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Part II Teaching-Learning Paths In Part II, we lay out a sequence of milestones for children ages 2-7 in the core areas of number (which includes whole number, relations, and op- erations) and geometry and measurement. We call this sequence a teaching- learning path. A teaching-learning path consists of the significant steps in learning in a particular topic; each new step in the learning path builds on the earlier steps. These paths are based on research that shows that young children generally follow particular paths when learning number-relations- operations and geometry-measurement (Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; Ginsburg, 1983). Of course, learning is a continuous process, but to overview the process, we have identified four related steps organized by age/grade. The four steps move from children 2 and 3 years old, to children age 4 or in prekindergarten, to children in kindergarten, to children in Grade 1. Grade 1 is included to indicate how the knowledge from the earlier step is usedâand vital for doing wellâin Grade 1. For our purposes, we define the core mathematical ideas as those that are mathematically central and coherent, consistent with the thinking of children who have had adequate mathematical experiences, and generative of future learning. Thus, they are foundational mathematically and devel- opmentally. They are achievable for children of these ages. That is, they are consistent with childrenâs ways of thinking, developing, and learning when they have experience with mathematics ideas. In addition, they are interesting to children. The committee recommends that all children learn this mathematics by the end of kindergarten. In Chapter 2, we discussed why these core ideas are important math- ematically. Here we focus on how they develop in children who have op- 121

122 MATHEMATICS LEARNING IN EARLY CHILDHOOD portunity to learn them as the ideas become increasingly sophisticated and interconnected over these years. Relationships among the ideas as well as some of childrenâs common errors are also discussed. Vital ideas for Grade 1 are briefly overviewed to indicate how younger childrenâs knowledge is developed and extended into Grade 1. As noted in Chapter 2, we are building on earlier efforts to articulate appropriate mathematics content for young children. In 1989, the Na- tional Council of Teachers of Mathematics (NCTM) issued Curriculum and Evaluation Standards for School Mathematics. This document described 13 curriculum standards for the grade band K-4 (as well as for the grade bands 5-8 and 9-12). Although these standards have been influential, they do not describe the mathematics to be learned in detail and did not give guidance by grade level, nor for children younger than kindergarten. In 2000, NCTM released Principles and Standards for School Math- ematics (PSSM) after an extensive process of revision of the 1989 standards. Prekindergarten (pre-K) was included this time, in the grade band pre-Kâ2. PSSM described five content standardsânumber and operations, algebra, geometry, measurement, and data analysis and probabilityâand five pro- cess standardsâproblem solving, reasoning and proof, communication, connections, and representationsâfor each of four grade bands (pre-Kâ2, 3-5, 6-8, 9-12), covering all of school mathematics from pre-K through the end of high school. Although PSSM discussed the mathematics to be learned at the grade bands in greater detail than the 1989 standards did, it still did not specify what was to be learned at individual grade levels. Recognizing the need for more in-depth attention to prekindergarten, early childhood educators and mathematics educators convened in 2000 and publish a conference report on the development of mathematics stan- dards for young children. The resulting book, Engaging Young Children in Mathematics: Findings of the 2000 National Conference on Standards for Preschool and Kindergarten Mathematics Education (Clements, Sarama, and DiBiase, 2004), contains 17 recommendations for early childhood mathematics education. They concern equity, programs, teaching, teachers and their development, assessment, appropriate mathematics for young children, and broader efforts to inform stakeholders and encourage col- laboration in early childhood education and addressing the need for age/ grade level standards. That report grouped the mathematics content for early childhood into four topic areas: number and operations, geometry, measurement, and algebra, patterns, and data analysis. In 2002, the National Association for the Education of Young Children (NAEYC) and NCTM approved a joint position statement, âEarly Child- hood Mathematics: Promoting Good Beginnings.â The statement includes 10 research-based recommendations to guide practice and 4 policy recom- mendations. The statement includes sample charts of learning paths related to a number goal and a geometry goal with activity examples.

TEACHING-LEARNING PATHS 123 In 2006, NCTM released Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (hereafter Cur- riculum Focal Points). These were developed in response to inconsistency in placement of topics by grade level in the United States and the lack of focus (âa mile wide and an inch deepâ) typical of U.S. mathematics cur- ricula. Although much shorter than PSSM (and developed over a much shorter time), this report gives grade-level recommendations for each indiÂ vidual grade from pre-K to grade 8. These grade-level recommendations do not specify a full curriculum but rather describe the most significant mathematical concepts and skills at each grade level. There are three focal points at each grade level, each of which is a coherent cluster of skills and ideas, sometimes cutting across NCTMâs five content strands. Curriculum Focal Points recommends that instruction at a grade level should devote the vast majority of attention to the content identified in the three focal points (p. 6). At pre-K and kindergarten, the three focal points concern number and operations, geometry, and measurement. In addition to the three focal points at each grade level, Curriculum Focal Points describes connections, which consist of related content, includ- ing contexts and material to receive continuing development from previous grade levels. At pre-K, the connections concern data analysis, number and operations, and algebra. At kindergarten, the connections concern data analysis, geometry, and algebra. Collectively, these previous reports form the basis for the descriptions of foundational and achievable mathemat- ics content in this report. The current report provides guidance on the two most critical mathematical areas during early childhood: number and operation and geometry and measurement, and as will be discussed later, number and operations is the area where young children need to spend the most time. Meaningful learning experiences in these content areas provide young children with the foundation that is necessary for them to be suc- cessful in later mathematics. SUPPORTING LEARNING IN MATHEMATICS Our view of children is one of powerful and intrinsically motivated mathematics learners who, in a supportive physical and social environment, spontaneously learn some aspects of mathematics and make connections and extensions. However, children need adult guidance to help them learn the many culturally important aspects of mathematics, such as language and counting. In preschools and care centers, all children will bring to each mathematical topic area some initial competencies and knowledge on which to build. The major teaching challenge is to build a mathematical learning and teaching environment in which children will learn at least the basics of each topic area. This will enable them to practice and build on their own knowledge, with guidance from adults, peers, and family members, and

124 MATHEMATICS LEARNING IN EARLY CHILDHOOD be supported to move through learning paths to learn the foundational and achievable content identified in this report. These teaching and learn- ing environments need to be consistent with the process goals outlined in Chapter 2, and they need to support children to be active in thinking about and discussing mathematical ideas. Children require significant amounts of time to develop the founda- tional mathematical skills and understandings they have the desire and po- tential to learn and that they will need for success at school. Although some children have a sufficiently enriched home environment and enough math- ematically focused interactions with family members so that they develop many of the necessary foundational mathematical understandings and skills at home, others do not. For the sake of equity, preschool programs should help children develop foundational mathematical understandings and skills; high-quality preschool programs that devote sufficient time to mathematics are able to do so (see Chapter 7). Even children who learn mathematical ideas at home will benefit from a consistent high-quality program experi- ence in the preschool and kindergarten years. It is therefore critical that suf- ficient time is devoted to mathematics instruction in preschool programs so that children develop the foundational mathematical skills and understand- ings described here. Time must be allocated not only for the more formal parts of mathematics instruction and discussions that occur in the whole group or in small groups, but also for children to elaborate and extend their mathematical thinking by exploring, creating, and playing. The time that is allotted for mathematics in early childhood programs must be allocated across various topics. The typical description of math- ematics content is divided into the five strands of number and operations, algebra, geometry, measurement, and data analysis and probability. These are used to describe and categorize all of school mathematics, from pre-K through high school, and these strands are intended to receive different amounts of emphasis at different grade levels. The committee is concerned that inclusion of all five strands for young children has led some programs and teachers to spread their mathematics time equally across these different content areas, thus spreading mathemati- cal experiences too thinly and not going deeply enough into the core foun- dations that children need to establish firmly. It is important to concentrate on number and operations and on geometry and measurement in the early childhood period, with a greater portion of time spent on number and oper- ations. Number is critically important to all of later mathematics. Geometry and measurement play an important supporting role in the development of number concepts and are themselves important to later mathematics. In addition, research on programs that result in positive learning gains for children indicate that children need sufficient time working with these ideas in order to achieve a level of proficiency that prepares them for continuing

TEACHING-LEARNING PATHS 125 success in mathematics. Of course, many activities overlap these topic areas and could be counted in either if there is a balanced focus on both. The time spent on number and operations and on geometry and measurement can also include connections to data analysis and patterns, as listed in Cur- riculum Focal Points and discussed in the chapters of Part II. The kind of learning involved in various number and operation compo- nents and in various aspects of geometry and measurement is different, as we describe. Major themes of these variations in the kinds of learning are the need for achieving fluency, the use of patterns, generalizing, and extend- ing. All of these require many repeated experiences with the same numbers and related similar tasks. This is part of what makes learning mathematics require so much time focused on mathematical content. Mathematics is a participant sport. Children must play it frequently to become good at it. They do need frequent modeling of correct performance, discussion about the concepts involved, and frequent feedback about their performance. Both modeling and feedback can come from other students as well as from adults, and feedback also sometimes comes from the situ- ation. All children must have sustained and frequent times in which they themselves enact the core mathematical content and talk about what they are doing and why they are doing it. In mathematics learning, effort cre- ates ability. REFERENCES AND BIBLIOGRAPHY Clements, D.H., and Sarama, J. (2007). Early childhood mathematics learning. In F.K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 461-555). New York: Information Age. Clements, D.H., and Sarama, J. (2008). Experimental evaluation of a research-based preschool mathematics curriculum. American Educational Research Journal, 45, 443â494. Clements, D.H., Sarama, J., and DiBiase, A. (2004). Engaging Young Children in Mathemat- ics: Findings of the 2000 National Conference on Standards for Preschool and Kinder- garten Mathematics Education. Mahwah, NJ: Erlbaum. Fuson, K.C. (1992a). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R.T. Putnam, and R.A. Hattrup (Eds.), The Analysis of Arith- metic for Mathematics Teaching (pp. 53-187). Hillsdale, NJ: Erlbaum. Fuson, K.C. (1992b). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 243-275). New York: Macmillan. Ginsburg, H.P. (1983). The Development of Mathematical Thinking. New York: Academic Press. National Association for the Education of Young Children and National Council of Teachers of Mathematics. (2002). Early Childhood Mathematics: Promoting Good Beginnings. A joint position statement of the National Association for the Education of Young Chil- dren and National Council of Teachers of Mathematics. Available: http://www.naeyc. org/about/positions/pdf/psmath.pdf [accessed August 2008]. National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.

126 MATHEMATICS LEARNING IN EARLY CHILDHOOD National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2006). Curriculum Focal Points for Prekinder- garten through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: Author.