Students Can’t Stop Flipping Equations: A Game Just Fixed 87% of the Mistakes

For more than four decades, mathematics educators have wrestled with one of the most stubborn misconceptions in the entire algebra curriculum: the reversal error. Give a university student the statement “there are six times as many students as professors” and ask for an equation using S for students and P for professors, and a substantial fraction will confidently write P = 6·S instead of the correct S = 6·P. The mistake has been documented among high schoolers just beginning formal algebra, among undergraduates in engineering, and even among physics students who can otherwise handle sophisticated mathematics. Now a research team spanning Spain and Sweden reports that a gamified interactive educational tool, deployed across eight structured levels on Android tablets, reduced these reversal errors by 87 percent in students who had already completed secondary education, while nearly eliminating two related classes of algebraic mistakes.

The study, published in the Journal of New Approaches in Educational Research, was led by Aida Moreno-Rus and Mercedes Ventura as co-first authors, together with Noelia Ventura-Campos, Dana G. Stefanescu and Zoe Falomir. Fifty-two university students aged 18 to 26, drawn from the sciences, health sciences, social sciences and humanities, were initially screened with two computerized detection tasks. The first, a time-limited task, gave participants just three seconds to judge whether an equation correctly matched a problem statement, a constraint designed to expose automatic, unreflective mistranslations. The second allowed unlimited time for students to construct their own equations using an on-screen interface. Only students who persisted in making reversal errors, either under time pressure alone or in both conditions, were selected for the intervention.

Reversal errors are deceptively simple. Cognitive theories dating to Clement’s classic 1982 work propose two main mechanisms. The word-order matching account holds that students translate text linearly, mimicking the sequence of words rather than modeling the underlying relation: “six times as many students as professors” becomes 6·S = P. The static comparison account holds that students form a correct mental image, one professor for every six students, but then encode that image as a symbolic shortcut such as 1·P = 6·S, which simplifies to the same reversed equation. Later scholarship, including work by Jankvist and Niss, has argued that the idiom “times as many as” itself poses a combined syntactic and semantic obstacle, particularly in languages such as Spanish where the phrase “veces más que” can blur the boundary between multiplication and addition.

Crucially, the intervention targeted not just reversal errors but the full family of translation mistakes observed in the screening tasks. Operator errors arose when students confused comparative idioms, writing addition where multiplication was required or vice versa. Structural errors appeared when students placed a number as a divisor or subtrahend of the variable itself, as in 9/P = S instead of P/9 = S, reflecting a shallow understanding of what a variable represents. Hybrid errors combined reversal and operator confusions in a single incorrect equation. Before the intervention, the researchers found that half of the participating students reported translating problem statements literally, following the order of the words without reorganizing the mathematical relationships, and a quarter admitted reading the statements hastily without identifying which quantity was being compared to which.

The gamified tool, built in Unity3D and running on ordinary Android tablets, deliberately avoided the narrative structure and fictional worlds of commercial video games and serious games. Instead, it offered a guided learning experience organized as a path across an island, with eight levels of ten exercises each. Each exercise unfolded through interactive phases: reading and understanding a comparative word problem, formulating an equation with a calculator-style interface, and then receiving immediate feedback. When a reversal error was detected, the tool first applied an arithmetic validation strategy, substituting concrete numerical values into the student’s equation so that the failed equality became visible. If the error persisted, a second corrective layer engaged: a visual representation using containers of different sizes as physical metaphors for the relative magnitudes of the two quantities, prompting the student to decide which variable needed to be multiplied, divided, added to or subtracted from so that both containers reached the same size.

From level four onward, the scaffolding deliberately faded. The tool stopped displaying the relative container sizes explicitly and instead asked participants to determine for themselves which quantity should be larger, choose the appropriate operator and decide which variable it should apply to. The designers also introduced a coin-based reward system at this point, granting one virtual coin per correct answer and up to fifty coins in total, a motivational layer added after pilot studies showed concentration flagging around level three. An inverse task, in which students constructed a word problem from an equation they had written, drew on Piagetian ideas about reversible thinking to deepen structural understanding. Explanatory messages addressed the semantics of phrases like “times more than” and “times less than,” directly targeting the operator confusions documented in prior research.

The quantitative results were striking. After roughly three weeks of training in two to three supervised sessions per week, correct responses increased significantly in both the timed task, with a large effect size of r = .87, and the untimed task, with r = .86. Reversal errors fell by 87 percent, operator errors dropped by 87.5 percent, and hybrid errors vanished entirely. Half of the intervention participants earned the maximum fifty coins, and the overwhelming majority exceeded the forty-five-coin threshold tied to a small financial incentive. Wilcoxon signed-rank tests confirmed that these gains were statistically robust at p < .001 across both assessment formats.

Equally notable were the affective findings. Drawing on flow theory, the researchers measured enjoyment, relaxation and concentration at the end of each level on six-point Likert scales, and administered a five-item retrospective flow questionnaire after the final level. Enjoyment remained stable and high throughout, while relaxation and concentration rose significantly as students progressed, particularly after the coin rewards were introduced. A principal component analysis of the three emotional measures yielded a single factor explaining 83 percent of the variance, interpreted as a unified dimension of positive emotional engagement. The average perceived flow score reached 4.57 out of 6, and flow correlated strongly with sustained emotional engagement, r = 0.661. Notably, students who had made reversal errors only under time pressure and those who made them consistently reported equally high engagement, suggesting the tool works across ability profiles.

Qualitative responses collected after the training illuminated how the learning happened. Forty-four percent of participants said they had come to understand that an equation expresses an equality between two expressions, with one student explaining that they now understood equations “in terms of quantities, not in the order of the sentence.” Others reported improved interpretation of comparative idioms, strategic self-correction through repeated feedback, and growing fluency and confidence. Many described the tool as having a metacognitive function, acting as a catalyst for recognizing their own error patterns and adjusting strategies accordingly, a hallmark of self-regulated learning. When asked about pedagogical value, nearly a third of participants said the tool was directly applicable in classrooms, particularly for introducing equations and equalities to younger learners.

The authors are careful about the limits of the work. The sample was small, purposively selected and non-probabilistic; the emotional data relied on self-report; and the design cannot fully disentangle the contribution of the gamified motivational layer from the instructional core. Future validation in authentic secondary school classrooms, with larger and more diverse samples, remains necessary, as does investigation of whether the cognitive and emotional gains persist over time. Nevertheless, the study represents one of the few instructional interventions ever shown to substantially reduce a misconception that has resisted traditional abstraction-based teaching since it was first documented in 1979. The message for educators is twofold: misconceptions rooted in the translation of everyday language into algebra are remediable with the right combination of visual grounding, immediate feedback and faded scaffolding, and the emotional states that sustain learning, enjoyment, calm and focused attention, can be designed for rather than left to chance.

Subject of Research: Reducing reversal errors in comparative algebra word problem solving through a gamified interactive educational tool

Article Title: A gamified interactive educational tool to support algebraic thinking: reducing reversal errors in comparative word problem solving

Article References: Moreno-Rus, A., Ventura, M., Ventura-Campos, N., Stefanescu, D. G., & Falomir, Z. (2026). A gamified interactive educational tool to support algebraic thinking: reducing reversal errors in comparative word problem solving. Journal of New Approaches in Educational Research, 15(1), Article 17. https://doi.org/10.1007/s44322-026-00066-z

Image Credits: AI Generated

DOI: 10.1007/s44322-026-00066-z

Keywords: reversal error, algebraic thinking, comparative word problems, gamification, educational technology, flow theory, mathematics education, equation translation, emotional engagement, interactive learning environment, gamified, interactive