Slope and Line of Best Fit: A Transfer of Knowledge Case Study Courtney Nagle Penn State Erie, The Behrend College Stephanie Casey Eastern Michigan University Deborah Moore-Russo University at Buffalo This paper brings together research on slope from mathematics education and research on line of best fit from statistics education by considering what knowledge of slope students transfer to a novel task involving determining the placement of an informal line of best fit. This study focuses on two students who transitioned from placing inaccurate to accurate lines of best fit during a task-based interview. The analysis focuses on describing shifts in slope reasoning that accompanied the change to accurately placed lines, and investigates factors that may have influenced the shift in reasoning. The results have implications for the teaching of both slope and the line of best fit. One of the primary goals of education is to facilitate learning experiences that promote the use of ideas in settings beyond the specific conditions of the learning environment (Evans, 1999; Lobato, 2006). As a result, many researchers have turned their attention to describing students’ transfer of knowledge to new contexts (Lobato, 2012; Schwartz & Martin, 2004). The mathematical concept of slope is a foundational topic that is introduced via linear functions in introductory algebra, but it is also useful when studying quadratic and exponential functions in precalculus, derivatives in calculus, and the line of best fit (LOBF) in statistics. Past research has investigated the transfer of slope to new problem situations (e.g., building a wheelchair ramp—Lobato & Siebert, 2002) and new academic contexts (e.g., in secondary versus post-secondary mathematics settings—Nagle, Moore-Russo, Viglietti, & Martin, 2013), but research has yet to explore students’ transfer of slope from a mathematical to a statistical problem context. In light of concerns that mathematics and statistics education are becoming increasingly disconnected (Groth, 2015), investigating students’ transfer of knowledge from a mathematical to a statistical setting is of particular interest. The LOBF is a statistical model used when data from two quantitative variables exhibit a linear association and is one of the most basic representations to introduce the direction of association between two variables. Describing the transfer of knowledge about slope to the statistical setting of finding the LOBF is particularly relevant given the importance of these topics and the natural transfer afforded by these closely related concepts and their proximity in the mathematics curriculum. We investigate the intricacies of the process involved in transfer of knowledge by focusing on factors that mediate and influence the transfer of prior knowledge of slope to the new task of placing an LOBF. Literature Review Slope and Covariational Reasoning Slope, or the constant rate of change of a linear function, is a focal point of the middle school mathematics curriculum (National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010; Ozer & Sezer, 2014; Zou, 2014). Despite its essential role, students are often restricted by procedural notions of slope (Nagle et al., 2013; Zahner, 2015; Zaslavsky, Sela, & Leron, 2002), and many students do not make connections between graphical interpretations of slope as a property of a line and analytic interpretations of slope as the rate of change of a linear function (Harel, Behr, Lesh, & Post, 1994; Stump, 2001a,b). Students’ difficulties with slope may stem from the multitude of ways that it can be conceptualized. Research suggests there are 11 ways that slope is presented (see Table 1; Moore- Russo, Conner, & Rugg, 2011) and that all of these conceptualizations are expected to be covered in the K–12 curriculum (Nagle & Moore-Russo, 2014b; Nagle et al., 2013; Stanton & Moore-Russo, 2012). In this manuscript, five conceptualizations are discussed. Linear Constant (LC) refers to the notion that slope is constant for a linear relationship, sometimes referred to as what makes a line “straight.” A Functional Property (FP) conceptualization of slope involves interpreting slope as a rate of change of two covarying quantities, whereas a Behavior Indicator (BI) conceptualization relates the sign of slope to the increasing or decreasing behavior of the linear relationship. Physical Property (PP) is used when relating the magnitude of slope to the steepness of a line. Any of these conceptualizations may be used in conjunction with a Real-World Situation (RW) conceptualization, which interprets slope in light of a real-world context. 13 School Science and Mathematics