Mathematics for Young Children Douglas H. Clements State University of New York at Buffalo A version of this paper has been published as: "Clements, D. H. (1999). Playing math with young children. Curriculum Administrator, 35(4), 2528." Note Time to prepare this material was partially provided by two National Science Foundation Research Grants, ESI9730804, "Building Blocks—Foundations for Mathematical Thinking, PreKindergarten to Grade 2: Researchbased Materials Development" and ESI9814218, "Planning for Professional Development in PreSchool Mathematics: Meeting the Challenge of Standards 2000." Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation. Mathematics for Young Children Is mathematics for young children just "getting ready" for school? Or is it one more pressure on children who will be deprived of their childhoods? Neither, if done well. Good early mathematics is broader and deeper than early practice on "school skills." Quality mathematics is a joy, not a pressure. It is the sum of the experiences children have from birth related to number, space, and patterns. It includes the parent placing cereal in a toddler’s hands, saying, "Here are two pieces. One, two!" It includes drawing a "treasure map" of the backyard. It includes noticing that two chants for skipping rope have the same pattern. Everything around us can be better understood with mathematics. Preschool is a good time for children to become interested in counting, sorting, building shapes, measuring, and estimating. Quality mathematics throughout early childhood (preschool to grade 2) does not involving pushing elementary arithmetic onto younger children. Instead, it allows children to experience mathematics as they play in and explore their world. To provide a clearer vision of quality early childhood mathematics, we start by asking several questions. Why do we need early childhood mathematics? Can very young children really "do" mathematics? Is it motivating for them? What types of mathematical experiences are most appropriate? Why Do We Need Early Childhood Mathematics? Should we provide mathematics education to children as young as preschoolers? In part, the question is moot. Many preschoolers are in school or other care settings. For example, the National Center for Educational Statistics 1995 surveys showed that 12.9 million infants, toddlers, and preschool children were receiving some type of care and education on a regular basis from persons other than their parents. Several states are instituting public preschool education. Unfortunately, most instruction that children receive does little to promote mathematics education, beyond rote counting and numeral recognition. This lack of attention to mathematics learning occurs despite the fact that many children, especially from minority and lowincome groups, later experience considerable difficulty in school mathematics. This deficit contrasts with research showing that these same children possess considerable competence in "informal mathematics" . In addition, gaps in mathematical knowledge between these and other children can be narrowed considerably by educational interventions (shown by work from the "Investigations in Number, Data, and Space" and the University of Chicago School Mathematics projects). Children who experience highquality early childhood education have a better chance of achieving to high levels than those who do not. So, early childhood mathematics education makes sense as social policy. Can Very Young Children Really "Do" Mathematics? Is it Motivating for Them? Still, some might argue that mathematics education for preschoolers seems rushed, almost unnatural. However, research shows that many mathematics concepts, at least in their intuitive beginnings, are developed before school. Infants spontaneously use the ability to recognize and discriminate small numbers of objects. For example, if three pictures showing 1, 2, or 3 dots are hung in front of a sixmonthold child and three drumbeats are sounded, the child will most likely focus on the picture with three dots! Before they enter school, many children develop early abilities in number and geometry, from accurate counting of objects to finding their way through their environment to making shapes. They use mathematical ideas in everyday life and develop informal mathematical knowledge that is surprisingly complex and sophisticated . With proper guidance, they can bring these ideas to an explicit level of awareness—crucial for mathematical understanding. Finally, young children like doing mathematics. For example, lowincome, inner city AfricanAmerican and Latino children exhibit a spontaneous interest in big mathematical ideas . They are selfmotivated to investigate patterns, shapes, measurement, and what numbers mean and how they work. There may be no better time to introduce substantial mathematics. What Types of Mathematical Experiences are Most Appropriate? Such introductions should be in harmony with the children’s ways of learning. Children’s interests and play should be the source of their first mathematical experiences. These experiences become mathematical as they are represented. Young children represent their ideas by talking, but also through models and graphics. From the motoric and singsong beginnings of "patacake" stem the geometric patterns of a "fence" built from unit blocks and the gradual generalization and abstraction of patterns throughout the child’s day ("See, my drawing uses an AABAAB pattern like your blocks did this morning!"). Especially in the years before first grade, quality learning is often incidental and informal. This does not mean unplanned or unsystematic. The child’s environment should encourage the elicitation of premathematical activities. Parents, teachers, and caregivers must be attuned to such beginning mathematics to help children become aware of their activity and how it connects to mathematics. This requires sensitive observation and thoughtful questioning at critical junctures ("Did you try this?") Observing children comparing the size of two rugs, a teacher may quietly ensure that connecting cubes, string, and other objects that might be used for measuring are close by. In such environments, young children can invent their own mathematical ideas and strategies. Alex, a fiveyearold girl whose brother, Paul, was age three, wandered into the room and announced: Alex: When Paul is six, I’ll be eight; when Paul is nine, I’ll be eleven; when Paul is twelve, I’ll be fourteen [she continues until Paul is 18 and she is 20]. Adult: My word! How on earth did you figure all that out? Alex: It’s easy. You just go "threeFOURfive" [saying the "four" very loudly, and clapping hands at the same time, so that the result was very strongly rhythmical, and had a softLOUDsoft pattern], you go "sixSEVEN [clap]eight," you go "nineTEN [clap!]eleven",…. . Alex had put together two things she knew about: counting and songs she sang rhythmically while jumping rope. This creation made sense to her…far more so than if an adult had tried to teach her an "add two algorithm." As schooling becomes more formal, children’s interests, ideas, and strategies should remain at the center of early childhood mathematics education. One study showed that primary grade students benefitted from using their own invented algorithms before standard algorithms. Compared to students who learned standard algorithms, students who first invented their own performed significantly better on assessments of baseten concepts and were more able to extend their knowledge to new situations, such as calculating change from $4.00 for $1.86 . Other studies show that the opposite approach—direct early teaching of traditional algorithms—can actually be harmful. Davis provides a final example of the power of children’s invention of mathematics. A third grade teacher in Weston, CN was teaching subtraction with borrowing. She wrote the following problem and said:
Kye invented a correct, creative, and very usable algorithm. What Resources Exist for Early childhood Mathematics? — New Developments New standards for early childhood mathematics. Given the need for new approaches to early childhood mathematics, it is heartening to know that there is a lot of work being done in the area. First, the National Council of Teachers of Mathematics (NCTM) is currently revising its curriculum Standards to include preschoolers (the author is on the PreKgrade 2 writing group). The draft includes many of the ideas discussed here, especially that mathematical experiences, "if appropriately connected to children’s worlds, challenge young children to explore ideas in more sophisticated and rich ways than previously believed possible. These ideas include number as well as quantitative relationships, shape, space, symmetry, and patterns." New curriculum development projects. The National Science Foundation (NSF) has recently funded several projects that are developing curriculum materials for early childhood mathematics. Our own project, "Building Blocks—Foundations for Mathematical Thinking, PreKindergarten to Grade 2: Researchbased Materials Development," illustrates these principles. Building Blocks is being developed at the State University of New York at Buffalo and Wayne State University (where Julie Sarama is the codirector). Our basic approach is finding the mathematics in, and developing mathematics from, children’s activity. We wish to help children extend and mathematize their everyday activities, from building blocks to art to songs to puzzles (this is the first meaning behind the Building Blocks name). Thus, we will design activities based on children’s experiences and interests, with an emphasis on supporting the development of mathematical activity. Mathematization emphasizes representing activity — creating models of activity with mathematical objects, such as numbers and shapes, and mathematical actions, such as counting or transforming shapes. Our materials will embody these actionsonobjects in a way that mirrors the theory and researchbased mathematical activity, or mental actions—children’s cognitive building blocks (e.g., creating, copying, uniting, and disembedding both units and composite units—the second meaning of the name). To accomplish this, we will be creating computer environments to supplement a wide range of offcomputer activities. Research has convinced us that computers can both be developmentally appropriate and mathematically powerful for young children . A simple illustration of providing experiences with mathematical actionsonobjects builds on young children’s experiences with and love of puzzles. Children will fill in puzzle outlines using an extended set of pattern blocks (a popular manipulative in which shapes have standard side lengths and angle measures). Different combinations are encouraged. On the computer, they may make a combination of 2 green triangles by gluing, then duplicate this unit to fill the outline. That is psychologically different from covering it with 20 separate triangles.
In these ways, the materials will develop basic mathematical building blocks—ways of knowing the world mathematically (and the third meaning of the name)—organized into two areas: (a) spatial and geometric competencies and concepts and (b) numeric and quantitative concepts, based on the considerable research in that domain. Developers Herbert P. Ginsburg, Robert Balfanz, and Carole Greenes are working on "Investigating the Big Ideas: A Mathematics Program for Preschool and Kindergarten Children for 4 to 5yearold children." At the core of Investigating the Big Ideas is a two year sequence of explorations and learning activities aimed at extending and elaborating on preschoolers’ informal mathematical interests and abilities, and developing their understanding of core mathematical concepts in an enjoyable and exciting way. The big ideas of mathematics will include what numbers mean, how numbers work ((e.g., properties such as onetoone correspondence of spoken numbers and objects and the notion of "one more" in the counting sequence), putting together and taking apart (e.g., adding and subtracting), parts of a whole, different ways to say the same thing (equivalence). patterns and combinations, measurement, and the nature of shapes and spatial relations. Beth Casey, Anne Goodrow, Michael Schiro, and Martha Bronson of Boston College are developing "Culturally Meaningful Adventure Stories: A medium for Teaching Kindergarten Geometry and Spatial Skills." This integrated approach addresses language arts as well as mathematics kindergarten competencies. The teachers will pose mathematics problems to the children that arise out of the adventures confronting multicultural characters within the stories. The kindergartners will solve the problems using manipulatives and other handson materials. The curriculum will be designed to help kindergartners draw on their "spatial sense" as well as their linguisticbased reasoning skills as a route to math learning across the curriculum. These nascent projects constitute the future of mathematics education at the early childhood level. Because the "2000" version of NCTM’s Standards will be the first to address mathematics education for preschoolers and to consider early childhood as a separate age group, these curricula will be the first to directly address the new standards. By collaborating with educators at this stage in the development to find the best way to effectively implement these programs, the developers hope to be able to make a significant impact on the future of mathematics education for young children. Planning for professional development. Finally, Julie Sarama and myself are working on a second NSFfunded project, "Planning for Professional Development in PreSchool Mathematics: Meeting the Challenge of Standards 2000." We are developing a plan for teacher enhancement and development for the diverse population of early childhood educators, especially those caring for and teaching disadvantaged students, historically underrepresented in mathematics. The goal will be substantive staff development supporting the implementation of standardsbased curricula and teaching approaches. We continue to look for administrators who wish to collaborate with us on achieving this goal. Final Words The most powerful mathematics for a preschooler is often not that acquired sitting down in a group lesson, but educed by the teacher from the children’s own selfdirected, intrinsicallymotivated activity. Young children can and should engage in mathematical thinking. All young children possess a substantive informal mathematics. Instruction should build upon and extend children’s daily activities, interests, and questions, bringing the mathematics in such activity to the fore. This approach ensures that mathematical content will be meaningful for very young children. Mathematics education for primary grade children should build on these early beginnings, mixing more formal approaches with a continuing emphasis on children’s interests and mathematical inventions. With such support, young children can and do think like mathematicians: making conjectures, solving problems, and looking for patterns in number and in space. References Author Information Douglas H. Clements, Professor of Mathematics and Computer Education at SUNY/Buffalo was a kindergarten teacher for five years. He has conducted research and published widely in the areas of the learning and teaching of geometry, computer applications in mathematics education, the early development of mathematical ideas, and the effects of social interactions on learning. He has codirected several NSF projects, producing Logo Geometry, several units and all the software in Investigations in Number, Data, and Space, and over 66 referred research articles, many from NSF (RTL) grants. Active in the NCTM, he is editor and author of the NCTM Addenda (to the Standards) materials and is on the preschool to grade 2 writing team for the revision of the Standards, NCTM’s Standards 2000 project. He was chair of the Editorial Panel of NCTM’s research journal, the Journal for Research in Mathematics Education. . In his current NSFfunded project, Building Blocks—Foundations for Mathematical Thinking, PreKindergarten to Grade 2: Researchbased Materials Development, he and Julie Sarama are developing mathematics software and activities for young children.
