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7 Standards, Curriculum, Instruction, and Assessment In this chapter, we address the topic of effective mathematics curricu- lum and teachingâwhat is known about how teachers can effectively sup- port childrenâs 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 childrenâs 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 mathemat- ics. 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, 225

226 MATHEMATICS LEARNING IN EARLY CHILDHOOD model or demonstrate skills, and use other teaching strategies in which they take the lead. Teacher-initiated learning experiences are determined by the teacherâs goals and direction, but they should also reflect childrenâs 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 interac- tions with objects and with peers (Epstein, 2007, p. 2). Child-initiated expe- rience emanates primarily from childrenâs 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 controlâalthough 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 âdirect instruction,â implying that teachers are entirely in control and children are passive recipients of information. The term is also used pejo- ratively 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 Mathemat- ics Advisory Panel (2008) uses the term explicit instruction to refer to the many ways that teachers can intentionally structure childrenâs experiences so that they support learning in mathematics. Throughout the day and across various contextsâwhole group, small group, centers, play, and routinesâteachers 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;

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 227 National Association for the Education of Young Children, 1997). To be effective, intentional teaching requires that teachers use formative assess- ment 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 dichoto- mies 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 mean- ingful 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, lan- guage 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 intel- lectually 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 childrenâs interests and prior knowledge. In this report, we use the terms, âprimary focus on mathematicsâ and âfocused mathematics timeâ to refer to dedicated time for a learning experience with mathematics as the primary goal. STANDARDS FOR CHILDRENâS MATHEMATICS LEARNING State standards for studentsâ learning have had an increasingly impor- tant 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

228 MATHEMATICS LEARNING IN EARLY CHILDHOOD 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 va- riety 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 Mathemat- ics, 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 math- ematical concepts and skills (i.e., number and operations, geometry and measurement) that should be addressed during childrenâs 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 statesâ mathematics standards, the committee commissioned two content analyses of current standards for young children: one at the prekindergarten level (here termed âearly learning standardsâ) 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 docu- ments 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).

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 229 Currently, 17 states have developed monitoring systems to ensure that stan- dards 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 math- ematics content and process areas in the PSSM. These categories include the three content areas emphasized in this report and in the Curriculum Focal Pointsânumber and operations and geometry and measurement. After the mathematics standards items from a stateâs 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 rela- tive emphasis given to each area of mathematics. Table 7-1 presents these results. TABLE 7-1â 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).

230 MATHEMATICS LEARNING IN EARLY CHILDHOOD These data show a focus on the area of number and operations; on av- erage, 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 ad- dition, 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 stan- dards); however, there is considerable variation among statesâfrom 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 frequentâthe number word list, 1-to-1 counting correspondences, and written number symbolsâand 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 childrenâs knowledge of properties of shapes (40 percent) and spatial rea- soning (25 percent) (e.g., knowledge related to spatial location and direc- tion), although, again, there was considerable variability among states. Some important aspects of geometry for young children receive little atten- tion, including transformation and visualization of shapes. In the measurement standards, areas most often emphasized are mea- surement of objects (34 percent of the standards), comparing objects (27 percent), and understanding of concepts related to time (27 percent). Again there was variabilityâfor 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 ap- proximately 50 percent of the U.S. school population and therefore influ- ence 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.

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 231 TABLE 7-2â 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 stateâs 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 operaÂ tion, (2) geometry, (3) measurement, (4) algebra, and (5) data analysis/Â probability (Clements, 2004; National Council of Teachers of ÂMathematics, 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.

232 MATHEMATICS LEARNING IN EARLY CHILDHOOD 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 âtotal setâ 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 operationsâ40 percent of a stateâs 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, identify- ing 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 com- mon 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 committeeâsuch 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

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 233 specific standards by process strand (Georgia, New York, and Texas). No- tably, 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 stan- dards 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 (âMathematical Reasoningâ in the California document and âMathematical Process Standardâ 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 stan- dards 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 stan- dards 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 math- ematics as a whole and relative emphasis given to various topics in math- ematics emerges from these analyses of standards. Thus, while some common topics could be identified, when taken as a whole, the state stan- dards 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 math- ematics instruction takes place. We then focus specifically on what is known about mathematics teaching and learning practices in preschool and kinder- garten classroomsâwhen it occurs, how often, and in what contexts. Results from several large studies of prekindergarten (pre-K) and kin- dergarten 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

234 MATHEMATICS LEARNING IN EARLY CHILDHOOD Early Development and Learning (NCEDL) and on the Early Childhood Longitudinal Study-Kindergarten (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 kin- dergarten classrooms across the United States and offer a unique window on childrenâs 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 representa- tive 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 Academicsâ Snapshot (Ritchie et al., 2001). How Children Spend Their Time in Prekindergarten and Kindergarten Results from both of the NCEDL studies (the MS and the SWEEP) in- dicate 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 â 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. â 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 observa- tions 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).

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 235 reading, oral language and phonemic awareness activities, writing, math- ematics, science, social studies, aesthetics, and fine and gross motor activi- ties. Each area was broadly defined so that time spent in dramatic play, block areas, coloring with markers, talking with teachers about things out- side 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 ac- tivity in 39 percent of the observed intervals. What were children doing during this noninstructional time? In pre- school 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 con- versation, 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 lan- guage and literacy activities than any otherâ14 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 in- structional 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 indi- vidual 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 Childrenâs Learning Opportunities in

236 MATHEMATICS LEARNING IN EARLY CHILDHOOD Early Childhood and Elementary Classrooms (CLASS framework; Pianta, La Paro, and Hamre, 2007). The CLASS framework captures three broad domains of classroom interactionsâemotional supports, classroom organi- zation, and instructional supportsâas well as more specific dimensions in each domain. The CLASS framework was derived from basic, theory-driven research on classroom environments and research on effective teaching practices, and it aligns well with a variety of conceptualizations of effec- tive teaching and empirical evidence on effective practices (see Hamre and Pianta, 2007, for a more detailed discussion). Emotional Supports in Preschool and Kindergarten NCEDL results indicate that across preschool and kindergarten, chil- dren, on average, experienced moderately positive interactions with teachers in moderately well-managed classrooms (La Paro et al., 2008). Approxi- mately one-third of children in this study were in classrooms characterized by high-quality emotional supports in both pre-K and kindergarten. Teachersâ emotional support may have direct links to studentsâ learn- ing (e.g., Connor, Son, and Hindman, 2005), as well as indirect links in which emotional support fosters engagement, which in turn leads to greater achievement (Rimm-Kaufman, Early, and Cox, 2002). Childrenâs social and emotional functioning in the classroom is increasingly recognized as an indicator of school readiness (Blair, 2002; Denham and Weissberg, 2004; Raver, 2004) and a potential target for intervention (Greenberg, Weissberg, and OâBrien, 2003; Zins et al., 2004). Children who are more motivated and connected to others in the early years of schooling are much more likely to establish positive trajectories of development in both social and academic domains (Hamre and Pianta, 2001; Pianta, Steinberg, and Rollins, 1995; Silver et al., 2005). Furthermore, there is some evidence that emotional sup- ports may be particularly important for supporting the academic develop- ment of students with social and emotional difficulties (Hamre and Pianta, 2005). Recent nonexperimental research in elementary classrooms suggests that there may be direct links between emotional supports and studentsâ mathematics knowledge (Pianta et al., 2008). Classroom Organization and Management Classrooms function best and provide the most opportunities to learn when students are well behaved, consistently have things to do, and are in- terested and engaged in learning tasks (Pianta et al., 2005). In short, children are better regulated in well-regulated classroom environments. In the NCEDL studies, this dimension of classrooms was measured using the CLASS. In general, the quality of classroom organization and management

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 237 in the early childhood classrooms observed in the NCEDL studies was moderately positive. The typical classroom was characterized by a mix of productive periods, with children engaged in learning, and other periods, in which significant behavior problems disrupted learning or teachers failed to actively engage children in learning opportunities. Instructional Supports Of greatest concern are results suggesting very low levels of instruc- tional supports across pre-K and kindergarten, as measured by both the CLASS and ECERS-R. In the CLASS, instructional supports include the three dimensions of support for concept development, quality of feedback, and language modeling. Interactions between adults and children are the key mechanism through which these instructional supports are provided to children in the early years of schooling. A child gets more out of an activity if the teacher is either directly interacting with the child in an intentional way or if the childâs participation in the activity has been sufficiently sup- ported by the teacher prior to the start of the activity, so that the child, in playing, is more intentional in the purpose of the activity (the section below called âResearch on Effective Mathematics Pedagogyâ is a more detailed discussion of instructional supports). Mathematics Practices in U.S. Preschools and Kindergartens Little is known about the math-specific learning opportunities that are provided to children in early childhood settings. This may reflect, in part, the focus on early literacy and language development that has consumed much of early childhood policy and research attention for the past decade. Although more recent attention has focused on early childhood matheÂ matics (Clements and Sarama, 2007a), there is not yet detailed, national- level information on the typical mathematical practices to which children are exposed. In this section, we again draw on the NCEDL MS Study and on data from the nationally representative ECLS-K cohort. We begin with a more detailed analysis of observational data from pre-K classrooms in the NCEDL MS Study and end with a description of kindergarten Âteachersâ self- reported mathematics practices from the ECLS-K. Although the ECLS-K is nationally representative, the information about mathematics instruction is limited. Mathematics Instruction in Prekindergarten The most relevant NCEDL MS data come from observations conducted during visits to pre-K classrooms in the fall and the spring. The average

238 MATHEMATICS LEARNING IN EARLY CHILDHOOD amount of time focused on mathematics content in the pre-K classrooms was minimal (6.5 percent in the fall, and 6.7 percent in the spring). More detailed analysis of the actual activities that took place during this mathematics time suggests that, for about half of the time, mathemat- ics content occurred during whole-group activities (49 percent in the fall and 48 percent in the spring). Free choice/center time was the second most common mathematics setting (31 and 29 percent in the fall and spring re- spectively), with small group instruction third (11 and 12 percent). Another important question is whether mathematics is taught alone or in conjunction with other content. Data indicate that mathematics content co-occurred with other academic content during the majority of the time (61 percent in the fall and 55 percent in the spring). About 20 percent of the time, when mathematics co-occurred with something else, it was with an art or music activity (aesthetics), and between 15 and 18 percent of the time it was with a fine motor activity. Other academic content occurred simultaneously with mathematics about 11 percent of the time for reading (a combination of being read to, prereading, and letter-sound), 13 percent for social studies, and 11 percent for science. These findings indicate that, when they do teach mathematics, early childhood education programs rely on integrated or embedded mathematics experiences a majority of the time, rather than including activities with a primary focus on mathematics. The selection of materials and activities such as puzzles, blocks, games, songs, and fingerplays seem to constitute mathematics for many teachers ( Â Clements and Sarama, 2007a). Using the Emerging Academics Snapshot, researchers found that teach- ersâ interactions with children during mathematics content were likely to be either encouraging or didactic in nature. Encouraging was coded when teachers provided feedback about effort and persistence, including praise, personal comments, and general statements that helped children stay en- gaged in their work. Didactic was coded when teachers focused on giving instructions, asking questions with one correct answer, and engaging chil- dren in instruction focused on mastering a discrete set of materials. Less often, teachers spent time scaffolding while delivering mathematics content. Scaffolding was coded when teachers showed an awareness of an individual childâs needs and responded in a manner that supported and expanded the childâs learning. â Note that math was coded when a child was verbally counting, counting with 1-to-1 correspondence, skip counting, identifying written numerals, matching numbers to pictures, making graphs, playing counting games (e.g., dice, dominoes, Candyland, Chutes and Lad- ders), keeping track of how many days until a special event, counting marbles in a jar, playing Concentration or Memory with numbers, working on mathematics worksheets, identifying shapes, talking about the properties of shapes (e.g., how many sides), finding shapes in the room, identifying same and different (e.g., big/little, biggest), sorting (by color, size, shape), discerning patterns (red, blue, red, blue), or measuring for cooking or size.

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 239 To summarize, in the state-funded pre-K classrooms observed in the NCEDL MS Study, mathematics was often taught in conjunction with art, music, and fine motor activities, suggesting that perhaps teachers were integrating mathematics with activities that they assumed would heighten childrenâs engagement and were making use of manipulatives. However, the committee thinks that the integration of mathematics with other activi- ties may or may not be effective in supporting childrenâs development of mathematics knowledge, depending on the integrity of, and emphasis on, the mathematical ideas. It is also evident that mathematics, like literacy, was often taught in a manner in which teachers focused on student performance of a discrete skill or display of factual knowledge. Children were less often exposed to instruction that was conversational, interactive, and focused on understanding and problem solving. Mathematics Instruction in Kindergarten The ECLS-K cohort is a nationally representative sample of 22,000 students in approximately 1,000 classrooms across the United States. This cohort of students entered the study in 1998-1999 as they began kindergar- ten and will be followed through eighth grade. The most relevant ECLS-K data for our purposes are items from a survey of kindergarten teachers who reported how often their students were exposed to classroom instruction in mathematics, including (1) broad exposure to mathematics, (2) instructional emphasis on specific mathematics concepts and skills, and (3) exposure to specific mathematics instructional strategies and activities. The committee commissioned a reanalysis of these teacher survey items because existing published analyses did not provide sufficient detail on mathematics (Hamre et al., 2008a). For the purposes of our analysis, the items were organized conceptually according to the NCTM Content Standards (National Council of Teachers of Mathematics, 2000) into the a Â reas of number and operations, geometry, measurement, algebra, and data analysis and probability. The vast majority of teachers (81 percent) indicated that mathematics instruction is a part of their daily classroom routine, with over half of the teachers (65 percent) reporting that they provide more than 30 minutes of mathematics instruction each day. Teachers also indicated the frequency with which they taught a list of 27 specific mathematics concepts and skills. By far, teachers reported concepts and skills associated with number and operations to be the most common emphasis of mathematics instruction. However, in contrast to the recommendations in this report for focusing on learning paths in a few key areas, concepts and skills associated with all of the NCTM standards were the emphasis of mathematics instruction to some extent in a given academic week. Specific to number and operations, the most common concepts and

240 MATHEMATICS LEARNING IN EARLY CHILDHOOD skills teachers reported teaching were correspondence between number and quantity, writing all numbers between 1 and 10, and reading two-digit numbersâall of these were frequently (77, 55, and 52 percent, respectively) reported to be the emphasis of instruction three or more times per week. Counting by 2s, 5s, and 10s was fairly common, with 44 percent of teachers reporting this to occur at least three times per week. Instruction was slightly less often focused on ordinal numbers (35 percent reported at least three times per week), adding single-digit numbers (40 percent), and subtracting single-digit numbers (28 percent). Research on childrenâs number and operations learning discussed in previous chapters suggests that such emphases are out of balance. For example, time dedicated to skip countingâespecially if involving only verbal countingâmight be bet- ter used to address concepts, strategies, and skills related to addition and subtraction. As for measurement concepts and skills, the most commonly endorsed items were identification of relative quantity (e.g., most, least, more, less), ordering objects by size or other properties, and sorting objects into sub- groups according to a rule, all of which were reported to be the emphasis of instruction once a week or more for 56-76 percent of teachers. Measure- ment concepts and skills received less frequent emphasis but still were re- portedly the focus of instruction at least once a month for most classrooms, as were using measurement instruments accurately, telling time, estimating quantities, and recognizing the value of coins and currency. Geometry, algebra, and data analysis/probability consisted of the fewest survey items. The lone geometry skill in the survey, recognizing and naming geometric shapes, was reported to be the emphasis of instruction once per week or more by more than 66 percent of teachers. Similarly, related to algebra, over two-thirds of teachers (72 percent) reported teaching copying, making, and extending a pattern at least once a week. Under data analysis and probability, over half of the teachers emphasized reading simple graphs once per week or more, while simple data collection and graphing was less often emphasized (54 percent reported doing this two to three times per month or less). The majority of teachers (59 percent) noted that estimating probability was a skill to be taught at a higher grade level. Another set of survey items asked teachers about the extent to which they used various instructional activities or strategies. In numbers and operations, the most common math-related activity reported by teachers was verbal counting, which happened on a daily basis in more than 79 percent of the kindergarten classrooms. Another relatively common activity involved use of counting manipulatives to learn basic operations, with 66 percent of teachers reporting this to occur three or more times per week. The use of geometric manipulatives was also fairly common, with 45 per- cent of teachers reporting their use three or more times a week. In contrast,

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 241 work with rulers, measuring cups, spoons, and other measuring instruments was fairly infrequent, with two-thirds of teachers (66 percent) reporting use of them three times per month or less. Generalized teaching strategies and activities are defined as those that can apply to a variety of the NCTM mathematics standards. The most prominent generalized strategy was calendar-related activities, which oc- curred on a daily basis in over 90 percent of the classrooms surveyed, this despite the fact that mathematics educators do not consider most calendar activities to be useful early childhood mathematics instruction and have serious questions about the efficacy of âdoing the calendarâ every day (see Box 7-1). More than half of the teachers reported using the following strategies and activities twice a week or more: playing mathematics-related games, explaining how a mathematical problem is solved, doing mathematical BOX 7-1 How Using the Calendar Does Not Emphasize Foundational Mathematics Many preschool and kindergarten teachers spend time each day on the cal- endar, in part because they think it is an efficient way to teach mathematics. Al- though the calendar may be useful in helping children begin to understand general concepts of time, such as âyesterdayâ and âtoday,â or plan for important events, such as field trips or visitors, these are not core mathematical concepts. The main problem with the calendar is that the groups of seven days in the rows of a calen- dar have no useful mathematical relationship to the number 10, the building block of the number system. Therefore, the calendar is not useful for helping students learn the base 10 patterns; other visual and conceptual approaches using groups of 10 are needed because these patterns of groups of 10 are foundational. Time spent on the calendar would be better used on more effective mathemat- ics teaching and learning experiences. âDoing the calendarâ is not a substitute for teaching foundational mathematics. 1 11 21 31 41 51 61 71 81 91 101 111 2 12 22 32 42 52 62 72 82 92 102 112 3 13 23 33 43 53 63 73 83 93 103 113 4 14 24 34 44 54 64 74 84 94 104 114 5 15 25 35 45 55 65 75 85 95 105 115 6 16 26 36 46 56 66 76 86 96 106 116 7 17 27 37 47 57 67 77 87 97 107 117 8 18 28 38 48 58 68 78 88 98 108 118 9 19 29 39 49 59 69 79 89 99 109 119 10 20 30 40 50 60 70 80 90 100 110 120

242 MATHEMATICS LEARNING IN EARLY CHILDHOOD worksheets, solving mathematical problems in small groups or with a partner, working on mathematical problems that reflect real-life situations, working in mixed achievement groups on mathematics activities, and using computers to learn mathematics. A somewhat different pattern was evident for using music to understand mathematics, using creative movement or creative drama to understand mathematical concepts, completion of math- ematical problems on the chalkboard, and engaging in peer tutoring. A quarter or more of the teachers indicated that they never asked students to do these activities, whereas another quarter or more used these activities at least one to two times per week. Mathematics Practices Across Diverse Preschool Settings Findings from the few smaller scale studies that examined mathematics in early childhood settings show a similar pattern. In one study, teachers in two states from a range of preschool settings, including family and group child care providers, were surveyed about their mathematics instruction (Sarama, 2002; Sarama and DiBiase, 2004). Most teachers reported using manipulatives (95 percent), number songs (74 percent), and games (71 percent). Only 33 percent used software, and 16 percent reported using mathematical worksheets. Teachers reported a preference for children to ex- plore mathematics activities and engage in free play rather than participate in large group lessons or do mathematical worksheets. The mathematics topics teachers reported were counting (67 percent), sorting (60 percent), numeral recognition (51 percent), patterning (46 percent), number concepts (34 percent), spatial relations (32 percent), making shapes (16 percent), and measuring (14 percent). The least popular topics were geometry and measurement. In an observational study of New Jersey preschools, teachers were found to provide little support for childrenâs mathematical skill develop- ment and seldom used mathematics terminology (Frede et al., 2007). Of particular interest is that over 40 percent of the classrooms in this study were rated as good to excellent quality on the ECERS-R measure of the en- vironmental quality of early childhood programs. Apparently, mathematics teaching and learning is relatively rare even in classrooms that are otherwise judged to be of high quality. RESEARCH ON EFFECTIVE MATHEMATICS INSTRUCTION The majority of research that is focused specifically on mathematics taught in early childhood examines the effectiveness of a particular math- ematics curriculum (e.g., Clements and Sarama, 2008a; Sophian, 2004; Starkey, Klein, and Wakeley, 2004). Although much of this work meets very

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 243 high empirical standards, it is often difficult to derive information about specific types of effective instructional practices from general information on whether or not a curriculum is successful. Nevertheless, this research base does provide some guidance on effective mathematics instruction. (Curriculum research is discussed later in the chapter.) There is also a large body of research on effective instruction in early childhood that is not specific to mathematics. The general principles of ef- fective instruction that emerge from this research can and should be taken into consideration when designing mathematics instruction for young chil- dren. Both of these bodies of research are briefly reviewed below. Taken together, they provide guidance on effective instruction, although further research on strategies specific to mathematics is needed. The large body of research on effective instruction informed the de- velopment of the CLASS system for observation described briefly in the previous section. Since the domain of the CLASS most closely associated with the development of mathematics knowledge and skill is instructional supports, we begin with a discussion of various kinds of instructional sup- ports as defined in the CLASS. We then move to discussion of other aspects of instruction that are important for supporting learning in mathematics. Instructional Supports The theoretical foundation for the CLASS conceptualization of instruc- tional supports comes primarily from research on childrenâs cognitive and language development (e.g., Catts et al., 2002; Fujiki, Brinton, and Clarke, 2002; Romberg, Carpenter, and Dremlock, 2005; Taylor et al., 2003; Vygotsky, 1991; Wharton-McDonald and Pressley, 1998). This literature highlights the distinction between simply learning facts and gaining usable knowledge, which is built on learning how facts are interconnected, orga- nized, and conditioned on one another (Mayer, 2002; National Research Council, 1999). A childâs cognitive and language development is contingent on the opportunities adults provide to express existing skills and scaffold more complex ones (Davis and Miyake, 2004; Skibbe, Behnke, and Justice, 2004; Vygotsky, 1991). The development of metacognitive skills, including childrenâs awareness and understanding of their own thinking processes as well as their executive function skills, are also critical to their academic de- velopment (Blair, 2002; Veenman, Kok, and Blote, 2005; Williams, Blythe, and White, 2002). The CLASS assessment system has been validated, both in terms of its factor structure (Hamre et al., 2008b) and in relation to preschool childrenâs language, literacy, and mathematics knowledge and social and emotional development (Burchinal et al., 2008; Howes et al., 2008; Mashburn et al., 2008). Children in classrooms that score higher on the instructional dimen-

244 MATHEMATICS LEARNING IN EARLY CHILDHOOD sions of concept development and quality of feedback, as measured by the CLASS, display greater gains in mathematics knowledge over the course of the year, although the effect sizes are small (between .10 and .20; Mashburn et al., 2008). These two dimensions of instructional support are discussed in greater detail below. Promoting Conceptual Development Concept development describes the instructional behaviors, conversa- tions, and activities that teachers use to help stimulate studentsâ higher order thinking skills (Pianta et al., 2007), which refers not only to the acquisition of knowledge, but also to the ability to access and apply this knowledge in new situations (Mayer, 2002). The four key elements of high- quality concept development are (1) analysis and reasoning, (2) creating, (3) integration, and (4) connections to the real world. In classrooms that fall at the high end of concept development, teachers not only plan activities in ways that will stimulate higher order thinking, but also they take advantage of the moment-to-moment opportunities in their daily interactions to push children toward deeper thinking. In con- trast, classrooms that are low on conceptual development lack instructional opportunities or focus instruction solely on remembering facts or on simple tasks that require only recognition or recall. Providing Scaffolding and Feedback In order for students to get the most benefit from instructional opportu- nities, they need feedback about their learning. Feedback refers to a broad range of interactions through which the teacher provides some information back to the students about their performance or effort. There are five Âmajor types of feedback interactions described in the CLASS: (1) scaffolding, (2) feedback loops, (3) prompting of thought processes, (4) provision of information, and (5) encouragement or affirmation. Feedback is a key ele- ment of formative assessment, which is discussed in greater detail later in this chapter. Scaffolding.â Teachers scaffold childrenâs learning by providing hints and as- sistance that enable them to perform at a higher level than they might be able to do on their own. This may occur during a whole-group or small-group discussion or individually during center time or childrenâs play (scaffolding is also discussed in the section on formative assessment in this chapter). Feedback loops.â Effective feedback is also characterized by sustained exchanges with a child (or group of children), leading them to a better or

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 245 deeper understanding of a particular idea. This is in contrast to a teacher who might give a single hint to a child but then move on, even if the child does not seem to understand. Prompting thought processes.â This feedback strategy asks students to ex- plain their thinking or actions. Prompting thought processes helps to identify children who may have completed an activity or answered a question cor- rectly, but who cannot yet clearly articulate their reasoning. By having a child articulate his or her thought process, the teacher discovers erroneous thinking and can intervene. This learning opportunity is in contrast to one in which the teacher just tells the child that he or she was correct or incorrect. Providing information.â In the context of instructional interactions, chil- dren often give the wrong answer or action. Each instance provides an opportunity for effective feedback by expanding on childrenâs answers and actions, clarifying incorrect answers, or providing very specific information about the correct answer. These are all in contrast to a teacher who simply tells students they are wrong. Encouragement and affirmation.â Another form of feedback consists of strategies that can motivate children to sustain their efforts and engage- ment. Simple recognition statements, such as âYou are working really hard on that puzzleâ reinforce studentsâ effort and encourage persistence. This may be especially important in the area of mathematics, in which older children in the United States have been found to assume that mathematics achievement is a product of ability rather than effort (National Mathemat- ics Advisory Panel, 2008). Young children may need help to learn that effort leads to improved results in learning mathematics. The Importance of Math Talk In a mathematics context, teachersâ use of language can facilitate con- nections between numbers, words, and ideas. In an elegant demonstration of the importance of mathematical language for young children, Klibanoff and colleagues (2006) showed that children exposed to more math talk in their preschool classrooms displayed greater gains in mathematical knowl- edge from October to April. The authors transcribed an hour of teachersâ utterances, including circle time, and coded the transcripts for the number of mathematical inputs in the following categories: counting, cardinality, equivalence, nonequivalence, number symbols, conventional nominative (as in naming an address or phone number), ordering, calculation, and placeholding. There was a wide range of mathematical inputs among the 26 classrooms (a range from 1 to 104, with an average of 28). References

246 MATHEMATICS LEARNING IN EARLY CHILDHOOD to cardinality were the most common, accounting for 48 percent of all inputs. Many of the inputs, such as equivalence, nonequivalence, ordering, calculation, and placeholding, were rare, each accounting for less than 5 percent of all inputs. After controlling for childrenâs prior performance, those in classrooms with a higher number of mathematics inputs displayed better performance in April on a short (15-item), multiple-choice test of general mathematical knowledge. Klibanoff et al. (2006) found only a small correlation between teachersâ syntactic complexity and frequency of math talk (r = .18). And only math talk, not syntactic complexity, was associated with gains in mathematical skills. As the authors point out, this is the first study to exam- ine the specific effects of math talk on childrenâs knowledge, and research is needed to understand more about the direct role of math talk in early childhood classrooms. In general, the amount and kind of language that occurs in the class- room among teachers and children is frequently related to outcomes for children. Correlational research with preschoolers demonstrates that, dur- ing large-group times, teachersâ explanatory talk and use of cognitively challenging vocabulary are related to better learning outcomes for children (Dickinson and Tabors, 2001). The use of open-ended questions also has the potential to increase the math talk in a classroom or in a home. Effective teachers make greater use of open-ended questions than less effective teachers. They ask children âWhy?â and âHow do you know?â They expect children, as young as pre- school, to share strategies, explain their thinking, work together to solve problems, and listen to each other (Askew et al., 1997; Carpenter et al., 1998, 1999; Clarke et al., 2001; Clements and Sarama, 2007a, 2008a; Cobb et al., 1991; Thomson et al., 2005). As the questions become internal, children can increasingly become self-sustaining mathematical learners who carry and use a mathematical lens for seeing and understanding their world. Examples of such open-ended mathematical questions are â¢ Where do you see this (mathematical idea) in our classroom? â¢ Tell me how you figured out (this mathematical idea). â¢ What is (insert mathematical idea, such as adding or subtracting)? â¢ What happens when I break this apart/put these together? â¢ How does this compare with something else? (Which one is smaller/ larger? Longer/shorter?) â¢ Where are the units? What are the units (that children are familiar with)? â¢ Do you see a pattern? What is the pattern? â¢ How can I describe this idea for myself or for someone else (such as, can you draw a picture, describe it in words, or use your body)?

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 247 Grouping as an Instructional Strategy As described previously, data from the NCEDL studies and the ECLS-K indicate that mathematics is taught in a whole group most of the time, es- pecially in kindergartens, where little time is allocated for centers or small groups. The almost nonexistent use of small groups in early childhood programs, documented in these studies, is of concern given that small-group instruction has been found to be an effective context for enhancing young childrenâs learning (Dickinson and Smith, 1993; Karweit and Wasik, 1996; Morrow, 1988). Various mathematics curricula that use small groups as one of several or as the main instructional strategy have shown substantial positive effects (e.g., Clements, 2007; Clements and Sarama, 2008a; Preschool Curriculum Evaluation Research Consortium, 2008; Sarama et al., 2008; Starkey et al., 2006). The results suggest that small-group work can significantly increase childrenâs scores on tests aligned with that work (Klein and Starkey, 2004; Klein, Starkey, and Wakeley, 1999), and can transfer to knowledge and abilities that have not been taught (Clements and Sarama, 2007a). Guide- lines in these curricula generally suggest four children with a teacher as the small-group size, although teachers have been observed using group sizes of two (for low achievers, for children with special needs, or to introduce an idea or activity for the first time) to six (usually for efficiencyâs sake; often used for easily managed activities). Whole groups can also be effective for supporting mathematics learning. In one program, children as young as kindergarten engaged in teacher- and peer-scaffolded mathematics learning, problem solving, and discussion dur- ing whole-class instruction (Fuson and Murata, 2007). Based on teaching- learning paths, the program successfully enabled teachers to individualize mathematics in large-group activitiesâa promising strategy to give math- ematics needed attention in the already packed schedule of half-day kin- dergarten. French and Song (1998) document extensive use of whole-group instruction to good effect in Korean preschools. Effective whole-group interactions include brief demonstrations and discussions, problem solving in which children talk to and work with the person next to them (other chil- dren and possibly adults), and physically active activities, such as marching around the room while counting (Clements, 2007; Clements and Sarama, 2007a, 2008a). Box 7-2 provides an example. Play as a Teaching and Learning Context A highly motivating learning context for young children is child-initiated play (Wiltz and Klein, 2001). Preschool children engage in different types of play that have potentially different benefits for learning and develop-

248 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 7-2 Mathematics Activities with Different Size Groups The Building Blocks Program dedicates several weeks to shape composition. One theme is puzzles. In a whole-group setting, the teacher asks the children what puzzles they like to solve at home and at school. She discusses various types of puzzles and what puzzles are, showing some examples, telling the chil- dren she will put them all out in the mathematics centers. She then introduces a new kind of puzzle: outline puzzles that can be completed with geometric shapes (e.g., pattern blocks or tangram pieces). She solves a simple puzzle with the children, using their ideas as to solutions. Later, with small groups of four children, the teacher introduces several of the outline puzzles. She carefully observes childrenâs solutions to these, evaluating where each child is in the learning trajectory for shape composition. Based on these observations, she provides individuals with puzzles at different levels of the learning trajectory (or mathematics teaching-learning paths), individualizing the challenge for each child. Meanwhile, the teacherâs assistant observes and discusses childrenâs work with the puzzles in the mathematics center, as well as supervising those in other centers, allowing the teacher to concentrate on the small-group work. One special center involves a series of computer activities, the Piece Puzzler series, in which children also solve puzzles by manipulating pattern blocks or tangram pieces to complete similar outline puzzles. They use icons of the geometric motions to slide, turn, and flip the shapes into place. Individualized help and feedback are offered to them immediately. For example, if they put on too large a shape, covering the puzzle and also other areas, the computer activity makes the shape transparent and shows them that it covers too much (something difficult to show with physical manipulatives). Also, the computer activity automatically adjusts the levels of the puzzle to match the childrenâs development along the learning trajectory. ment. Among the typically observed play experiences in an early childhood classroom are constructive play, such as block building; play with table toys (manipulatives, puzzles, Lego blocks); pretend play; mathematical play; and games, including ones in which mathematics is a secondary focus, as well as ones in which mathematics is the primary focus. (Of course children also engage in outdoor play, rough-and-tumble play, and other forms of play that have benefits as well.) Block Building Play, especially block play, provides valuable opportunities for children to explore and engage in mathematical activity on their own (Ginsburg, 2006; Hirsch, 1996). Young children enjoy playing with blocks, and there is evidence that they naturally engage in mathematical play with them (Seo

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 249 and Ginsburg, 2004). However, mathematics learning is enhanced if teach- ers engage children in a discussion of mathematical principles during block play (Clements and Sarama, 2007a), such as introducing new terminology (e.g., edges, faces) and commenting on childrenâs rotation of objects during construction. The provision of these supports by teachers during play en- hances childrenâs learning during the specific interaction as well as in future play sessions, when the child may incorporate these new ideas. Research also indicates that teachers should incorporate planned, sys- tematic block building into their curriculum, which they rarely do (Kersh, Casey, and Young, 2008). Preschoolers who are provided such scaffold- ing display significant increases in the complexity of their block building (Gregory, Kim, and Whiren, 2003). Important to our teaching-learning paths approach (also called learning trajectories), the teachersâ scaffolding was based on professional development aimed at helping them recognize developmental progressions in the levels of complexity of block building. Teachers learned to provide verbal scaffolding based on those levels but not to directly assist children or engage in any block building themselves. Interventions that incorporate full teaching-learning pathsâthat is, a goal, a developmental progression, and matched activitiesâappear to be effective in developing childrenâs skills. Groups of kindergartners who experienced such a learning trajectory improved in block-building skill more than con- trol groups who received an equivalent amount of block-building experi- ence during unstructured free play sessions (Kersh et al., 2008). One longitudinal study indicated that block building may help lay a foundation for mathematics achievement in later years (Wolfgang, ÂStannard, and Jones, 2001). More specifically, block building has been linked to improved spatial skills, although most of these studies are correlational (ÂBrosnan, 1998; Serbin and Connor, 1979). Similarly, in a preschool popu- lation, two types of block-building skills were associated with two mea- sures of spatial visualization: block design and analyzing and reproducing abstract patterns (Caldera et al., 1999). In an experimental study, children who received instruction on spatial-manipulation improved in spatial visu- alization skills, whereas the control group did not (Sprafkin et al., 1983). Sociodramatic Play One particularly valuable form of play is mature sociodramatic playâ pretend play that lasts 10 minutes or more and involves a theme, props, roles, rules for roles, and language interaction. An example would be four children playing grocery store with play food, a cash register, and shopping carts, and different children playing the roles of store manager, cashier, and customers. Rules restrict the behavior of each playerâfor example, only the cashier can use the register. A Vygotskian-based curriculum, Tools of

250 MATHEMATICS LEARNING IN EARLY CHILDHOOD the Mind, uses this type of mature sociodramatic play as a primary format for childrenâs learning and development (Bodrova and Leong, 2007). In this approach, teachers scaffold childrenâs play skills by engaging them in preparing written play plans and reflecting after play is finished. Teachers work with children to make play more complex over time and to encourage the use of sophisticated vocabulary. Studies of Tools of the Mind show positive impacts on language and early literacy (Barnett et al., 2006, 2008) and on self-regulation and execu- tive functioning (Diamond et al., 2007). The latter is relevant for mathemat- ics learning, as executive function and self-regulation are important for academic success. Executive function has been found to predict academic outcomes in school independent of intelligence or family background (Blair and Razza, 2007). Importantly, the approaches used in Tools of the Mind have been shown to be effective with children from low-income families. Practice During Play Learning many early mathematics skills, such as counting, requires large amounts of practice to become fluent. Play can be an excellent context for children to practice developing abilities. For example, 3- and 4-year- old children will repeatedly attempt to build a block tower or string a set of beads in a pattern until they have mastered the skill to their personal satisfaction. Many mathematics competencies, such as counting, require repeated, often massive amounts of experience, as well as demonstrations, modeling, or scaffolding from adults (Fuson, 1988). Practice is important for consolidating skills, but such practice can be done in the meaningful and motivating context of childrenâs play and with teachersâ assistance as needed. For example, after a walk in the park, children can return to the classroom and examine their collections of leaves, trying to find out who has the most. The teacher can help the children to count their leaf collec- tions, which they choose to do again and again. After repeatedly counting the separate collections, they can work as a group to count the total. Childrenâs play and self-selected activities can provide valuable contexts for mathematics teaching and learning experiences. Capitalizing on their everyday experience is likely to motivate and help them see the relevance in mathematics, as well as lead to complex child-centered projects that include mathematics. Early childhood education has a strong tradition of teachersâ observation of childrenâs play for the purpose of determining how best to respond to support their learning. Teachers can and should be intentional in supporting and mathematizing childrenâs play experiences. However, using only âteachable momentsâ during child-initiated play is unlikely to lead to an effective, comprehensive mathematics program (Ginsburg, Lee, and Boyd, 2008).

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 251 Mathematical Play These examples bring us to another type of play, mathematical play, or play with mathematics itself (Sarama and Clements, 2009; see Steffe and Wiegel, 1994). The following features of mathematical play may be im- portant for supporting learning: (a) it is a solver-centered activity with the solver in charge of the process; (b) it uses the solverâs current knowledge; (c) it develops links between the solverâs current schemes while the play is occurring; (d) it will, via âc,â reinforce current knowledge; (e) it will assist future problem solving/mathematical activity as it enhances future access to knowledge; and (f) these behaviors and advantages apply irrespective of the solverâs age (Holton et al., 2001). Games One recent study provides evidence that board games in which young children count on (1 or 2) along a number list (squares with numbers on them) can be an effective instructional tool for developing their numerical knowledge (Siegler and Ramani, in press). In an experiment conducted in a Head Start program, children played a board game, similar to Chutes and Ladders, four times (for 15 to 20 minutes) over a two-week period. The game used numbered squares for the experimental group and colored squares for the control group. Children using the numbered squares said the numbers on the squares as they moved their token one or two spaces. At the end of the intervention, children who played the number game demonstrated increased knowledge of four different number skills: numerical magnitude comparison, number line estimation, numeral identification, and counting. The gains were still apparent nine weeks later (Siegler and ÂRamani, in press). To achieve such gains through play, however, requires that important math- ematical structures are used by children within the game. Using Concrete Materials and Manipulatives Using concrete materials, such as puzzles and matching games, with task selection and scaffolding adjusted to childrenâs strategies, is effective in moving them through mathematics teaching-learning paths (Clements and Sarama, 2007a). Manipulatives, such as small blocks, cubes, beads, and pegs, are ubiquitous in high-quality early childhood classroom environ- ments. There is evidence suggesting that the use of manipulatives enhances mathematical knowledge for young children (Clements and Sarama, 2007a). This is an area in which there has been a fair amount of mathematics- specific research (Clements and McMillen, 1996), although most work in this area has focused on elementary school children (e.g., Greabell, 1978;

252 MATHEMATICS LEARNING IN EARLY CHILDHOOD Prigge, 1978). Concrete objects are needed for preschoolers to learn non- verbal and counting strategies for addition and subtraction. In fact, children need objects to solve larger number problems until about age 5Â½ (Jordan, Huttenlocher, and Levine, 1992). The manipulatives give meaning to the task, count words, and order (Clements and Sarama, 2007a). That is, at a certain level, number is an adjective rather than a noun for childrenâ â5 kittensâ is meaningful, but â5â as an abstract quantity is not. Pictures can be useful in several ways, such as to illustrate concepts, and young children can learn to interpret pictures (Scott and Neufeld, 1976). However, manipulatives can be more effective than pictures for teaching certain mathematical concepts, because pictures are not manipula- ble, that is, they cannot be acted on extensively and flexibly (Clements and M Â cMillen, 1996; Gerhardt, 1973; Prigge, 1978; Sowell, 1989; Â Stevenson and McBee, 1958). For example, in one study children benefited more from using pipe cleaners than pictures to make nontriangles into triangles ( Â Martin, Lukong, and Reaves, 2007). They merely drew on top of the pic- tures, but they transformed the pipe cleaners. The suggestion that manipulatives and other materials are effective should not be interpreted to mean that young children should always be provided with manipulatives or that simply providing these manipulatives is sufficient. Rather, teachers should be thoughtful about the most appropri- ate manipulative for a specific lesson. Once children have mastered a task using manipulatives, they can often solve simple arithmetic tasks without them (Grupe and Bray, 1999). Using Computers As all-purpose tools, computers can also constitute quite different environments that support mathematics teaching and learning. They can provide effective experiences, ranging from complex problem solving to practice with concepts and skills, managed at the childrenâs level of thinking and at the level of individual tasks. The computer aids the metacognitive aspect of spatial activity, en- abling the child to go beyond the physical world limitations (Clements and Â Battista, 1991; Johnson-Gentile, Clements, and Battista, 1994). For example, children can cut shapes and put them together in new ways. They become aware of and describe the geometric motions they use to solve geometric puzzles (Sarama and Clements, 2009; Sarama, Clements, and Vukelic, 1996)âthat is, doing physical puzzles, they move shapes intui- tively. However, on the computer, they choose the geometric motionâslide, flip, or turnâthat they need. This helps them become explicitly aware of those motions and intentional in their use. Children as young as age 3 have been shown to benefit from focused

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 253 computer activities (Clements, 2003). Connected representations in practice or tutorial computer environments help them form concepts that are inter- related and thus mutually reinforcing. Computer environments can also foster deeper conceptual thinking, including a valuable type of âcognitive playâ (Steffe and Wiegel, 1994). That is, children will pose problems for themselves and explore the computer objects or shapes with the same playful attitudeâand the same beneficial learningâfound in other types of play. Several characteristics of effective computer software can guide its creation and selection (Clements and Sarama, 2005, 2008b; Sarama and Clements, 2002a, 2006): â¢ Actions and graphics should provide a meaningful context for children. â¢ Reading level, assumed attention span, and way of responding should be appropriate for the age level. Instructions should be clear, such as simple choices in the form of a picture menu. â¢ After initial adult support, children should be able to use the soft- ware independently. There should be multiple opportunities for success. â¢ Feedback should be informative. â¢ Children should be in control. Software should provide as much manipulative power as possible. â¢ Software should allow children to create, program, or invent new activities. It should have the potential for independent use but should also challenge. It should be flexible and allow more than one correct response. As with using manipulatives, initial adult support and active mentor- ing has significant positive effects on childrenâs learning with computers (Sarama and Clements, 2002b). Effective teachers closely guide childrenâs learning of basic tasks; then they encourage experimentation with open- ended problems. These teachers are frequently encouraging, questioning, prompting, and demonstrating, without offering unnecessary help or limit- ing childrenâs opportunity to explore. The teachers redirect inappropri- ate behaviors, model strategies, and give children choices. Whole-group discussions that help children communicate about their solution strategies and reflect on what theyâve learned are also essential components of good teaching with computers. Using Movement Another context for learning mathematics is teachersâ use of movement to engage children. There is evidence suggesting that young children benefit

254 MATHEMATICS LEARNING IN EARLY CHILDHOOD from engaging in self-directed movement during instruction, particularly in learning spatial concepts (Poag, Cohen, and Weatherford, 1983; ÂRieser, Garing, and Young, 1994). During a mapping activity, for example, chil- dren are more likely to benefit from actually taking a tour around the classroom than simply thinking about the classroom and being asked to represent it abstractly (Ginsburg and Amit, 2008). Book Reading Book reading is used frequently in early childhood settings. Earlier studies have produced equivocal results with relation to the effect of book reading on mathematics achievement (Hong, 1996). However, several re- cent studies provide evidence that this can be an effective learning context for mathematics instruction (Casey, Kersh, and Young, 2004; Casey et al., 2008; Young-Loveridge, 2004). Young-Loveridge (2004) provides evidence that children exposed to a seven-week pull-out mathematics program, using storybooks, rhymes, and games, made greater gains pre- to posttest on mathematical knowledge than did children not receiving this program. Casey and colleagues (2008) provide evidence that mathematics content (spatial and number skills) delivered in a storytelling context produced greater mathematics learning than delivering the content alone. Notably, the approach, Storytelling Sagas, is based on a series of specially written mathematics storybooks for preschool through Grade 2 that are primarily mathematical and secondarily for literacy. However, the approach demon- strates the important role of language in childrenâs mathematics learning. In this study, researchers compared an intervention that taught a specific set of geometry skills in a storytelling context and alone. Kindergarten children who learned the geometry content in a storytelling context appeared to gain more knowledge, as assessed on both near- and far-transfer tasks. The authors suggest that the storytelling context engages children in the content in ways that more decontextualized instruction does not. FORMATIVE ASSESSMENT A core instructional principle of early childhood education is that teach- ing must be child-centered and âdevelopmentally appropriateâ Â (Copple and Bredekamp, 2009). To promote genuine and enthusiastic learning, the teacher must be sensitive to the individual childâs emotions and must establish a trusting and supportive relationship with him or her. But child- centered and developmentally appropriate teaching requires cognitive as well as emotional sensitivity: to support mathematics learning, the teacher â Pull-out programs remove children from the regular classroom for some portion of the day to give specialized instruction.

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 255 must acquire an understanding of the childâs current mathematical perfor- mance and knowledge. Formative assessment is the process of gaining insight into childrenâs learning and thinking in the classroom and using that information to guide instruction (Black and Wiliam, 1998b) and improve it (Black and Wiliam, 2004). According to the National Mathematics Advisory Panel (2008), âTeachersâ regular use of formative assessments improves their studentsâ learning, especially if teachers have additional guidance on using the assess- ment results to design and individualize instructionâ (p. 47). Formative assessment does not involve formal testing conducted out- side the classroom (with results usually left there); however, it can provide teachers a way in which to track childrenâs progress toward high-quality early learning standards. Formative assessment entails the use of several methodsâobservation, task, and flexible interviewâto collect information about childrenâs thinking and learning and then adapt teaching methods to help them learn. It is often inseparable from teaching and usually not dis- tinctly identified as assessment. Teachers assess children all the time, often unaware that they do so. But formative assessment can also be more delib- erate and organized than is usually the case. This section provides guidance about how teachers can use formative assessment to improve classroom teaching practices so that studentsâ learning needs are best met. Rationale for Formative Assessment The need for sound formative assessment is evident from a variety of theoretical perspectives. Approaches that stress the need to capitalize on the teachable moment (Dodge, Colker, and Heroman, 2002) require teachers to understand when that moment occursâthat is, when the child is ready to learnâand then to exploit it so as to help the child undertake further learning. Using observation to identify the teachable moment is one use of formative assessment (Seo, 2003). Early childhood educators often draw on Vygotskyâs theory to advo- cate effective scaffolding. Scaffolding in turn involves first determining the childâs âactual developmental levelâ so that one can help the child reach his or her potential âthrough problem solving under adult guidance or in collaboration with more capable peersâ (Vygotsky, 1978, p. 86). Determin- ing both actual and potential developmental level, as well as the scaffold- ing useful to help the child traverse this âzone of proximal development,â requires formative assessment. Piagetâs theory stresses the distinction between overt performance and underlying thought (Piaget and Inhelder, 1969). To illustrate: A child says that the sum of 3 apples and 2 apples is 6 apples. The incorrect response is clearly important and needs to be corrected, but even more important is the method used to obtain the response. The child may have got it by

256 MATHEMATICS LEARNING IN EARLY CHILDHOOD faulty memory (âI just knew itâ), faulty calculation (the child miscounts the objects in front of him or her), or faulty reasoning (âI know that 3 and 2 is more than 4 and 6 is 2 more than 4â). Identifying and promoting underly- ing thought requires formative assessment. Contemporary cognitive theories often stress establishing a link be- tween the childâs informal knowledge and what is to be taught (Baroody, 1987; National Research Council, 1999; Resnick, 1989, 1992). The child brings to the task of learning a body of prior knowledgeâan âeveryday mathematicsâ that is often relatively powerful and sometimes a source of misconceptions. In either case, the teacher needs to understand the childâs current cognitive state (the everyday mathematics) in order to adjust in- struction to it. Sometimes the everyday mathematics can serve as a fruit- ful basis for further development; the childâs learning may in part involve mathematizing what she or he already knows. Sometimes the teacher needs to employ methods to help the child abandon everyday concepts in favor of more accurate notions, as when the child believes that the symbol = means âget an answerâ instead of an equivalence relation (Seo and Ginsburg, 2003), or that a long, skinny scalene triangle is not an acceptable triangle (Clements, 2004). Those who practice behavior modification also need to employ for- mative assessment to acquire an accurate account of the childâs current behavior so they know what to shape. Careful observation of behavior and decisions about appropriate reinforcement can also be conceptualized under the rubric of formative assessment. In brief, many theoretical approaches advocate getting information about the childâs current behavior, thinking, and learning so that effective teaching can be implemented. It is hard to imagine a theory of teaching that would advocate ignorance of the childâs mind or behavior. Three Kinds of Formative Assessment Formative assessment is a very natural and commonplace activity for teachers, who do it all the time without necessarily knowing that what they do is assessment. Here we discuss three major kinds of formative assess- ment: everyday observations, tasks, and interviews (see Box 7-3). These ev- eryday practices of observation, presenting tasks, and interviewing involve an informal, often unplanned, implementation of formative assessment, which is so bound up with everyday teaching that it often goes unrecog- nized. Yet the three types of formative assessment can be rigorous, focused, and deliberate. The early childhood assessment systems discussed here include widely used integrated programs as well as mathematics-specific programs: Big Math for Little Kids, Building Blocks, Core Knowledge, Creative Curriculum, High/Scope, and Number Worlds.

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 257 BOX 7-3 Formative Assessment Teachers often use everyday observations to make inferences about childrenâs abilities. The teacher sees that Juanita often spontaneously names shapes as she places them on the table. She can identify large and small objects and red and green objects as rectangles, and she even knows the name for a trapezoid. The teacher concludes that she can see the differences among various shapes, un- derstands that color and size are irrelevant, and even knows some shape names. She is now ready to learn to mathematize her knowledgeâthat is, to analyze the properties of shapes so that she will understand explicitly what defines a rectangle and other shapes. Teachers also give children specific tasks to elicit their understanding. The teacher has seen that Juanita spontaneously names rectangles and trapezoids but has never seen her name a square. So the teacher shows her a large red square and a small green one and asks what they are. Juanita says that the red one is a square but that the green one is not. Having given this specific task, the teacher now concludes that Juanita knows the name for square but applies it in a rather unusual way. The teacher is puzzled because Juanita was able to identify small green rectangles as rectangles, but she cannot ignore size or color in the case of squares. Why does she do this? To find out, the teacher goes a step further. She wants to know how Juanita thinks about squares. What makes something a square? Whatâs the role of color and size? Can she talk about it? So she interviews Juanita. She asks her why she said this large red object was a square, whereas this small green one was not. Juanita says that color does not matter, but that squares have to have 4 sides the same length and have to be big. Why do they have to be big? She does not know. The teacher concludes that her lessons on shape should include specific at- tention to issues of size and when it is relevant or not relevant. And the teacher has a clue about how to proceed. She will put Juanita in a situation of cognitive conflict, which, according to Piaget (1985), is a major impetus to cognitive growth (Limon, 2001). She points out to Juanita that color and size do not matter for rectangles. Why should they matter for squares? Juanita looks puzzled. But then she quickly agrees that of course squares can be small, too. Her expression says: How silly to think otherwise! Assessments like these take place in many classrooms. Teachers observe their children, set them brief everyday tasks, and question them about their thinking. They do these classroom assessments on the fly, spontaneously, and without special preparation. Sometimes children learn a good deal, and sometimes they donât. Observation Observation involves several components. One is obtaining useful in- formation. The teacher needs to observe relevant aspects of an individual childâs mathematical behavior. For example, she needs to observe that, in

258 MATHEMATICS LEARNING IN EARLY CHILDHOOD free play, the child is not only comparing the lengths of two blocks but also makes the mistake of failing to use a common baseline. This is not easy to do when the teacher must observe and supervise a room full of young children who have many needs and who exhibit complex patterns of be- havior. There is an enormous amount of behavior taking place at any one moment in the classroom day. Nevertheless, it is possible for teachers to focus observations on at least a few children in order to provide activities that promote further mathematics learning. A second important component of observation is interpretation of the evidence. The observer needs to understand what the behavior means. In the example above, the childâs failure to use a common baseline in comparing the length of blocks indicates a common misconception of a fundamental idea underlying measurement (Clements, 2004; Piaget and Inhelder, 1967). Teachers need to be aware of and understand this misconception in order to interpret behavior accurately. Observation is very theoretical. To interpret everyday behavior, the teacher needs to be familiar with the development of mathematical thinking, as well as with the mathematics about which the child is thinking. Teachers need to receive professional development about learning in early childhood to be able to effectively interpret their observa- tions of childrenâs mathematical thinking. A third component of observation is careful evaluation of evidence. Suppose the teacher sees a child spontaneously place a red and a blue isosceles triangle into one collection. But the child does not place a red skinny irregular (scalene) triangle into that same collection. Does this evidence suggest that the child has an understanding of triangles? On one hand, maybe the child did not see the small triangle and, had she seen it, perhaps would have placed it with the others, thus demonstrating at least some understanding of what defines a triangle. On the other hand, maybe she did see it and decided not to include it with the others because it was so strange looking, not an isosceles triangle, thus revealing that she had a narrow concept of a triangle. The evidence is inconclusive, and one can- not make a firm conclusion; both interpretations are possible. Evaluation of evidence requires skills of critical thinking that do not come easily and often need to be taught (Kuhn, 2005). How well do teachers observe mathematical behavior? Research on this issue appears to be lacking. But there are reasons to be pessimistic about the likelihood that they make useful and insightful observations. Teachers sel- dom have time to observe behavior during free play; they tend to have their hands full with management and discipline (Kontos, 1999). Also, teachers may not know what to look for. As Piaget said, observation requires knowl- edge of what is to be observedâin this case, mathematical thinking: âif they are not on the lookout for anything . . . they will never find anythingâ (Piaget, 1976a, p. 9). In addition, as Chapter 8 discusses, early childhood

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 259 teachers have little training in either early mathematics education or math- ematics, especially in the analysis of mathematical behavior. Organized systems of observation.â Teachers need guidance in the ob- servation of mathematical behavior. Their college or university education may have provided some useful experience in observation. And although popular textbooks on the subject (e.g., Boehm and Weinberg, 1997; ÂCohen, Stern, and Balaban, 1997) discuss general issues of observation, they do not discuss in any depth the observation of mathematical behavior in particular. Several widely used curricula offer guidance in observation of math- ematical behavior. They may provide checklists for observation of various topics with directions about which behaviors should be recorded. For ex- ample, one form instructs teachers to record their observations of a childâs knowledge of number and operations. The checklist specifically focuses on counting aloud in the correct order and grouping objects. A checklist like this is broad and provides teachers with little guidance for assessing chil- drenâs mathematical knowledge. Rather, teachers should use the checklist as a start to assessing where children are on the mathematics teaching-learning path. Ideally, teachers would use follow-up questions and various tasks in conjunction with observation to ascertain the childâs level of mathematical knowledge. At best, the observations give only an extremely crude idea of the childâs interests and provide very little information about his or her knowledge. Other widely used early childhood mathematics assessment systems offer the opportunity for the teacher to collect interesting anecdotes about individual children. For example, teachers create a personal log of each childâs actions and abilities with spaces for writing numerous anecdotesâ brief reports on individual childrenâs classroom behavior and work samples that highlight their developing abilities. Again, we note that observation of mathematical behavior requires training and theoretical background. These assessment systems do not seem to provide evidence concerning the quality of observations or their usefulness. Other curricula and their related assessment systems stress the analysis of various products of learning activities. The Reggio Emilia group in Italy uses âpedagogical documentation to capture learning moments through ob- servation, transcriptions and visual representations that provoke reflection and inspire teachers, children, and parents to consider the significance of the interactions taking place, and the next steps to be taken in teaching and learningâ (MacDonald, 2007, p. 232). Strengths and weaknesses of observation.â Observation can be an ex- tremely powerful method. It may provide insight into the childâs spontane-

260 MATHEMATICS LEARNING IN EARLY CHILDHOOD ous interests and everyday competence in the absence of adult pressure or constraint. Observation deals with behavior in âauthenticâ situations, like block play or snack time. The teacher may learn from careful observation that the child possesses a competence that is not expressed when he or she is tested or given instruction. At the same time, no single method of assessment is perfect, always accurate, or completely informative, and observation has some limita- tions. Sometimes, the teacher can wait indefinitely before observing truly important behavior. Sometimes, the childâs behavior does not express the true extent of her or his competence. As Piaget said: âHow many inex- pressible thoughts must remain unknown so long as we restrict ourselves to observing the child without talking to him?â (Piaget, 1976a, pp. 6-7). Thus, observation may show that the child does not seem to sort objects by common shapes, putting triangles into one group and rectangles into another. Instead, he or she places them all into one messy collection and tells stories about them. Does this mean that the child does not understand the difference between triangles and rectangles? The observer will never know without explicitly asking the child to sort them. This is a task, the next type of assessment. Task Sometimes to find out about a childâs learning, thinking, or perfor- mance, one presents him or her with some kind of task, a simple problem to solve. The teacher may ask, âWhat do you see in this picture book? What is the clown doing?â Or the teacher may say, âWhat do you call this thing [a triangle]?â The childâs response may give an indication of his or her competence. If the child says, âThe clown is juggling three balls,â then the teacher may learn something about the accuracy of his or her counting skills. If in response to the question about the triangle the child says, âI donât know,â then the teacher has learned that the child may not be able to produce the correct word or apply it to at least a certain kind of triangle. Yet from Vygotskyâs perspective, the childâs response to this task may be an indication only of current developmental level. The teacher therefore goes on to provide a little scaffolding, asking, âWhat shape is it?â The child then answers, âa triangle,â and this indicates his level of potential development. In brief, tasks are initiated by the teacher to learn about the childâs performance with respect to a particular topic of interest to the teacher. Basically, the teacher wants to know whether the child can do somethingâ count, recognize a triangle, or make a patternâperhaps with a little help. Evidence about how well teachers employ tasks in the classroom ap- pears to be lacking. Yet it may be relatively easy for the teacher to ask the

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 261 child to respond to simple tasks with which the child is engaged during free play (âWhat is that block called?â) or to ask questions about the topic of the teacherâs instruction (âWhich animal is first in line?â). Of course the teacher must interpret the childâs response with accuracy and is therefore faced with some of the same difficulties as discussed in the case of observa- tion. The teacher must understand the development of mathematical think- ing to appreciate the meaning of the childâs response. Organized systems of tasks.â Some early childhood curricula present a series of tasks as the basis of their assessment system. For example, an item might instruct the teacher to use manipulative counters (e.g., blocks) to create different groups of objects containing between one and four items and to arrange the groups in different configurations (e.g., straight line, random grouping). The teacher would need to be sure that the child had several opportunities to correctly count, and she would record whether the child counted correctly and, for incorrect counts, the kinds of mistakes the child made. The task employed, namely to count a given number of objects, is com- mon both in the research literature and in some tests at this age level (e.g., Ginsburg and Baroody, 2003). What appears to be lacking, however, is any indication of how to interpret the results. What does it mean, for example, if the child can count a randomly arranged set of 3 but not 4, which he or she can count if it is placed in a line? Several primary mathematics cur- ricula have a large collection of tasks with a clear theoretical basis (Case and Okamoto, 1996; Griffin, 2004). Strengths and weaknesses of tasks.â The strength of this method is that it provides information about the childâs performance on a task in which the teacher has an interest. The teacher is attempting to teach something about pattern and needs to know whether the child is âgetting itâ so that she can take the next appropriate instructional step. There is some evidence that, at least at the elementary school level, frequent monitoring of student behavior can improve performance (Fuchs et al., 1999). But there are at least two basic weaknesses in using tasks. The first is also its strength, namely that the teacherâs interests determine the choice of task. The teacher is trying to teach pattern, but the child may in fact be more interested in or dealing with another topic, like the shapes of the objects intended to comprise the pattern. Because children do not always learn what teachers teach, teachersâ questions about what they are trying to teach do not necessarily reveal what the child is learning. Second, the childâs behavior may indicate success or failure on the task but does not necessarily reveal how the child construes or solves the task. As Piaget pointed out, it is not enough to ascertain the childâs answer; one must in addition learn

262 MATHEMATICS LEARNING IN EARLY CHILDHOOD how the child got it. It is possible for the correct answer to be the result of a mechanical process devoid of understanding and for an incorrect answer to be the result of insightful thinking. Flexible Interview A constructivist and child-centered perspective demands that the teacher go beyond observation and tasks to probe the childâs thinking. Observation and tasks can provide useful information about performance, but the flex- ible interview is needed to dig below the surface to learn what the child is thinking. A truly child-centered, cognitively sensitive approach requires ask- ing how the child solved the problem, how she got the answer, and why she said what she did. This kind of questioning originated in Piagetâs âclinical interview methodâ (Piaget, 1976a), which we term âflexible interviewing,â so as to avoid any connotation of the âclinical interviewâ devoted to the investigation and cure of pathological phenomena. Flexible interviewing involves several steps (Ginsburg, 1997). First, the interviewer notices what seems to be an important child behavior worthy of further investigation. Sometimes this occurs in the course of naturalistic observation of everyday classroom activities. More frequently it occurs when the child gives an interesting response to a task. In either event, the interviewer follows up in various ways. He may rephrase the initial ques- tion, ask the child to talk about how she or he solved the problem, or request that the child expand on an answer or justify it. Occasionally the interviewer may challenge a childâs response and ask her to prove why it is not correct. The essential questions include: âHow did you figure it out? How did you know? How did you get the answer? Tell me more about it. How do you know you are right?â In general, the rationale is that, if the goal is to learn what the child is thinking, the teacher must engage in flexible interviewing, asking the child to elaborate on his or her ways of interpreting and approaching a problem. Note that the flexible interview involves elements of both the task and ob- servation. The interviewer frequently begins with a simple task for the child to solve (âWhat do you call this figure?â) and then follows up on the childâs response (âWhy do you think it is a triangle?â). And as the child seems to be thinking about the problem or provides an answer, the interviewer care- fully observes the childâs behavior to determine, for example, whether he points to a certain object or looks confused or seems to whisper his thought aloud. Indeed, Piaget maintained that the interview method combines the best of observation and task. Flexible interviewing involves a good deal of skill and mental agility. It requires the same kind of observational sensitivity, critical thinking, and interpretive skills discussed in connection with observation and task. It also

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 263 requires the interviewer to think on her feet, to improvise, and to come up with the right follow-up question on the spot. How frequently and well do teachers employ the flexible interview in the classroom? Research on the issue seems to be lacking. At the same time, flexible interviewing, although difficult, is a natural form of human interaction in which the participants attempt to make sense of problems and how they can be solvedââclinical interviewing is a species of naturally occurring mutual inquiryâ (diSessa, 2007, p. 534). Asking a person why he or she said or did something is an entirely familiar form of discourse and not necessarily artificial or lacking in ecological validity. Organized systems.â Few curricula provide extensive guidance in flexible interviewing. D.M. Clarke and colleagues (Clarke et al., 2001) have used a developmental trajectory theory as the basis for development of an exten- sive collection of âtask-based interviewsâ for children beginning at age 5. The collection of interview items is intended to form the basis for a comprehensive program of professional development, as well as to serve as a formative assessment tool for the teacher. âThe [theoretical] framework of growth points provides a means for understanding young childrenâs mathematical thinking in general, the interview provides a tool for assess- ing this thinking for particular individuals and groups, and the professional development program is geared towards developing further such thinkingâ (p. 2). In many respects, the work is a model for what should be done in this area. To date, few early childhood curricula provide guidance on flex- ible interview. Big Math for Little Kids (Ginsburg, Greenes, and Balfanz, 2003), however, includes extensive guidance on flexible interviewing for each major topic. The Number Worlds curriculum (Griffin, 2007) offers an assessment system that largely involves a series of tasks (boldly called âtestsâ), some of which include flexible interview follow-ups. For example, âHow many more smiley faces does the hexagon have than the triangle has? How did you figure that out?â (p. 72). After these instructions, an example of a possible child response is presented: â2 more; I counted to 3 and there were 2 left that I didnât countâ (p. 72). In general, the focus on flexible interviewing, even though it is at the very heart of a child-centered approach, is limited in current curricula. Strengths and weaknesses.â The flexible interview can provide basic and often surprising information about childrenâs knowledge. It sometimes shows that the child who seems to know something really doesnât, and the child who doesnât seem to know something really does. This kind of information can help teachers overcome preconceptions they might have about childrenâs abilities. For example, teachers may expect low-income children to be more capable of procedural than conceptual knowledge.

264 MATHEMATICS LEARNING IN EARLY CHILDHOOD The results of a flexible interview may help to disabuse the teachers of this preconception. The flexible interview allows the interviewer to make sense of puzzling observations of everyday behavior or responses to tasks. The benefits ac- cruing from this knowledge may be considerable: Understanding the childâs perspective can provide a sensitive guide to instruction. If the childâs wrong response was the result of a misinterpretation of the question, the teaching solution is different from what is needed if the response resulted from a basic misunderstanding. Also, use of the method entails secondary benefits. Flexible interviewing requires that teachers talk a great deal with children. Furthermore, flexible interviewing not only promotes the teacherâs language but also requires it from the child. Flexible interviewing stresses to the child the importance of talking about oneâs thinking, justifying oneâs conclusions, and in general engaging in mathematical communication, which as we have seen is one of the main goals of mathematics education at all levels (National Council of Teachers of Mathematics, 2000; National Research Council, 2001a). Indeed, the very process of being interviewed may have a salutary effect on the child. There is some evidence with older children that self-explanation (providing an explanation of material recently studied) promotes increased understanding (Chi et al., 1994). Similarly, the requirement to explain oneâs thinking might help one to examine, organize, and in the process even improve it. Interviewing can be hard to do well, especially when very young chil- dren are involved. As noted, it demands interpretative skill, creativity, and flexibility in questioning. It is easy to ask misleading or uninformative ques- tions and distort results; it requires considerable skill and sophistication to do really well. It is hard for young children to be aware of their mental processes or to describe them in words (Flavell, Green, and Flavell, 1995; Kuhn, 2000; Piaget, 1976). The strength of the methodâits flexibility and sensitivity to the individualâis at the same time its weakness. Some General Remarks In general, childrenâs developmental characteristics make it difficult, although not impossible, to assess their learning, thinking, or performance. They can be shy, uncooperative, nonverbal, impatient, noncommunicative, and so on. Their self-regulation skills are imperfectly developed (Bronson, 2000), and they are egocentric (Piaget, 1955). The result is that assessment of the child at any one point in time may be inaccurate. But that does not mean teachers should not attempt to assess. It means that assessment needs to be done as sensitively as possible. Similarly, it is hard to diagnose a 2- year-old childâs hearing, but there is a moral obligation to do it as well as possible.

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 265 Similarly, because of the natural fluctuation and rapid development of childrenâs behavior, a single assessmentâwhether done by observation, task, or interviewâmay not provide accurate information. It is necessary to assess young children frequently and to base educational decisions on multiple sources of information (National Research Council, 2001b, 2008). Formative assessment should be complementary to program evaluation, which is conducted outside the classroom (see Box 7-4). Also, it is possible and sometimes desirable to blend the three methods. Thus, the teacher can observe in the natural setting and at the same time give the children simple tasks and even interview them. The importance of teachersâ understanding of their students cannot be overemphasized. According to the National Research Council report Adding It Up: Helping Children Learn Mathematics, âinformation about students is crucial to a teacherâs ability to calibrate tasks and lessons to studentsâ current understanding. . . . In addition to tasks that reveal what students know and can do, the quality of instruction depends on how teach- ers interpret and use that information. Teachersâ understanding of their studentsâ work and the progress they are making relies on . . . their ability to use that understanding to make sense of what the students are doingâ (National Research Council, 2001a, pp. 349-350). Teachersâ understanding of their students is the key, or at least one key to successful teaching. Finally, although formative assessment shows great promise, the meth- ods of assessment have not been clearly linked to instructional interven- tions. In fact, there seem to be few if any research studies that investigate the power of formative assessment to improve student achievement (exceptions include Black and Wiliam, 1998a, 1998b; Heritage, Kim, and ÂVendlinkskil, 2008). One of these studies suggests that, although elementary school teachers are reasonably skilled in interpreting student behavior, they have difficulty linking the assessment to subsequent teaching (Heritage, Kim, and Vendlinski, 2008). Clearly, further research and development are required. Development is needed to create links between assessment and instruction, and research is needed to investigate the effectiveness of those links. All of this should be easier to do in the teaching-learning paths described in this report because they keep the teacher situated in an organized set of goals with directionality both for individual children and for the class. RESEARCH ON THE EFFECTIVENESS OF MATHEMATICS CURRICULUM Although this chapter addresses the topics of pedagogy and curriculum separately, in practice there is often no clear distinction between the two. This is especially true in early childhood education. Early childhood cur- riculum has traditionally emphasized the process of teaching and learning rather than the content of what children are learning (National Association

266 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 7-4 Comments on Program Evaluation Programs for young children, like those for older ones, need to be held ac- countable. People want their children to receive the best early childhood math- ematics education possible. There is no dispute as to the necessity for evaluation of programs, but the evaluation has to be as fair, sound, and based on scientific evidence and theory as much as possible. Current evaluations are informative but limited. Several obstacles need to be overcome to improve the quality of evaluation efforts. First, it is hard to assess young children. Just as in the case of formative assessment, observation alone is insufficient, and the adult must employ some form of task or interview. But as pointed out earlier, even when a friendly adult does the assessment on a 1-to-1 basis, young children can be shy, uninterested, uncooperative, or inconsistent. Conditions like these require highly trained adult assessors who can engage children and approach the assessment with sensitivity and intelligence. This in turn âcreates significant feasibility issues for large-scale accountability initiatives. Relatively large numbers of assessors must be trained and supervised. Quality assurance is another major challenge: the consistency, credibility and integrity of child assessment reports must be established and monitoredâ (National Early Childhood Accountability Task Force, 2007, p. 23). Second, and even more important, there are few psychometrically valid as- sessment instruments to use in the evaluation of early mathematics education programs. Current instruments either focus on a narrow aspect of early mathemat- ics, like number (e.g., Ginsburg and Baroody, 2003), or lack extensive psycho- metric support. A useful assessment should cover a broad array of mathematical knowledge, from number to pattern to space. Also, it should examine the âproduc- tive disposition,â that is, the âhabitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and oneâs own efficacyâ (National Research Council, 2001a, p. 5). And the evaluation it should be easy to administer and enjoyable to take. Such an instrument with sound psychometric qualities is not yet available. Because evaluations are only as valuable as the measures they employ, current evaluations must be considered of limited value. Finally, it is as important to assess program quality, including teaching, to assess the childrenâs performance. At present there are few psychometrically sound measures of early mathematics teaching or program quality (for an in-depth discussion on this topic, see National Research Council, 2008). Just as early mathematics education has been neglected for many years, so have the methods needed to evaluate it. In view of the former, the latter should come as no surprise. As a result, considerable research and development need to be conducted to create evaluation methods appropriate for examining the quality of programs and their success in educating children. for the Education of Young Children, 1997). Given this view of curricu- lum, research and debate have focused on which curriculum model is most effective in supporting childrenâs short-term and long-term development (Epstein, Schweinhart, and McAdoo, 1996). Many early childhood educators are not comfortable with defining

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 267 curriculum as a written plan or specifying scope and sequence in advance. This concern grows out of the strong tradition of emergent curriculum in early childhood education (Jones and Nimmo, 1994). According to this perspective, the focus should be on children, not on curriculum. Advocates of emergent curriculum believe that childrenâs interests and needs should determine what goes on in a classroom rather than a predetermined plan. They also assume that a planned scope and sequence cannot be responsive to childrenâs individual and cultural variations. Emergent curriculum is often implemented using the project approach (Katz and Chard, 1989), in which children and teachers engage in an in- tensive investigation of a topic of interest. Sometimes people refer to the project approach as a curriculum model, but it is more akin to a teaching strategy or context. In recent years, advocates of the project approach have been more specific about how state standards can be incorporated and met during the planning and implementation of a project (Helm and Katz, 2000). To help children achieve learning goals, educators have begun to emphasize intentional teaching in an emergent curriculum or project ap- proach (Epstein, 2007). During the past 15 years, early childhood practice in the United States (and throughout the world) has been influenced by the Reggio Emilia ap- proach (Edwards, Gandini, and Forman, 1998). The approach is not a curriculum, nor is it a model. It is a coherent set of principles and practices that reflect a sociocultural perspective on learning and development. A key element of the approach is serious project work involving small groups of children collaborating with teachers to undertake investigations, theorizing, representing, revisiting experiences, and revising conceptualizations. Project work often arises from real rather than contrived situations. For example, one school needed a new table and the carpenter asked for measurements, a project documented in a book called Shoe and Meter (Reggio Children, 1997). The children worked together to figure out how to measure the table. They tried measuring using their various body parts but were dis- mayed to discover that each personâs foot was a different length. Finally, they chose one childâs foot to be their standard length. Then, they held his foot up to the ruler and determined how it compared, and so on. In the past decade in the United States, there has been an explosion in commercially published early childhood curriculum resources. In 2007, the PreK Now website listed 27 research-based curricula for preschool children (see http://www.preknow.org). Some of these curricula are Âcomprehensiveâ designed to address all domains of childrenâs learning and development. These comprehensive programs tend to be organized into units, often called themes, based on childrenâs predictable interests, but they are also broad enough to connect many different experiences and achieve multiple goals. Such themes usually include such topics as weather, animals, or construc- tion. Comprehensive curricula are sometimes integrated curricula, in which

268 MATHEMATICS LEARNING IN EARLY CHILDHOOD one topic or experience is designed to meet goals across subject matter areas, such as reading a book that includes scientific information. Some comprehensive curricula have a limited number of themes, six to nine, allowÂing for more in-depth attention to the topic. Others change the topic weekly. In the past (and today as well), teacher-developed preschool âcur- riculumâ was often theme-based, consisting of a series of activities related to the changing seasons, holidays, and events in childrenâs lives, such as visits to the firehouse. Often newly available curriculum resources are designed to provide instruction focusing on language, literacy, and/or mathematics. In some of these resources, learning and instruction are devoted to a single content domain, such as mathematics or literacy skills. Sometimes, a curriculum resource focuses on only one aspect of one domain rather than on an entire domain, such as phonological awareness or social-emotional development. These resources require teachers to figure out how to offer a coherent cur- riculum that covers all important learning goals. Little research is available on the extent to which preschool programs use specific curriculum. The six-state study of prekindergarten conducted by the National Center for Early Development and Learning provides some evidence about curriculum use in state-funded preschool programs (Early et al., 2005). Only 4 percent of teachers reported having no curriculum, 14 percent used a locally developed curriculum, and 9 percent used a state cur- riculum. The most widely used curricula are High/Scope, with 38 percent of classrooms, and Creative Curriculum, accounting for 19 percent (National Center for Early Development and Learning Prekindergarten Study, 2005). These two curricula are also the most widely used in Head Start programs (U.S. Department of Health and Human Services, 2006). There is increasing agreement over many early childhood teaching practices, often called developmentally appropriate practice (see previous section on effective instruction; see also Copple and Bredekamp, 2006, 2009). Developmentally appropriate practice as defined by the National Association for the Education of Young Children (Copple and Bredekamp, 2009) calls for teachers to make decisions that are informed by knowledge of child development and learning, knowledge about individual children, and knowledge about the social and cultural context in which they live. The concept is that teachers adapt the curriculum and teaching strategies for the age, experience, and abilities of individual children to help them make learning progress. Despite the support for developmentally appropriate practice in the field, there is less acceptance of the need for a written curriculum, especially if that curriculum provides a planned sequence of teaching and learning op- portunities (Lee and Ginsburg, 2007). Yet such a curriculum organized by research-based teaching-learning paths, such as those described in Chapters 5 and 6, or at least some learning path organization of the mathematics

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 269 activities over the year, is needed to ensure that all children have a chance to learn the topics in the learning path. Such systematic opportunities are needed to help improve mathematical outcomes for all young children. Mathematics Curriculum A limited amount of research is available on the effectiveness of specific mathematics curricula or curricular approaches. As described earlier, most early childhood programs do not include primary mathematics experiences or focused mathematics time but rather rely on integrated mathematics experiences in which mathematics is a secondary goal and often incidental (Preschool Curriculum Evaluation Research Consortium, 2008). However, incidental mathematics instruction appears to be less effective than activi- ties with a primary focus on mathematics, although this evidence is only correlational (Starkey et al., 2006). In addition, reliance on incidental or integrated mathematics may con- tribute to the fact that little time is spent on math. For example, in the Preschool Curriculum Evaluation Research (PCER) Study, conducted by the U.S. Department of Education, a literacy-oriented curriculum (Bright Beginnings, available at http://www.brightbeginningsinc.org/) and a devel- opmentally focused one (Creative Curriculum, available at http://www. teachingstrategies.com/) engendered no more mathematics instruction than a control group (Aydogan et al., 2005). Other research (Farran et al., 2007) found a negligible time devoted to mathematics in a literacy-oriented com- prehensive curriculum. It is important to note, however, that in response to changing stan- dards and current research, the developers of Creative Curriculum have recently added a mathematics component to their approach (Copley, Jones, and Dighe, 2007). In addition, the High/Scope curriculum (Hohmann and W Â eikart, 2002) is developing a more challenging focused mathematics com- ponent (Schweinhart, 2007). Large effect sizes support the strategy of designing a mathematics cur- riculum built on comprehensive research-based principles, including an emphasis on hypothesized teaching-learning paths (Clarke, Clarke, and Horne, 2006; Clements and Sarama, 2007b, 2008a; Thomas and Ward, 2001; Wright et al., 2002). Most of these studies also emphasized key devel- opmental milestones in the main teaching-learning paths, promoting deep, lasting learning of critical mathematical concepts and skills. Teaching-learning paths or learning trajectories are useful instruc- tional, as well as theoretical, constructs (Bredekamp, 2004; Clements and Sarama, 2004; Simon, 1995; Smith et al., 2006). The developmental pro- gressionsâlevels of understanding and skill, each more sophisticated than the lastâare essential for high-quality teaching based on understanding both mathematics and learning. Early childhood teachersâ knowledge of

270 MATHEMATICS LEARNING IN EARLY CHILDHOOD young childrenâs mathematical development is related to their studentsâ achievement (Carpenter et al., 1988; Peterson, Carpenter, and Fennema, 1989). In one study, the few teachers that actually led in-depth discussions in reform mathematics classrooms saw themselves not as moving through a curriculum, but as helping students move through levels of understanding (Fuson, Carroll, and Drueck, 2000). Furthermore, research suggests that professional development focused on developmental progressions increases not only teachersâ professional knowledge but also their studentsâ motiva- tion and achievement (Clarke, 2004; Clarke et al., 2001, 2002; Fennema et al., 1996; KÃ¼hne, van den Heuvel-Panhulzen, and Ensor, 2005; Thomas and Ward, 2001; Wright et al., 2002). Thus, teaching-learning paths can facilitate developmentally appropriate teaching and learning for all children (see Brown et al., 1995). A few words of caution are in order in interpreting findings about mathematics curriculum research. In the early childhood context, random- ized control trials in mathematics may tend to overstate effect sizes because teaching some mathematics will always be more effective than teaching no or almost no mathematics (which is usually what the control classrooms are doing). Comparing the large effect sizes of the mathematics PCER study (Starkey et al., 2006) with the results of no significant differences for most of the literacy PCER studies does not mean that mathematics curricula are effective while literacy curricula are not. Preschools have had a decade of focus on literacy, so the control groups in those studies were doing a lot of literacy as well as the experimental groups. Curricular research does have great potential to advance understanding of effective instructional strate- gies, but only if this research is conducted with this explicit goal in mind. The inclusion of observational measures, both of fidelity to the curriculum and generalized instructional processes, greatly enhances the ability of the research to speak to specific teaching strategies that may be most important for student learning. For example, Clements and Sarama (2008a) included extensive obser- vation using the Classroom Observation of Early Mathematics Environment and Teaching (COEMET) and Fidelity of Implementation during a ran- domized control trial of two mathematics curriculaâBuilding Blocks and Preschool Mathematics Curriculum (PMC; Klein, Starkey, and Ramirez, 2002)âand a control condition. The results indicate that research-based mathematics preschool curricula can be implemented with good fidelity, if teachers are provided ongoing training and support. Using data from the COEMET the researchers identified instructional strategies that significantly predicted gains in childrenâs mathematical knowl- edge over the course of the year: (1) the percentage of time the teacher was actively engaged in activities, (2) the degree to which the teacher built on and elaborated childrenâs mathematical ideas and strategies, and (3) the degree to which the teacher facilitated childrenâs responding. Examples are provided

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 271 in Box 7-5. In addition, the researchersâ inclusion of multiple curricula also facilitates generalization beyond the effects of a specific curriculum to the broader approaches that may be embedded in it. The ability of curricular research to inform effective practice would also be enhanced if individual curricula more clearly defined the instructional approaches embedded in them. Often curricula distinguish themselves in terms of content (e.g., covering geometry or not) and generalized approach (e.g., whole-group versus small-group instruction) more than in the instruc- tional strategies that are endorsed and supported in the activities. Thus, any findings that one curriculum is more effective than another provides little knowledge about specific teaching strategies that may be useful. Improving Mathematics Outcomes for Children in Poverty The limited amount of time devoted to the subject of mathematics may account for why Head Start children make little or no gain in mathematics. For example, using randomized assignment, the Head Start Impact Study found no significant impacts for the early mathematics skills of 3- or 4- year-olds (U.S. Department of Health and Human Services, 2005). Other examples include control groups from experiments (Clements and Sarama, 2007b; Clements and Lewis, 2009; Starkey et al., 2006). The control group in one study, for example, made small gains in number, but little or no gain in geometry (Clements and Sarama, 2007b) and Head Start children made no significant gain in any area of mathematics during the school year (control classrooms continued using their schoolâs mathematics activities, which were informed by a mixture of influences ranging from commercially published curricula to homegrown materials based on state standards). Research demonstrates that interventions with a primary focus on mathe- matics have the potential to increase the mathematics achievement of children living in poverty and those with special needs (Campbell and Silver, 1999; Clements and Lewis, 2009; Fuson, Smith, and Lo Cicero, 1997; Â Griffin, 2004; Griffin, Case, and Capodilupo, 1995; Ramey and Ramey, 1998), which can be sustained into first (Magnuson et al., 2004) to third grade (Gamel-McCormick and Amsden, 2002). For example, both the Building Blocks and Big Math for Little Kids curricula significantly and substantially increase the mathematical knowledge of children from low-income families (e.g., ÂClements and Lewis, 2009; Clements and Sarama, 2007b, 2008a). The success, even in comparison to other curricula, is probably due to the shared core of learning trajectories (teaching-learning paths) emphasized in the cur- riculum and the professional development that ensures that teachers spend time teaching appropriate mathematics topics during the year. Another example, the Rightstart program (Griffin, Case, and Siegler, â Now published as Number Worlds (Griffin, 2004, 2007).

272 MATHEMATICS LEARNING IN EARLY CHILDHOOD BOX 7-5 Examples of Low-Quality and High- Quality Mathematics Teaching 1. The teacher was actively engaged. Consider a situation in which the teacher has put out a mathematics center with play dough. A nonengaged teacher talks for An engaged teacher works with several several minutes exclusively to another children at the center until she observes adult in the room. they âhave the ideaâ of the activity. She keeps her eye on the center and encourages children to keep building. Another works with children in another In another room, the teacher works with area of the room, but neither she nor children in another area of the room, the aide visits the math center. while the aide visits the center and helps or acknowledges the childrenâs mathematics work. 2. The teacher built on and elaborated childrenâs mathematical ideas and strategies. Consider a situation in which children are to put some dinosaurs on a play scene and describe what they did. One child put out dinosaurs, but then just pointed. A teacher who did not build on or A teacher who does build on or elabo- elaborate childrenâs ideas, merely rate childrenâs ideas says, âWhat do says, âOK.â you have there?â The child does not respond. Another teacher asks, âWhat are the âWhat are these two dinosaurs doing?â dinosaurs doing?â âFighting!â says the âFighting.â âHow many are in your child, as he picks up the dinosaurs pond?â âTwo.â âWhat are they going to and loudly demonstrates the fighting. see? On the hill here? âA T-rex. One The teacher says, âThatâs enough T-rex.â âWow! Four dinosaurs, two here of that!â excuses this child and calls and two on the pond, are seeing that another child to the activity. Tyrannosaurus Rex. Iâll bet they are scared!â 3. The teacher facilitated childrenâs responding. Consider a situation in which the teacher asks one child to figure out how many 1 more than 3 is. One teacher who does not facilitate A teacher who does facilitate childrenâs childrenâs responding says to a child responding says, âCan you show me 3 who does not answer, âSomeone else to get started?â The child says âfour.â can answer.â Once another child gave The teacher asks, âCan you teach us the correct answer the teacher moves how you did that?â After the child ex- on to the next task. plains, the teacher asks, âDid anybody do it a different way?â SOURCE: COEMET.

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 273 1994), which uses small-group games and active experiences with different models of number, led to substantial improvement in childrenâs knowledge of number. Children in the program were better able to employ reasonable problem-solving strategies and solve arithmetic problems even more dif- ficult than those in the program. (Core Knowledge includes a mathematics sequence developed by Sharon Griffin based on this work.) Program children also passed five far-transfer tests that were hypoth- esized to depend on similar cognitive structures (e.g., balance beam, time, money). The foundation these children received supported their learning of new, more complex mathematics through Grade 1. In a 3-year longitudinal study in which children received consistent experiences from kindergarten through the primary grades, children gained and surpassed both a second low-income group and a mixed-income group that showed a higher ini- tial level of performance and attended a magnet school with an enriched mathematics curriculum. The children also compared favorably with high- income groups from China and Japan (Case, Griffin, and Kelly, 1999). On a more limited scale, a study of 8 classrooms with 112 children found that a 6-week focused mathematics intervention was successful in improving Head Start childrenâs mathematical skills as well as their interest in math- ematics (Arnold et al., 2002). Teachers in experimental classrooms were provided with a choice of math-relevant activities to use during circle time, with small groups, and during routines and transitions, while the control classrooms did typical activities. Experimental group children scored signifi- cantly higher on the Test of Early Mathematics Ability (TEMA-2) and also reported that they enjoyed mathematics more than the control children. Teachers, too, reported that they increased their knowledge and enjoyment in implementing mathematics activities. Notably, boys showed substantial gains compared with girls, and African American and Puerto Rican children gained more than white children. Like other mathematics interventions, this study includes several variables, making it impossible to determine which particular teaching and learning experiences make the most difference to children. At the very least, the study indicates once again that more inten- tional teaching of mathematics leads to better mathematics outcomes. PRINCIPLES TO GUIDE MATHEMATICS CURRICULUM AND PEDAGOGY Based on an extensive review of research on the current state of early mathematics education and effective practices, we present a set of principles to guide early childhood mathematics curriculum and instruction. Research points specifically to the following key indicators of an effective mathemat- ics program at the preschool level (e.g., Clarke et al., 2002; Clements and Sarama, 2007b, 2008a; Thomson et al., 2005; Wood and Frid, 2005):

274 MATHEMATICS LEARNING IN EARLY CHILDHOOD â¢ Uses research to specify a comprehensive set of cognitive concepts, processes, and teaching-learning paths to design developmentally sequenced activities and help teachers collect data by observation and interaction with children and use those data to modify planning and teaching strategies. Tasks are sequenced, but teachers need to adapt for particular studentsâ conceptual development rather than rigidly following a prescribed curriculum. â¢ Emphasizes mathematization of childrenâs experiences, including redescribing (i.e., with more specific and often mathematical lan- guage), reorganizing, abstracting, generalizing, reflecting on, and giving language to what is first understood on an intuitive, informal level (premathematical foundations). â¢ Builds an awareness of the need for direct, formal development of childrenâs concepts in mathematics together with an instructional focus on mathematics. This includes explicit plans for mathemat- ics as a separate area of the program and ability to plan based on teaching-learning paths. â¢ Uses a variety of instructional methods, such as a combination of small groups, the whole group, play, routines and transitions, and computer activities. Uses teachable moments as they occurâin general, has the ability to make connections between mathematical ideas, between activities, between mathematics and other subjects, and everyday life. â¢ Uses an âassisted performanceâ approach to instruction that sup- ports problem solving and inquiry processes in mathematics ac- tivities. Uses a variety of question types to encourage children to explain their thinking and to listen attentively to individual children and understand their level of thinking along mathematical teaching- learning paths. â¢ Engages and focuses childrenâs thinking through introductions and activities. Draws out key mathematical ideas at the conclusion of an activity or period of study and helps children consolidate and con- nect their knowledge. â¢ Across the program, teachers show an interest in mathematics and have high but realistic expectations and clear goals and an ability to communicate these clearly. Engages and cultivates childrenâs interests and motivation to learn mathematics. â¢ Uses classroom-based formative assessment to make adjustments to teachersâ instructional practices so that they better under- stand childrenâs learning needs and facilitate their mathematical development.

STANDARDS, CURRICULUM, INSTRUCTION, AND ASSESSMENT 275 SUMMARY Young children in early childhood classrooms do not spend much time engaged in mathematics content. Time spent on mathematics increases somewhat in kindergarten. The time that is spent engaged in mathematics is typically of low instructional quality (La Paro et al., 2008) and, more often than not, is conducted as a part of whole-class activities or embedded in center time or free play. Early childhood teachers rarely teach mathematics in small groups. They report that they are much more likely to use embed- ded mathematical strategies or do the calendar, which they consider to be teaching mathematics, rather than provide experiences with a primary focus on mathematics in which they scaffold childrenâs progress along important mathematics teaching-learning paths. Formative assessment has consider- able potential to provide teachers with meaningful methods for assessing childrenâs mathematical knowledge and improving their instruction to meet childrenâs needs. On a more optimistic note, the early childhood education field is ac- tively working to improve the teaching of mathematics. The National As- sociation for the Education of Young Children and the National Council of Teachers of Mathematics (2002) issued a joint position statement calling for more and better mathematics curriculum and teaching in early child- hood programs. Head Start has launched a new mathematics professional development initiative. In addition, the reauthorization of Head Start calls for research-based curriculum and practices. The time is right to enhance young childrenâs mathematics experiences not only to improve school readi- ness, but also to lay a foundation for lifelong understanding and enjoyment of mathematics. The challenges as well as the advances in research and policies aimed at improving young childrenâs mathematics learning speak to the need for extensive professional development around young childrenâs mathematicsâthe focus of the next chapter. REFERENCES AND BIBLIOGRAPHY Archibald, S., and Goetz, M. (2008). A Professional Development Framework for Pre- Kindergarten Education. Paper commissioned by the Committee for Early Childhood Mathematics, Mathematics Science Education Board, Center for Education, Division of Behavioral and Sciences and Education, National Research Council, Washington, DC. Arnold, D.H., Fischer, P.H., Doctoroff, G.L., and Dobbs, J., (2002). Accelerating math devel- opment in Head Start classrooms. Journal of Educational Psychology, 94(4), 762-770. Askew, M., Brown, M., Rhodes, V., Johnson, D., and Wiliam, D. (1997, September). Effective Teachers of Numeracy. Paper presented at the British Educational Research Association Annual Conference (September 11-14, University of York). Available: http://www.leeds. ac.uk/educol/documents/000000385.htm [accessed July 2008]. Aubrey, C. (1997). Childrenâs early learning of number in school and out. In I. Thompson (Ed.), Teaching and Learning Early Numbers (pp. 20-29). Philadelphia, PA: Open Uni- versity Press.

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