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The current paper presents an introduction to a special issue focusing on mathematical flexibility, which is an important aspect of mathematical thinking and a cherished, but capricious, outcome of mathematics education. Mathematical flexibility involves the flexible, creative, meaningful, and innovative use of mathematical concepts, relations, representations, and strategies. In this introduction we discuss the most relevant theoretical, methodological, and educational considerations related to mathematical flexibility, which form the background of the empirical studies presented in the special issue. Collectively, these studies provide a broader understanding of the mathematical flexibility, its subcomponents, influences, and malleability.

Mathematical flexibility is an important aspect of mathematical competence (

This special issue presents current research aimed at advancing our scientific insight into mathematical flexibility. In this introduction, we review some major theoretical, methodological, and educational issues in the domain of mathematical flexibility and forecast the empirical studies in this special issue, that each address some combination of these three aspects of the study of mathematical flexibility

In our consideration of mathematical flexibility we focus on two main theoretical issues: first, the variety of definitions of mathematical flexibility, and second, the relations between mathematical flexibility and cognitive and non-cognitive factors.

Mathematical flexibility is defined in various ways in the literature, ranging from rather narrow to quite broad. A common conceptualization focuses on the use of various strategies to solve mathematical problems. Within this strategy-focused conceptualization there are two different perspectives on what constitutes mathematical flexibility. In the first, narrow, perspective, flexibility only refers to the knowledge and use of various strategies (

However, this focus on strategy use does not present a full picture of what mathematical flexibility entails (

In this special issue we did not start from an

Theoretical models and research findings suggest that mathematical flexibility is related to several cognitive and non-cognitive factors.

Within the cognitive factors, a distinction can be made between domain-specific and domain-general factors. One relevant domain-specific factor is conceptual understanding, and the question is how strategic flexibility is related to students’ understanding of the underlying mathematical principles (

Another relevant domain-specific factor is overall mathematical performance. Although several studies showed that students who show more mathematical flexibility in a particular task domain tend to have higher levels of overall mathematical ability (

Besides domain-specific factors, there are also domain-general cognitive factors that are likely related to mathematical flexibility. There is ample research evidence that executive functions (shifting, inhibition, and working memory) are important for developing mathematical proficiency (

Turning to the relation with non-cognitive factors, we first consider the role of gender. Not all studies report gender differences in students’ flexibility, but if they do it is in favor of boys (

Affective factors may also be related to mathematical flexibility. One affective factor is mathematics anxiety, which is known to be negatively related to mathematics performance on routine tasks (

Despite their importance (

An important question, of course, is how to assess mathematical flexibility. Given the variety of conceptualizations of the construct (

Since solution strategies are crucial for many conceptualizations of mathematical flexibility, the identification of the strategies students use is often pivotal to its measurement. A common approach is to let students solve a set of tasks, for instance arithmetic problems or algebra exercises, and infer their solution strategies from their written work (e.g.,

Moreover, if it is possible to designate certain strategies as innovative or efficient shortcut strategies for particular tasks (e.g., using the compensation strategy on problems such as 837 – 399 = _), charting to what extent and on what tasks students use these strategies gives information about whether students make adaptive strategy choices with respect to task characteristics (

However, strategies that are efficient from the perspective of the task (having the fewest and/or easiest computation steps) are not necessiraly efficient for individual students, since it requires understanding of relations between numbers and between operations and/or good mastery of certain (sub)procedures that are part of that strategy (

The issue of appropriateness of a particular choice is also relevant in the history of research on representational flexibility. Initially, methodological approaches in this domain mostly conceptualized and operationalized representational flexibility as choosing the representation that provides the best match to the characteristics of the task to be solved. But researchers gradually also started to broaden their methodological scope, for instance, by including data-gathering and/or data-analytic methods that allow the inclusion of students’ abilities, preferences, and affects interacting with these representations, as well as the context in which such interaction takes place (

Beyond examinations of strategy or representational choice, other methods have been developed for examining other aspects of mathematical flexibility. For example, in examining aspects of students’ abilities that support flexible strategy choice,

In sum, there is wide variety in the methodologies used to measure specific aspects of mathematical flexibility, each of which provides a specific lense with which the topic is studied.

Although mathematical flexibility is deemed an important objective of mathematics education curricula, fostering it can be challenging. Mathematical flexibility is something that develops over time, not easily nor quickly (

Research has indicated several instructional factors that may promote mathematical flexibility, including stimulating children to invent, reflect, and discuss strategies or representations (

Some studies have compared direct instruction of different strategies or representations to implicit instruction with prompts to discover strategies but without explicit instruction (

Furthermore, we already discussed research suggesting that the socio-mathematical context affects the likelihood that students show mathematically flexible behavior. An educational implication is that classrooms probably differ in the extent to which mathematical flexibility is valued. Factors such as the extent to which accuracy is valued over speed, the extent to which mental computation is valued over written computation, or the extent to which finding mathematically elegant approaches or solutions are valued may be factors that teachers can consider in order to stimulate mathematical flexibility.

An important consideration is the interaction between individual factors and flexibility learning gains. One such factor is prior knowledge: as discussed earlier, research findings suggest that developing flexibility is very difficult for students with low prior knowledge. Indeed, students’ prior knowledge has been found to be related to students’ flexibility gains (

The proposed special issue provides a broad view of state-of-the-art research on mathematical flexibility. Mathematical flexibility is examined in different populations (including primary school students, secondary school students, and adults), in different domains (including whole-number arithmetic, fractions, proportional reasoning, and algebra), and with different methodologies (verbal reports, online tests, choice/no-choice design). Importantly, the papers also take on different theoretical perspectives. At the core of most of the studies involved in this special issue is the common conceptualization of mathematical flexibility as the ability to know and use different strategies and to select the most appropriate one for a given problem. However, this focus is broadened in several ways: by addressing reasoning about procedures, formulas, or magnitudes, by zooming in on the complex relationship between strategy use and accuracy, by attempts to measure whether children know more than they show, by focusing on multiple representations of a problem situation, by taking the affective side of mathematical flexibility into account, and finally by using a quite different conceptualization of mathematical flexibility: the flexible understanding of number. Furthermore, the studies shed light on the roles of domain-general and domain-specific skills play in mathematical flexibility as well as on the effects of an intervention. Collectively, the special issue aims to provide a broader understanding of the influences, subcomponents, and malleability of mathematical flexibility. We believe that this collection of empirical papers provides new insights and provokes new questions regarding the nature of mathematical flexibility. We hope it is of interest to researchers interested in mathematical cognition and learning and mathematics education alike.

The work was partly funded by a personal grant from the Dutch Research Council (NWO) for Hickendorff: 016.Veni.195.166/6812. These funding sources did not play a role in the content of the article.

The authors have declared that no competing interests exist.

The authors have no additional (i.e., non-financial) support to report.