Teachers who teach mathematics and statistics in upper elementary grades must have not only strong general teaching skills but also strong content knowledge, strong knowledge of mathematicsspecific pedagogy, and much more, including cultural knowledge about their individual students, school policies, and how to collaborate with other teachers. Only with this knowledge will teachers of mathematics be able to meaningfully support the learning of each and every student.
Building on the standards presented in Chapters 2 and 3, we put forth in this chapter particular elaborations of the standards for the knowledge, skills, dispositions, and actions of wellprepared beginning upper elementary grades teachers of mathematics as well as requirements for preservice programs to ensure the effective preparation of candidates to develop those necessary knowledge, skills, dispositions, and actions. Additionally, the chapter includes commentary and examples about those standards relevant to upper elementary grades. The chapter concludes with standards regarding how programs may achieve these relevant standards. The elaborations in this chapter focus on those standards with specific upper elementary considerations; therefore, although all the standards in Chapters 2 and 3 apply to upper elementary mathematics teacher candidates, not all require elaboration.
Table 5.1 lists the elaborations of the standards presented in Chapters 2 and 3 as they relate to preparing teachers of mathematics for upper elementary grades.
Table 5.1. Elaborations of Candidate and Program Standards for Teachers of Mathematics for Upper Elementary Grades 

Part 1. Candidate Knowledge, Skills, and Dispositions 

UE.1. Mathematics Concepts and Connections to Mathematical Practices 
Wellprepared beginning teachers of mathematics at the upper elementary level understand foundational mathematics concepts that they will teach, and they connect those concepts to mathematical practices as well as to the mathematics of PreK–2 and the middle level curriculum. [Elaboration of C.1.1 and C.1.2] 
Wellprepared beginning teachers of mathematics at the upper elementary level develop pedagogical knowledge and practices to cultivate students’ mathematical proficiency, including such components as conceptual understanding, procedural fluency, problemsolving ability, and facility with the mathematical processes essential for learning. [Elaboration of C.2.2 and C.2.3] 

Wellprepared beginning teachers of mathematics at the upper elementary level effectively use technology tools, physical models, and mathematical representations to build student understanding of the topics at these grade levels. [Elaboration of C.1.6 and C.2.3] 

UE.4. Assessment to Promote Learning and Improve Instruction 
Wellprepared beginning teachers of mathematics at the upper elementary level learn to use both formal and informal assessment tools and strategies to gather evidence of students’ mathematical thinking in ways appropriate for young learners, such as the use of observations, interviews, questioning, paperandpencil and computerbased tasks, and digital records, including audio and video. [Elaboration of C.3.1] 
Wellprepared beginning teachers of mathematics at the upper elementary level nurture students' proficiency with, and making sense of, mathematical ideas, processes, and practices. [Elaboration of C.3.1 and C.3.2] 

Wellprepared beginning teachers of mathematics at the upper elementary level understand their roles as ethical advocates for elementarygrades students to have access to and advance in mathematics that cultivates positive mathematics identities and connects to students' mathematical thinking and lived experiences; these teachers build partnerships with families and communities and work to eliminate institutional and curricular barriers to learning. [Elaboration of C.4.1] 

Part 2. Program Characteristics 

UE.7. Mathematical Content Preparation of Upper Elementary Grades Teachers of Mathematics 
Effective programs preparing teachers of mathematics at the upper elementary level include coursework and other experiences focused on key mathematical ideas and skills that are pivotal in those grades. [Elaboration of P.2.1] 
UE.8. Mathematics Methods Coursework for Upper Elementary Grades Teachers of Mathematics 
Effective programs preparing teachers of mathematics at the upper elementary level include at least one mathematics methods course, or the equivalent experience of 3 semester units, focused particularly on mathematics teaching and learning in upper elementary grades. [Elaboration of P.3.1, P.3.2, P.3.3, and P.3.4] 
UE.9. Clinical Experiences for Upper Elementary Grades Teachers of Mathematics 
Effective programs preparing teachers of mathematics at the upper elementary level develop and systematically use a collection of clinical settings that support beginning teachers’ work with diverse learners and curricula and within differing institutional contexts. [Elaboration of P.4.2 and P.4.3] 
Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by WellPrepared Beginning Teachers of Mathematics in the Upper Elementary Grades
This section provides additional detail, commentary, and examples of the knowledge, skills, and dispositions wellprepared beginning teachers who teach mathematics in upper elementary grades have, organized by the general standards described in Chapter 2. Standards specific to teaching mathematics in upper elementary grades are defined and described, along with additional commentary and examples about the general standards in Chapter 2. The elaborations in this chapter focus on those standards with specific considerations for upper elementary grades. Although all the standards in Chapters 2 and 3 apply to mathematicsteacher candidates for upper elementary grades, not all require elaboration.
Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing Mathematics Teachers for Upper Elementary Grades
In this section we recommend what preservice programs need to do to prepare their students to meet the requirements specified in the previous section. Standards specific to upperelementarygrades mathematics teacher preparation programs are defined and described, along with additional commentary and examples related to the general standards presented in Chapter 3. Programs preparing candidates to teach a broader range of grades, such as PreK–5, should also meet the recommendations in this document for the PreK–2 preparation. The other standards are listed for the sake of completeness.
Closing Remarks
Wellprepared beginning teachers of mathematics for upper elementary grades must understand mathematics and use mathematical practices and processes, develop strong mathematical dispositions, and use mathematical tools and technology. They must learn to plan and implement effective instruction, analyze their teaching practices, and collaborate with colleagues, families, and community members. Wellprepared beginners must understand students’ mathematical thinking, their use of strategies and mathematical practices, and the development of their mathematical dispositions. Candidates must understand and be committed to an advocacy role for every mathematics learner.
Effective programs preparing teachers of mathematics at the upper elementary level develop candidates’ abilities to use highleverage, effective mathematics teaching practices (NCTM, 2014a) that require deep understanding of the mathematics they are expected to teach. Programs must include carefully designed opportunities for candidates to learn effective mathematicsspecific pedagogy, to learn about students as mathematics learners, and to participate in practicebased clinical experiences that are carefully designed and sequenced. Highquality programs designed on the basis of these standards will support beginning teachers’ growth toward teaching excellence.