The review followed the PRISMA 2020 guidelines, the international standard for reporting systematic reviews, and was conducted using a carefully constructed search of the Scopus database supplemented by Google Scholar. The researchers formulated their central research question using the PICO protocol, which structures questions around population, intervention, comparison, and outcome. Their search equation combined keywords such as argumentation, mathematical argumentation, argumentation skills, teaching strategies, didactic strategies, instructional strategies, teaching methods, and pedagogical approaches using Boolean operators. The initial Scopus search on August 30, 2025 yielded 134 documents, comprising 96 journal articles, 15 book chapters, 14 conference papers, 8 reviews, and 1 book. After applying language filters that restricted inclusion to Spanish, Portuguese, and English publications, reducing the records to 130, and screening 138 total records against inclusion and exclusion criteria, 12 studies met the requirements and were selected for detailed analysis.

The inclusion criteria were demanding. Studies had to address the teaching of argumentation at primary, secondary, or tertiary education levels and report measurable outcomes related to argumentative skills. Studies were excluded if they addressed argumentation only peripherally without focusing on didactic strategies, if they pertained to disciplines other than mathematics education, or if they were review articles themselves. The temporal delimitation starting in 2019 reflects the authors’ interest in capturing the most recent trends in a field that has witnessed accelerated growth in publications, driven in part by the consolidation of key theoretical frameworks and the expansion of empirical research contexts. The methodological quality of the included studies was assessed using the Mixed Methods Appraisal Tool, though in accordance with the tool’s recommendations, no numerical score was assigned to any study since summing or averaging scores across criteria is explicitly discouraged.

Four main categories of instructional strategies emerged inductively from the collaborative analysis carried out by both authors. The first category comprises technology-mediated approaches, including the use of GeoGebra, Scratch, WhatsApp, Argunet, and educational video games. The second encompasses dialogic and debate-oriented models such as DAIM, ACODESA, and peer feedback. The third integrates historical and contextual approaches that weave the history of mathematics into the teaching of proof. The fourth involves gamification and STEAM-based strategies designed to enhance motivation and mathematical communication. These categories did not come from a predetermined classification scheme but rather surfaced from the analysis of the included studies themselves, giving the review an empirical grounding that many syntheses lack.

The co-citation analysis conducted on the reference lists of the 12 included studies using VOSviewer version 1.6 revealed the theoretical pillars that sustain current research on the teaching of mathematical argumentation. Four clusters were identified, each reflecting a distinct theoretical orientation. Among the most prominently cited authors are Conner, Yackel, Pedemonte, Krummheuer, Cobb, Knipping, Osborne, Boero, and Singletary. Toulmin’s model constituted the predominant analytical framework across the reviewed studies, appearing in at least seven of them. Sociocultural perspectives and socio-mathematical norms were also extensively drawn upon to account for the dialogical dimension and co-construction processes. The pragma-dialectical framework and other dialectical models emerge as alternative approaches that emphasize negotiation and critique in argumentation. More recently, approaches related to STEAM, gamification, and motivational theories such as Keller’s ARCS model have been incorporated to enhance engagement, motivation, and creativity in teaching and learning processes.

The theoretical backbone of the review draws on some of the most influential contributions to the debate on argumentation and proof. Duval’s work establishes a clear distinction between arguing and proving: an argument is not necessarily defined by the validity of the reasoning that supports it but rather by its semantic coherence, whereas a proof is constructed through a chain of deductive reasoning governed by the laws of formal logic. Duval emphasizes that no matter how elaborate an argument may be, it does not constitute a proof, thereby highlighting the cognitive gap between these two processes. Boero and colleagues examined the connections between argumentation and proof through the construct of cognitive unity, understood as the cognitive continuity between conjecture formulation and proof construction. This construct posits that the elements, intuitions, and arguments produced during conjecturing can be retrieved and reorganized within a formal proof, though the authors emphasize that this transition is not automatic and requires teacher mediation to guide the reorganization of students’ reasoning.

The findings of the reviewed studies indicate a progressive transition among students from empirically based arguments to more deductive forms of reasoning, fostered by various instructional interventions. In this process, the use of counterexamples, refutations, and critical debate has become consolidated as a set of key elements for enhancing the quality, coherence, and validity of arguments. Similarly, the integration of digital technologies fostered collaborative environments and provided visual resources that supported mathematical argumentation, though their effectiveness was contingent upon teacher mediation. Strategies such as guided discussion, deliberate questioning, and the use of contingencies reported by Solar and colleagues and by Zhuang and Conner align with Boero’s claim that the transition from conjecture to proof is not automatic and requires teacher intervention to reorganize students’ reasoning. Meadows and Caniglia explicitly draw on cognitive unity to link geometric constructions with formal demonstration, representing the most direct application of this construct among the reviewed studies.

Regarding the educational level and population of the included studies, the majority focused on teacher education contexts, either with pre-service teachers or with in-service teachers, accounting for seven of the twelve studies. Three studies were conducted with university students in non-teacher education programs, and two studies focused on secondary school students. No study in the reviewed sample addressed mathematical argumentation at the primary education level, a striking gap given that argumentation is increasingly recognized as a fundamental competency across all educational stages. The countries with the highest representation are the United States and Spain, each contributing three publications, followed by Colombia, Peru, and Turkey, each with two publications. For collaborative studies involving researchers from more than one country, all authors’ affiliations were considered in this analysis.

Several notable trends in instructional strategies for teaching mathematical argumentation were identified across the reviewed literature. One such trend is the strategic use of errors and unanticipated student responses as catalytic resources for initiating argumentative discussion and supporting the collective construction of arguments. Another notable trend is the use of open-ended tasks and non-routine problems, which act as catalysts for discussion and the development of justifications. Technological mediation and digital learning environments have gained importance, with tools such as GeoGebra, WhatsApp, Argunet, and Scratch or educational video games being employed. Gamification and game-based learning approaches have been designed to enhance students’ motivation and mathematical communication. The centrality of scientific debate and the co-construction of meaning also stand out as key axes of argumentative practice. Finally, some studies integrate the history and epistemology of mathematics to enrich the teaching of proof and to connect cognitive and cultural dimensions.

The review emphasizes the need for longitudinal and cross-cultural research and offers insights for teacher education and curriculum design. Several limitations must be acknowledged: the inclusion of studies published only in English, Spanish, and Portuguese introduces a language bias that may have excluded relevant research produced in other academic traditions. The reliance on Scopus as the primary database, supplemented by Google Scholar, may have introduced a publication bias, as indexed databases tend to overrepresent studies with statistically significant or otherwise favorable results. Gray literature such as doctoral dissertations, institutional research reports, and non-indexed policy documents was not included, which may limit the comprehensiveness of the synthesis. The coding and data extraction process was conducted collaboratively rather than independently, and no formal inter-rater reliability coefficient was calculated, which may affect the replicability of the categorization of instructional strategies. The temporal restriction to studies published between 2019 and 2025 excludes foundational research on mathematical argumentation that predates this period. Despite these constraints, the synthesis makes clear that teaching students to argue mathematically requires the integration of cognitive, social, and technological dimensions, and that innovative instructional approaches foster engagement and reasoning even as most studies report limitations related to small sample sizes and short intervention periods. Future research should prioritize large-scale, longitudinal, and cross-cultural studies to strengthen the evidence on sustainable impact, and the limited focus on primary education stands out as an area warranting systematic investigation.

Subject of Research: Mathematics Education

Article Title: Teaching strategies to foster mathematical argumentation: a systematic review (2019–2025)

Article References: Ríos-Cuesta, W., & Ospina-Marulanda, L. P. (2026). Teaching strategies to foster mathematical argumentation: a systematic review (2019–2025). Discover Education, 5(1), Article 850. https://doi.org/10.1007/s44217-026-02056-x

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02056-x

Keywords: mathematical argumentation, systematic review, mathematics education, teaching strategies, Toulmin model, cognitive unity, GeoGebra, gamification, Teaching, strategies, foster, mathematical

The starting point was sobering. Before any new instruction, 80 students completed an initial mathematical representation test with five written items scored on a 0-to-4 rubric across three indicators: visual, symbolic, and verbal. Combined achievement across the three indicators was only 34.53 percent. The symbolic indicator was weakest at 17.66 percent, followed by the verbal indicator at 40.63 percent and the visual indicator at 45.31 percent. In other words, translating a situation, a graph, or a table into an algebraic expression was where students struggled most, even though they could often recognize a parabola when they saw one.

Classroom observations reinforced the test results. Quadratic functions were typically introduced through the general form f(x) = ax² + bx + c and its algebraic characteristics, such as maximum or minimum values and intersections with the coordinate axes. Instruction emphasized symbol manipulation over meaning. On the symbolic items, none of the 80 students completed the graph-based item in full. Although 13 students, 16.25 percent, correctly derived the quadratic formula, none went on to determine the corresponding pattern of function values that the item also required. On the contextual item, which asked students to model a ball’s trajectory and decide whether it would pass through a goal, only 7.5 percent answered correctly.

Semi-structured interviews with 12 purposively selected students revealed how students framed the concept. Three described a function purely as a computational tool, with one student saying, “A function is just a formula, Ma’am, used for calculation.” Two recalled a memorized set-based definition without understanding its meaning. The remaining seven, the largest group, identified a quadratic function primarily by the presence of the x² term or the shape of a parabola rather than by any relationship between changing quantities. “If there is x², then it is a quadratic function,” one student explained. Graphical work showed a related pattern: 58 of the 80 students, 72.5 percent, produced graphs that were not parabolic, and 29 students, 36.25 percent, connected plotted points with straight-line segments, treating quadratic situations with strategies characteristic of linear graphs.

Perhaps most striking, none of the twelve interviewed students identified a pattern of change as a defining feature of a quadratic function, regardless of their written performance. Four of them had already derived a correct quadratic formula on the graph-based item yet still could not identify the underlying pattern when asked directly. Other students judged a quadratic data set to be linear “because the values kept increasing,” or said the numbers seemed random. This suggests that pattern recognition functions as a distinct competency rather than a natural extension of symbolic proficiency, a dissociation that became the specific starting point for the instructional design.

The learning sequence integrated three tools, each matched to a distinct representational purpose. A physical walking activity opened instruction because it grounded the idea of changing quantities in an embodied experience: students walked first at a constant pace and then at an accelerating pace, producing an intuitive contrast between constant and non-constant change before any numerical or symbolic representation appeared. The PhET Moving Man simulation followed immediately, reproducing the same motion context under controlled conditions with position = 0, velocity = 0, and acceleration = 2, allowing students to generate stable position data that the walking activity alone could not consistently provide. GeoGebra came last, after students had constructed a symbolic generalization, so they could manipulate sliders for the coefficients a, b, and c and observe how each shapes the resulting graph.

The teaching experiment involved a separate group of 80 students, 40 per school, organized into ten heterogeneous groups of four per class. Each cycle ran as a single 135-minute session covering a contextual walking activity, the PhET simulation, a table-completion task, a GeoGebra exploration, and practice items. Cycle 1 exposed two obstacles. In the walking activity, conducted without a structured measurement procedure, 8 of 10 groups produced data too irregular to discern a consistent pattern of first or second differences. In the PhET activity, only 4 of 10 groups independently derived the correct symbolic generalization, and 3 of 10 groups required teacher assistance to identify the constant second difference. Four groups were also confused by position values that were not whole numbers.

The researchers revised the design in response. Students were instructed to walk in stages and stop briefly to measure distance at each one-second interval, and a generalization row represented by x was added to the existing table to help students move from particular values to general symbolic forms. Two brief prerequisite reviews were also added: decimal rounding before the simulation, and number classification before the GeoGebra slider activity. In Cycle 2, only 3 of 10 groups produced walking data too irregular to show a trend, only 1 of 10 groups required assistance identifying the constant second difference, and 8 of 10 groups derived the correct symbolic generalization independently. All 10 groups concluded that the coefficients a, b, and c were real numbers, compared with 8 of 10 in Cycle 1.

One group’s worksheet illustrates the mechanism. Using the simulation, the group recorded position values of 0, 1.085, 4, and 8.752, then rounded them to 0, 1, 4, and 9. From this sequence they computed first differences of 1, 3, and 5, identified a constant second difference of 2, correctly predicted the values 16 and 25 for the following two seconds, and concluded that f(x) = x². This episode shows that second differences did not function merely as a computational step; they provided an intermediate representation through which students could connect numerical growth with symbolic structure. Overall, 7 of 10 groups in Cycle 2 reached representational coordination, consistently linking the numerical pattern, its symbolic generalization, and the underlying motion context without teacher support, compared with 4 of 10 in Cycle 1. Mean quiz scores rose from 5.05 out of 8 (63.13 percent) to 6.73 out of 8 (84.13 percent), with correct answers increasing from 25 percent to 55 percent of students.

The researchers caution that the two cycles occurred in different classes at different schools, so the improvement cannot be attributed solely to the design revisions, and the study examined only the first meeting of a longer sequence. Still, the findings yield three concrete design principles: begin with contrastive situations that allow students to compare linear and quadratic change; use second differences as a representational bridge between numerical patterns and symbolic generalization; and formalize the general form f(x) = ax² + bx + c, with a ≠ 0, only after students have recognized the structure of change in values. The lesson for teachers is that learning quadratic functions should not begin directly from formulas or graphing procedures, but from activities that help students construct meaning through connected representations, with the formula emerging as a symbolic generalization of a pattern students have already seen.

Subject of Research: Supporting students’ understanding of quadratic functions as patterns of change through representational transitions

Article Title: Supporting students’ understanding of quadratic functions as patterns of change through representational transitions

Article References: Siregar, H. M., Turmudi, T., Nurjanah, N., Darhim, D., & Nurlaelah, E. (2026). Supporting students’ understanding of quadratic functions as patterns of change through representational transitions. Discover Education, 5(1), Article 1177. https://doi.org/10.1007/s44217-026-02298-9

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02298-9

Keywords: mathematics education, quadratic functions, representational transitions, design research, patterns of change, PhET simulation, GeoGebra, Indonesia, Supporting, students, understanding, quadratic

The study, led by researchers at Universidad Técnica de Manabí together with collaborators from the Autonomous Decentralized Government of Chone Canton and Universidad Laica Eloy Alfaro de Manabí, was designed as a descriptive, exploratory, non-experimental investigation with a quantitative approach. Its target population consisted of 1,817 third-year upper secondary students aged 15 to 17, from which a purposive sample of 843 students—46.40 percent of the reference population—was drawn from 21 institutions. Every selected student completed the diagnostic assessment, a 100 percent participation rate within the sample, though the authors caution that because selection depended on accessibility, institutional coordination, and student availability, the results cannot be generalized beyond the schools included.

The instrument itself was a five-task diagnostic test intended to probe the multidimensional construct of logical-mathematical reasoning, which the researchers define as encompassing conceptual understanding, procedural application, interpretation of mathematical representations, and contextualized problem solving. Each task was scored on an ordinal scale from 0 to 4 points using general criteria from an analytical rubric, and the five tasks carried equal weight. Summing the scores produced an overall result between 0 and 20 points, which mapped onto five performance bands: low (0–4), regular (5–9), good (10–14), very good (15–18), and excellent (19–20). The same scoring criteria and cut-off points were applied uniformly across all 21 institutions, and the archived structure of the test and its scoring rules are documented in the study’s supplementary material.

According to the available documentation, the test and rubric were reviewed by three specialists with experience in mathematics education, instrument design, and educational assessment, and a preliminary administration was conducted with secondary students outside the final sample to check the clarity of instructions and tasks. However, the authors are unusually candid about the limits of their records: the complete original rubric was not retained, individual task-level scores were unavailable, and the database contained only aggregated overall classifications. As a result, the team could not retrospectively estimate the internal consistency of the instrument or calculate inter-rater agreement, and they explicitly state that the performance categories should be interpreted as operational classifications rather than as fully psychometrically validated measures.

The statistical machinery behind the analysis was correspondingly careful. Because no student reached the excellent level, that empty category was excluded, leaving a 21-by-4 contingency table with 84 cells and 60 degrees of freedom. Assumption checks revealed that 57.14 percent of cells had expected frequencies below 5 and 28.57 percent below 1, violating the conditions for the standard asymptotic chi-square approximation. The researchers therefore turned to a Monte Carlo simulation, generating 100,000 tables under the null hypothesis of independence while preserving the marginal totals. The resulting association between institution and performance level was emphatically significant: χ²(60) = 392.62 with a Monte Carlo p-value of 0.00001 and a Cramer’s V of 0.394, indicating a substantial relationship between which school a student attended and where they landed on the performance scale.

Because the performance categories are ordinal, the team also applied a Kruskal–Wallis test with tie correction, coding the levels from low = 1 to very good = 4. This analysis corroborated the chi-square findings, yielding H(20) = 208.51, p < 0.001, with an overall effect size of ε² = 0.229. To locate where the differences actually lived, the researchers then ran all 210 possible pairwise comparisons among the 21 institutions using Dunn’s test with tie correction, adjusting significance values with the Holm procedure to control the error inflation inherent in so many simultaneous tests. Fifty-four of the 210 comparisons remained statistically significant after adjustment, while 156 did not—a pattern suggesting that the overall variability was concentrated in specific institutional contrasts rather than spread uniformly across every pair of schools.

Those contrasts were vivid. At Colón Arteaga García, no students were classified at the low level, and participants split evenly between the regular and good levels; the school posted higher performance ranks than 18 other institutions, differing significantly from all but Bejucal and Santa Mariana de Jesús. Bejucal was the only institution with students at the very good level and outperformed 13 schools. Santa Mariana de Jesús, 5 de Mayo, San Cayetano de Chone, and Eugenio Espejo all showed percentages at the good level above the overall figure of 5.93 percent. At the other end of the spectrum, El Pueblito de Río Grande had every single participant concentrated at the low level, while Oswaldo Castro Intriago and Carlos María Castro each recorded 96.67 percent of students there, followed by Gonzalo Abad Grijalva at 96.00 percent and Aníbal Andrade Solórzano at 95.00 percent. Adjusted standardized residuals confirmed these patterns, with the largest contributions to the overall association coming from the very good level at Bejucal and the low and good levels at Colón Arteaga García.

What the study deliberately does not do is explain why these gaps exist. No data were collected on students’ sex, socioeconomic status, prior academic achievement, previous experience with digital tools, device access, connectivity, or teacher characteristics, and the hierarchical clustering of students within schools was not modeled with multilevel procedures. The authors are emphatic that the results should not be used to construct rankings of institutional quality, since unequal sample sizes and the absence of explanatory variables make causal interpretation impossible. Variables such as school climate, teacher training, and resource availability are plausible but untested explanations, and the observed differences are presented strictly as exploratory evidence of institutional heterogeneity rather than as a verdict on any school’s effectiveness.

The assessment also has a complicated relationship with technology. It was administered in person, in a session of roughly 60 minutes supported by two collaborating teachers per institution, before any GeoGebra-supported classroom activities took place. GeoGebra—the free dynamic mathematics software that integrates algebraic, graphical, geometric, and numerical representations—was subsequently used in activities guided by shared general objectives, guiding questions, and achievement criteria, with students manipulating parameters, formulating conjectures, and comparing strategies across representations. But because the study included no comparison group, no equivalent pretest–posttest measurement, and no standardized implementation checklist, the authors state plainly that the software’s effectiveness cannot be evaluated and that no changes in reasoning can be attributed to it. The GeoGebra activities serve purely as the pedagogical context that followed the diagnosis, and the documentation does not even allow the number or duration of sessions to be reconstructed uniformly across the 21 schools.

That restraint reflects a broader lesson from the literature the team reviewed. Meta-analyses of GeoGebra research, including a 2025 synthesis of two decades of studies on dynamic visualization, report favorable average outcomes for mathematics learning but reveal considerable heterogeneity tied to research design, educational level, activity duration, mathematical content, and—critically—teacher mediation. From constructivist and sociocultural perspectives, dynamic representations foster exploration and discussion only when accompanied by guiding questions, feedback, and opportunities to explain and contrast strategies, while social cognitive theory suggests that teachers’ self-efficacy, outcome expectations, and institutional support shape how such tools are actually integrated. In other words, the software is never the active ingredient on its own; the interaction among tool, task, teacher, student, and context determines the outcome.

The practical implications, the authors argue, begin with diagnosis itself. Given that 93.48 percent of tested students sat in the two lowest performance bands, the participating institutions should consider conducting initial assessments before planning instructional sequences, organizing differentiated support, and separately monitoring conceptual understanding, procedural application, interpretation of representations, and problem solving—something the present instrument, with its single overall score, could not do. Future work, they suggest, should employ experimental or quasi-experimental designs with comparison groups, instruments with documented validity and reliability, calibrated rubrics with inter-rater agreement, and multilevel models that respect the clustered structure of classroom data. Longitudinal studies could track the stability and transfer of learning, while research on teacher professional development could examine how technological self-efficacy and dialogic teaching influence the use of dynamic visualizations. For now, the study stands as a rare, methodologically transparent baseline from a region often absent from the global mathematics education literature—and as a warning that the gap between what students can do and what curricula assume is wider, and more unevenly distributed, than many had hoped.

Subject of Research: Diagnostic assessment of logical-mathematical reasoning performance among secondary school students in Chone Canton, Ecuador

Article Title: Diagnostic performance in logical mathematical reasoning varies across secondary education institutions in Chone Canton Ecuador

Article References: Avellán-Avellán, L. M., Arteaga-Solórzano, R. A., Párraga-Alava, R. C., Vargas-Zambrano, P. A., Moreira-Borja, G. H., Moreira-Menéndez, H. O., Zambrano-Arteaga, R. I., & Avellán-Avellán, M. C. (2026). Diagnostic performance in logical mathematical reasoning varies across secondary education institutions in Chone Canton Ecuador. Discover Education, 5(1), Article 992. https://doi.org/10.1007/s44217-026-02186-2

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02186-2

Keywords: logical-mathematical reasoning, mathematics education, diagnostic assessment, secondary education, Ecuador, GeoGebra, institutional variability, chi-square test, Kruskal-Wallis test, Monte Carlo simulation, educational inequality, Chone Canton

Yet the mathematics underpinning these new standards remains locked behind graduate-level abstraction, and a team of researchers at the Universidad de La Laguna in Spain argues that this is a problem education must solve now. In a study published in the Journal of New Approaches in Educational Research, Édgar Pérez-Ramos, Candelaria Hernández-Goya, and Pino Caballero-Gil present an interactive GeoGebra-based activity designed to introduce high school students to the core ideas of lattice-based post-quantum cryptography. The work is notable not only for its subject matter but for its results: after a single session, students with no prior cryptography experience showed dramatic gains in both motivation and conceptual understanding, with some test scores improving by more than 40 percentage points.

The mathematical heart of the activity is the lattice, an object the researchers describe as a multidimensional infinite grid. In its simplest two-dimensional form, a lattice is the collection of all points reachable by adding and subtracting whole-number multiples of a finite set of vectors. This definition deliberately builds on concepts students already meet in secondary school: the dot product of two vectors, orthogonality, Euclidean distance, and the so-called taxi or Manhattan distance, which measures travel along a rectangular grid rather than a straight line. The dot product links the lattice-generating vectors to the cosine of the angle between them, creating a natural bridge to trigonometry, while the taxi distance gives students a tangible way to think about how lattice points relate to one another in space.

Why do these geometric ideas matter for security? Modern lattice-based schemes rest on computational problems that are believed to be intractable even for quantum computers, including the Shortest Vector Problem, the Closest Vector Problem, and the Module Learning With Errors Problem. Finding the shortest vector from a lattice point to an arbitrary target looks trivial in the flat Cartesian plane, but the real cryptographic schemes operate in dimensions up to 1024, with coefficients drawn modulo large prime numbers such as q = 3329 in CRYSTALS-Kyber. Crucially, the less orthogonal the vectors defining a lattice are, the harder the associated problem becomes. A student who can see this relationship in two dimensions has grasped the seed of an idea that protects encrypted communications at planetary scale.

The researchers chose GeoGebra, an open-source mathematics platform combining dynamic geometry, algebra, and data visualization, as the vehicle for the lesson. The publicly available activity, titled A Brief Introduction to Lattice Theory, lets students drag vectors on screen and watch the resulting lattice reshape itself in real time. They can verify that the only null lattice consists entirely of null vectors, discover that non-null vectors with infinite coefficients generate an infinite lattice, and explore what happens when generating vectors are perpendicular versus skewed. Short-answer and open-ended questions are woven throughout, pushing students beyond passive observation toward constructing their own lattices and articulating the rules that govern them.

The design is grounded in three established frameworks from mathematics education research. The Theory of Didactical Situations structures the session into action, formulation, validation, and institutionalization phases: students first manipulate vectors independently, then formulate hypotheses about how vector choices shape the lattice, test those conjectures through GeoGebra’s immediate feedback, and finally formalize their discoveries with the teacher using concepts like basis, orthogonality, and scalar products. APOS Theory describes the cognitive progression from concrete actions to internalized processes, reified objects, and integrated schemas, mapping how a lattice transforms from something students click on into a mathematical object with intrinsic properties. The Theory of Instrumental Genesis adds a dual perspective, treating GeoGebra both as a tool students learn to operate and as an artifact that actively reshapes their mathematical thinking.

One unconventional element is the use of ChatGPT to provide an accessible, natural-language definition of a lattice at the start of the session. The researchers are careful to frame this as a didactical support tool rather than a replacement for instruction: prompts are carefully designed, and outputs are systematically checked by teachers against established academic sources. The activity also connects the abstract material to the physical world through an aerial view of Barcelona, whose famously regular urban grid is overlaid with lattice points at street intersections. Students are challenged to navigate from point A to point B using only lattice points, discovering that there is no single straight-line solution but multiple valid routes, a vivid demonstration of the taxi distance and the non-unique structure of lattice problems.

To evaluate the activity, the team ran experiments with 102 secondary school students in Spain, spanning the final year of compulsory education and both years of the Baccalaureate program. The cohort was 43.14 percent female and 53.92 percent male, with specializations split across technology, natural sciences, social sciences, and arts. A baseline survey revealed that 94.12 percent of participants had no prior experience with cryptography, making the group an ideal test bed for whether digital tools can open access to advanced content. The study followed a pre-post design: a short pre-test, a 30-minute introductory lecture on post-quantum cryptography, roughly one hour with the GeoGebra activity, and then a post-test. Ethical approval was obtained from the university ethics committee, with informed consent from students and their legal guardians, and the dataset was made publicly available on Zenodo.

The statistical results were striking. Using paired two-sample Student’s t-tests at a significance level of 0.05, the researchers rejected both null hypotheses. Students’ self-assessed knowledge of post-quantum cryptography rose from a median of 1 to a median of 3 on a five-point scale, and reported interest increased significantly. Correct answers on the question of what post-quantum cryptography is based on jumped from 3.92 percent to 82.35 percent, while understanding of its mathematical foundations climbed from 23.53 percent to 82.35 percent. The gains extended to prerequisite mathematics: correct responses on the conceptual definition of vectors rose from 40.20 percent to 81.37 percent, vector addition from 32.35 percent to 81.37 percent, and scalar multiplication from 33.33 percent to 84.31 percent. Notably, the largest improvements appeared in questions addressed through dynamic visualization and direct manipulation, suggesting that hands-on exploration with immediate visual feedback consolidates learning more effectively than definitions and examples alone.

The timing of this work is significant beyond the classroom. Spain’s latest PISA results showed its lowest mathematics score since the assessment began, with students dropping eight points compared to the previous edition, creating what the researchers call an urgent need for innovative and useful mathematics materials. The activity aligns with Spain’s current education law, LOMLOE, and its evaluation criteria on selecting digital tools and forms of representation, as well as basic knowledge areas covering dynamic geometry software and computational thinking. It also echoes the Informatics for All Coalition’s vision of citizens as critical users and active creators of technology rather than mere consumers. Existing resources for learning lattice cryptography, such as CrypTool, SageMath materials, and IBM Quantum’s online course, target advanced learners and professionals, leaving a gap at earlier educational stages that this study directly addresses.

The researchers see the activity as a promising model for teacher training initiatives seeking to bring advanced topics into classrooms through technology, and they plan to extend it with more advanced content, including the determinant of a lattice’s fundamental parallelepiped and the Learning With Errors problem. Further workshops with students and teachers are planned to gather additional data and refine the approach, with an eye toward tailoring activities to individual differences in prior knowledge for more equitable outcomes. As the 2035 deadline for the post-quantum transition approaches, the study makes a compelling case that the mathematics protecting tomorrow’s secrets can, with the right interactive tools, be taught today in an ordinary high school classroom.

Subject of Research: Teaching lattice-based post-quantum cryptography to secondary school students through an interactive GeoGebra activity

Article Title: An interactive tool for teaching lattice-based post-quantum cryptography

Article References: Pérez-Ramos, É., Hernández-Goya, C., & Caballero-Gil, P. (2026). An interactive tool for teaching lattice-based post-quantum cryptography. Journal of New Approaches in Educational Research, 15(1), Article 18. https://doi.org/10.1007/s44322-026-00068-x

Image Credits: AI Generated

DOI: 10.1007/s44322-026-00068-x

Keywords: post-quantum cryptography, lattice-based cryptography, GeoGebra, mathematics education, quantum computing, NIST standards, secondary education, CRYSTALS-Kyber, interactive learning, vectors, taxi distance, digital security