal college education, let alone a course in ECME. A second reason is that although most teacher training institutions offer their students many courses in literacy, most require only one course in mathematics education. Moreover, this "math methods" course typically does not focus on ECME, perhaps because few, if any, mathematics programs for which preschool teachers needed to be trained previously existed.

The nation's laudable efforts to introduce programs of early education on a wide scale, particularly for children from low-income backgrounds, will not succeed unless child care providers and preschool educators receive sound training in ECME. To do this, however, the widespread confusion about what ECME entails must be clarified.

WHAT IS MATHEMATICS EDUCATION FOR YOUNG CHILDREN?

Since the early part of this century, early childhood professionals have been rethinking their approach to ECME (Clements, Sarama, & DiBiase, 2004). The current view, echoing a position that was influential in earlier periods of our history (Balfanz, 1999), proposes that teaching mathematics to young children is both developmentally appropriate and desirable. The joint position statement of the National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM) asserts that "high-quality, challenging, and accessible mathematics education for 3- to 6-year-old children is a vital foundation for future mathematics learning" (2002, p. 1). High-quality ECME has several basic characteristics.

Early Mathematics Is Both Broad and Deep

The content of ECME encompasses a wide variety of topics, including number, shape, measurement, pattern, logic, operations, and spatial relations. In turn, each entails several interesting subtopics. The topic number is sometimes unfortunately called "numeracy." Number covers matters such as the counting words (e.g., "One, two, three . . ."), the ordinal positions (e.g., "First, second, third . . ."), and the idea of cardinal value (e.g., "How many are there?"). Shape includes not only simple plane figures (e.g., circle, triangle) but also hexagons and octagons (if young children can say and understand "brontosaurus," they can do the same for "octagon"), solids (e.g., cubes, cylinders), and symmetries in two and three dimensions. Spatial relations includes ideas such as position (e.g., in front of, behind), navigation (e.g., "First go three steps to the left"), and mapping (e.g., creating a schematic representing the location of objects in the classroom). Early mathematics is broad in scope and must deal with more than the topic of number and simple shapes.

Early mathematics is also deep. Consider for example the topic of enumeration — that is, counting a set to determine its number. A child sees on his or her right a haphazard arrangement including a red block, a small stuffed dog, and a penny. How many items are in this set? To answer correctly, the child must appreciate several basic ideas. The first is that different objects can all be counted as part of one set. One can count blocks, dogs, and pennies or big things and small things. One can even count fantastical images like five red unicorns. A second idea is that the number words (i.e., "One, two, three . . .") must be associated once and only once with each object in the set. The red block is "one," the dog is "two"; it is impossible to say both "one" and "two" when referring to the red block. The third idea is that the final number in the sequence (i.e., "three") does not refer to the penny alone. Even though one might say "three" while pointing to the penny, this number word describes not that solitary object but instead the cardinal value of the group as a whole. One could have started the count with the penny, in which case it would have been "one," not "three." Therefore, any object in the group could have been "one," "two," or "three," but the group as a whole has three objects — regardless of the order in which they are counted.

The breadth and depth of early mathematics represents a challenge and an opportunity for those who teach it. One aspect of the challenge is that the teacher must have a deep understanding of the fundamental ideas of number, shape, pattern, logic, and so forth (Ma, 1999; Shulman, 1987). Teaching early mathematics is more than imparting a few rote skills or trivial ideas. There is an opportunity to build on children's deep interest in these ideas.

Children Have an Interest in and Knowledge of Early Mathematics

For many years, early childhood educators' views were heavily influenced by an interpretation of Piaget's (1952) theory suggesting that young children's cognitive immaturity prevents them from learning mathematics. As a result, it was decided that there is no point in attempting to "push development" by providing specific instruction in abstract mathematics concepts. Rather, because Piaget emphasized that children learn from their own actions, developmentally appropriate practice was seen as giving young children the opportunity to play and freely explore their surroundings. This general approach was adapted by NAEYC's 1986 standards (Bredekamp, 1987) and still seems to dominate the thinking of many early childhood professionals.

The NAEYC has since changed its position (Bredekamp & Copple, 1997), partly on the basis of new psychological research showing that young children develop an "everyday" or "informal" mathematics with several important characteristics. One is that young children have a spontaneous interest in mathematical ideas. Naturalistic research has shown, for example, that young children enjoy saying the counting words up to relatively large numbers, such as 100 (Irwin & Burgham, 1992), and even want to know what the "largest number" is (Gelman, 1980). Also, during free play, young children spend a good deal of time determining which tower is higher than another, creating and extending interesting patterns with blocks, exploring shapes, creating symmetries, and so forth (Seo & Ginsburg, 2004). Much of this activity is spontaneous, occurring without adult guidance. Indeed, adults are often unaware of the mathematics involved when children do these things.

The second point is that young children are competent in a wider range of mathematical abilities (Ginsburg, Klein, & Starkey, 1998) than Piaget's (1952) theory might lead one to believe. From an early age, children seem to understand basic ideas of addition and subtraction (Brush, 1978), ratios (Hunting, 1999), and spatial relations (Clements, 1999). They can spontaneously develop various methods of calculation (Groen & Resnick, 1977), such as counting on from the larger number (Baroody & Wilkins, 1999).

A third point is that when given instruction, young children are ready to learn some rather complex mathematics. Thus, children can be taught — and enjoy learning — interesting aspects of addition (Zur & Gelman, 2004) and symmetry (Zvonkin, 1992), among other topics (Greenes, 1999). In brief, young children are eager mathematicians. Although their thought is in some respects limited and different from that of adults, young children deal with mathematical ideas in everyday play, are curious about the subject, know something about it, and can learn complex mathematics when it is taught.

Adult Guidance

Free play is not enough to promote early mathematical thinking. Children do indeed learn some mathematics from free play, but it does not afford the extensive and explicit examination of mathematical ideas that only adult guidance can provide. Free play can provide a useful foundation for learning, but a foundation is only an opportunity for building a structure. Of course, young children should engage in a great deal of free play: In addition to being enjoyable, it promotes learning. At the same time, adult guidance is necessary to build a structure on the foundation of children's everyday mathematics. As Dewey (1976, p. 281) put it, "Guidance is not external imposition. It is freeing the life-process for its own most adequate fulfillment."

The question is not whether adults should guide the children but how. The NAEYC/NCTM (2002) position statement suggests that adults should deliberately introduce mathematical concepts, methods, and language and should help children examine mathematical concepts in depth. All of this must be done in ways that are developmentally appropriate — for example, by building on children's spontaneous mathematical interests and informal knowledge. Clearly, developmentally appropriate teaching should not involve extensive use of written materials or other methods used at higher grades, should not entail a "push-down" curriculum (i.e., a curriculum originally designed for older children), and should not continually engage the children in instruction. At the same time, the adult needs to play an active role in teaching early mathematics to individuals, small groups, and even entire classrooms.