The position of the National Council of Teachers of Mathematics (NCTM) on strategy flexibility is firm. As cited in Star et al. (2022), the NCTM (2014) has emphasised that “All students need to have a deep and flexible knowledge of a variety of procedures, along with an ability to make critical judgments about which procedures or strategies are appropriate for use in particular situations.” In addition to this, research has seen more papers published around strategy flexibility in different content areas of mathematics such as algebra and calculus (e.g., Maciejewski and Star, 2016; Maciejewski and Star, 2019; Rittle-Johnson and Star, 2007; Shaw et al., 2020; Star et al., 2009; Star et al., 2022; Torbeyns et al., 2009; Xu et al., 2017). Within the research area of algebra, flexibility in performing arithmetic tasks and linear equation solving has been dominating (e.g., Hickendorff, 2022; Jóelsdóttir and Andrews, 2025; Star et al., 2022; Xu et al., 2017). Among different demographic groups and geographical locations of students, the concept has also been studied (Verschaffel, 2024). The literature review by Verschaffel (2024) showed that though more has been done about the construct of strategy flexibility, there are “many aspects of it that are still not well-understood” and that there is the need for more studies.

Theoretical explanation of strategy flexibility has been underpinned on the theory of adaptive expertise (Hatano and Inagaki 1986). It is not only in this study that you would see this but in the broader literature on the concept of strategy flexibility in mathematics (e.g., Baroody and Dowker, 2003; Baroody, 2013; Hatano and Oura, 2003; Heinze et al., 2009). Adaptive expertise theory originally distinguishes between routine expertise and adaptive expertise. According to Hatano and Inagaki (1986), routine expertise is characterised by efficient implementation of well-practiced procedures and adaptive expertise on the other hand involves the ability to modify or generate strategies in response to new problem situations (Baroody, 2013; Baroody and Dowker, 2003). Based on the theory, adaptive experts are not just fast or more accurate when they execute procedures but they also understand when and why to use particular strategies and can adapt their approach when confronted with unfamiliar or complex tasks (Schwartz et al., 2005; Crawford et al., 2005).

Within this theoretical lens, strategy flexibility in mathematics aligns closely with the adaptive dimension of expertise. Baroody and Dowker (2003) identifies flexibility as the underlying principle of adaptive expertise because it requires coordinating conceptual understanding with procedural knowledge to make informed strategic decisions. Star (2007) also characterises strategic flexibility as a form of deep procedural knowledge, in which learners know multiple strategies and can evaluate their relative appropriateness. This perspective is supported empirically by studies showing that flexible strategy use correlates with richer conceptual understanding and more sophisticated procedural knowledge (McMullen et al., 2017; Rittle-Johnson et al., 2012; Schneider et al., 2011). Thus, strategy flexibility is not merely about knowing and choosing multiple strategies but the ability to discern which one best fits a given situation (Star et al., 2022).

Central to the above perspective is the idea of situational appropriateness which concerns the ability to choose, from one’s repertoire of strategies, the one that aligns best with the structure and demands of the problem situation (Dover and Shore, 1991; Star and Newton, 2009). Yet determining what strategy counts as “most appropriate” is often subtle and context-dependent. In the perspective of Verschaffel et al. (2009), appropriateness may reflect efficiency (fewest steps or quickest execution), reliability (the method least prone to error), or, in more advanced mathematical thinking, “elegance,” which is a valued but difficult-to-define aesthetic criterion. Thus, while the importance of flexible, informed strategy choice is well emphasised, identifying and enacting situationally appropriate strategy is very difficult part of mathematics education. Hence the need for more intervention research into this area as Verschaffel (2024) has recommended.

Algebra has proven to be a fruitful domain for research on strategy flexibility and the trend continues recently (e.g., Star et al., 2015; Star et al., 2022; Xu et al., 2017). The focus on algebra make sense, since many view it as the first point at which students experience the abstract thinking that underly the power of mathematics (Susac et al., 2014). Much of what we know of strategy flexibility here is based on a surprisingly set of simple algebraic problem-solving tasks particularly involving arithmetic and linear equation solving (e.g., Hopkins et al., 2025; Lemaire and Lecacheur, 2010; Newton et al., 2010; Star and Seifert, 2006; Star et al., 2022; Torbeyns et al., 2009; Xu et al., 2017; Threlfall, 2009; Hickendorff, 2022). In many mathematics curricula around the world, students are introduced to linear equation solving the most at all times and levels as the path to transition to complex algebraic reasoning and problem-solving (Otten et al., 2019). It has been reported that the task of solving linear equations is core to algebraic understanding and rather remains difficult aspect of algebraic thinking for pretertiary students (Wati and Fitriana, 2018), hence the need for strategic flexibility has prompted substantial research attention so far. Researchers have thus gravitated toward linear equation-solving, a domain where a standard algorithm is universally taught and derivations from it are easy to identify, which also make this concept so ripe for flexibility research.

From the above point, it follows that given linear equations, standard algorithm(s) exist but most at times there are other alternative strategies that are arguably more optimal (Buchbinder et al., 2015; Star and Seifert, 2006). Decades of research have shown that students often default to familiar procedures even when more efficient or elegant alternatives exist (e.g., Newton et al., 2010). Studies comparing students’ use of inverse-operations, balancing, and intuitive structural shortcuts (e.g., Star and Newton, 2009; Star et al., 2015) have generated important insights precisely because linear equations are simple, well-structured, and cognitively manageable. Research in this area (e.g., Lemaire and Siegler, 1995; Star et al., 2015) has illuminated important principles about flexibility, but tasks involving linear equation solving remain far removed from the demands of complex algebraic reasoning. While this evidence forms the backbone of what is currently known about strategy flexibility in mathematics, it reflects mathematical situations that are among the least complex that students encounter.

The focus of strategy flexibility on narrow algebra tasks such as linear equation solving becomes even more evident when we consider what is missing in research on solving higher-order polynomial equations. Even quadratic equations (with degree of 2) which are possibly the first form of non-linear equations that offer meaningful strategic variation, appear only occasionally in the literature on strategy flexibility (e.g., Ernvall-Hytönen et al., 2024; Küttel et al., 2025), despite the wealth of different strategies they support when solving them (e.g., factoring, completing the square, or applying the quadratic formula). Beyond quadratic equations, the literature remains almost silent. Equations of the form of cubic or quartic, systems of nonlinear equations, or more structurally intricate polynomial equations rarely appear in strategy flexibility literature. Solving these equations demand deeper structural thinking and offer multiple, truly variable strategies that could serve to be precisely the kind of problem-solving environments where strategy flexibility should matter most. Taken together, the existing literature paints a picture of a research field that has generated valuable conclusions, but largely within highly simplified algebraic contexts. Linear equations and arithmetic tasks have provided a productive starting point, but they represent only a narrow slice of the algebraic terrain students encounter at the pretertiary level. The relative absence of studies involving higher-order polynomial equations suggests an important opportunity to extend the research on strategy flexibility into the richer, more complex algebraic spaces where mathematics structure becomes more nuanced, strategies proliferate, and choices matter in genuinely consequential ways.

Problems and problem-solving lie at the heart of the history of mathematics (Ernest, 1998; Torres-Peña et al., 2024). Thus, the intersection between history and problem-solving flexibility can be argued for. As echoed in the experiences of McGinn and Boote (2003) in their self-study with solving questions with ancient mathematics sources such as Plimpton 322. At first glance they perceived such problem seemed computationally trivial but upon deeper engagement with it, it revealed an underlying pedagogical purpose. The value in the problem lies “less in obtaining an answer” and more in experiencing a range of methods and understanding the process of solving it. This self experience of McGinn and Boote also suggest that when learners solve a problem using ancient methods alongside modern ones, they gain “greater appreciation for each of the methods” and begin to see how different strategies can coexist, complement one another, or illuminate the problem from distinct angles. Importantly, these comparisons allowed students to recognise precursors of modern methods in earlier methods, therefore helping them to “learn about solving problems in mathematics” (De Vittori, 2022) and to use strategies flexibly across related problems.

The study by Rizos and Gkrekas (2023) also share much insights about how problem-solving in the context of historical sources could lead to flexible skills. In their study, when students encountered historically grounded, open-ended problems such as the classical “hundred fowls” problem,” they were compelled to recognise that mathematics problems do not always possess a single prescribed method or a uniquely “appropriate” solution strategy. Students exposed to these historical problems “understood, at least to some extent, that a problem can have different solutions” and began to lose their certainty that mathematics is strictly about fixed procedures that leads to predetermined results (Fauvel, 1991). Instead, the historical context acted as “an environment that can activate the students strategic thinking and in parallel give them the ability to come out with their different strategies.” In the study by Zimmermann (2004), when students analyse these historical methods, they gain insight into heuristics that are broader and more adaptable than the narrow procedural methods typically emphasised in standard textbooks.

Swetz (1986) further emphasises that solving problems in historical context foster problem-solving skills precisely because they reveal both the continuity and evolution of mathematics methods. Students got the chance to investigate how mathematicians across cultures and centuries approached similar problems in different ways. As a result, they get a richer sense of how strategies emerged, adapted, and sometimes became obsolete. As Meavilla and Flores (2007) notes, comparing modern solution processes with original ones “illustrates the evolution of solution processes,” which prompts students to question their default methods and consider alternative approaches. Swetz (1986) example of Liu Hiu’s Sea Island Mathematical Classic demonstrates this vividly that problems that today fall under trigonometry were historically solved using proportions. Such contrasts invite students to explore why a strategy works, when it is efficient, and how different representations can lead to viable, even elegant solutions. This could become that precise comparison that activates flexible strategy thinking as students come to see that they “could be innovative” in problem solving, not bound to the teacher taught methods alone.

Research on strategy flexibility in algebra has demonstrated that students often rely on familiar procedures even when alternative strategies are available (Star and Seifert, 2006; Newton et al., 2010; Star et al., 2015). Exposing students to historical methods may offer a distinctive pathway for addressing this issue. When students engage with solution methods developed in different historical contexts, they are exposed to strategies grounded in alternative representations, assumptions, and problem structures (Swetz, 1986; Meavilla and Flores, 2007). This exposure changes the idea of “just” solving equations towards identifying the structure inherent in equations as learners must interpret the mathematical logic underlying each method rather than merely apply a memorised algorithm. Studies drawing on historical problem-solving contexts show that such engagement encourages learners to compare methods, identify structural similarities and differences, and evaluate the efficiency and suitability of strategies across problems (McGinn and Boote, 2003; Rizos and Gkrekas, 2023; Zimmermann, 2004). These processes align closely with the theory of adaptive expertise, which emphasises flexible transfer, strategic choice, and responsiveness to problem structure rather than routine performance. By situating algebraic problem solving within historically grounded methods, students are provided with opportunities to develop adaptive expertise through sustained structural reasoning and meaningful strategic comparison. However, despite the theoretical alignment between historical approaches and adaptive expertise, empirical studies examining this connection in complex algebraic contexts remain scarce, motivating the present study.

The history of mathematics present diverse original strategies for solving algebraic equations. It also exposes complete accounts of how each strategy evolved to become what we know of today. Looking at the history of cubic equations, we can see that different methods were invented to solve different forms of the same equation. This historical context is appropriate for helping students improve on their strategy flexibility. Thus, in the present study our focus is to exploit the history of cubic equations as a context for developing students strategy flexibility in solving cubic equations in senior high school algebra. The premise of this study follows that, in the history, diverse methods for solving cubic equations can be found, but these methods are largely absent from standard school mathematics curricula. Focus has only been on standard methods of factorisation, synthetic division, factor theorem and remainder theorem, which limits students strategy repertoire and choice. While most of the earlier strategies such as those of del Ferro, Tartaglia, and Cardano are very robust, when students are exposed to them it can make them get hold of more choices of innovative methods of solving cubic equations. In addition, when students are taught how these methods emerged; their limitations and strengthens, it will strengthen their ability to make situational strategy choices at the right time. Moving students through the history of cubic equations, thus will help them understand why and how each method matters. In this sense, the history of cubic equations is a lens through which students can appreciate the value of innovative methods, that they are not taught. The purpose of this study is to determine whether students who are taught cubic equations through a historical context will demonstrate improved strategy flexibility. Accordingly, we propose the following hypotheses:

Prior to the commencement of the study, sample size was determined using power analysis computed in G*Power (Faul et al., 2009; Kang, 2021). The results based on significance level (α) of 0.05, a statistical power (1–β) of 0.80, and an estimated medium effect size (Cohen’s d = 0.50) indicated that sample size of 128 student (control [n = 64; males = 50, females = 14] and experimental group [n = 64; males = 46, females = 18] combined) is deemed appropriate (males = 94, females = 30). The medium effect size estimate was based on conventional benchmarks set by Cohen (1998), which were adopted as widely used reference guidelines in educational research and interpreted as heuristic indicators rather than rigid cut-off values, acknowledging that practical significance should be considered in relation to the study context. Participants ages range between 11 and 17 years. Participants were enrolled in two senior high schools located in the Ashanti region of Ghana. The number of students were conveniently selected from 4 science classes (two classes from each school) to form the experimental and control groups, because the number of students in one intact class (as recommended by quasi-experiments) cannot make up exactly the number of students in either of the groups. Form 2 science students were selected purposely because, by this level, they have typically covered standard methods for solving cubic equations (factor theorem, synthetic division, long division, and factorisation). Conversations with the class teacher indicated that standard methods were first introduced in Form 1. At this stage, students are cognitively matured enough to engage with more advanced and innovative methods inspired by the history of mathematics, which made them suitable candidates for the study. In addition to the students been at the same academic level and taking the same course, results of independent sample t-test of the pretest scores showed that the groups did not differ in terms of initial strategy flexibility [ t = − 323 , p = 0.747 > 0.05 ] . This confirmed that the groups are homogenous and thus could start the study at comparable levels.

In this study, our operationalisation of strategy flexibility aligns more to the practical flexibility dimension of procedural flexibility in Xu et al. (2017). This dimension treats strategy flexibility as the spontaneous use of innovative method on the first attempt. Previous studies had also measured strategy flexibility this way (e.g., Star et al. 2015). For example, Star et al. (2015) instructed students to identify the innovative first step for the solution of an equation and their responses were taken as knowledge of innovative strategies. While in the literature, scholars commonly define “innovative” methods as those that can be used to solve problems following the fewest steps and the greatest degree of algorithmic simplicity (e.g., Heinze et al., 2009; Star and Rittle-Johnson, 2008; Star and Newton, 2009), for the sake of this study, an “innovative” method is one that shortens the solution process or uses structural insight that is history-inspired. So, a method for solving cubic equations is deemed innovative, if it is especially: (i) history-inspired, (ii) not well emphasised in mathematics curricula (iii) enables more structural exploitation, and (iv) uses cleaner transformations. Any method apart from these is deemed standard. Typical curriculum order of methods for working cubic equations by looking for rational roots using factor theorem, and then using long division or synthetic division to reduce the cubic into quadratic × linear factors is defined as standard methods taught in school algebra.

Accordingly, an essay-type test is designed with 9 cubic equations as the measurement instrument for strategy flexibility. In the test, different categories (Type A to Type C) of cubic equation based on their structural forms were given such that each could be solved using a standard method but where a more innovative method could also be used. We followed the approach by Xu et al. (2017) where they measured students use of innovative strategy in the Phase One of their Tri-phase Flexibility Assessment. Students were instructed to solve each cubic equation as quickly and accurately as they could, which encouraged them to choose and apply an innovative method (Star et al., 2022). If a student used an innovative method for a given cubic equation, that student is said to have demonstrated strategy flexibility. Type A equations were of the form of x 3 + bx + c = 0 . The innovative method here involved students to easily use the substitution x = u + v . Type B were of the form of x 3 + a x 2 + c = 0 . The innovative method involved setting x = y − b / 3 to remove the x 2 term before applying the cubic formula. Type C were of the form of the general cubic x 3 + b x 2 + cx + d = 0. The innovative method involves using the substitution x = y − b / 3 to eliminate the x 2 term or simplify coefficients before applying the cubic formula. All these first steps that are found innovative as in this study were those developed by ancient mathematicians: del Ferro, Tartaglia, and Cardano for cubic equations of the form x 3 + bx + c = 0 , x 3 + a x 2 + c = 0 , and x 3 + b x 2 + cx + d = 0 respectively. Emphasis has been placed here on these since they are the methods that have historical significance when teaching cubic equations. The equations were set that none of them when solving would end up into the irreducible case “casus irreducibillis” where students have to find real roots using complex numbers. Note that the order in which the equations have been given in the test reflects the order of increasing task difficulty as it was done in Xu et al. (2017) and was seen to be a good practice. Type C equations were the most challenging. The 9 cubic equations are listed in the Table 1.

To enrich the data collected using the test instrument, an interview guide was developed to collect qualitative data. The interview guide focuses on students who participated in the history-integrated lessons from the experimental group. The interview was conducted from 10 students to provide further explanations to why there was improved strategy flexible among the experimental group. According to Guest et al. (2006), this sample size was seen to be suitable because theoretical data saturation occurred after the 10th interviewee. On the interview guide, two questions were asked. These questions are: (i) How did learning to solve cubic equations through its history help you solve cubic equations? (ii) Did the history-integrated lessons change the way you think about solving problems? How? Responses taken from students on these questions were recorded on smart phone and analysed thematically to add more depth to the findings of the study.

Because the purpose was to investigate if participating in the history-integrated lessons improved students strategy flexibility, we compare the strategy flexibility scores between the two groups. The independent sample t-test was used for this. We analysed the pretest data to establish baseline equivalence between the experimental and control groups. Baseline equivalence ensures that any post-test differences in flexibility cannot be attributed to pre-existing disparities in strategy flexibility. Levene’s test for equality of variances was conducted to assess whether the variability in pretest flexibility scores was statistically similar for the two groups. Based on this outcome, the independent samples t-test for equal variances was used. After this, the post-test data was analysed to examine whether there exists difference in the mean strategy flexibility scores between the experimental group and the control group. To complement the comparative tests, descriptive statistics (means, standard deviations) were also computed to provide a clear picture of group differences. Additionally, effect sizes (Cohen’s d) were computed to quantify the magnitude of the effect of history-integrated lesson on strategy flexibility. The effect size helped us to understand the practical significance beyond the p-value interpretation.

To further explain and enrich the results obtained from the quantitative data, interview data were collected and analysed using thematic analysis (Ahmed et al., 2025). First the audio recorded interviews were transcribed carefully one after the other (participant by participant). After this, we begun the analysis by following the six steps offered by Braun and Clarke (2006). The analysis begun with repeated reading of the transcripts to achieve familiarity with the data. Initial codes were then generated inductively to capture meaningful features of the responses. These codes were subsequently organised into broader themes that reflected recurring ideas or insights. The themes were reviewed and refined to ensure coherence, internal consistency, and accurate representation of the dataset. Finally, each theme was clearly defined and supported with illustrative quotations (Braun and Clarke, 2021, 2006; Braun et al., 2023). This was done to allow the qualitative findings to deepen and contextualise the quantitative results.