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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity 7 Standards, Curriculum, Instruction, and Assessment In this chapter, we address the topic of effective mathematics curriculum and teaching—what is known about how teachers can effectively support children’s learning of important foundational mathematics content. We begin the chapter with a description and analysis of current state standards for early learning. Standards are intended to influence the development of curriculum and assessment tools, and therefore they have the potential to serve as a bridge between what research says about children’s learning and the kinds of teaching and learning that actually occur. Next, the chapter provides an overview about the state of mathematics teaching and learning experiences in early childhood settings and reviews the literature on effective practices for teaching young children mathematics. Following this is a discussion of formative assessment, an essential and often overlooked element of effective instruction. The chapter concludes with a discussion of research on effective curricula. DEFINITIONS To enhance understanding of the content of this chapter, we first define some of the most frequently used early childhood education terminology. Teacher-Initiated and Child-Initiated Experiences Early childhood practices are often described as either teacher-initiated or child-initiated. Teacher-initiated or teacher-guided means that teachers plan and implement experiences in which they provide explicit information,
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity model or demonstrate skills, and use other teaching strategies in which they take the lead. Teacher-initiated learning experiences are determined by the teacher’s goals and direction, but they should also reflect children’s active engagement (Epstein, 2007). Ideally, teacher-initiated instruction actively involves children. Indeed, when appropriately supportive and focused, teacher-initiated instruction can lead to significant learning gains (French and Song, 1998; Howes et al., 2008). In practice, however, most teacher-initiated instruction is associated with the passive engagement of children (Pianta et al., 2005). By contrast, child-initiated or child-guided means that children acquire knowledge and skills through their own exploration and through interactions with objects and with peers (Epstein, 2007, p. 2). Child-initiated experience emanates primarily from children’s interests and actions with support from teachers. For child-initiated learning to occur, teachers organize the environment and materials and provide the learning opportunities from which children make choices (Epstein, 2007). Teachers thoughtfully observe children during child-initiated activity, gauging their interactions and the provision of new materials, as well as reorganization of the environment, to support their continued learning and development. During optimal child-initiated experience, teachers are not passive, nor are children entirely in control—although this ideal is not always realized in practice. For example, classroom observational research reveals that teachers tend to spend little time with children during free play (Seo and Ginsburg, 2004), or they focus their interactions on behavior management rather than on helping children learn (Dickinson and Tabors, 2001; Kontos, 1999). Instruction and Intentional Teaching In early childhood education, the term instruction is most often used to mean “direct instruction,” implying that teachers are entirely in control and children are passive recipients of information. The term is also used pejoratively to refer to drill and practice on isolated skills. Direct instruction is more accurately defined as situations in which teachers give information or present mathematics content directly to children. The National Mathematics Advisory Panel (2008) uses the term explicit instruction to refer to the many ways that teachers can intentionally structure children’s experiences so that they support learning in mathematics. Throughout the day and across various contexts—whole group, small group, centers, play, and routines—teachers need to be active and draw on a repertoire of effective teaching strategies. This skill in adapting teaching to the content, type of learning experience, and individual child with a clear learning target as a goal is called intentional teaching (Epstein, 2007;
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity National Association for the Education of Young Children, 1997). To be effective, intentional teaching requires that teachers use formative assessment to determine where children are in relation to the learning goal and to provide the right kind and amount of support for them to continue to make progress. Intentional teaching is useful to get beyond the dichotomies that arise when teaching is characterized as either teacher-directed or child-initiated. Integrated and Focused Curriculum Early childhood curriculum is often integrated across content domains or subject matter disciplines. Integration is the blending together of two or more content areas in one activity or learning experience (Schickedanz, 2008). The purpose of an integrated curriculum is to make content meaningful and accessible to young children. Integration also enables more content to be covered during the limited school day. Integration typically occurs in two ways. One approach is to add a mathematics content goal to a storybook reading. In this situation, language and literacy goals related to storybook reading are primary, and mathematics learning is secondary. Another way of integrating curriculum is to use a broad topic of study, a theme (such as animals or plants), or a project of interest to children through which mathematics content goals are addressed. Projects are extended investigations into a topic that intellectually engages and interests children, such as how to create a garden or build a house (Katz and Chard, 1989). In both of these approaches to integration, mathematics learning is a secondary objective, rather than the primary focus of attention. In this report, we use both integrated learning experience and secondary focus on mathematics (which some studies have referred to as embedded mathematics) to reflect the teaching/exposure to mathematics content as an ancillary activity. By contrast, focused curriculum or primary focus on mathematics refers to experiences in which mathematics is the major learning goal. A focused mathematics curriculum should also be meaningful and connect to children’s interests and prior knowledge. In this report, we use the terms, “primary focus on mathematics” and “focused mathematics time” to refer to dedicated time for a learning experience with mathematics as the primary goal. STANDARDS FOR CHILDREN’S MATHEMATICS LEARNING State standards for students’ learning have had an increasingly important role in education over at least the past decade, particularly in K-12 education. More recently, standards have begun to play a role in early
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity childhood education as well. Standards have great potential for shaping instruction, curricula, and assessment; however, the impact of standards on learning depends heavily on the content and specific learning goals laid out in them. The number of states with published early learning standards has grown over the past eight years from 27 in 2002 to 49 as of 2008. To inform their early learning standards in mathematics, states have used a variety of National Council of Teachers of Mathematics (NCTM) resources, including Principles and Standards for School Mathematics (2000) (14 states) and Early Childhood Mathematics: Promoting Good Beginnings, issued by NCTM and National Association for the Education of Young Children (NAEYC). Engaging Young Children in Mathematics (2004) is also a widely recognized guide for state early learning standards. Curriculum Focal Points (National Council of Teachers of Mathematics, 2006), the most recent set of guidelines provided by NCTM, was developed after most states had already established their standards. The Curriculum Focal Points provides guidance about the most significant mathematical concepts and skills (i.e., number and operations, geometry and measurement) that should be addressed during children’s early education. Curriculum Focal Points also has a clear emphasis on the PSSM process standards, which are essential for meaningful and substantive mathematics learning. The process strands of communication, reasoning, representation, connections, and particularly problem solving allow children to understand their mathematics learning as a coherent and connected body of knowledge (National Council of Teachers of Mathematics, 2006). Curriculum Focal Points does not, however, provide the kind of in-depth coverage of what children should know and can do that this report does. In order to gain a more systematic understanding of the content of states’ mathematics standards, the committee commissioned two content analyses of current standards for young children: one at the prekindergarten level (here termed “early learning standards”) and one at the kindergarten level (Reys, Chval, and Switzer, 2008; Scott-Little, 2008). Early Learning Standards Many states developed early learning standards to improve classroom instruction and professional development; they also serve as a component of accountability systems. The age levels addressed in the standards documents vary across states. In 17 states the standards targeted children ages 3 to 5, 12 states targeted 3- and 4-year-olds, and 11 states targeted children finishing prekindergarten or starting kindergarten. State-funded prekindergarten programs are the most common target audience for the early learning standards (42 states), which are usually required to implement the standards (39 states) (Scott-Little et al., 2007).
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity Currently, 17 states have developed monitoring systems to ensure that standards are being implemented, and 4 others are in the process of developing such a system. States also report that they intend for the early learning standards to be used in child care (39 states), Head Start (38 states), the Individuals with Disabilities Education Act (26 states), and Even Start (27 states) programs, although the use of the standards in these programs is typically voluntary. For the early learning standards it was possible to evaluate how much emphasis each state has given to mathematics across all of the standards as a whole. On average, states devoted 15 percent of the total number of early learning standards to mathematics, although there was wide variation across states (from a low in New Mexico of only 4 percent to a high in Colorado of 54 percent). In the content analysis of the mathematics early learning standards (Scott-Little, 2008), each standard was first coded into 1 of the 10 mathematics content and process areas in the PSSM. These categories include the three content areas emphasized in this report and in the Curriculum Focal Points—number and operations and geometry and measurement. After the mathematics standards items from a state’s document were coded, the total number of items in each area was summed. Because the total number of items varied from state to state, the total for each area was divided by the total number of mathematics items to produce a percentage that was comparable across documents. In effect, the percentage represents the relative emphasis given to each area of mathematics. Table 7-1 presents these results. TABLE 7-1 Percentages of States Early Learning Mathematics Standards That Fall in Each of the PSSM Areas PSSM Area Mean SD Min. Max. Content Numbers and operations 32.3 9.8 9 50 Algebra 19.0 8.8 0 50 Geometry 17.8 7.9 0 44 Measurement 15.8 8.7 0 50 Data analysis 5.3 5.8 0 17 Process Problem solving 3.7 6.2 0 25 Communication 1.4 3.6 0 4 Reasoning 1.3 3.1 0 13 Representation 0.6 1.8 0 11 Connections 0.4 1.3 0 7 Other 2.5 3.4 0 15 NOTE: PSSM = Principles and Standards for School Mathematics, n = 49 states. SOURCE: Scott-Little (2008).
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity These data show a focus on the area of number and operations; on average, states devoted 32 percent of their mathematics standards to this area, and all states had at least some standards in this area. Geometry received less emphasis than number in the early learning standards (18 percent), and measurement accounted for 16 percent of standards in mathematics. In addition, there was much greater overall emphasis on the content standards areas than on the process standards areas (see Table 7-1). A more detailed analysis was conducted of all standards in each of the three content areas that are the focus of this report (as well as the NCTM Curriculum Focal Points): (1) number and operations, (2) geometry, and (3) measurement. Table 7-2 provides the details of the results for each area. In the area of number and operations, states have most often addressed number sense (an average of 24 percent of the number/operations standards); however, there is considerable variation among states—from 11 states with no standards in this area, to 4 states for which number sense accounted for 100 percent of their number and operations standards. Three other core areas of number were relatively frequent—the number word list, 1-to-1 counting correspondences, and written number symbols—and each is addressed by 11 to 14 percent of the standards. Cardinality and the three basic kinds of addition/subtraction situations received minimal attention. In the geometry early learning standards, there was an emphasis on children’s knowledge of properties of shapes (40 percent) and spatial reasoning (25 percent) (e.g., knowledge related to spatial location and direction), although, again, there was considerable variability among states. Some important aspects of geometry for young children receive little attention, including transformation and visualization of shapes. In the measurement standards, areas most often emphasized are measurement of objects (34 percent of the standards), comparing objects (27 percent), and understanding of concepts related to time (27 percent). Again there was variability—for example, 2 states had no measurement standards at all, and 15 states had no standard related to comparisons of objects and the concept of time (see Table 7-2). Kindergarten Standards The committee also commissioned an analysis of the 10 states with the largest student populations that publish kindergarten-specific mathematics standards: California, Florida, Georgia, Michigan, New Jersey, New York, North Carolina, Ohio, Texas, and Virginia (Reys, Chval, and Switzer, 2008). These states were selected for analysis because they represent approximately 50 percent of the U.S. school population and therefore influence the intended curriculum for a substantial population of students. Given their size, these 10 states are also likely to influence textbook development and materials that are produced by commercial curriculum publishers.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity TABLE 7-2 Classification of State Mathematics Early Learning Standards by Content Area and Focal Area Content/Focal Area Mean% SD Minimum % Maximum % Number and Operations Number sense 24.1 26.6 0 100 1-to-1 correspondence 13.8 10.3 0 43 Number word list 13.1 10.2 0 50 Written number symbols 11.4 11.6 0 40 Perceptional comparisons 9.6 10.3 0 50 Combining/taking apart 7.3 9.6 0 33 Cardinality 5.4 7.2 0 25 Estimation 4.7 8.4 0 33 Change 3.9 7.9 0 33 Ordinal numbers 3.8 6.6 0 25 Counting comparisons 2.2 9.0 0 60 Additive comparisons 0.6 2.1 0 11 Place value 0.2 1.6 0 11 Geometry Properties of shapes 39.6 17.9 0 100 Spatial reasoning 25.3 23.2 0 100 Analyzing and comparing shapes 13.3 15.8 0 67 Location and directionality 12.2 15.5 0 50 Composing/decomposing shapes 6.6 10.7 0 40 Symmetry 1.6 5.3 0 25 Transformation of shapes 1.5 6.0 0 33 Visualization of shapes 0.0 0.0 0 0 Measurement Measurement of objects 33.9 25.3 0 100 Comparing objects 27.1 26.0 0 100 Time 26.9 23.3 0 100 Measurable attributes 12.7 16.0 0 50 Composing objects 0.0 0.0 0 0 NOTE: For number and operations n = 49 states; for geometry n = 48 as one state had no geometry standards; for measurement n = 47 as two states had no measurement standards. Percentages represent the number of a state’s standards in a focal area divided by the total number of standards in the content area (content areas are number and operation, geometry, and measurement). SOURCE: Scott-Little (2008). The kindergarten learning standards for each state were coded into the five PSSM mathematical content areas or strands: (1) number and operation, (2) geometry, (3) measurement, (4) algebra, and (5) data analysis/probability (Clements, 2004; National Council of Teachers of Mathematics, 2000). Results allow an examination of which of these mathematical strands are emphasized across and within states. Relative emphasis devoted to each strand was calculated as a percentage of standards in that strand within the total number of mathematics standards.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity There was considerable variability across the 10 states studied. The total number of mathematics standards varied widely, from 11 in Florida to 74 in Virginia (average number of standards was 29). Of the “total set” of 103 specific standards identified in the analysis, only 1 standard was common to all 10 states (extending a pattern) and another 3 standards were common to 9 states. Only 20 percent of the 103 standards were common to 6 or more states. In kindergarten (as with the early learning standards), the greatest emphasis across all the mathematics standards is placed on number and operations—40 percent of a state’s mathematics standards on average (with a range from a low of 27 percent to a high of 56 percent among states). Geometry and measurement each receive less emphasis than number (19 and 21 percent, respectively), although, again, variability is high (from 9 to 45 percent across states for geometry and from 11 to 28 percent for measurement). In the number strand, the heaviest emphasis is placed on counting. Areas of emphasis (meaning at least 6 of the 10 states had standards in this focal area) include counting objects, reading and writing numerals, identifying ordinal numbers, comparing the relative size of groups of objects, and modeling and solving problems using addition and subtraction. Consistent with the theme of state variability, however, no single number/operations standard appeared in all 10 state documents. In both geometry and measurement, few learning standards were common across the states; only 6 topics (of 43 total across geometry and measurement) appeared in the documents of 6 or more states. In geometry, these topics were identifying and naming two-dimensional (2-D) shapes and knowing the relative position of objects. In measurement, the most common topics were comparing the weight of objects; sort, compare, and/or order objects; compare length of objects; and know days of the week. Taken together, the three focal areas emphasized by the committee (number, geometry, and measurement) account for 80 percent of the content of the kindergarten standards across the 10 states. However, many states also include some specific standards that would not be considered core or primary mathematics by the committee—such as knowing the names of the months, parts of the day, seasons, ordering events by time, comparing time, understanding the concept of time, identifying the time of everyday events to the nearest hour, and measurement of weight, capacity, and temperature. Process strands were addressed quite differently by different states, so no systematic analyses could be done. Specifically, three states make no mention of process standards at the kindergarten level (Florida, North Carolina, and Virginia), and three other states include identification of
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity specific standards by process strand (Georgia, New York, and Texas). Notably, although these strands are specified for kindergarten, these process standards are very similar, if not identical, at each grade, K-8. Two states (Arizona and Massachusetts) provide a general description of process standards in the introductory material of their K-6 or K-8 document. These descriptions emphasize the importance of the process strands outlined in the PSSM (National Council of Teachers of Mathematics, 2000). The California and Ohio documents include process standards organized within one strand (“Mathematical Reasoning” in the California document and “Mathematical Process Standard” in the Ohio document) for each grade. The California document lists process standards that are common across kindergarten and Grade 1. Likewise, the Ohio document includes a list of process standards that are common to Grades K-2. Summary A total of 49 states have early learning standards in mathematics; on average, states devote the greatest emphasis to the area of number (32 percent of the standards on average). Specific emphasis within the areas of number, geometry, and measurement showed considerable state-to-state variation. According to our analysis for the 10 largest states, the greatest emphasis in kindergarten is also placed on number (40 percent of the standards on average). However, there is also considerable variation in content of the specific standards across all of the areas. In fact, of the 103 total standards across the 10 states, 47 are unique to just 1 or 2 state documents. A pattern of wide variation across states in emphasis given to mathematics as a whole and relative emphasis given to various topics in mathematics emerges from these analyses of standards. Thus, while some common topics could be identified, when taken as a whole, the state standards do not communicate a clear consensus about the most important mathematical ideas for young children to learn. THE CLASSROOM CONTEXT We begin with a description of the classroom context in which mathematics instruction takes place. We then focus specifically on what is known about mathematics teaching and learning practices in preschool and kindergarten classrooms—when it occurs, how often, and in what contexts. Results from several large studies of prekindergarten (pre-K) and kindergarten classrooms paint a detailed picture of how young children spend their time in these settings and the quality of their learning experiences. We draw particularly on two studies conducted by the National Center for
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity Early Development and Learning (NCEDL) and on the Early Childhood Longitudinal Study-Kindergarten1 (ECLS-K). The NCEDL conducted two major studies of state-funded pre-K and kindergarten classrooms: the six-state Multi-State Study of Preschool (MS) and the five-state State-wide Early Education Programs (SWEEP) Study (Early et al., 2005). While neither of these studies included a nationally representative sample, as of 2001-2002, almost 80 percent of all children in the United States who were participating in state-funded prekindergarten were in one of these 11 states (Early et al., 2005). When combined, these two studies provide observational data on over 700 preschool and 800 kindergarten classrooms across the United States and offer a unique window on children’s classroom experiences. It is important to note that classrooms were included in these studies only if they received state pre-K funding, so the results are not representative of the larger segment of schooling opportunities for 4-year-olds. State-funded pre-K classrooms are a small subset of early childhood classrooms, generally with greater funding and tighter regulation and monitoring, than the larger set of early childhood classrooms. The studies must be interpreted in this context. In both studies, classrooms were observed using a variety of measures to capture the content and quality of learning opportunities and materials afforded to children, including the Early Childhood Environment Rating Scale (ECERS-R; Harms, Clifford, and Cryer, 1998), Classroom Assessment Scoring System (CLASS; Pianta, La Paro, and Hamre, 2008), and Emerging Academics’ Snapshot (Ritchie et al., 2001).2 How Children Spend Their Time in Prekindergarten and Kindergarten Results from both of the NCEDL studies (the MS and the SWEEP) indicate that children in state pre-K programs spend a great deal of time not engaged in any type of instructional activity. Using the Emerging Academics Snapshot, both NCEDL studies recorded the proportion of time spent in all major areas of curriculum, assessing the amount of time students spent in 1 Material in this section is based on a paper prepared for the committee by Hamre et al. (2008), which included a review of the published literature related to these studies as well as some reanalysis of the data conducted for this report. 2 During pre-K, observation days lasted from the beginning of class until the end of class in part-day rooms and until nap in full-day rooms. In pre-K, observers stayed with the children all day, including lunch, outside time, and special activities. In kindergarten, the observations were slightly different because the days were generally longer. Snapshot and CLASS observations lasted the entire day, but no observations were made during lunch, recess, or nap. For this reason, pre-K and kindergarten Snapshot percentages of time spent are discussed separately. More information about these studies can be found on the NCEDL website (http://www.fpg.unc.edu/~ncedl/) and in several published articles (Clifford et al., 2005; Howes et al., 2008; Pianta et al., 2005).
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity reading, oral language and phonemic awareness activities, writing, mathematics, science, social studies, aesthetics, and fine and gross motor activities. Each area was broadly defined so that time spent in dramatic play, block areas, coloring with markers, talking with teachers about things outside school, and singing songs were included in one of these areas. During the preschool day, the average student spent 44 percent of the time engaged in none of these curriculum activities. Data from kindergarten classrooms revealed that the average student was not engaged in any instructional activity in 39 percent of the observed intervals. What were children doing during this noninstructional time? In preschool classrooms, much of the time (22 percent) was spent engaged in routine activities, such as transitioning, waiting in line, or washing hands. Some time (11 percent) was also spent in meals and snacks (Early et al., 2005). Importantly, routine, meal, and snack times could be included as instructional time if, for example, teachers and children engaged in a conversation, sang a song, or played a number game during a transition. But few preschool or kindergarten teachers appeared to take advantage of the learning opportunities that arose during transitional periods or employed strategies for getting the most out of this time in the classroom. Which types of instructional opportunities are young children exposed to most often? Of all content areas, young children spent more time in language and literacy activities than any other—14 percent of the observed day in preschool and 28 percent of the observed day in kindergarten (La Paro et al., 2008). None of the other major areas occurred much more than 10 percent, on average, in any given day. Pre-K children in the NCEDL studies were exposed to mathematics content in only 6 percent of the observations, and kindergarten children were exposed to mathematics an average of 11 percent of the day. Another relevant question concerns the use of various instructional contexts, such as free choice/center time or whole-group instruction. Data from the NCEDL studies suggest there is a major shift in the preferred instructional context from preschool to kindergarten. Children in preschool classrooms spent an average of 33 percent of the school day in free choice or center time, compared with only 6 percent of the day in kindergarten classrooms. Once in kindergarten, both whole-group instruction and individual time, in which children work independently at desks, becomes much more frequent. Across kindergarten and preschool, teachers rarely made use of small-group instruction. Quality of Teacher-Child Interactions in Preschool and Kindergarten The NCEDL data also provide a window into the quality of teacher-child interactions and instruction to which young children are exposed, using the CLASS Framework for Children’s Learning Opportunities in
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