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Limitations and future directionsDespite the many advances reported here, some gaps remain in our understanding of effective strategy interventions for fractions and algebra. For example, our knowledge of negatives is limited compared to their positive counterparts. Yet, errors with negatives prevent many students from being successful in algebra (Booth et al., 2014). Further, as noted above, more research is needed on ways to promote regular use of flexible strategies in algebra, such as asking students to solve the same problem in two ways once they have learned multiple strategies. Although there has been renewed interest in fraction knowledge recently, much of this research has focused on identifying challenges to learning fractions and finding effective interventions for faulty strategies (e.g., Fuchs et al., 2017; Siegler et al., 2013). Given the difficulties that children and adults have with fractions, this focus is a reasonable and laudable one. Yet, research on flexibility with fractions is scarce and so future directions should include ways to promote flexibility with fractions. Much of the research reported here has focused on conceptual and procedural knowledge of fractions and algebra, such as the ability to order fractions or to solve linear equations. While this knowledge is clearly important, additional research is needed to understand and support problem solving with fractions and algebra. “Problem solving” sometimes refers to a student calculating an answer, even to a routine exercise, as opposed to studying a worked example (see Newton & Sword, 2018). I am instead referring to opportunities for students to apply knowledge to a new situation, often presented in the form of word problems. Jitendra, Harwell, Dupuis, and Karl (2017) have conducted extensive research on this kind of problem solving, with a focus on proportional reasoning. Proportions and proportional reasoning represent a bridge between fractions and algebra, making it a critical mathematical milestone in late middle school. Their works suggests that schemabased instruction, where students are taught to identify and represent the underlying structure of a problem, has been effective with general and special education students. The work of researchers like Empson and Levy (2011) builds on the premise that problem solving is important as a means for understanding fractions (not simply as an application of fraction knowledge). Future research should extend work presented in this chapter by considering the role of problem solving in learning fractions and algebra. For example, is problem solving compatible with the use of worked examples? If so, how can they work together to support learning? Related, increased efforts to cross the boundaries of psychology and education are needed. Obersteiner, Dresler, Bieck, and Moeller (2019) recently reported on findings in these fields and in neuroscience, leading to recommendations for fraction instruction. Moreover, research in these fields should come together to support development and understanding of effective comprehensive interventions, such as those described by Fuchs et al. (2017) and McNeil et al. (2019). Interventions that are successful in short laboratory studies or controlled classroom studies sometimes fall short in the context of an ongoing and complicated classroom, for a variety of reason, including lack of implementation by teachers (Star et al., 2015). Attention, prior knowledge, fidelity, and many other factors can prevent even empirically supported practices from making the same impact in the classroom. Yet, for research on fractions and algebra to make a difference with students, the findings must be integrated into regular instruction. 
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