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Flexibility with AlgebraSimilar to fractions, flexibility in algebraic thinking is also a characteristic of experts. Star and Newton (2009) interviewed experts about their strategy choices when solving algebra problems, and they had a tendency to value and use cognitively efficient strategies. They typically justified strategy choices by saying a particular method was “easier.” When prompted to elaborate, “easier” usually involved fewer steps but, more importantly, it referred to less need for written computation. On the other hand, students struggling to learn algebra may be more concerned with methods that they can successfully use (Newton et al., 2010). This focus on accuracy may or may not involve the most efficient method. For example, a student might prefer clearing the denominator for problems such as ^{1}/s(x+6) = 4 not because it is the method with the fewest steps, but because he or she is not confident about distributing the fraction without errors. This same student may use the distributive property for a problem such as 3(x + 6) = 15 because it is most familiar and the student maybe confident in using it successfully. These students are more likely to value efficiency when they are equally confident in different problemsolving strategies. Flexibility in algebra is predicted by both procedural and conceptual knowledge (Schneider, RittleJohnson, & Star, 2011). However, students with both low and high prior algebra knowledge seem to appreciate efficiency in problem solving, even when they do not use more efficient methods (Newton et al., 2019). The third recommendation described above, learning and choosing from multiple strategies, is particularly relevant for promoting procedural flexibility with algebra (Star et al., 2015). By comparing and using different ways to solve the same problem, students are encouraged to notice problem features that make a particular strategy more efficient in some cases. InterventionsComparing. As noted above, comparing two worked examples presented side by side can help students draw conclusions about important ideas or strategies (Star et al., 2015). In particular, comparing a general strategy to a more efficient alternative can highlight when one strategy might be more efficient than the other one. Relative to studying examples sequentially, comparing two examples side by side led to improved procedural knowledge and flexibility in a study involving seventh graders learning to solve equations (RittleJohnson & Star, 2007). The intervention in this study was specifically designed to help students attend to structural features that might make one method more efficient in some cases. For example, when solving 3(x + 5) + 4(x + 5) = 14, students might notice the parentheses and then distribute as a first step to solving the equation. An alternative approach would rely on noticing that each set of parentheses contains the same expression; therefore, a first step can to be combine the two like expressions. The compare condition included prompts to help students notice these features and consider when the alternative method might be a good problemsolving choice. Students in another condition studied the same worked examples, but sequentially (on separate pages) and with prompts that avoided comparisons. The fact that both groups were exposed to both multiple solution methods highlights comparison as an effective intervention for accurate and efficient equationsolving strategies during early algebra learning. In a followup study of seventh and eighth graders, RittleJohnson and Star (2009) explored the effects of different types of comparisons in algebra. In one condition, students compared different solution methods for the same problem. Students in another condition compared different problem types solved using the same solution method. The third condition involved comparing equivalent problems using the same method. These researchers found that comparing different methods for solving the same problem was more effective for conceptual knowledge and flexibility than the other two types of comparisons. Some interventions have targeted representational flexibility in the context of algebra. Nistal, Van Dooren, and Verschaffel (2014) tested an intervention targeted at improving students’ representational flexibility, conceptualized as the ability to make adaptive choices when choosing representations (e.g., table, graph, formula), to solve linear function problems. The intervention was individualized, in that it involved providing students with feedback on their accuracy using different representations to solve problems at pretest. It also included asking students to reflect on their representational choices and compare them to other (example) students’ choices and reasons for making those choices. It was emphasized that the best choice may be depend on both the problem as well as the student (given not everyone has the same proficiency with different representations). Students in the intervention group learned and used representations to a similar extent as students in a control group. However, students who received the intervention solved problems with more speed and accuracy at posttest. Further analyses indicate that this improvement was a result of an improved representational flexibility. In other words, students chose representations for themselves that led to better accuracy at posttest compared to pretest. Generating More than One Strategy. Comparing two worked examples, presented side by side, can lead to increased flexibility with algebra by providing opportunities to learn about and evaluate different solution strategies (Star et al., 2015a). However, “a specific learner’s flexibility in using different methods ... depends upon the familiarity of the specific problem at hand” (Atkinson et al., 2000, p. 185). One recommendation to help students gain fluency with a particular strategy is to interleave worked examples with practice problems. However, even when students have knowledge of more efficient strategies, they do not always use them regularly (Newton et al., 2019). The ability to generate multiple solution methods to a single algebra problem seems to be a key skill that helps explain the gap between knowledge (e.g., being able to recognize valid alternatives) and use of efficient strategies for solving algebra problems. Asking students to generate a new way to solve an equation led to increased flexibility in a study of sixth grade students with no prior experience in algebra, compared to a control group that was asked to solve a new problem in the same way (Star & Seifert, 2006). Despite solving fewer problems overall, the two groups had similar levels of equationsolving accuracy at posttest. In this case, students were generating alternative solution strategies through invention. Another possibility is to ask students to generate more than one way to solve a problem by recalling methods they learned through studying worked examples. Research is needed to confirm whether this kind of activity can effectively support students learning to flexibly use a variety of problemsolving strategies. 
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