Conceptualizing "Homework" in Flipped Mathematics Classes

Introduction Within flipped instruction, teachers invert or "flip" the settings in which teacher lecture and student practice occur. In the last few years, teachers' implementation of flipped instruction has increased dramatically. Flipped instruction has now reached the point where more than two-thirds of teachers in the United States report flipping a lesson, if not an entire course (Smith, 2014). Although flipped instruction originated in the context of university-level sciences (Mazur, 1991), it has become especially popular in the context of secondary and postsecondary mathematics (Moore, Gillett, & Steele, 2014; Zack et al., 2015). The increase in flipped mathematics courses is perhaps due to the recent proliferation of online mathematics instructional videos (e.g., Khan Academy) and the longstanding tradition, particularly in the United States, of mathematics instruction characterized by lecture and exposition (Stigler & Hiebert, 1999). This lecture-based approach translates well to video-based delivery. Further, the more frequent use of lectures at the secondary and post-secondary levels than at the elementary level (Banilower et al., 2013) corresponds with greater utilization of flipped instruction in those settings (Yarbro, Arfstrom, McKnight, & McKnight, 2014). Thus, secondary and post-secondary mathematics classes in the United States constitutes a ripe site for empirical investigations of flipped instruction. The potential innovative power of flipped instruction can stem from videos themselves, from a different use of class time, or from a synergy between the two. Because research on flipped instruction is still limited (Uzunboylu & Karagozlu, 2015), we focus on the most defining component of flipped instruction: the instructional video. Teachers in flipped classrooms typically assign the instructional videos for homework. Thus, the videos replace the problem sets that are the typical form of homework in mathematics. We present a framework for flipped mathematics homework that categorizes types. We also draw on technology literature and mathematics education literature to discern quality for each type. Literature review Much of the existing literature related to flipped instruction involves expositional descriptions of how authors intend to implement it (e.g., Bergmann & Sams, 2012) but does not involve empirical, third-party observations of those implementations. In mathematics education, research is emerging but the early studies tend to focus on small numbers of flipped classes and on outcomes rather than on the nuances of the implementations. For example, Clark (2015) examined two flipped algebra classes and compared their performance on unit content tests and their survey responses to those of students in non-flipped algebra classes. Clark found no significant differences with respect to performance and only moderate differences in the survey results, with slight preference indicated toward flipped instruction. This study, however, did not give attention to the specific ways in which flipped instruction was implemented. Further, the study did not account for the fact that other implementations of flipped instruction could vary widely from Clark's. Similarly, DeSantis and colleagues (2015) compared one instance of flipped geometry instruction with non-flipped instruction and found no significant differences with regard to mathematical performance. They did find a slight difference in student opinions, in this case with students preferring non-flipped instruction. Others (Love, Hodge, Grandgenett, & Swift, 2014; Sahin, Cavlazoglu, & Zeytuncu, 2015) found more positive results for flipped instruction in university mathematics courses. However, as with the previously cited studies, the details and variability of flipped instruction were not accounted for in the study design. This may be because there is not yet a robust framework for flipped instruction in mathematics. …