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.

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This cross-national study examined students' evaluation of strategies for solving linear equations, as well as the extent to which their evaluation criteria were related to their use of strategies and/or aligned with experts' views about which strategy is the best. A total of 792 middle school and high school students from Sweden, Finland, and Spain participated in the study. Students were asked to solve twelve equations, provide multiple solving strategies for each equation, and select the best strategy among those they produced for each equation. Our results indicate that students' evaluation of strategies was not strongly related to their initial preferences for using strategies. Instead, many students' criteria were aligned with the flexibility goals, in that a strategy that takes advantages of task context was more highly valued than a standard algorithm. However, cross-national differences in strategy evaluation indicated that Swedish and Finnish students were more aligned with flexibility goals in terms of their strategy evaluation criteria, while Spanish students tended to consider standard algorithms better than other strategies. We also found that high school students showed more flexibility concerns than middle school students. Different emphases in educational practice and prior knowledge might explain these cross-national differences as well as the findings of developmental changes in students' evaluation criteria.