This chapter consists of elaborations and examples of the standards in Chapter 2, describing the knowledge, skills, dispositions, and actions that wellprepared beginning Early Childhood mathematics teachers need to develop, followed by elaborations and examples of the standards in Chapter 3, describing characteristics needed for PreK to Grade 2 preservice programs to ensure the effective preparation of their candidates (see Table 4.1). The elaborations in this chapter are focused on those standards involving specific early childhood considerations; therefore, although all the standards in Chapters 2 and 3 apply to the mathematical preparation of early childhood teacher candidates, not all require elaboration.
Although young children are ready and eager to learn mathematics, many early childhood teachers have not been provided substantive opportunities for learning to engage children in rich experiences in domains other than literacy (Institute of Medicine [IOM] & National Research Council [NRC], 2015; NRC, 2001b, 2007). Early childhood educators teach children from birth to age 8 years, an especially critical developmental period of learning for mathematics. These early years form the cognitive foundations of mathematical thinking. Later school success in elementary school depends on preschool children's knowledge of mathematics (Duncan et al., 2007). Further, later reading achievement as well as early reading skills can be predicted from early performance in mathematics (Lerkkanen, RaskuPuttonen, Aunola, & Nurmi, 2005). The quantitative, spatial, and logicalreasoning competencies of mathematics establish an early cognitive foundation for thinking and learning across subjects.
Given the importance of mathematics to academic success (National Mathematics Advisory Panel, 2008), all children need to develop robust knowledge of mathematics in their earliest years. The development of this knowledge among children as well as the formation of their beliefs and dispositions toward mathematics is dependent upon and related to the capabilities and dispositions of their teachers (Tsamir & Tirosh, 2009). Given the limitations in the present preparation of the early childhood workforce in the domain of mathematics education (IOM & NRC, 2015), that teacher preparation programs attend to the standards set forth in this document is imperative. Effective teacher preparation programs at the early childhood level include focused, sustained, and substantial attention to developing candidates’ knowledge of learning trajectories, including deep understanding of early mathematics, knowledge of children’s development, and highleverage pedagogical skills and practices for teaching mathematics. This attention occurs in the context of a comprehensive program of professional preparation for teachers of young children (e.g., addressing socialemotional development, whole child, and curricula that are integrated throughout the day) but with a specific focus on mathematics, as supported by organizations such as the NAEYC (e.g., the standard of “understanding content knowledge and resources in academic disciplines” including mathematics, NAEYC, 2012, p. 36).
Table 4.1. Elaborations of Selected Candidate and Program Standards for Early Childhood Teachers of MathematicsEffective teacher preparation programs at the early childhood level not only meet the expectations outlined in Chapters 2 and 3 but also demonstrate particular attention to the elaborations of selected standards listed below. 

Part 1. Candidate Knowledge, Skills, and Dispositions 

Wellprepared beginning teachers of mathematics at the early childhood level have deep understandings of the mathematical concepts and processes important in early learning as well as knowledge beyond what they will teach. [Elaboration of C.1.1] 

EC.2. Positive Attitudes Toward Mathematics and Productive Dispositions Toward Teaching Mathematics

Wellprepared beginning teachers of mathematics at the early childhood level demonstrate positive attitudes toward mathematics as a discipline and productive dispositions toward the teaching and learning of mathematics. [Elaboration of C.1.3] 
EC.3. Mathematics Learning Trajectories: Paths for Excellence and Equity 
Wellprepared beginning teachers of mathematics at the early childhood level understand learning trajectories for key mathematical topics, including how these learning trajectories connect to foundational knowledge, curriculum, and assessment frameworks. [Elaboration of C.1.4] 
EC.4. Tools, Tasks, and Talk as Essential Pedagogies for Meaningful Mathematics 
Wellprepared beginning teachers of mathematics at the early childhood level intentionally plan for and use tools, tasks, and talk as pedagogies for young children’s engagement in meaningful mathematics. [Elaboration of C.2.2 and C.2.3] 
EC.5. Understanding Young Children’s Mathematical Thinking Informs Teaching 
Wellprepared beginning teachers of mathematics at the early childhood level elicit and analyze young children’s mathematical thinking to inform classroom interactions and instructional decisions. [Elaboration of C.2.3] 
EC.6. Collaboration With Families Enhances Children’s Mathematical Development 
Wellprepared beginning teachers of mathematics at the early childhood level collaborate with families in a mutually respectful, reciprocal manner to enhance and connect children’s inschool and outofschool mathematical development. [Elaboration of C.2.5] 
Wellprepared beginning teachers of mathematics at the early childhood level are conversant in the developmental progressions that are the core components of learning trajectories and strive to see mathematical situations through children’s eyes. [Elaboration of C.3.1] 

EC.8. Creating Positive Early Childhood Learning Environments 
Wellprepared beginning teachers of mathematics at the early childhood level create mathematical learning environments characterized by exploration, reasoning, and problem solving; they draw upon children’s mathematical, cultural, and linguistic strengths thereby developing conceptual understanding and positive mathematical identities. [Elaboration of C.4.2 and C.4.3] 
Part 2. Program Characteristics 

EC.9. Mathematics Content Preparation of Early Childhood Teachers

Effective programs preparing teachers of mathematics at the early childhood level require at least one mathematics content course (or equivalent professionallearning experiences) focused on key mathematical ideas and processes that are important in early mathematics, including problem solving, number, operations, spatial thinking, shapes, measurement, and early algebraic thinking. [Elaboration of P.2] 
EC.10. Mathematics Methods Experiences for Early Childhood Teachers

Effective programs preparing teachers of mathematics at the early childhood level require at least one mathematics methods course (equivalent of 3 semester units) focused on mathematics teaching, children’s mathematical thinking, and development of mathematics learning at the early childhood level. [Elaboration of P.2 and P.3] 
EC.11. Clinical Experiences in Mathematics for Early Childhood Teachers 
Effective programs preparing teachers of mathematics at the early childhood level provide clinical experiences specific to mathematics focused on children’s mathematical thinking and mathematics instruction with diverse learners in preschool and primary settings. [Elaboration of P.1 and P.4] 
Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by WellPrepared Beginning Early Childhood Teachers of Mathematics
This section provides additional detail, commentary, and examples of the knowledge, skills, and dispositions wellprepared early childhood mathematics teachers possess, organized by the general standards in Chapter 2.
Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing Early Childhood Teachers of Mathematics
This section provides additional detail of what preservice programs need to do to effectively prepare candidates to teach early childhood mathematics. Although young children are ready and eager to learn (NRC, 2001b), many early childhood teachers are not eager and prepared to engage children in rich experiences in domains other than literacy (Brenneman, StevensonBoyd, & Frede, 2009; IOM & NRC, 2015; NRC, 2007). Teachers of young children historically have not been prepared to teach domainspecific knowledge to young children (Isenberg, 2000). Effective programs ensure that they follow all the recommendations in Chapter 3 to provide guidance in enacting researchbased policies and practices for the preparation of all early childhood teachers of mathematics. Here we provide brief elaboration of three of the program standards, along with commentary on the other two program standards.
Closing Remarks
Effective programs for preparing teachers of mathematics at the early childhood level develop candidates’ abilities to use highleverage, effective mathematics teaching practices (NCTM, 2014a) that require deep understandings of the mathematics candidates are expected to teach. The teaching they are expected to enact on a daily basis with young learners often stands in sharp contrast to what many candidates experienced themselves as learners of mathematics (Isenberg, 2000). They often describe their own experiences as being teachercentered instruction emphasizing memorization of facts and procedures with little to no emphasis on understanding, problem solving, reasoning, and application. In addition, many early childhood teachers report high levels of mathematics anxiety and avoidance. Effective programs consciously break the insidious cycle that currently exists in which early childhood teachers pass their own anxieties and superficial knowledge of mathematics on to their children. Instead, effective programs prepare their candidates to engage in ambitious teaching, teaching in which teachers "aim to teach all kinds of children to not only know academic subjects, but also to be able to use what they know in working on authentic problems in academic domains” (Lampert, Boerst, & Graziani, 2011, p. 1). In addition, effective programs instill in their candidates positive and productive dispositions toward mathematics teaching and learning, which they will, in turn, pass on to their students.