Understanding the development of flexibility in struggling algebra students

The ability to flexibly solve problems is considered an important outcome for school mathematics and is the focus of this paper. The paper describes the impact of a three-week summer course for students who struggle with algebra. During the course, students regularly compared and contrasted worked examples of algebra problems in order to promote flexible use of solution strategies. Assessments were designed to capture both knowledge and use of multiple strategies. The students were interviewed in order to understand their rationales for choosing particular strategies, as well as their attitudes toward instruction that emphasized multiple strategies. Findings suggest that students gained both knowledge of and appreciation for multiple strategies, but they did not always use alternate strategies. Familiarity, understandability, efficiency, and form of the problem were all considerations for strategy choice. Practical and theoretical implications are discussed. Development of flexibility 3 Exploring the Development of Flexibility in Struggling Algebra Students For decades, researchers in the fields of mathematics education and cognitive psychology have been interested in the relationship between procedural and conceptual knowledge (e.g., Byrnes & Wasik, 1991; Hiebert, 1986; Rittle-Johnson, Siegler, & Alibali, 2001). Researchers have explored how these two types of knowledge are linked, including which develops first and whether one is necessary for the other. Yet, some suggest the dichotomy is not particularly useful for describing mathematical knowledge and should be abandoned (Wu, 1999). Recent arguments have emerged that critically examine the procedural/conceptual knowledge framework, with an emphasis on challenging how the two knowledge types have been defined and measured (Baroody, Feil, & Johnson, 2007; Blinded, 2008; Star, 2005, 2007). In particular, Star (2005, 2007) argued that procedural knowledge is often conceived of and measured in ways that are consistent with rote memorization, whereas conceptual knowledge often encompasses both superficial and deep knowledge. Star argued that procedural knowledge can also be deep, as it must be in order to flexibly apply solution methods. The ability to flexibly solve problems is considered an important outcome for school mathematics (National Advisory Panel, 2008) and is the focus of this paper. According to the National Council of Teachers of Mathematics (NCTM, 2006), students should have the ability to be strategic and efficient when solving simple equations. The National Research Council (NRC, 2001) argues for procedural fluency, which they define as "knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently" (p. 121). Apparently, flexibility is a critical component of procedural fluency, but it is currently unclear how flexibility should be situated within the conceptual/procedural knowledge framework. Is flexibility part of procedural knowledge (Star, 2005) or part of Development of flexibility 4 conceptual knowledge (Baroody, Feil, & Johnson, 2007), or does flexibility result from the integration of conceptual and procedural knowledge (Baroody & Dowker, 2003)? We adopt the latter view in this paper, where we see flexibility as resulting from a merging of both procedural and conceptual knowledge. Of course, flexibility is not a new construct. Its roots can be found in the seminal work of Krutetskii (1976) and Wertheimer (1959), and for years it has been considered an important element of adaptive expertise (Baroody & Dowker, 2003; Hatano & Inagaki, 1986). More recently, it has received significant attention from European researchers, who have been interested in children’s flexibility with arithmetic (Blöte, Klein, & Beishuizen, 2000; Blöte, Van der Burg, & Klein, 2001; Klein, Beishuizen, & Treffers, 1998; Torbeyns, Verschaffel, & Ghesquiere, 2006; Verschaffel, Luwel, Torbeyns, & Van Dooren, 2007). Perhaps because of these varying interests, definitions of flexibility have also varied. According to Verschaffel and colleagues (Verschaffel et al., 2007), some researchers equate flexibility with adaptability, while others make a distinction between the two constructs. Some researchers use the term to describe the ease with which a person can switch between various solution methods, whether or not the methods are optimal, while others suggest that a tendency to choose the optimal strategy in a given situation is a critical characteristic of flexibility. In the current study we take the latter perspective, suggesting that flexibility develops slowly from knowledge of multiple procedures to the adaptive use of them (Blöte, Van der Burg, & Klein, 2001; Star & Seifert, 2006). It is important to note here that in this paper, strategy is used almost interchangeably with method or procedure (Siegler, 1996). However, trivial differences in solving an equation can technically result in a different procedure (e.g., adding a negative 5 as opposed to subtracting 5), whereas a strategy captures the essence of the procedure (e.g., using the quadratic formula). Development of flexibility 5 Prior Research on Flexibility The literature on flexibility has begun to provide suggestions on both how to help students develop this capacity and also how researchers can assess it. Several recent studies found that students gained knowledge of multiple strategies for solving problems before they began to use them regularly (e.g., Blöte et al., 2001; Star & Rittle-Johnson, 2008), which suggests it may be necessary to prompt students in order for them to fully demonstrate their knowledge. As an example, Blöte and colleagues showed that young students can learn multiple ways of solving addition and subtraction problems, but the tendency to use alternative methods lags behind their knowledge and preference for them (Blöte et al., 2001). In their study with 206 second graders, students involved in an instructional program that encouraged them to create and share solution methods showed higher levels of flexibility and conceptual knowledge than those involved in a program that emphasized skill and traditional solution methods. Related to this work, Klein and colleagues (Klein et al., 1998) demonstrated positive effects of explicit flexibility instruction in their study of 275 second graders. These researchers compared the impact of arithmetic instruction that promoted flexibility from the beginning, to instruction that emphasized procedural skill prior to promoting flexibility. Both groups of students demonstrated flexibility at posttest; however, promoting flexibility from the beginning of instruction seemed to be more effective. Students who learned flexibility and skill simultaneously outperformed those who focused first on skill alone. Furthermore, their findings suggest that early exposure to multiple solution methods does not hinder procedural competence; the groups performed similarly on an arithmetic skills assessment. Work by Star and Seifert (2006) also suggests that exposure to multiple ways of solving problems does not hinder accuracy. In their experimental investigation with middle school Development of flexibility 6 students, a treatment group (n = 18) was asked to find two different ways to solve the equations presented to them, while a control group (n = 18) was asked to solve a series of equations in one way. While both groups were similarly accurate in their equation solving, the treatment group proved more flexible in their strategy use. Using a related instructional intervention, research by Rittle-Johnson and Star (2007; see also Star & Rittle-Johnson, 2009) demonstrated that students who are learning algebra can gain conceptual knowledge, procedural knowledge, and flexibility simultaneously. By comparing and contrasting two different worked solutions to the same problem, students in a treatment group (n = 36) gained more on measures of flexibility and procedural knowledge compared to a control group (n = 34) that examined the same two methods but with isomorphic problems. Students in each group gained similarly with regard to conceptual knowledge. However, the study was conducted over four instructional days (within 45-minute classes), with only two of these involving the experimental intervention. Given this brief intervention, the significant differences are surprising, but it is not clear how flexibility with algebra might develop over time. Finally, Star and Rittle-Johnson (2008) examined the development of flexibility by comparing direct instruction to a more discovery-oriented approach. In their study of 132 rising seventh graders learning to solve linear equations, they found that the different types of instruction had similar effects on accuracy of equation solving but differential effects on flexibility. Namely, prompting students to solve problems in more than one way was most effective for increasing students’ use of multiple methods, whereas direct demonstration of efficient methods was most effective at increasing students’ use of efficient methods. But although the treatments impacted flexibility differently in terms of use, they were similarly effective in increasing flexibility in terms of knowledge (i.e., knowing that multiple methods Development of flexibility 7 exist and that some are more efficient than others.) This distinction is important and lends support to the notion that flexibility develops on a continuum. The Current Study The current study builds on prior research on flexibility and addresses some of the limitations of that research. First of all, most prior studies have focused on elementary and middle school classrooms; studies examining flexibility in classroom settings at the high school level are lacking. Teachers in mathematics classrooms at this level are often faced with pressures to cover large amounts of material in a limited amount of time, gradi