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COPYRIGHT 2002 ERIC Clearinghouse on Elementary and Early Childhood Education
Abstract
The National Council of Teachers of Mathematics emphasizes that young children need play-based opportunities to develop and deepen their conceptual understanding of mathematics. From a social-constructivist perspective, learning is more likely to occur if adults or more-competent peers mediate children's learning experiences. Emphasizing both the developmental and the curricular perspectives, this article focuses on the role of the teacher in guiding preschool children's mathematical learning while they play with everyday materials. Professional growth in three areas was identified as critical in teachers' learning to guide young children's learning of mathematical concepts. First is the ability to recognize children's demonstrated understanding of mathematical concepts, second is the ability to use mathematical language to guide their progress from behavioral to representational understanding of mathematical concepts, and third is the ability to assess systematically children's understanding of mathematical concepts. Checklists tracing the development of three fundamental mathematical concepts--one-to-one correspondence, classification, and seriation--are suggested as tools for teachers to monitor preschool children's learning of mathematical concepts and plan appropriate learning experiences within children's zones of proximal development. Creating an environment that is mathematically empowering and mediating children's experiences in this environment establish the foundation for constructing, modifying, and integrating mathematical concepts in young children.
Introduction
Laura has just finished reading the story "Goldilocks and the Three Bears" to her preschool class. She announces that it is now time for free play. Four-year-old Rachel looks around the room for a while and walks over to the dramatic play/housekeeping center. Today this center is equipped with dolls, other soft toys, cups, plates, plastic silverware, plastic food items, a table, chairs, and some dress-up clothes. Rachel picks up an oversized shirt and slips her feet into "mummy shoes." She then brings out three stuffed bears of different sizes from the collection and places them around the table. As she seats the bears on three chairs, she mutters under her breath, "You are Papa Bear" (picking the largest bear), "you are the Mummy Bear" (picking the medium-sized bear), "and you are Baby Bear" (picking the smallest bear). Rachel then walks to the shelf and pulls out one plate and places it before Papa Bear; she walks back to the shelf to get a second plate and places it before Mama Bear; and then she makes one last trip to pick up a plate to place before Baby Bear. Next Rachel walks to the shelf and picks up a collection of spoons of different sizes. She is now joined by 5-year-old Tiffany, who tells her that the biggest bear needs the biggest spoon, the medium bear the medium spoon, and baby bear the smallest spoon. "Remember, like the bears story Ms. Laura read us." Rachel looks at Tiffany and then at the spoons, then randomly places a spoon before each bear. Tiffany immediately takes over and rearranges the spoons according to the size of the bears. Rachel watches for a few seconds and then walks away.
Although observing a play episode like this one would not be unusual in many preschool classrooms, it had a particularly strong impact on how Laura understood her students' mathematical knowledge. As a new member of the local chapter of the National Council of Teachers of Mathematics (NCTM), Laura became particularly interested in the development of mathematical concepts in her students. She realized that the most remarkable growth of mathematical knowledge occurs between the pre-kindergarten and grade 2 levels and that it was especially important at this stage to focus on guiding children's development of fundamental mathematical concepts. Yet the lack of an agreed-on math curriculum for preschool made it difficult for Laura to decide which concepts were the most appropriate for her preschool children. Like many other teachers, Laura struggled to make sense of the development of her students' mathematical learning and relate it to her instructional decisions (Franke & Kazemi, 2001). She wrote in her journal:
Teaching math has always been outside of my "comfort zone." Many commercial and teacher-made math games, including sets of animals, fruit, vehicles, shapes; board counting game; board classification games; and various spinners and large dice, are useful in reinforcing one-to-one correspondence, classification, and seriation. However, while used randomly and in isolation, these games may not help children fully grasp the math concepts they are built on. I have to go beyond providing some form of mathematical learning; I really need to have a well-thought-out math curriculum. I have tried math activities that I hoped would promote learning. I graphed with the children on a large mat. I had them each take off a shoe and decide by color where it should be placed. It was an activity that seemed to me that would be fun and hands-on, but the children were restless and bored. I set out small manipulatives with similar attributes and let the children explore and sort in bowls. I encouraged them to bring in leaf collections for the science table and discussed color and shape. Although the children were exploring the materials, I was challenged to find a way to assess what the children were learning and how to further develop their knowledge.
As is evident from this journal entry, Laura felt the need for a strong conceptual framework that would take into consideration the developmental characteristics of preschool children and would indicate environments that would foster children's natural mathematical abilities. Such a framework could help Laura to decide which mathematical concepts were appropriate for her students and the order in which they should be taught. Laura realized that these decisions needed to be based on her knowledge of the development of mathematical concepts and on an appropriate assessment of children's mathematical knowledge. She also realized that preschool programs needed to expand and deepen the conceptual knowledge that young children have already developed by 3 years of age (Payne, 1990). The NCTM's (2000) new standards emphasize that all preschool children need opportunities to explore their world and experience mathematics through play. Knowing that, however, left Laura with more questions than answers. She wrote in her journal:
How do I use play and play materials to enhance children's learning of primary math concepts? As a facilitator of learning, how can I engage the children in activities that would enable them to further construct mathematical concepts? What is the order in which math concepts develop? What are the primary math concepts and skills that preschool children need to develop in order to build a solid foundation for their later success in math in school? How do I ensure that I provide opportunities for each child as an individual to learn at his or her own rate? What kind of an ongoing assessment will be most helpful in planning developmentally appropriate math curriculum? How can I further expand the children's math knowledge and skills by improving my own practice and develop my knowledge in teaching mathematics?
Laura's observation of the play episode that involved Rachel and Tiffany helped her focus her work on the following specific questions:
* What mathematical concepts did Rachel and Tiffany exhibit during their play?
* How can I guide their learning so that their understanding of these concepts progresses to a higher level?
* Are other children in my class at the same stage as Rachel with regard to some of these concepts?
With these questions in mind, Laura began her master's project. Because we had a research interest in early learning of mathematical concepts, we became Laura's supervisors. At that time, our own research was at the stage of developing a series of teacher-friendly assessment tools that would facilitate curriculum planning in the content area of mathematics. This project was an exciting opportunity for Laura to deepen her understanding of young children's learning of mathematics. For us, Laura's project was as an opportunity to implement and document the use of these tools in a preschool classroom and to receive her feedback on their appropriateness and usefulness for an ongoing assessment of young children's development of primary math concepts. As Laura's supervisors, we were able to document, through observations and analysis of her journal entries, how her thinking about young children's learning of mathematical concepts developed, and how her understanding about the need to align curriculum, instruction, and assessment grew. In this article, we will focus on the main areas of growth in Laura's professional development that we believe could be helpful to other preschool teachers' growth.
Learning to Recognize Children's Demonstrated Understanding of Mathematical Concepts
The first and most important stage in Laura's professional growth was her enhanced ability to identify children's demonstrated understanding of mathematical concepts. Her observation of Rachel and Tiffany's reenactment of the story of "Goldilocks and the Three Bears" directed Laura's attention to the "impressive informal mathematical strengths " (Baroody, 2000, p. 61) that young children bring to the classroom. She saw that in this episode Rachel demonstrated her behavioral knowledge, that is, knowing how to enact procedures and roles and to implement several mathematical concepts (Katz & Chard, 2000). For example, choosing only the bears from a larger collection of dolls and plush toys demonstrated her behavioral knowledge of the mathematical concept of classification. Providing a plate for each bear and a bear for each chair demonstrated her knowledge of one-to-one correspondence; ordering the bears in size from biggest to smallest showed her behavioral knowledge of seriation. Tiffany also demonstrated her behavioral knowledge of double seriation by rearranging the spoons to correspond with the size of the bears after Rachel had placed the spoons randomly. More important, however, Tiffany demonstrated her ability to verbalize what needed to be done so that each bear received the appropriate size of spoon. Laura's raised awareness of the mathematical context of the interaction between the two children helped her to recognize the different stages they had reached in the development of their knowledge of seriation. She also became aware that young children express their mathematical knowledge in a variety of contexts that are not necessarily related to "math activities." As a result, she could plan individually appropriate learning experiences for them as well as joint experiences where they could learn from each other. She could also encourage informal mathematical learning by creating a math-rich environment and engaging children in mathematical conversations as they interact with the environment.
Learning to Use Language to Guide Children's Construction of Mathematical Concepts
The next stage of Laura's professional growth was marked by a change in her understanding of the role of teachers in preschool children's learning of mathematical concepts. Traditionally, the emphasis in preschool settings has been on how concepts are acquired, not on what should be taught. Kagan (quoted in Jacobson, 1998, p. 12) pointed out, "We've approached [early education] more from developmental perspectives and not from curricular perspectives. We need both."
The constructivist paradigm based on Piaget's theory of cognitive development has long provided the theoretical framework for educational practice in which children acquired concepts through active involvement with the environment and constructed their own knowledge as they explored their surroundings. Applying this theory to mathematics has led to the use of manipulative materials that enable young children to count, engage in active learning, and develop concepts (Kaplan, Yamamoto, & Ginsberg, 1989). The teacher has been seen to take the role of providing a variety of materials and arranging an environment that is rich in materials and choices. However, in the revised version of the principles of developmentally appropriate practice (Bredekamp & Copple, 1997), the National Association for the Education of Young Children (NAEYC) leaders acknowledged that the emphasis on providing a variety of choices in the classroom and avoiding teaching children specific skills has been misinterpreted. As a result, in preschool settings, manipulative materials were typically used in a nonsystematic way that permitted double randomization: one to do with the appearance of the manipulative material per se and the other determined by variations in the readiness of the children to register them (Feuerstein & Feuerstein, 1991). This randomization may have prevented real conceptual learning from occurring for a number of children who could have otherwise been included in planned activities for learning. Although high-quality learning in the preschool years is often informal, this informality does not imply an unplanned or unsystematic program. Mathematics learning in preschools should be thought provoking, should include opportunities for active learning, and should be rich in mathematical language. More recently, the NCTM's (2000) standards addressed the issue of mathematical content, mathematical process, and the importance of introducing young children to the language and conventions of mathematics.
Thus the role of the teacher in active learning has been seen more recently as being crucial. The teacher is the facilitator who creates a learning environment that is mathematically empowering (NCTM, 1991). The theoretical framework that informed this change was Vygotsky's (1978, 1986) social-constructivist theory of cognitive development. In this theory, learning is more likely to occur if adults or older children mediate young children's learning experiences (Baroody, 2000). Vygotsky believed in a learning continuum characterized by the distance between a child's ability to solve a problem independently and his or her "maximally assisted" problem-solving ability under adult or more-experienced peer guidance. He called this area where real learning occurs the "zone of proximal development" (ZPD). The role of the teacher, therefore, is to provide "scaffold assistance" (Berk & Winsler, 1995), which entails a continual modification of the tasks so as to provide the appropriate level of challenge that enables the child to learn. The adult changes the quality of the support over a teaching session, adjusting the assistance to fit the child's level of performance (Berk & Winsler, 1995). Children learn through meaningful, naturalistic, active learning experiences. The adult must build on this knowledge and take the children to higher levels of understanding.
Having embraced the Vygotskian view of learning, Laura began to realize she must decide what further opportunities--not only materials, but more important, interactions--she needed to provide for Rachel, Tiffany, and the rest of the children in her classroom. Only then could she develop and expand their understanding of mathematics meaningfully. She wrote in her journal:
I need to make the physical environment in my classroom more math rich. The furniture is child sized and easily adaptable to accommodate cooperative work. There is adequate and comfortable space on the partially carpeted floor to explore, construct, and work with concrete materials. Math materials and manipulatives are stored in clear bins on picture-labeled, open shelves and are in easy reach of the children. It is my intent now to increase the children's mathematical comprehension by assisting their construction of knowledge in one-to-one correspondence, classification, and seriation.
To guide children's learning of the concepts demonstrated during the free play episode, Laura began to see the need to become involved in a variety of situations that create a common language related to mathematics (Franke & Kazemi, 2001). For example, we were able to observe her daily discussions with children that involved comparisons of opposites during choice time. The children and the teacher talked about which blocks were bigger or smaller, which blocks fit into the shelves the best: small, medium, or large. They also made it a daily habit to discuss order: who was the first person in line, who was the second person in line, who was the last person in line or the caboose, the snack person. Language allows the acquisition of new information as well as the appropriation of complex ideas and processes (Bodrova & Leong, 1996). Open-ended questioning can encourage expanded thinking. "What else?" and "I wonder what would happen if" can draw children's attention to new ways of thinking and interacting. Kamii (1982) explains that it is important to allow children who are constructing their own mathematical knowledge to do so without the teacher reinforcing the "right-ness" or correcting the "wrong-ness" of the child's answer. Disagreement with peers can help the child reexamine the correctness of his or her own thinking. Social interactions through group games are an excellent source of constructing new mathematical...
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