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POSITION STATEMENT

Early Childhood : Promoting Good Beginnings

A joint position statement of the National Association for the of Young Children (NAEYC) and the National Council of of Mathematics (NCTM). Adopted in 2002. Updated in 2010.

Position solid foundation for success in . In elemen- tary and , children need mathemat- The National Council of Teachers of Mathemat- ical understanding and skills not only in math ics (NCTM) and the National Association for the courses but also in , social studies, and Education of Young Children (NAEYC) affirm that other subjects. In high school, students need high-quality, challenging, and accessible mathe- mathematical proficiency to succeed in course matics education for 3- to 6-year-old children is a work that provides a gateway to technological vital foundation for future mathematics . and [1–4]. Once out In every early childhood setting, children should of school, all need a broad range of basic experience effective, -based mathematical understanding to make informed and teaching practices. Such high-quality - decisions in their jobs, households, communities, room practice requires policies, organizational and civic lives. supports, and adequate resources that enable Besides ensuring a sound mathematical teachers to do this challenging and important foundation for all members of our society, the work. nation also needs to prepare increasing numbers of young people for work that requires a higher The challenges proficiency level [5, 6]. The National Commission on Mathematics and Science Teaching for the Throughout the early years of life, children notice 21st Century (known as the Glenn Commission) and explore mathematical dimensions of their asks this question: “As our children move toward world. They compare quantities, find patterns, the day when their decisions will be the ones navigate in space, and grapple with real problems shaping a new America, will they be equipped such as balancing a tall block building or sharing with the mathematical and scientific tools needed a bowl of crackers fairly with a playmate. Math- to meet those challenges and capitalize on those ematics helps children make sense of their world opportunities?” [7, p. 6] outside of school and helps them construct a

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1 Early Childhood Mathematics

Since the 1970s a of assessments of In 2000, with the growing evidence that the U.S. students’ performance has revealed an over- early years significantly affect mathematics learn- all level of mathematical proficiency well below ing and attitudes, NCTM for the first time includ- what is desired and needed [5, 8, 9]. In recent ed the prekindergarten year in its Principles and years NCTM and others have addressed these Standards for School Mathematics (PSSM) [19]. challenges with new standards and other re- Guided by six overarching principles—regarding sources to improve , and equity, curriculum, teaching, learning, assess- progress has been made at the elementary and ment, and technology—PSSM describes for each middle school levels—especially in that mathematics content and process area what chil- have instituted reforms [e.g., 10–12]. Yet achieve- dren should be able to do from prekindergarten ment in mathematics and other areas varies through second grade. widely from state to state [13] and from to school district. There are many en- couraging indicators of success but also areas of NCTM Principles for School continuing concern. In mathematics as in Mathematics literacy, children who live in poverty and who are members of linguistic and ethnic minority groups Equity: Excellence in mathematics education demonstrate significantly lower levels of achieve- requires equally high expectations and ment [14–17]. strong for all students. If progress in improving the mathematics Curriculum: A curriculum is more than a col- proficiency of Americans is to continue, much lection of activities; it must be coherent, greater attention must be given to early math- focused on important mathematics, and well ematics experiences. Such increased awareness articulated across the grades. and effort recently have occurred with respect to Teaching: Effective mathematics teaching re- early foundations of literacy. Similarly, increased quires understanding of what students know energy, time, and wide-scale commitment to the and need to learn and then challenging and early years will generate significant progress in supporting them to learn it well. mathematics learning. Learning: The opportunity is clear: Millions of young Students must learn mathematics children are in care or other early educa- with understanding, actively building new tion settings where they can have significant knowledge from experience and prior knowl- early mathematical experiences. Accumulating edge. research on children’s capacities and learning Assessment: Assessment should support the in the first six years of life confirms that early learning of important mathematics and fur- experiences have long-lasting outcomes [14, 18]. nish useful information to both teachers and Although our knowledge is still far from com- students. plete, we now have a fuller picture of the math- Technology: Technology is essential to teach- ematics young children are able to acquire and ing and learning mathematics; it influences the practices to promote their understanding. the mathematics that is taught and enhances This knowledge, however, is not yet in the hands students’ learning. of most early childhood teachers in a form to ef- Note: These principles are relevant across all fectively guide their teaching. It is not surprising grade levels, including early childhood. then that a great many early childhood programs have a considerable distance to go to achieve The present statement focuses on children high-quality mathematics education for children over 3, in large part because the knowledge age 3-6. base on mathematical learning is more robust for this age . Available evidence, however,

Copyright © 2002 National Association for the Education of Young Children

2 NAEYC/NCTM Joint Position Statement indicates that children under 3 enjoy and benefit Recognition of the importance of good begin- from various kinds of mathematical explorations nings, shared by NCTM and NAEYC, underlies and experiences. With respect to mathematics this joint position statement. The statement de- education beyond age 6, the recommendations scribes what constitutes high-quality mathemat- on practice presented here remain ics education for children 3–6 and what is nec- relevant. Further, closely connecting curriculum essary to achieve such quality. To help achieve and teaching for children age 3–6 with what is this goal, the position statement sets forth 10 done with students over 6 is essential to achieve research-based, essential recommendations to the seamless mathematics education that chil- guide classroom1 practice, as well as four recom- dren need. mendations for policies, systems changes, and other actions needed to support these practices.

In high-quality mathematics education 8. provide ample time, materials, and for 3- to 6-year-old children, teachers and support for children to engage in play, a other key professionals should context in which they explore and manipulate mathematical ideas with keen interest 1. enhance children’s natural interest in math- ematics and their disposition to use it to make 9. actively introduce mathematical concepts, sense of their physical and social worlds methods, and language through a range of ap- propriate experiences and teaching strategies 2. build on children’s experience and knowl- edge, including their family, linguistic, cultural, 10. support children’s learning by thoughtfully and community backgrounds; their individual and continually assessing all children’s math- approaches to learning; and their informal- ematical knowledge, skills, and strategies. knowledge To support high quality mathematics edu- 3. base mathematics curriculum and teaching cation, institutions, program developers, practices on knowledge of young children’s and policy makers should cognitive, linguistic, physical, and social- emotional development 1. create more effective early childhood teach- er preparation and continuing professional 4. use curriculum and teaching practices that development strengthen children’s problem-solving and reasoning processes as well as representing, 2. use collaborative processes to develop well communicating, and connecting mathematical aligned systems of appropriate high-quality ideas standards, curriculum, and assessment 5. ensure that the curriculum is coherent and 3. design institutional structures and policies compatible with known relationships and se- that support teachers’ ongoing learning, team- quences of important mathematical ideas work, and planning 6. provide for children’s deep and sustained 4. provide resources necessary to overcome interaction with key mathematical ideas the barriers to young children’s mathematical proficiency at the classroom, community, insti- 7. integrate mathematics with other activities tutional, and system-wide levels. and other activities with mathematics

1 Classroom refers to any group setting for 3- to 6-year-olds (e.g., child care program, family child care, , or public school classroom).

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3 Early Childhood Mathematics

Recommendations 2. Build on children’s experience and knowl- edge, including their family, linguistic, Within the classroom cultural, and community backgrounds; To achieve high-quality mathematics edu- their individual approaches to learning; cation for 3- to 6-year-old children, teach- and their informal knowledge. 2 ers and other key professionals should Recognizing and building on children’s individ- 1. Enhance children’s natural interest in ual experiences and knowledge are central to mathematics and their disposition to use it effective early childhood mathematics educa- to make sense of their physical and social tion [e.g., 20, 22, 29, 30]. While striking similari- worlds. ties are evident in the mathematical issues that Young children show a natural interest in and interest children of different backgrounds [31], enjoyment of mathematics. Research evidence it is also true that young children have varying indicates that long before entering school chil- cultural, linguistic, home, and community expe- dren spontaneously explore and use mathemat- riences on which to build mathematics learning ics—at least the intuitive beginnings—and their [16, 32]. For example, number naming is regular mathematical knowledge can be quite complex in Asian languages such as Korean (the Korean and sophisticated [20]. In play and daily activi- word for “eleven” is ship ill, or “ten one”), while ties, children often explore mathematical ideas English uses the irregular word eleven. This and processes; for example, they sort and clas- difference appears to make it easier for Korean sify, compare quantities, and notice shapes and children to learn or construct certain numeri- patterns [21–27]. cal concepts [33, 34]. To achieve equity and educational effectiveness, teachers must know Mathematics helps children make sense of the as much as they can about such differences physical and social worlds around them, and and work to build bridges between children’s children are naturally inclined to use math- varying experiences and new learning [35–37]. ematics in this way (“He has more than I do!” “That won’t fit in there—it’s too big”). By capi- In mathematics, as in any knowledge domain, talizing on such moments and by carefully plan- learners benefit from having a variety of ways ning a variety of experiences with mathemati- to understand a given concept [5, 14]. Building cal ideas in mind, teachers cultivate and extend on children’s individual strengths and learn- children’s mathematical sense and interest. ing styles makes mathematics curriculum and instruction more effective. For example, some Because young children’s experiences fun- children learn especially well when instruc- damentally shape their attitude toward tional materials and strategies use to mathematics, an engaging and encouraging convey number concepts [38]. climate for children’s early encounters with mathematics is important [19]. It is vital for Children’s confidence, competence, and inter- young children to develop confidence in their est in mathematics flourish when new expe- ability to understand and use mathematics— riences are meaningful and connected with in other words, to see mathematics as within their prior knowledge and experience [19, 39]. their reach. In , positive experiences At first, young children’s understanding of a with using mathematics to solve problems mathematical concept is only intuitive. Lack of help children to develop dispositions such as explicit concepts sometimes prevents the child curiosity, imagination, flexibility, inventiveness, from making full use of prior knowledge and and persistence that contribute to their future connecting it to school mathematics. There- success in and out of school [28]. fore, teachers need to find out what young children already understand and help them 2 Teachers refers to adults who care for and educate begin to understand these things mathematical- groups of young children.

Copyright © 2002 National Association for the Education of Young Children

4 NAEYC/NCTM Joint Position Statement ly. From ages 3 through 6, children need many opment and her sensitivity to the individual experiences that call on them to relate their child’s frustration tolerance and persistence knowledge to the vocabulary and conceptual [45, 46]. frameworks of mathematics—in other words, For some mathematical topics, researchers have to “mathematize” what they intuitively grasp. identified a developmental continuum or learn- Toward this end, effective early childhood ing path—a indicating how particular programs provide many such opportunities concepts and skills build on others [44, 47, 48]. for children to represent, reinvent, reorganize, Snapshots taken from a few such are quantify, abstract, generalize, and refine that given in the accompanying chart (pp. 19–21). which they grasp at an experiential or intuitive level [28]. Research-based generalizations about what many children in a given grade or age range can 3. Base mathematics curriculum and teaching do or understand are key in shaping curriculum practices on knowledge of young children’s and instruction, although they are only a start- cognitive, linguistic, physical, and social- ing point. Even with comparable learning op- emotional development. portunities, some children will grasp a concept All decisions regarding mathematics curricu- earlier and others somewhat later. Expecting lum and teaching practices should be grounded and planning for such individual variation are in knowledge of children’s development and always important. learning across all interrelated areas—cogni- With the enormous variability in young chil- tive, linguistic, physical, and social-emotional. dren’s development, neither policymakers nor First, teachers need broad knowledge of teachers should a fixed timeline for children children’s cognitive development—concept to reach each specific learning objective [49]. development, reasoning, and , In addition to the risk of misclassifying indi- for instance—as well as their acquisition of vidual children, highly specific timetables for particular mathematical skills and concepts. skill acquisition pose another serious threat, Although children display mathematical ideas especially when accountability pressures are at early ages [e.g., 40–43] their ideas are often intense. They tend to focus teachers’ attention very different from those of adults [e.g., 26, 44]. on getting children to perform narrowly defined For example, young children tend to believe skills by a specified time, rather than on laying that a long line of pennies has more coins than the conceptual groundwork that will serve a shorter line with the same number. children well in the long run. Such prescrip- Beyond cognitive development, teachers need tions often lead to superficial teaching and rote to be familiar with young children’s social, emo- learning at the expense of real understanding. tional, and motor development, all of which Under these conditions, children may develop are relevant to mathematical development. only a shaky foundation for further mathemat- To determine which puzzles and manipulative ics learning [50–52]. materials are helpful to support mathematical 4. Use curriculum and teaching practices that learning, for instance, teachers combine their strengthen children’s problem-solving and knowledge of children’s with the reasoning processes as well as represent- knowledge of fine7 motor development [45]. ing, communicating, and connecting math- In deciding whether to let a 4-year-old struggle ematical ideas. with a particular or to offer a clue, the teacher draws on more than Problem solving and reasoning are the heart of an understanding of the cognitive demands in- mathematics. Teaching that promotes profi- volved. Important too are the teacher’s under- ciency in these and other mathematical pro- standing of young children’s emotional devel- cesses is consistent with national reports on

Copyright © 2002 National Association for the Education of Young Children

5 Early Childhood Mathematics

mathematics education [5, 19, 53] and recom- The big ideas or vital understandings in early mendations for early childhood practice [14, childhood mathematics are those that are 46]. While content represents the what of early mathematically central, accessible to children childhood mathematics education, the process- at their present level of understanding, and es—problem solving, reasoning, communica- generative of future learning [28]. Research and tion, connections, and representation—make it expert practice indicate that certain concepts possible for children to acquire content know and skills are both challenging and accessible edge [19]. These processes develop over time to young children [19]. National professional and when supported by well designed opportu- standards outline core ideas in each of five nities to learn. major content areas: number and operations, geometry, measurement, (including pat- Children’s development and use of these terns), and data analysis [19]. For example, the processes are among the most longlasting and idea that the same pattern can describe differ- important achievements of mathematics educa- ent situations is a “big idea” within the content tion. Experiences and intuitive ideas become area of algebra and patterning. truly mathematical as the children reflect on them, represent them in various ways, and con- These content areas and their related big ideas, nect them to other ideas [19, 47]. however, are just a starting point. Where does one begin to build understanding of an idea The process of making connections deserves such as “counting” or “symmetry,” and where special attention. When children connect does one take this understanding over the number to geometry (for example, by count- early years of school? Articulating goals and ing the sides of shapes, using arrays to under- standards for young children as a develop- stand number combinations, or measuring the mental or learning continuum is a particularly length of their classroom), they strengthen useful strategy in ensuring engagement with concepts from both areas and build knowledge and mastery of important mathematical ideas and beliefs about mathematics as a coherent [49]. In the key , research- system [19, 47]. Similarly, helping children con- ers have at least begun to map out trajectories nect mathematics to other subjects, such as or paths of learning—that is, the sequence in science, develops knowledge of both subjects which young children develop mathematical as well as knowledge of the wide applicability understanding and skills [21, 58, 59]. The ac- of mathematics. Finally and critically, teaching companying chart provides brief examples of concepts and skills in a connected, integrated learning paths in each content area and a few fashion tends to be particularly effective not teaching strategies that promote children’s only in the early childhood years [14, 23] but progress along these paths. Information about also in future learning [5, 54]. such learning paths can support developmen- 5. Ensure that the curriculum is coherent tally appropriate teaching, illuminating various and compatible with known relationships avenues to understanding and guiding teachers and sequences of important mathematical in providing activities appropriate for children ideas. as individuals and as a group. In developing early mathematics curriculum, 6. Provide for children’s deep and sustained teachers need to be alert to children’s experi- interaction with key mathematical ideas. ences, ideas, and creations [55, 56]. To create In many early childhood programs, mathemat- coherence and power in the curriculum, how- ics makes only fleeting, random appearances. ever, teachers also must stay focused on the Other programs give mathematics adequate “big ideas” of mathematics and on the connec- time in the curriculum but attempt to cover tions and sequences among those ideas so many mathematical topics that the result [23, 57].

Copyright © 2002 National Association for the Education of Young Children

6 NAEYC/NCTM Joint Position Statement is superficial and uninteresting to children. 7. Integrate mathematics with other activities In a more effective third alternative, children and other activities with mathematics. encounter concepts in depth and in a logical Young children do not perceive their world as if sequence. Such depth and coherence allow it were divided into separate cubbyholes such children to develop, construct, test, and reflect as “mathematics” or “literacy” [61]. Likewise, on their mathematical understandings [10, effective practice does not mathematics 23, 59, 60]. This alternative also enhances to one specified period or time of day. Rather, teachers’ opportunities to determine gaps in early childhood teachers help children develop children’s understanding and to take time to mathematical knowledge throughout the day address these. and across the curriculum. Children’s everyday Because curriculum depth and coherence activities and routines can be used to introduce are important, unplanned experiences with and develop important mathematical ideas [55, mathematics are clearly not enough. Effective 59, 60, 62–67]. For example, when children are programs also include intentionally organized lining up, teachers can build in many opportu- learning experiences that build children’s nities to develop an understanding of mathe- understanding over time. Thus, early childhood matics. Children wearing something red can be educators need to plan for children’s in-depth asked to get in line first, those wearing blue to involvement with mathematical ideas, includ- get in line second, and so on. Or children wear- ing helping families extend and develop these ing both something red and sneakers can be ideas outside of school. asked to head up the line. Such opportunities to build important mathematical vocabulary Depth is best achieved when the program fo- and concepts abound in any classroom, and cuses on a number of key content areas rather the alert teacher takes full advantage of them. than trying to cover every topic or skill with equal weight. As articulated in professional Also important is weaving mathematics into standards, researchers have identified number children’s experiences with , language, and operations, geometry, and measurement science, social studies, , movement, , as areas particularly important for 3- to 6-year- and all parts of the classroom environment. For olds [19]. These play an especially significant example, there are books with mathematical role in building the foundation for mathemat- concepts in the corner, and clipboards ics learning [47]. For this reason, researchers and wall charts are placed where children are recommend that algebraic thinking and data engaged in science observation and record- analysis/ receive somewhat less ing (e.g., measuring and charting the weekly emphasis in the early years. The beginnings of growth of plants) [65, 66, 68–71]. Projects ideas in these two areas, however, should be also reach across subject-matter boundaries. woven into the curriculum where they fit most Extended investigations offer children excel- naturally and seem most likely to promote lent opportunities to apply mathematics as well understanding of the other topic areas [19]. as to develop independence, persistence, and Within this second tier of content areas, pat- flexibility in making sense of real-life problems terning (a component of algebra) merits special [19]. When children pursue a project or inves- mention because it is accessible and interesting tigation, they encounter many mathematical to young children, grows to undergird all alge- problems and questions. With teacher guid- braic thinking, and supports the development ance, children think about how to gather and of number, spatial sense, and other conceptual record information and develop representa- areas. tions to help them in understanding and using the information and communicating their work to others [19, 72].

Copyright © 2002 National Association for the Education of Young Children

7 Early Childhood Mathematics

Another rationale for integrating mathematics have emerged in their play. Teachers enhance throughout the day lies in easing competition children’s mathematics learning when they ask for time in an increasingly crowded curriculum. questions that provoke clarifications, exten- Heightened attention to literacy is vital but can sions, and development of new understandings make it difficult for teachers to give mathemat- [19]. ics and other areas their due. With a strong Block building offers one example of play’s interdisciplinary curriculum, teachers can still value for mathematical learning. As children focus on one area at times but also find ways build with blocks, they constantly accumulate to promote children’s competence in literacy, experiences with the ways in which objects mathematics, and other subjects within inte- can be related, and these experiences become grated learning experiences [73]. the foundation for a multitude of mathematical An important final note: As valuable as integra- concepts—far beyond simply sorting and seri- tion is within early childhood curriculum, it ating. Classic unit blocks and other construc- is not an end in itself. Teachers should ensure tion materials such as connecting blocks give that the mathematics experiences woven children entry into a world where objects have throughout the curriculum follow logical predictable similarities and relationships [66, sequences, allow depth and focus, and help 76]. With these materials, children reproduce children move forward in knowledge and skills. objects and structures from their daily lives The curriculum should not become, in the and create abstract designs by manipulating name of integration, a grab bag of any mathe- pattern, symmetry, and other elements [77]. matics-related experiences that seem to relate Children perceive geometric notions inherent to a theme or project. Rather, concepts should in the blocks (such as two square blocks as the be developed in a coherent, planful manner. equivalent of one rectangular unit block) and the structures they build with them (such as 8. Provide ample time, materials, and teacher symmetric buildings with parallel sides). Over support for children to engage in play, a time, children can be guided from an intuitive context in which they explore and manipu- to a more explicit conceptual understanding of late mathematical ideas with keen interest. these ideas [66]. Children become intensely engaged in play. A similar progression from intuitive to explicit Pursuing their own purposes, they tend to tack- knowledge takes place in other kinds of play. le problems that are challenging enough to be Accordingly, early childhood programs should engrossing yet not totally beyond their capaci- furnish materials and sustained periods of ties. Sticking with a problem—puzzling over it time that allow children to learn mathemat- and approaching it in various ways—can lead ics through playful activities that encourage to powerful learning. In addition, when sev- counting, measuring, constructing with blocks, eral children grapple with the same problem, playing board and card games, and engaging in they often come up with different approaches, dramatic play, music, and art [19, 64]. discuss, and learn from one another [74, 75]. These aspects of play tend to prompt and pro- Finally, the teacher can observe play to learn mote thinking and learning in mathematics and more about children’s development and inter- in other areas. ests and use this knowledge to inform curricu- lum and instruction. With teacher guidance, an Play does not guarantee mathematical develop- individual child’s play interest can develop into ment, but it offers rich possibilities. Significant a classroom-wide, extended investigation or benefits are more likely when teachers fol- project that includes rich mathematical learn- low up by engaging children in reflecting on ing [78–82]. In in which teachers and representing the mathematical ideas that are alert to all these possibilities, children’s

Copyright © 2002 National Association for the Education of Young Children

8 NAEYC/NCTM Joint Position Statement play continually stimulates and enriches math- game more mathematically powerful and more ematical explorations and learning. appropriate for children of differing develop- mental levels [55, 83]. 9. Actively introduce mathematical concepts, methods, and language through a range Use of materials also requires intentional plan- of appropriate experiences and teaching ning and involvement on the teacher’s part. strategies. Computer technology is a good example [84]. Teachers need to intentionally select and use A central theme of this position statement is research-based computer tools that comple- that early childhood curriculum needs to go ment and expand what can be done with other beyond sporadic, hit-or-miss mathematics. In media [59]. As with other instructional materi- effective programs, teachers make judicious als, choosing software and determining how use of a variety of approaches, strategies, and best to incorporate computer use in the day-to- materials to support children’s interest and day curriculum requires thoughtful, informed ability in mathematics. decision-making in order for children’s learning Besides embedding significant mathemat- experiences to be rich and productive. ics learning in play, classroom routines, and In short, mathematics is too important to be learning experiences across the curriculum, an left to chance, and yet it must also be con- effective early mathematics program also pro- nected to children’s lives. In making all of these vides carefully planned experiences that focus choices, effective early childhood teachers children’s attention on a particular mathemati- build on children’s informal mathematical cal idea or set of related ideas. Helping children knowledge and experiences, always taking chil- name such ideas as horizontal or even and odd dren’s cultural background and language into as they find and create many examples of these consideration [23]. categories provides children with a means to connect and refer to their just-emerging ideas 10. Support children’s learning by thought- [35, 37]. Such concepts can be introduced and fully and continually assessing all children’s explored in large- and small-group activities mathematical knowledge, skills, and strate- and learning centers. Small groups are particu- gies. larly well suited to focusing children’s attention Assessment is crucial to effective teaching on an idea. Moreover, in this setting the teacher [85]. Early childhood mathematics assess- is able to observe what each child does and ment is most useful when it aims to help young does not understand and engage each child in children by identifying their unique strengths the learning experience at his own level. and needs so as to inform teacher planning. In planning for new investigations and activi- Beginning with careful observation, assessment ties, teachers should think of ways to engage uses multiple sources of information gath- children in revisiting concepts they have ered systematically over time—for example, a previously explored. Such experiences enable classroom book documenting the graphs made children to forge links between previously by children over several weeks. Mathematics encountered mathematical ideas and new appli- assessment should follow widely accepted prin- cations [19]. ciples for varied and authentic early childhood assessment [85]. For instance, the teacher Even the way that teachers introduce and needs to use multiple assessment approaches modify games can promote important mathe- to find out what each child understands—and matical concepts and provide opportunities for may misunderstand. Child observation, docu- children to practice skills [55, 57]. For example, mentation of children’s talk, interviews, collec- teachers can modify any simple board game in tions of children’s work over time, and the use which players move along a path to make the

Copyright © 2002 National Association for the Education of Young Children

9 Early Childhood Mathematics

of open-ended questions and appropriate per- and ongoing professional development is an formance assessments to illuminate children’s urgent priority. In mathematics, as in literacy thinking are positive approaches to assessing and other areas, the challenges are formidable, mathematical strengths and needs [86, 87]. but research-based solutions are available [14, 92–95]. To support children’s mathematical Careful assessment is especially important proficiency, every early childhood teacher’s when planning for ethnically, culturally, and lin- professional preparation should include these guistically diverse young children and for chil- connected components: (1) knowledge of the dren with special needs or disabilities. Effective mathematical content and concepts most teachers use information and insights gathered relevant for young children—including in-depth from assessment to plan and adapt teaching understanding of what children are learning and curriculum. They recognize that even now and how today’s learning points toward young children invent their own mathematical the horizons of later learning [5]; (2) knowledge ideas and strategies and that children’s ideas of young children’s learning and development can be quite different from those of adults [44]. in all areas—including but not limited to cogni- They interpret what the child is doing and tive development—and knowledge of the issues thinking, and they attempt to see the situation and topics that may engage children at differ- from the child’s point of view. With this basis ent points in their development; (3) knowledge in thoughtful assessment, teachers are able to of effective ways of teaching mathematics to make informed decisions about what the child all young learners; (4) knowledge and skill in might be able to learn from new experiences. observing and documenting young children’s Reliance on a single group-administered test to mathematical activities and understanding; document 3- to 6-year-old children’s mathemati- and (5) knowledge of resources and tools that cal competence is counter to expert recom- promote mathematical competence and enjoy- mendations on assessment of young children ment [96]. [85, 88–91]. Educators must take care that as- Essential as this knowledge is, it can be sessment does not narrow the curriculum and brought to life only when teachers themselves inappropriately label children. If assessment have positive attitudes about mathematics. results exclude some children from challenging Lack of appropriate preparation may cause learning activities, they undercut educational both preservice and experienced teachers to equity. Teachers and makers fail to see mathematics as a priority for young need to stay in control of the assessment pro- children and to lack confidence in their ability cess, ensuring that it helps build mathematical to teach mathematics effectively [97]. Thus, competence and confidence. Well conceived, both preservice education and continuing well implemented, continuous assessment is an professional development experiences need to indispensable tool in facilitating all children’s place greater emphasis on encouraging teach- engagement and success in mathematics. ers’ own enjoyment and confidence, building Beyond the classroom positive mathematical attitudes and disposi- tions. To support excellent early mathematics Even graduates of four-year early childhood education, institutions, program develop- programs with state licensure usually lack ers, and policy makers should adequate preparation in mathematics. State 1. Create more effective early childhood legislatures often address their concern over teacher preparation and continuing profes- teachers’ weak background in mathematics by sional development. simply increasing the number of required math- Improving early childhood teacher preparation ematics courses needed for teacher licensure.

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10 NAEYC/NCTM Joint Position Statement

This remedy lacks research support [5, 92]. ticipation of staff who work in similar settings; Credit hours or yearly training requirements do content focused both on what and how to little or nothing unless the content and delivery teach; active learning techniques; and profes- of professional development are designed to sional development as part of a coherent pro- produce desired outcomes for teachers and gram of teacher learning [5, 99]. Innovative and children [93]. effective professional development models may use a variety of research-based approaches. In Teachers of young children should learn the addition, classroom-based inquiry, team teach- mathematics content that is directly relevant to ing by mathematics and early childhood educa- their professional role. But content alone is not tion specialists, discussion of case studies, and enough. Effective professional programs weave analysis of young children’s work samples tend together mathematics content, , and to strengthen teachers’ confidence and engage- knowledge of and family ment in early childhood mathematics [5, 97, 99, relationships [98]. When high-quality, well 100]. supervised field work is integrated throughout a training program, early childhood teachers Delivering this kind of ongoing professional can apply their knowledge in realistic con- development requires a variety of innovative texts. Courses or inservice training should strategies. For early childhood staff living in be designed to help teachers develop a deep isolated communities or lacking knowledgeable understanding of the mathematics they will trainers, distance learning with local facilita- teach and the habits of mind of a mathematical tors is a promising option. Literacy initiatives thinker. Courses, practicum experiences, and are increasingly using itinerant or school-wide other training should strengthen teachers’ abil- specialists; similarly, mathematics education ity to ask young children the kinds of questions specialists could offer resources to a number that stimulate mathematical thinking. Effective of early childhood programs. Partnerships professional development, whether preservice between higher education institutions and local or inservice, should also model the kind of flex- early childhood programs can help provide this ible, interactive teaching styles that work well support. Finally, school-district-sponsored pro- with children [92]. fessional development activities that include participants from community child care cen- Preservice and inservice professional develop- ters, family child care, and Head Start programs ment present somewhat differing challenges. In along with public school /primary preservice education, the major challenge is to teachers would build coherence and continuity build a sound, well integrated knowledge base for teachers and for children’s mathematical about mathematics, young children’s develop- experiences. ment and learning, and classroom practices [5]. Inservice training shares this challenge but also 2. Use collaborative processes to develop carries risks of superficiality and fragmentation. well aligned systems of appropriate high-quality standards, curriculum, and To avoid these risks, inservice professional assessment. development needs to move beyond the one- time workshop to deeper exploration of key In mathematics, as in other domains, the task mathematical topics as they connect with of developing curriculum and related goals and young children’s thinking and with classroom assessments has become the responsibility not practices. Inservice professional development only of the classroom teacher but also of other in mathematics appears to have the greatest educators and policy makers. State agencies, impact on teacher learning if it incorporates six school districts, and professional organizations features: teacher networking or study groups; are engaged in standards setting, defining de- sustained, intensive programs; collective par- sired educational and developmental outcomes

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11 Early Childhood Mathematics

for children below kindergarten age [13]. This the principles articulated by national groups trend represents an opportunity to improve concerned with appropriate assessment for early childhood mathematics education but young children [88–91]. also presents a challenge. The opportunity is District- or program-level educators are often to develop a coherent, developmentally ap- responsible for selecting or developing cur- propriate, and well aligned system that offers riculum. Decision makers can be guided by the teachers a framework to guide their work. The general criteria for curriculum adoption articu- challenge, especially at the preschool and kin- lated in the position statement jointly adopted dergarten levels, is to ensure that such a frame- by NAEYC and the National Association of Early work does not stifle innovation, put children Childhood Specialists in State Departments of into inappropriate categories, ignore important Education [85]. In addition, decision makers individual or cultural differences, or result in should insist that any mathematics curriculum narrowed and superficial teaching that fails to considered for adoption has been extensively give children a solid foundation of understand- field tested and evaluated with young children. ing [49]. 3. Design institutional structures and policies To avoid these risks, state agencies and others that support teachers’ ongoing learning, must work together to develop more coherent teamwork, and planning. systems of standards, curriculum, instruction, and assessment that support the development National reports stress the need for teacher of mathematical proficiency. To build coher- planning and collaboration [5, 7, 101, 102], yet ence between preschool and early elementary few early childhood programs have the struc- mathematics, the processes of setting stan- tures and supports to enable these processes to dards and developing early childhood curricu- take place regularly. Teachers of young children lum and assessment systems must include the face particular challenges in planning mathe- full range of stakeholders. Participants should matics activities. Early childhood teachers work include not only public school teachers and in diverse settings, and some of these settings administrators but also personnel from center- pose additional obstacles to teamwork and col- based programs and family child care, private laboration. Many early childhood programs, in and public prekindergarten, and Head Start, as or out of public school settings, have little or no well as others who serve young children and time available for teacher planning, either indi- their families. Families too should participate vidually or in groups. Team meetings and staff as respected partners. Relevant expertise development activities occur infrequently. should be sought from professional associa- The institutional divide between teachers in tions and other knowledgeable sources. child care, Head Start, or preschool programs As in all effective standards-setting efforts, and those in public kindergarten and primary early childhood mathematics standards should programs presents a barrier to the communi- be coupled with an emphasis on children’s cation necessary for a coherent mathematics opportunities to learn, not just on expectations curriculum. Without communication opportu- for their performance. Standards also should nities, preschool teachers often do not know be accompanied by descriptions of what young what kindergarten programs expect, and early children might be expected to accomplish elementary teachers may have little idea of the along a flexible developmental continuum [49]. content or pedagogy used in prekindergarten Standards for early childhood mathematics mathematics education. New strategies and should connect meaningfully but not rigidly structures, such as joint inservice programs with curriculum. Assessment also should align and classroom visits, could support these with curriculum and with standards, following linkages.

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12 NAEYC/NCTM Joint Position Statement

In addition, many programs have limited ac- To support effective teaching and learning, cess to specialists who might help teachers mathematics-rich classrooms require a wide as they try to adopt new approaches to early array of materials for young children to explore childhood mathematics. Administrators need and manipulate [45, 59, 107]. Equity requires to reexamine their allocation of resources and that all programs, not just those serving afflu- their scheduling practices, keeping in mind the ent communities, have these resources. value of investing in teacher planning time. Finally, resources are needed to support 4. Provide the resources necessary to over- families as partners in developing their young come the barriers to young children’s children’s mathematical proficiency. The grow- mathematical proficiency at the classroom, ing national awareness of families’ central role community, institutional, and system-wide in literacy development is a good starting point levels. from which to build awareness of families’ equally important role in mathematical de- A variety of resources, some financial and some velopment [108, 109]. Public awareness cam- less tangible, are needed to support imple- paigns, distribution of materials in ways similar mentation of this position statement’s recom- to the successful Reach Out and Read initia- mendations. Partnerships among the business, tive, computer-linked as well as school-based philanthropic, and government sectors at the meetings for families, Family Math Nights, national, state, and local levels will improve and take-home activities such as mathematics teaching and learning in all communities, games and manipulative materials tailored to including those that lack equitable access to the ages, interests, languages, and cultures of mathematics education. Universally available the children—these are only a few examples of early childhood mathematics education can the many ways in which resources can support occur only in the context of a comprehensive, families’ engagement in their young children’s well financed system of high-quality early mathematical learning [110, see also the online education, including child care, Head Start, and “Family Math” materials at www.lhs.berkeley. prekindergarten programs [103–106]. To sup- edu/equals/FMnetwork.htm and other resourc- port universal mathematical proficiency, access es at www.nctm.org/corners/family/index.htm]. to developmentally and educationally effective programs of early education, supported by adequate resources, should be available to all Conclusion children. A positive attitude toward mathematics and a Improvement of early childhood mathematics strong foundation for mathematics learning begin education also requires substantial investment in early childhood. These good beginnings reflect in teachers’ professional development. The all the characteristics of good early childhood mathematics knowledge gap must be bridged education: deep understanding of children’s with the best tools, including resources for dis- development and learning; a strong community of seminating models of effective practice, videos teachers, families, and children; research-based showing excellent mathematics pedagogy in knowledge of early childhood curriculum and real-life settings, computer-based professional teaching practices; continuous assessment in development resources, and other materials. the service of children’s learning; and an abiding In addition, resources are needed to support respect for young children’s families, cultures, teachers’ involvement in professional confer- and communities. ences, courses, summer institutes, and To realize this vision, educators, adminis- visits to model sites. trators, policy makers, and families must work together—raising awareness of the importance of mathematics in early education, informing

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others about sound approaches to mathematical 13. Education Week. 2002. Quality Counts 2002: Build- teaching and learning, and developing essential ing blocks for success: State efforts in early-child- resources to support high-quality, equitable hood education. Education Week (Special issue) 21 (17). mathematical experiences for all young children. 14. Bowman, B.T., M.S. Donovan, & M.S. Burns, eds. 2001. Eager to learn: Educating our preschoolers. Washington, DC: National Press. References 15. Denton, K., & J. West. 2002. Children’s reading and 1. Haycock, K., & S. Huang. 2001. Are today’s high mathematics achievement in kindergarten and first school graduates ready? Thinking K–16 5 (1): 3–17. grade. Washington, DC: National Center for Educa- 2. Haycock, K. 2001. Youth at the crossroads: Facing tion . high school and beyond. Thinking K–16 5 (1): 1–2. 16. Natriello, G., E.L. McDill, & A.M. Pallas. 1990. School- 3. Schoenfeld, A.H. 2002. Making mathematics work for ing disadvantaged children: Racing against catastro- all children: Issues of standards, testing, and equity. phe. New York: Teachers College Press. Educational Researcher 31: 13–25. 17. Starkey, P., & A. Klein. 1992. Economic and cultural 4. The Education Trust. 2001. Actions for communities influence on early mathematical development. In and states. Thinking K–16 5 (1): 18–21. New directions in child and family research: Shaping 5. Kilpatrick, J., J. Swafford, & B. Findell. 2001. Adding it Head Start in the nineties, eds. F. Lamb-Parker, R. up: Helping children learn mathematics. Washington, Robinson, S. Sambrano, C. Piotrkowski, J. Hagen, S. DC: National Academy Press. Randolph, & A. Baker, 440. Washington, DC: Adminis- 6. U.S. Department of Labor Bureau of Labor Statistics. tration on Children, Youth and Families (DHHS). 2000. The outlook for college graduates, 1998–2008. 18. Shonkoff, J.P., & D.A. Phillips, eds. 2000. From neu- In Getting ready pays off!, U.S. DOE, October 2000, & rons to neighborhoods: The science of early childhood BLS, Occupational Employment Projections to 2008, in development. Washington, DC: National Academy NAB, Workforce Economics 6 (1). Press. 7. Glenn Commission. 2000. Before it’s too late: A report 19. National Council of Teachers of Mathematics. 2000. to the nation from the National Commission on Math- Principles and standards for school mathematics. ematics and Science Teaching for the 21st Century. Reston, VA: Author. Washington, DC: U.S. Department of Education. 20. Seo, K.-H., & H.P. Ginsburg. 2004. What is develop- 8. Mullis, I.V.S., M.O. Martin, A.E. Beaton, E.J. Gon- mentally appropriate in early childhood mathemat- zalez, D.L. Kelly, & T.A. Smith. 1997. Mathematics ics education? In Engaging young children in math- achievement in the years: IEA’s Third ematics: Standards for early childhood mathematics International Mathematics and Science Study (TIMSS). education, eds. D.H. Clements, J. Sarama, & A.-M. Chestnut Hill, MA: Center for the Study of Testing, DiBiase, 91–104. Mahwah, NJ: Lawrence Erlbaum. Evaluation, and Educational Policy, . 21. Baroody, A.J. 2004. The role of psychological re- 9. Mullis, I.V.S., M.O. Martin, E.J. Gonzalez, K.D. Grego- search in the development of early childhood math- ry, R.A. Garden, K.M. O’Connor, S.J. Chrostowski, & ematics standards. In Engaging young children in T.A. Smith. 2000. TIMSS 1999 international mathemat- mathematics: Standards for early childhood mathemat- ics report. Boston: International Study Center, Boston ics education, eds. D.H. Clements, J. Sarama, & A.-M. College, Lynch School of Education. DiBiase, 149–72. Mahwah, NJ: Lawrence Erlbaum. 10. Fuson, K.C., W.M. Carroll, & J.V. Drueck. 2000. 22. Clements, D.H., S. Swaminathan, M.-A. Hannibal, & Achievement results for second and third graders J. Sarama. 1999. Young children’s concepts of shape. using the standards-based curriculum Everyday Journal for Research in Mathematics Education 30: Mathematics. Journal for Research in Mathematics 192–212. Education 31: 277–95. 23. Fuson, K.C. 2004. Pre-K to grade 2 goals and stan- 11. Mullis, I.V.S., M.O. Martin, E.J. Gonzalez, K.M. dards: Achieving 21st century mastery for all. In O’Connor, S.J. Chrostowski, K.D. Gregory, R.A. Engaging young children in mathematics: Standards for Garden, & T.A. Smith. 2001. Mathematics benchmark- early childhood mathematics education, eds. D.H. Cle- ing report: TIMSS 1999—. Chestnut Hill, ments, J. Sarama, & A.-M. DiBiase, 105–48. Mahwah, MA: International Association for the Evaluation of NJ: Lawrence Erlbaum. Educational Achievement. 24. Gelman, R. 1994. Constructivism and supporting 12. Riordan, J.E., & P.E. Noyce. 2001. The impact of two environments. In Implicit and explicit knowledge: An standards-based mathematics curricula on student educational approach, ed. D. Tirosh, 55–82. Norwood, achievement in Massachusetts. Journal for Research NJ: Ablex. in Mathematics Education 32: 368–98.

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25. Ginsburg, H.P., A. Klein, & P. Starkey. 1998. The de- 39. Bredekamp, S., & T. Rosegrant. 1995. Reaching po- velopment of children’s mathematical thinking: Con- tentials: Transforming early childhood curriculum and necting research with practice. In Handbook of child assessment. Volume 2. Washington, DC: NAEYC. psychology, Volume 4: Child psychology in practice, 40. Clements, D.H. 1999. Geometric and spatial thinking eds. W. Damon, I.E. Sigel, & K.A. Renninger, 401–76. in young children. In Mathematics in the early years, New York: John Wiley & Sons. ed. J.V. Copley, 66–79. Reston, VA: National Council of 26. Piaget, J., and B. Inhelder. 1967. The child’s concep- Teachers of Mathematics. tion of space. New York: W.W. Norton. 41. Starkey, P., & R.G. Cooper Jr. 1980. Perception of 27. Steffe, L.P. 2004. PSSM from a constructivist per- numbers by human . Science 210: 1033–35. spective. In Engaging young children in mathematics: 42. Starkey, P., E.S. Spelke, & R. Gelman. 1990. Nu- Standards for early childhood mathematics educa- merical abstraction by human infants. Cognition 36: tion, eds. D.H. Clements, J. Sarama, & A.-M. DiBiase, 97–128. 221–52. Mahwah, NJ: Lawrence Erlbaum. 43. Trafton, P.R., & A. Andrews. 2002. Little kids—Power- 28. Clements, D.H., & Conference Working Group. 2004. ful problem solvers: Math stories from a kindergarten Part one: Major themes and recommendations. In classroom. Portsmouth, NH: Heinemann. Engaging young children in mathematics: Standards 44. Steffe, L.P., & P. Cobb. 1988. Construction of arith- for early childhood mathematics education, eds. D.H. metical meanings and strategies. New York: Springer- Clements, J. Sarama, & A.-M. DiBiase, 7–76. Mahwah, Verlag. NJ: Lawrence Erlbaum. 45. Bronson, M.B. 1995. The right stuff for children birth 29. Copple, C.E. 2004. Math curriculum in the early to 8: Selecting play materials to support development. childhood context. 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54. Sophian, C. 2004. A prospective developmental 65. Hong, H. 1999. Using storybooks to help young perspective on early mathematics instruction. In children make sense of mathematics. In Mathematics Engaging young children in mathematics: Standards for in the early years, ed. J.V. Copley, 162–68. Reston, VA: early childhood mathematics education, eds. D.H. Cle- National Council of Teachers of Mathematics. ments, J. Sarama, & A.-M. DiBiase, 253–66. Mahwah, 66. Leeb-Lundberg, K. 1996. The block builder math- NJ: Lawrence Erlbaum. ematician. In The block book, ed. E.S. Hirsh. Washing- 55. Kamii, C.K., & L.B. Housman. 1999. Young children ton, DC: NAEYC. reinvent : Implications of Piaget’s theory. 2d 67. Shane, R. 1999. Making connections: A “number ed. New York: Teachers College Press. curriculum” for preschoolers. In Mathematics in the 56. Steffe, L.P. 1990. Mathematics curriculum design: early years, ed. J.V. Copley, 129–34. Reston, VA: Na- A constructivist’s perspective. In Transforming chil- tional Council of Teachers of Mathematics. dren’s mathematics education: International perspec- 68. Coates, G.D., & J. Franco. 1999. Movement, math- tives, eds. L.P. Steffe & T. Wood, 389–98. Hillsdale, NJ: ematics, and learning: Experiences using a family Lawrence Erlbaum. learning model. In Mathematics in the early years, ed. 57. Griffin, S., R. Case, & A. Capodilupo. 1995. Teaching J.V. Copley, 169–74. Reston, VA: National Council of for understanding: The importance of the central Teachers of Mathematics. conceptual structures in the elementary mathemat- 69. Copley, J.V. 2010. The young child and mathematics. ics curriculum. In Teaching for transfer: Fostering gen- 2d ed. Washington, DC: NAEYC. eralization in learning, eds. A. McKeough, J. Lupart, 70. Goodway, J.D., M. E. Rudisill, M.L. Hamilton, & & A. Marini. Mahwah, NJ: Lawrence Erlbaum. M.A. Hart. 1999. Math in motion. In Mathematics in 58. Clements, D.H. 2004. Linking research and cur- the early years, ed. J.V. Copley, 175–81. Reston, VA: riculum development. In Handbook of international National Council of Teachers of Mathematics. research in mathematics education, ed. L.D. English. 71. Kim, S.L. 1999. Teaching mathematics through Mahwah, NJ: Lawrence Erlbaum. musical activities. In Mathematics in the early years, 59. Sarama, J. 2004. Technology in early childhood ed. J.V. Copley, 146–50. Reston, VA: National Council mathematics: Building Blocks™ as an innovative of Teachers of Mathematics. technology-based curriculum. In Engaging young 72. Helm, J.H., S. Beneke, & K. Steinheimer. 1998. Win- children in mathematics: Standards for early child- dows on learning: Documenting young children’s work. hood mathematics education, eds. D.H. Clements, New York: Teachers College Press. J. Sarama, & A.-M. DiBiase, 361–76. Mahwah, NJ: 73. Balfanz, R. 2001. Developing and assessing young Lawrence Erlbaum. children’s mathematical knowledge. Washington, DC: 60. Griffin, S. 2004. Number Worlds: A research-based National Institute for Early Childhood Professional mathematics program for young children. In Engag- Development & NAEYC. ing young children in mathematics: Standards for early 74. Nastasi, B.K., & D.H. Clements. 1991. Research childhood mathematics education, eds. D.H. Clements, on cooperative learning: Implications for practice. J. Sarama, & A.-M. DiBiase, 325–42. Mahwah, NJ: Review 20: 110–31. Lawrence Erlbaum. 75. Yackel, E., P. Cobb, & T. Wood. 1991. Small group 61. Clements, D.H. 2001. Mathematics in the preschool. interactions as a source of learning opportunities in Teaching Children Mathematics 7: 270–75. second grade mathematics. Journal for Research in 62. Basile, C.G. 1999. The outdoors as a context for Mathematics Education 22: 390–408. mathematics in the early years. In Mathematics in 76. Pratt, C. 1948. I learn from children. New York: the early years, ed. J.V. Copley, 156–61. Reston, VA: Simon and Schuster. National Council of Teachers of Mathematics. 77. Ginsburg, H.P., N. Inoue, & K.-H. Seo. 1999. Young 63. Casey, M.B., R.L. Nuttall, & E. Pezaris. 1999. children doing mathematics: Observations of every- Evidence in support of a model that predicts how day activities. In Mathematics in the early years, ed. biological and environmental factors interact to J.V. Copley, 88–100. Reston, VA: National Council of influence spatial skills.Developmental Psychology 35 Teachers of Mathematics. (5): 1237–47. 78. Edwards, C., L. Gandini, & G. Forman. 1993. The 64. Hildebrandt, C., & B. Zan. 2002. Using group games hundred languages of children: The Reggio Emilia to teach mathematics. In Developing constructivist approach to early childhood education. Norwood, NJ: early childhood curriculum: Practical principles and Ablex. activities, ed. R. DeVries, 193–208. New York: Teach- 79. Helm, J.H., & L.G. Katz. 2001. Young investigators: ers College Press. The project approach in the early years. 2d ed. New York: Teachers College Press.

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80. Jones, E., & J. Nimmo. 1994. Emergent curriculum. 93. NAEYC. 2001. NAEYC standards for early childhood Washington, DC: NAEYC. professional preparation. Washington, DC: Author. 81. Katz, L.G., & S.C. Chard, 2000. Engaging children’s 94. Peisner-Feinberg, E.S., R. Clifford, M. Culkin, C. minds: The project approach. 2d ed. Stamford, CT: Howes, & S.L. Kagan. 1999. The children of the Cost, Ablex. Quality, and Outcomes Study go to school. Chapel Hill, 82. Malaguzzi, L. 1997. Shoe and meter. Reggio Emilia, NC: Frank Porter Graham Child Development Center, : Reggio Children. University of North Carolina at Chapel Hill. 83. Charlesworth, R. 2000. Experiences in math for 95. U.S. Department of Education. 1999. New teachers young children. Albany, NY: Delmar. for a new century: The future of early childhood profes- 84. Clements, D.H. 1999. Young children and technol- sional preparation. Washington, DC: Author. ogy. In Dialogue on early childhood science, math- 96. Copley, J.V., & Y. Padròn. 1999. Preparing teachers ematics, and , ed. G.D. Nelson, of young learners: Professional development of early 92–105, Washington, DC: American Association for childhood teachers in mathematics and science. In the Advancement of Science. Dialogue on early childhood science, mathematics, 85. NAEYC & the National Association of Early Child- and technology education, ed. G.D. Nelson, 117–29. hood Specialists in State Departments of Education. Washington, DC: American Association for the Ad- 1991. Guidelines for appropriate curriculum content vancement of Science. and assessment in programs serving children ages 3 97. Sarama, J., & A.-M. DiBiase. 2004. The professional through 8. Young Children 46 (3): 21–38. development challenge in preschool mathematics. In 86. Chittenden, E. 1991. Authentic assessment, evalua- Engaging young children in mathematics: Standards for tion, and documentation of student performance. In early childhood mathematics education, eds. D.H. Cle- Expanding student assessment, ed. V. Perrone, 22–31. ments, J. Sarama, & A.-M.DiBiase, 415–48. Mahwah, Alexandria, VA: Association for Supervision and Cur- NJ: Lawrence Erlbaum. riculum Development. 98. Baroody, A.J., & R.T. Coslick. 1998. Fostering chil- 87. Lindquist, M.M., & J.N. Joyner. 2004. Moving ahead dren’s mathematical power: An investigative approach in support of young children’s mathematical learn- to K–8 mathematics instruction. Mahwah, NJ: Law- ing: Recommendations to conference organizers and rence Erlbaum. participants. In Engaging young children in mathemat- 99. Copley, J.V. 2004. The early childhood collabora- ics: Standards for early childhood mathematics educa- tive: A professional development model to com- tion, eds. D.H. Clements, J. Sarama, & A.-M. DiBiase, municate and implement the standards. In Engaging 449–56. Mahwah, NJ: Lawrence Erlbaum. young children in mathematics: Standards for early 88. Horm-Wingerd, D.M., P.C. Winter, & P. Plofchan. childhood mathematics education, eds. D.H. Clements, 2000. Primary level assessment for IASA Title I: A call J. Sarama, & A.-M. DiBiase, 401–14. Mahwah, NJ: for discussion. Washington, DC: Council of Chief State Lawrence Erlbaum. School Officers. 100. Ball, D., & D. Cohen. 1999. Developing practice, 89. National Association of School Psychologists. 1999. developing practitioners: Toward a practice-based Position statement on early childhood assessment. theory of professional education. In Teaching as the Washington, DC: Author. learning profession, eds. L. Darling-Hammond & G. 90. National Education Goals Panel. 1998. Principles Sykes. San Francisco: Jossey-Bass. and recommendations for early childhood assessments 101. Bransford, J.D., A.L. Brown, & R.R. Cocking, eds. (submitted to NEGP by the Goal 1 Early Childhood 1999. How people learn. Washington, DC: National Assessments Resource Group, eds. L. Shepard, S.L. Academy Press. Kagan, & E. Wurtz). Washington, DC: U.S. Govern- 102. Darling-Hammond, L. 1990. Instructional policy ment Printing Office. into practice: “The power of the bottom over the 91. Neisworth, J.T., & S.J. Bagnato. 2001. Recommended top.” and Policy Analysis 12 practices in assessment. In DEC recommended (3): 339–47. practices in early intervention/early childhood special 103. Barnett, W.S., & L. Masse. 2001. Financing early education, eds. S. Sandall, M.E. McLean, & B.J. Smith, care and education in the : CEER policy 17–28. Longmont, CO: Sopris West. brief. New Brunswick, NJ: Center for Early Education 92. Conference Board of the Mathematical . Research. 2001. The mathematical education of teachers, part 104. Brandon, R.N., S.L. Kagan, & J.M. Joesch. 2000. one. Providence, RI: Mathematical Association of Design choices: Universal financing for early care and America. education. Seattle: University of Washington.

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105. Mitchell, A., L. Stoney, & H. Dichter. 2001. Financ- 108. Moll, L.C., C. Armanti, D. Neff, & N. Gonzalez. 1992. ing child care in the United States: An expanded Funds of knowledge for teaching: Using a qualitative catalog of strategies. 2d ed. Kansas City, MO: approach to connect homes and classrooms. Theory Ewing Marion Kauffman Foundation. into Practice 31: 132–41. 106. Office of Economic Cooperation and Development. 109. Starkey, P., & A. Klein. 2000. Fostering parental 2000. OECD country note: Early childhood educa- support for children’s mathematical development: tion and care policy in the United States of America. An intervention with Head Start families. Early Edu- Washington, DC: Office for and cation and Development 11: 659–80. Improvement. 110. Edge, D. 2000. Involving families in school math- 107. Clements, D.H. 2003. Teaching and learning geom- ematics: from Teaching Children Mathemat- etry. In A research companion to Principles and Stan- ics, Mathematics Teaching in the Middle School, and dards for School Mathematics, eds. J. Kilpatrick, W.G. Arithmetic Teacher. Reston, VA: National Council of Martin, & D.E. Schifter. Reston, VA: National Council Teachers of Mathematics. of Teachers of Mathematics.

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Learning Paths and Teaching Strategies in Early Mathematics

The research base for sketching a picture of chil- early and late in the 3–6 age range. These are, then, dren’s mathematical development varies consider- simply two points along the learning path that may ably from one area of mathematics to another. Out- have many steps in between. For each content area, lining a learning path, moreover, does not mean we the Sample Teaching Strategies column shows a few can predict with confidence where a child of a given of the many teacher actions that promote learning age will be in that sequence. Developmental varia- when used within a classroom context that reflects tion is the norm, not the exception. However, chil- the recommendations set forth in this NAEYC/NCTM dren do tend to follow similar sequences, or learning position statement. In general, they are helpful strat- paths, as they develop. This chart illustrates in each egies, with adaptations, across the age range. area some things that many children know and do—

Content Examples of Typical Knowledge and Skills Sample Teaching Area Strategies From Age 3 Age 6

Number and Counts a collection of one Counts and produces (counts Models counting of small operations to four items and begins out) collections up to 100 collections and guides chil- to understand that the last using groups of 10. dren’s counting in every- counting word tells day situations, emphasizing how many. that we use one counting word for each object:

   “One . . . two . . . three . . .” Models counting by 10s while making groups of 10s (e.g., 10, 20, 30 . . . or 14, 24, 34 . . . ).

Quickly “sees” and labels Quickly “sees” and labels Gives children a brief collections of one to three with the correct number glimpse (a couple of with a number. “patterned” collections seconds) of a small collec- (e.g., dominoes) and unpat- tion of items and asks how terned collections of up to many there are. about six items.

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19 Early Childhood Mathematics

Content Examples of Typical Knowledge and Skills Sample Teaching Area From Age 3 Age 6 Strategies

Number and Adds and subtracts non- Adds or subtracts using Tells real-life stories involv- operations verbally when numbers counting-based strategies ing numbers and a prob- are very low. For example, such as counting on (e.g., lem. Asks how many ques- when one ball and then adding 3 to 5, says “Five . . . , tions (e.g., “How many are another are put into the six, seven, eight”), when left?” “How many are there box, expects the box to numbers and totals do not now?” “How many did they contain two balls. go beyond 10. start with?” “How many were added?”). Shows children the use of objects, fingers, counting on, guessing, and checking to solve problems.

Geometry Begins to match and name Recognizes and names a Introduces and labels a wide and spatial 2-D and 3-D shapes, first variety of 2-D and 3-D variety of shapes (e.g., skin- sense only with same size and shapes (e.g., quadrilater- ny triangles, fat rectangles, orientation, then shapes als, trapezoids, rhombi, prisms) that are in a variety that differ in size and hexagons, spheres, cubes) of positions (e.g., a square orientation (e.g., a large in any orientation. or a triangle standing on a triangle sitting on its point corner, a cylinder “standing Describes basic features of versus a small one sitting up” or horizontal). shapes (e.g., number of on its side). sides or angles). Involves children in con- structing shapes and talk- ing about their features.

Uses shapes, separately, to Makes a picture by combin- Encourages children to create a picture. ing shapes. make pictures or models of familiar objects using shape blocks, paper shapes, or other materials. Encourages children to make and talk about models with blocks and toys. Challenges children to mark Describes object locations Builds, draws, or follows a path from a table to the with spatial words such simple maps of familiar wastebasket with masking as under and behind and places, such as the class- tape, then draw a map of builds simple but mean- room or playground. the path, adding pictures ingful “maps” with toys of objects appearing along such as houses, cars, and the path, such as a table or trees. easel.

Copyright © 2002 National Association for the Education of Young Children

20 NAEYC/NCTM Joint Position Statement

Content Examples of Typical Knowledge and Skills Sample Teaching Area From Age 3 Age 6 Strategies

Measure- Recognizes and labels Tries out various pro- Uses comparing words to ment measurable attributes cesses and units for model and discuss measuring of objects (e.g., “I need measurement and begins (e.g. “This book feels heavier a long string,” “Is this to notice different results than that block,” “I wonder if heavy?”). of one method or another this block tower is taller than (e.g., what happens when the desk?”). Begins to compare and we don’t use a standard sort according to these Uses and creates situations unit). attributes (e.g., more/ that draw children’s attention less, heavy/light; “This Makes use of nonstandard to the problem of measuring block is too short to be measuring tools or uses something with two different the bridge”). conventional tools such units (e.g., making garden as a cup or ruler in non- rows “four shoes” apart, first standard ways (e.g., “It’s using a teacher’s shoe and three rulers long”). then a child’s shoe).

Pattern/ Notices and copies simple Notices and discusses pat- Encourages, models, and dis- algebraic repeating patterns, such terns in arithmetic (e.g., cusses patterns (e.g., “What’s thinking as a wall of blocks with adding one to any num- missing?” “Why do you think long, short, long, short, ber results in the next that is a pattern?” “I need a long, short, long. . . . “counting number”). blue next”). Engages children in finding color and shape patterns in the environment, number patterns on calendars and charts (e.g., with the numerals 1–100), patterns in arithmetic (e.g., recognizing that when zero is added to a number, the sum is always that number).

Displaying Sorts objects and counts Organizes and displays Invites children to sort and and analyz- and compares the groups data through simple organize collected materials ing data formed. numerical representa- by color, size, shape, etc. Asks tions such as bar graphs them to compare groups to Helps to make simple and counts the number in find which group has the most. graphs (e.g., a pictograph each group. formed as each child Uses “not” language to help places her own photo in children analyze their data the row indicating her (e.g., “All of these things are preferred treat—pretzels red, and these things are NOT or crackers). red”). Works with children to make simple numerical summa- ries such as tables and bar graphs, comparing parts of the data.

Copyright © 2002 National Association for the Education of Young Children

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