Well-prepared beginning teachers of mathematics have solid and flexible knowledge of mathematical processes and practices, recognizing that these are tools used to solve problems and communicate ideas. |

The mathematical knowledge of well-prepared beginning teachers of mathematics includes ability to use mathematical and statistical processes and practices (NCTM, 2000; NGA & CCSSO, 2010; Shaughnessy, Chance, & Kranendonk, 2009) to solve problems. They use mathematical language with care and precision. These teachers can explain their mathematical thinking using grade-appropriate concepts, procedures, and language, including grade-appropriate definitions and interpretations for key mathematical concepts. They can apply their mathematical knowledge to real-world situations by using mathematical modeling to solve problems appropriate for the grade levels and the students they will teach. They are able to effectively use representations and technological tools appropriate for the mathematics content they will teach. They regard *doing mathematics* as a sense-making activity that promotes perseverance, problem posing, and problem solving. In short, they exemplify the mathematical thinking that will be expected of their students.

Well-prepared beginners recognize processes and practices when they emerge in their mathematical thinking and highlight these actions and behaviors when they observe them in others. Over time, beginning teachers can (a) better distinguish intricacies among the various processes and practices, determining those that are at the crux of a mathematical investigation and (b) see the interrelationships among the processes and practices.

Well-prepared beginners understand that mathematics is a human endeavor that is practiced in and out of school, across many facets of life. They know that mathematics has a history and includes contributions from people with different genders and cultural, linguistic, religious, and racial/ethnic backgrounds. Mathematics is based on constructed conventions and agreements about the meanings of words and symbols, and these conventions vary. Well-prepared beginners are aware that algorithms considered as standard in the United States differ from algorithms used in other countries and that some alternative algorithms have different, desirable properties that make them worth knowing. This idea is elaborated in later *standards* and *indicators* of this chapter as well as in Chapters 4–7.