Exploring students’ procedural flexibility in three countries

Abstract

Background In this cross-national study, Spanish, Finnish, and Swedish middle and high school students’ procedural flexibility was examined, with the specific intent of determining whether and how students’ equation-solving accuracy and flexibility varied by country, age, and/or academic track. The 791 student participants were asked to solve twelve linear equations, provide multiple strategies for each equation, and select the best strategy from among their own strategies. Results Our results indicate that knowledge and use of the standard algorithm for solving linear equations is quite widespread across students in all three countries, but that there exists substantial within-country variation as well as between-country variation in students’ reliance on standard vs. situationally appropriate strategies. In addition, we found correlations between equation-solving accuracy and students’ flexibility in all three countries but to different degrees. Conclusions Although it is increasingly recognized as an important construct of interest, there are many aspects of mathematical flexibility that are not well-understood. Particularly lacking in the literature on flexibility are studies that explore similarities and differences in students’ repertoire of strategies for solving algebra problems across countries with different educational systems and curricula. This study yielded important insights about flexibility and can push the field to explore the extent that within- and between-country differences in flexibility can be linked to differences in countries’ educational systems, teaching practices, and/or cultural norms around mathematics teaching and learning.

Which One Is the “Best”: a Cross-national Comparative Study of Students’ Strategy Evaluation in Equation Solving

“…For example, according to Xu et al (2017), if a student solved the equation in Table 1 using the two strategies shown and then identified Strategy 2 as the best one, then the student was recognized as having chosen the appropriate strategy for this problem. Specifically, in Xu et al (2017) (see also Star & Seifert, 2006;Star et al, 2022;Rittle-Johnson & Star, 2007), Strategy 2 is considered a more appropriate strategy in this particular task context because it reduces computational steps (Star & Seifert, 2006). A flexible problem solver should be able to solve equations such as the one shown in Table 1 using multiple ways and to identify that Strategy 2 is more appropriate (Liu et al, 2018;Xu et al, 2017).…”

A complicated relationship: Examining the relationship between flexible strategy use and accuracy

“…Among other possible strategies, some are arguably better than the standard algorithm, where better (or "situational ly appropriate"; Star et al, 2022) may mean that the strategy is more elegant and/or better matched to the structural features of the problem. To illustrate, for the above equation 3(x + 1) = 15, an arguably better strategy would involve dividing both sides of the equation by 3 as a first step.…”

“…Student-problems were coded as better-than-standard if their approach demonstrated more elegance and innovation than the standard approach, based on similar determinations in prior studies (e.g., Star & Seifert, 2006;Star et al, 2022). For example, for Question 2, if a student divided by 3 as a first step, this was considered a better-than-standard strategy because the strategy takes advantage of the structural features of the problem (15 is evenly divisible by 3) and can be solved in fewer steps.…”

“…Because our research questions were concerned with differences in accuracy between standard and better-thanstandard approaches, we restricted the analysis sample to student-problems that used either of these two types of strategies across both Parts 1 and 2. Given the well-established relationship between worse-than-standard strategies and inaccuracy, we were only interested in examining problem solving accuracy between standard or better strategies (e.g., Star et al, 2022). Student-problems that were not included in the subsequent analysis included those that were incomplete, left blank, or showed a strategy that was judged by coders to be worse than the standard strategy on at least one part of the assessment.…”

Mathematical Creativity in Elementary School Children: General Patterns and Effects of an Incubation Break

“…One factor that impacts students' ability to think creatively is their content and domain knowledge (Baer and Garrett, 2010). Although higher grades are not guaranteed to equate to greater content knowledge, students gain experience with mathematical concepts across grade level and this content knowledge builds upon each other and improves the ability to think flexibly about mathematical concepts (Star et al, 2022). Hence, greater experience may boost mathematical creativity, as it provides more content to draw from when asked to generate new ways to solve a problem.…”