Measures of Potential Flexibility and Practical Flexibility in Equation Solving

Abstract

Researchers interested in mathematical proficiency have recently begun to explore the development of strategic flexibility, where flexibility is defined as knowledge of multiple strategies for solving a problem and the ability to implement an innovative strategy for a given problem solving circumstance. However, anecdotal findings from this literature indicate that students do not consistently use an innovative strategy for solving a given problem, even when these same students demonstrate knowledge of innovative strategies. This distinction, sometimes framed in the psychological literature as competence vs. performance—has not been previously studied for flexibility. In order to explore the competence/performance distinction in flexibility, this study developed and validated measures for potential flexibility (e.g., competence, or knowledge of multiple strategies) and practical flexibility (e.g., performance, use of innovative strategies) for solving equations. The measures were administrated to a sample of 158 Chinese middle school students through a Tri-Phase Flexibility Assessment, in which the students were asked to solve each equation, generate additional strategies, and evaluate own multiple strategies. Confirmatory factor analysis supported a two-factor model of potential and practical flexibility. Satisfactory internal consistency was found for the measures. Additional validity evidence included the significant association with flexibility measured with the previous method. Potential flexibility and practical flexibility were found to be distinct but related. The theoretical and practical implications of the concepts and their measures of potential flexibility and practical flexibility are discussed.

Dutch sixth graders’ use of shortcut strategies in solving multidigit arithmetic problems

“…Similarly, previous studies have shown that what students do on a standard task (their strategic behavior) does not fully reflect what they know (their strategic knowledge) (e.g., Blöte et al 2001). Xu et al (2017) recently found this so-called potential flexibility (what students know) and practical flexibility (what students do) in the domain of equation solving to be distinct but related. The current study only measured practical flexibility, ignoring potential flexibility.…”

“…Strategic flexibility and adaptivity are considered important components of mathematical proficiency (e.g., Baroody 2003;Xu et al 2017). Although many different definitions and operationalizations exist in the literature, they converge on two central themes: the knowledge of different solution strategies and the ability to adapt them appropriately when solving a problem (Rathgeb-Schnierer and Green 2017).…”

Turning Potential Flexibility Into Flexible Performance: Moderating Effect of Self-Efficacy and Use of Flexible Cognition

“…The mathematical flexibility assessment ( Xu et al, 2017 ) consisted of 12 linear equations (see Table 1 ). Drawing on prior research on flexibility in linear equation solving (e.g., Star and Seifert, 2006 ; Rittle-Johnson and Star, 2007 ), the assessment included four types of equations, each of which can be solved using both a standard algorithm and a more innovative approach that takes advantage of specific numerical or structural features of the problem to arrive at a solution more efficiently.…”

How mathematics anxiety affects students' inflexible perseverance in mathematics problem‐solving: Examining the mediating role of cognitive reflection

“…Solving problems flexibly and adaptivelywhich requires students to know multiple strategies and to shift between strategies based on the problem context so that the problem can be solved in the most efficient wayhas been recognized as an important goal in mathematics learning and thinking (Heinze, Star, & Verschaffel, 2009;Heirdsfield & Cooper, 2002;National Research Council & Mathematics Learning Study Committee, 2001;Star & Rittle-Johnson, 2008;Star & Seifert, 2006;Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009;Xu et al, 2017). Students who develop mathematical flexibility are more capable of utilizing their procedural knowledge to solve novel problems, and they also show a greater understanding of mathematics concepts (Bl€ ote, Van der Burg, & Klein, 2001; Rittle-Johnson, Star, & Durkin, 2012).…”

“…Moreover, flexibility is a key character of the development of mathematics expertise (Baroody, 2003;Star & Newton, 2009). However, studies have found that too many students persevere in using a single, sometimes nonoptimal, strategy to solve many similar problems, perhaps only switching strategies to a more efficient one when explicitly prompted to do so (Hickendorff, 2018;Newton, Lange, & Booth, 2019;Star & Rittle-Johnson, 2008;Star & Seifert, 2006;Xu et al, 2017).…”