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Mathematics strategy interventionsIn mathematics, a strategy can be defined as “a general approach for accomplishing a task or solving a problem that may include sequences of steps to be executed, as well as the rationale behind the use and effectiveness of these steps” (Star et al., 2015, p. 26). For example, a student may always approach a quadratic equation by using the quadratic formula (which involves a particular sequence of steps), even though some quadratics can be solved in other ways (e.g., by factoring). Some experts prefer to first “clear the denominator” when they encounter a linear equation with a fraction in front of parentheses (e.g., '/s(x + 6) = 4 becomes x + 6 = 12), whereas a student struggling to learn algebra may generally prefer to use the distributive property whenever parentheses are involved, because it is familiar and reliable (Newton, Star, & Lynch, 2010; Star & Newton, 2009). When students hold misconceptions, their general approach may lead to errors. For example, a student may believe that common denominators are always necessary for operating with fractions, leading him or her to use that strategy to multiply two fractions. Studies identifying common errors can help illuminate misconceptions and poor strategies that need to be targeted as part of an intervention. Therefore, in the first part of this chapter I overview major errors documented in the literature, then I present some interventions designed to target these errors. As illustrated above, sometimes there are multiple valid strategies for solving mathematics problems. Therefore, in the second part of the chapter I focus on interventions that promote flexible problem solving. “Procedural flexibility includes identifying and implementing multiple methods to solve algebra problems, as well as choosing the most appropriate method” (Star et al., 2015, p. 2). As with the strategy of clearing the denominator described above, employing alternative strategies can make problems easier to solve in some way (e.g., fewer steps). Experts demonstrate this kind of flexibility (Star & Newton, 2009), but researchers have found that it is slow to develop in students (Newton, Lange, & Booth, 2019). Both erroneous and flexible strategy use can be viewed through Siegler’s Overlapping Waves Theory (OWT), which helps to explain how strategy use changes over time. Children typically know multiple ways to solve a given problem; with practice, they begin to use more accurate and efficient strategies (Fazio, DeWolf, & Siegler, 2016; Opfer & Siegler, 2007). The development of accurate, efficient strategies is not smooth or linear. For example, to find the total number of objects given two sets, a child might sometimes count all of the objects, whereas other times the child might use a memorized addition fact to find the total. With practice and feedback, correct strategies replace incorrect ones, and more efficient strategies replace less efficient ones. This chapter focuses on both erroneous and alternative strategies, specifically for fractions and early algebra, and it overviews interventions for addressing errors and promoting flexibility. Algebra has been identified as a gatekeeper and a major concern of researchers and educators interested in mathematics learning (National Mathematics Advisory Panel, 2008). Fractions represent critical prerequisite knowledge for algebra learning that cause difficulties for many children, therefore receiving increased attention in the literature in recent years (Siegler, Fazio, Bailey, & Zhou, 2013). Although whole number knowledge and basic arithmetic are foundational to both fractions and algebra, these topics have received significant attention in the strategy literature. Interested readers are encouraged to seek out that work (e.g., Baroody & Dowker, 2003; Carpenter, Fennema, & Franke, 1996; Shrager & Siegler, 1998; Vanbinst, Ghesquiere, & De Smedt, 2012; Verschaffel & De Corte, 1993; Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). Given space limitations, the current chapter is by no means an exhaustive review of the literature on strategies and strategy interventions related to fractions and algebra. Incorrect strategies 
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