Chapter 2: Candidate Knowledge, Skills, and Dispositions

Teaching is a complex enterprise, and teaching mathematics is particularly demanding. Thus, that initial preparation focused on teachers’ knowledge of the subject, on the curriculum, or on how students learn the subject is more effective than preparation focused on teachers' specific behaviors is not surprising (cf. Ball & Forzani, 2011; Philipp et al., 2007). As described in Accreditation Standards and Evidence: Aspirations for Educator Preparation (Council for the Accreditation of Educator Preparation (CAEP), 2013), teacher candidates must learn “critical concepts and principles of their discipline and, by completion, are able to use discipline-specific practices flexibly” (p. 2). This chapter includes standards and indicators to describe the specific knowledge, skills, and dispositions that well-prepared mathematics teacher candidates at all levels will know and be able to do upon completion of an initial preparation program. Additional expectations specific to a particular grade-band are provided in Chapters 4–7 as elaborations. We refer to those who are starting their careers after completion of a teacher preparation program as “well-prepared beginning teachers of mathematics” (“well-prepared beginners” for short).

Organization of This Chapter

This chapter includes four equally important and interrelated standards that describe the knowledge, skills, and dispositions that well-prepared beginners need to acquire. The first standard, “Knowledge of Mathematics Concepts, Practices, and Curriculum,” describes disciplinary knowledge involved in the teaching of mathematics. The second, “Knowledge and Pedagogical Practices for Teaching Mathematics,” describes research-based practices or strategies for effective mathematics teaching. The third, “Knowledge of Students as Learners of Mathematics,” describes what teachers need to know about their students’ mathematical knowledge, skills, representations, and dispositions, for both individual students and groups of students. The final standard in this chapter, “Social Contexts of Mathematics Teaching and Learning,” describes the knowledge and dispositions beginning teachers must have related to the social, historical, and institutional contexts of mathematics that affect teaching and learning, themes also woven into the first three standards.

As indicated in Table 2.1, each standard includes specific indicators, along with accompanying explanations. These standards and indicators apply to all well-prepared beginning teachers of mathematics from prekindergarten through high school.




C.1. Mathematics Concepts, Practices, and Curriculum

Well-prepared beginning teachers of mathematics possess robust knowledge of mathematical and statistical concepts that underlie what they encounter in teaching. They engage in appropriate mathematical and statistical practices and support their students in doing the same. They can read, analyze, and discuss curriculum, assessment, and standards documents as well as students’ mathematical productions.

C.1.1. Know Relevant Mathematical Content

C.1.2. Demonstrate Mathematical Practices and Processes

C.1.3. Exhibit Productive Mathematical Dispositions

C.1.4. Analyze the Mathematical Content of Curriculum

C.1.5. Analyze Mathematical Thinking

C.1.6. Use Mathematical Tools and Technology

C.2. Pedagogical Knowledge and Practices for Teaching Mathematics

Well-prepared beginning teachers of mathematics have foundations of pedagogical knowledge, effective and equitable mathematics teaching practices, and positive and productive dispositions toward teaching mathematics to support students’ sense making, understanding, and reasoning.

C.2.1. Promote Equitable Teaching

C.2.2. Plan for Effective Instruction

C.2.3. Implement Effective Instruction

C.2.4. Analyze Teaching Practice

C.2.5. Enhance Teaching Through Collaboration With Colleagues, Families, and Community Members

C.3. Students as Learners of Mathematics

Well-prepared beginning teachers of mathematics have foundational understandings of students’ mathematical knowledge, skills, and dispositions. They also know how these understandings can contribute to effective teaching and are committed to expanding and deepening their knowledge of students as learners of mathematics.

C.3.1. Anticipate and Attend to Students’ Thinking About Mathematics Content

C.3.2. Understand and Recognize Students’ Engagement in Mathematical Practices

C.3.3. Anticipate and Attend to Students’ Mathematical Dispositions

C.4. Social Contexts of Mathematics Teaching and Learning

Well-prepared beginning teachers of mathematics realize that the social, historical, and institutional contexts of mathematics affect teaching and learning and know about and are committed to their critical roles as advocates for each and every student.

C.4.1. Provide Access and Advancement

C.4.2. Cultivate Positive Mathematical Identities

C.4.3. Draw on Students’ Mathematical Strengths

C.4.4. Understand Power and Privilege in the History of Mathematics Education

C.4.5. Enact Ethical Practice for Advocacy

What Should Well-Prepared Beginning Teachers of Mathematics Know and Be Able to Do, and What Dispositions Should They Develop?

The guiding question for this chapter is “Recognizing that learning to teach is an ongoing process over many years, what are reasonable expectations for the most important knowledge, skills, and dispositions that beginning teachers of mathematics must possess to be effective?” Answering this question is difficult because some aspects of teaching will not be well learned initially, even though they may be critically important to student learning. What a beginner knows is also a significant equity issue because students with the greatest needs are often taught by teachers with the least experience (Kalogrides, Loeb, & Béteille, 2012; Oakes, 2008).

Mathematics teachers, from the very beginning of their careers, must robustly understand the mathematical content knowledge for the age groups or grades they may teach, along with the content taught to the age groups preceding and following those they teach—and in a different and deeper way than is often presented in textbooks, curriculum documents, or standards. Such knowledge affects their students’ learning (e.g., Hill, Rowan, & Ball, 2005; National Mathematics Advisory Panel, 2008).

Well-prepared beginners must be ready to teach each and every student in their first classrooms. Although pedagogical skills develop over time, beginners must have an initial repertoire of effective and equitable teaching strategies; for example, in selecting tasks, orchestrating classroom discussions, building on prior knowledge, and connecting conceptual understanding and procedural fluency (NCTM, 2014a). All teachers, including well-prepared beginners, must hold positive dispositions about mathematics and mathematics learning, such as the notions that mathematics can and must be understood, and that each and every student can develop mathematical proficiency, along with a commitment to imbue their students with similar beliefs and dispositions.

To teach effectively, one must hold knowledge of learners and learning, both general pedagogical knowledge and knowledge specific to the learning and teaching of mathematics. Understanding mathematical learners includes knowing about their backgrounds, interests, strengths, and personalities as well as knowing how students think about and learn mathematics, including possible misconceptions and creative pathways they may take in learning (Ball & Forzani, 2011; Clements & Sarama, 2014; Sztajn, Confrey, Wilson, & Edgington, 2012). Well-prepared beginners must understand—at least at an initial level—how to assess the understandings and competencies of their students and use this knowledge to plan and modify instruction using research-based instructional strategies (e.g., Ball & Forzani, 2011; Shulman, 1986).

Mathematics teaching and learning are influenced by social, historical, and institutional contexts. Beginning teachers must be aware of learners’ social, cultural, and linguistic resources; know learners’ histories; and recognize how power relationships affect students’ mathematical identities, access, and advancement in mathematics (e.g., Gutiérrez, 2013b; Martin, 2015; Strutchens et al., 2012; Wager, 2012). For example, classroom dynamics and social interactions strongly influence students’ emerging mathematical identities, which in turn affect the students’ learning opportunities. In short, well-prepared beginners must be ethical advocates for every student.

Closing Remarks

In this chapter, we presented four equally important standards (with sets of indicators) to describe the knowledge, skills, and dispositions that well-prepared beginners have upon completion of their preparation programs. These standards are not discrete lists but are interrelated and interdependent. For example, one cannot support the mathematical learning for each student, without having knowledge of content, pedagogical skill, and awareness of social contexts. Setting such high expectations for beginning teacher of mathematics is critical to the success of Pre-K–12 students. For candidates to reach these goals requires strong preparation programs, the focus of Chapter 3.