The Common Core State Standards for Mathematics (CCSSM) states that high school students should be able to recognize patterns of growth in linear, quadratic, and exponential functions and construct such functions from tables of data (CCSSI 2010). Accordingly, many high school curricula include a method that uses finite differences between data points to generate polynomial functions. That is, students may examine differences between successive output values (called first differences), successive differences of the first differences (second differences), or successive differences of the (*n* - 1)th differences (*n*th-order differences), and rely on the following:

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### Kathryn Rhoads and James A. Mendoza Alvarez

### Keith Weber and Kathryn Rhoads

Understanding what mathematics teachers know, what they need to know about mathematics, and how that knowledge is learned are important goals in mathematics education. Research on mathematics teacher knowledge can be divided into two categories: (a) what knowledge mathematics teachers have or need to have to teach effectively (e.g., Hill, Rowan, & Ball, 2005; Kahan, Cooper, & Bethea, 2003), and (b) how teachers' mathematical knowledge for teaching can be developed (e.g., Bell, Wilson, Higgins, & McCoach, 2010; Proulx, 2008). This book describes research of the second type. To date, research in this area has focused primarily on how mathematical knowledge develops in university or researcher-led teacher preparation or professional development programs. This book is novel in that it concerns how and what teachers learn through the process of teaching itself. In his contribution to this book, Ron Tzur (chapter 3) lays out three reasons why this research is essential. First, he argues, teacher preparation programs simply do not contain enough time for teachers to learn all they need to know, so teachers' learning through teaching is essential. Second, for teachers to develop knowledge of how students think about mathematics and how students receive mathematical lessons, teachers must have classroom experience. Third, the experiences that teachers encounter when teaching have the potential to give rise to meaningful changes in their beliefs and practice.