Mathematics Learning in Early Childhood Paths Toward Excellence and Equity (2009) / Chapter Skim
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7 Standards, Curriculum, Instruction, and Assessment
7 Standards, Curriculum, Instruction, and Assessment
Pages 225-288
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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.
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. Instruction and Intentional Teaching In early childhood education, the term instruction is most often used to mean "direct instruction," implying that teachers are entirely in control and children are passive recipients of information.
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. To be effective, intentional teaching requires that teachers use formative assessment to determine where children are in relation to the learning goal and to provide the right kind and amount of support for them to continue to make progress.
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is also a widely recognized guide for state early learning standards. Curriculum Focal Points (National Council of Teachers of Mathematics, 2006)
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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)
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Cardinality and the three basic kinds of addition/subtraction situations received minimal attention. In the geometry early learning standards, there was an emphasis on children's knowledge of properties of shapes (40 percent)
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. The kindergarten learning standards for each state were coded into the five PSSM mathematical content areas or strands: (1)
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In kindergarten (as with the early learning standards) , the greatest emphasis across all the mathematics standards is placed on number and operations -- 40 percent of a state's mathematics standards on average (with a range from a low of 27 percent to a high of 56 percent among states)
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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)
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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)
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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.
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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 effective teaching and empirical evidence on effective practices (see Hamre and Pianta, 2007, for a more detailed discussion)
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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.
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, 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.
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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.
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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.
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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 occurred 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 "doing the calendar" every day (see Box 7-1)
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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 mathematics curriculum (e.g., Clements and Sarama, 2008a; Sophian, 2004; Starkey, Klein, and Wakeley, 2004)
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The general principles of effective instruction that emerge from this research can and should be taken into consideration when designing mathematics instruction for young children. Both of these bodies of research are briefly reviewed below.
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This may occur during a whole-group or small-group discussion or individually during center time or children's play (scaffolding is also discussed in the section on formative assessment in this chapter)
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showed that children exposed to more math talk in their preschool classrooms displayed greater gains in mathematical knowledge from October to April. The authors transcribed an hour of teachers' 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)
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As the authors point out, this is the first study to examine the specific effects of math talk on children's 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 classroom among teachers and children is frequently related to outcomes for children.
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. 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)
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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.
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, the teachers' 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.
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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 teachers' observation of children's play for the purpose of determining how best to respond to support their learning.
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Using Concrete Materials and Manipulatives Using concrete materials, such as puzzles and matching games, with task selection and scaffolding adjusted to children's strategies, is effective in moving them through mathematics teaching-learning paths (Clements and Sarama, 2007a)
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. Using Computers As all-purpose tools, computers can also constitute quite different environments that support mathematics teaching and learning.
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Whole-group discussions that help children communicate about their solution strategies and reflect on what they've learned are also essential components of good teaching with computers. Using Movement Another context for learning mathematics is teachers' use of movement to engage children.
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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 teaching must be child-centered and "developmentally appropriate" (Copple and Bredekamp, 2009)
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; however, it can provide teachers a way in which to track children's progress toward high-quality early learning standards. Formative assessment entails the use of several methods -- observation, task, and flexible interview -- to collect information about children's thinking and learning and then adapt teaching methods to help them learn.
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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.
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They do these classroom assessments on the fly, spontaneously, and without special preparation. Sometimes children learn a good deal, and sometimes they don't.
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Teachers need to receive professional development about learning in early childhood to be able to effectively interpret their observations of children's mathematical thinking. A third component of observation is careful evaluation of evidence.
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At best, the observations give only an extremely crude idea of the child's 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.
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260 MATHEMATICS LEARNING IN EARLY CHILDHOOD ous interests and everyday competence in the absence of adult pressure or constraint. Observation deals with behavior in "authentic" situations, like block play or snack time.
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The teacher is attempting to teach something about pattern and needs to know whether the child is "getting it" 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)
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, which we term "flexible interviewing," so as to avoid any connotation of the "clinical interview" devoted to the investigation and cure of pathological phenomena. Flexible interviewing involves several steps (Ginsburg, 1997)
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have used a developmental trajectory theory as the basis for development of an extensive collection of "task-based interviews" 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.
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Furthermore, flexible interviewing not only promotes the teacher's language but also requires it from the child. Flexible interviewing stresses to the child the importance of talking about one's thinking, justifying one's 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)
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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.
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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.
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. To help children achieve learning goals, educators have begun to emphasize intentional teaching in an emergent curriculum or project approach (Epstein, 2007)
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. 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)
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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)
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Curricular research does have great potential to advance understanding of effective instructional strategies, 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.
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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.
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2. The teacher built on and elaborated children's mathematical ideas and strategies.
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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 mathematics program at the preschool level (e.g., Clarke et al., 2002; Clements and Sarama, 2007b, 2008a; Thomson et al., 2005; Wood and Frid, 2005)
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Engages and cultivates children's interests and motivation to learn mathematics. • Uses classroom-based formative assessment to make adjustments to teachers' instructional practices so that they better under stand children's learning needs and facilitate their mathematical development.
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Formative assessment has considerable potential to provide teachers with meaningful methods for assessing children's mathematical knowledge and improving their instruction to meet children's needs. On a more optimistic note, the early childhood education field is actively working to improve the teaching of mathematics.
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, Engaging Young Children in Math ematics: Standards for Early Childhood Mathematics Education (pp.
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, Engaging Young Children in Mathemat ics: Standards for Early Childhood Mathematics Education (pp.
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Journal of Research in Mathematics Education, 38(2)
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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. Flavell, J.H., Green, F.L., and Flavell, E.R.
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Journal for Re search in Mathematics Education, 31(3)
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, Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (pp.
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, Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (pp.
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. Toward formative assessment: The use of pedagogical documentation in early elementary classrooms.
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. Early Childhood Mathematics: Promoting Good Beginnings.
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Paper commissioned by the Committee for Early Childhood Mathematics, Mathematics Science Education Board, Center for Edu cation, Division of Behavioral and Sciences and Education, National Research Council, Washington, DC. Rieser, J.J., Garing, A.E., and Young, M.F.
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. Early Childhood Mathematics Education Research: Learning Trajectories for Young Children.
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. What is developmentally appropriate in early child hood mathematics education?
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Children's percep tions of high and low quality classrooms. Early Childhood Research Quarterly, 16(2)
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