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In this study, I examined the degree to which experienced teachers were aware of the relationship between prior knowledge and new learning. Interviews with teachers revealed that they were explicitly aware of when students made connections between prior knowledge and new learning, when they applied their prior knowledge to new contexts, and when they developed their prior knowledge as a result of applying that knowledge to new contexts. However, teachers were not explicitly aware of backwardtransfer effects. Results from this study have implications for future research on backward transfer, as well as for teacher professional development.

Even in simple mathematical situations, there is an array of different mathematical features that students can attend to or notice. What students notice mathematically has consequences for their subsequent reasoning. By adapting work from both cognitive science and applied linguistics anthropology, we present a focusing framework, which treats noticing as a complex phenomenon that is distributed across individual cognition, social interactions, material resources, and normed practices. Specifically, this research demonstrates that different centers of focus emerged in two middle grades mathematics classes addressing the same content goals, which, in turn, were related conceptually to differences in student reasoning on subsequent interview tasks. Furthermore, differences in the discourse practices, features of the mathematical tasks, and the nature of the mathematical activity in the two classrooms were related to the different mathematical features that students appeared to notice.

How can research have a larger impact on educational practice? What kinds of research can have the greatest impact on educational practice? These are perennially thought-provoking questions for mathematics education researchers (e.g., Battista et al., 2007; Boerst et al., 2010; Heck et al., 2012; Heid et al., 2006; Herbel-Eisenmann et al., 2016; Langrall, 2014; Silver, 2003) as well as educational researchers more broadly (Kane, 2016; Snow, 2016). In recent years, educational researchers have lamented the failure of educational research to have a transformative effect on educational practice despite repeated reform efforts. One might be tempted to adapt a motto of the Reformation, Ecclesia reformata, semper reformanda, to describe the history of education: reformed and always reforming. Payne (2008) systematically reported the persistence of failure in urban schools despite “so much reform.” However, the failed impact of educational research on practice goes far beyond urban schools (Bryk, Gomez, Grunow, & LeMahieu, 2015). We are forced to ask, how can the field of educational research improve its impact on practice?

In our first editorial (Cai et al., 2017), we highlighted the long-standing, critical issue of improving the impact of educational research on practice. We took a broad view of impact, defining it as research having an effect on how students learn mathematics by informing how practitioners, policymakers, other researchers, and the public think about what mathematics education is and what it should be. As we begin to dig more deeply into the issue of impact, it would be useful to be more precise about what impact means in this context. In this editorial, we focus our attention on defining and elaborating exactly what we mean by “the impact of educational research on students' learning.”

In our last editorial, we considered the impact of research on students' learning. In clarifying our perspective, we answered the question of “impact of research on what” to include both cognitive and noncognitive outcomes in students as well as long-term impact on students that goes well beyond short-term cognitive impact. A natural next step is to examine the conditions under which students can achieve such broad goals. We will devote the next set of editorials to exploring ways in which researchers can design their work to increase its impact on students' opportunities to achieve these goals.

In our May editorial (Cai et al., 2017), we argued that a promising way of closing the gap between research and practice is for researchers to develop and test sequences of learning opportunities, at a grain size useful to teachers, that help students move toward well-defined learning goals. We wish to take this argument one step further. If researchers choose to focus on learning opportunities as a way to produce usable knowledge for teachers, we argue that they could increase their impact on practice even further by integrating the implementation of these learning opportunities into their research. That is, researchers who aim to impact practice by studying the specification of learning goals and productively aligned learning opportunities could add significant practical value by including implementation as an integral part of their work.

We began our editorials in 2017 seeking answers to one complex but important question: How can we improve the impact of research on practice? In our first editorial, we suggested that a first step would be to better define the problem by developing a better understanding of the fundamental reasons for the divide between research and practice (Cai et al., 2017). This sparked subsequent editorials in which we delved deeper into some of the many complicated facets of this issue. In our March (Cai et al., 2017b) editorial, we argued that impact needs to be defined more broadly than it often has been, notably, to include cognitive and noncognitive outcomes in both the near term and longitudinally. This led us to focus our May (Cai et al., 2017a) editorial on the ways that research might have a greater impact on the learning opportunities that help students reach broader learning goals. We argued that it is not enough to identify learning goals–it is also necessary to conduct research that breaks those learning goals into subgoals that can be appropriately sequenced. We highlighted research on learning trajectories as an example of this sort of work but also emphasized the need to work at a grain size that is compatible with teachers' classroom practice. Finally, in our July (Cai et al., 2017c) editorial, we argued that the implementation of learning opportunities in the classroom is an integral element of research that has an impact on practice.

In the past year, we have used this space to tackle a chronic and important concern in mathematics education: how to increase the impact of research on practice. Because of the unique nature of this issue of JRME, we pause to address the critical idea of replication in educational research. In later issues, we will continue our primary theme and consider how the ideas raised in this editorial can further our understanding of the relationships between research and practice.

In our November 2017 editorial (Cai et al., 2017), we presented a vision of a future in which research has a significant impact on practice. In the world we described, researchers and teachers work together, sharing similar goals and incentive structures. A critical feature of this brave new world is the existence of an online professional knowledge base comprising “useful findings and artifacts that are continuously refined over time, indexed by specific learning goals and subgoals, and that assist teachers and researchers in implementing learning opportunities in their classrooms” (p. 469). Moreover, we argued that teacher—researcher partnerships are a necessary condition for greater impact on practice.

In our March editorial (Cai et al., 2018), we considered the problem of isolation in the work of teachers and researchers. In particular, we proposed ways to take advantage of emerging technological resources, such as online archives of student data linked to instructional activities and indexed by learning goals, to produce a professional knowledge base (Cai et al., 2017b, 2018). This proposal would refashion our conceptions of the nature and collection of data so that teachers, researchers, and teacher-researcher partnerships could benefit from the accumulated learning of ordinarily isolated groups. Although we have discussed the general parameters for such a system in previous editorials, in this editorial, we present a potential mechanism for accumulating learning into a professional knowledge base, a mechanism that involves collaboration between multiple teacher-researcher partnerships. To illustrate our ideas, we return once again to the collaboration between fourth-grade teacher Mr. Lovemath and mathematics education researcher Ms. Research, who are mentioned in our previous editorials(Cai et al., 2017a, 2017b).