Students Say Proof Matters, Yet Most Stay Silent in Geometry Class

Ask high school students whether mathematical argumentation matters, and the answer is an emphatic yes. Ask whether they actually argue in geometry class, and the picture changes dramatically. A new survey study of 437 Vietnamese high school students has uncovered a striking pattern: teenagers overwhelmingly value reasoning and proof, yet report surprisingly few opportunities to practice them in the classroom. The research, published in Discover Education by a team at Can Tho University, offers one of the most detailed quantitative portraits to date of how students experience mathematical argumentation, and it raises uncomfortable questions for geometry teachers everywhere.

The study focused on a region of Vietnam that has been largely absent from the international mathematics education literature. Between September and December 2025, researchers administered an in-person questionnaire to students in twelve classrooms across nine schools in An Giang Province and Can Tho City. The instrument was built around a well-known framework from argumentation theory: Toulmin’s Argumentation Model, which dissects any argument into six interlocking parts, namely data, claim, warrant, backing, qualifier, and rebuttal. In geometry, this structure maps naturally onto proof. The data are the givens, the claim is the statement to be proven, and the warrant is the theorem or property that connects the two. Crucially, the researchers used Toulmin’s model as a conceptual lens rather than a rigid measurement template, acknowledging that the model has known limitations when applied to the dynamic, social process of classroom discussion.

The questionnaire began with fifteen items rated on a five-point Likert scale, grouped provisionally into perceptions and attitudes, use of argumentation components, and difficulties in geometric reasoning. Statistical screening quickly reshaped that structure. Four items fell away, and an exploratory factor analysis on the remaining eleven items revealed something the researchers had not anticipated: instead of three tidy domains, the data organized themselves into four distinct factors. The first, Participation in Mathematical Argumentation, captured students’ enjoyment of reasoning tasks, their confidence in presenting arguments, their willingness to engage in discussion, and their critical evaluation of claims. The second, Use of Reasoning Components, measured how often students identified given information, deployed prior knowledge as a warrant, and explained why each step of a proof was valid. The third factor reflected the Perceived Importance of Mathematical Reasoning, and the fourth captured Difficulties in Geometric Reasoning, such as trouble recalling or correctly applying theorems.

Together these four factors explained 61.3 percent of the total variance, with reliability coefficients ranging from acceptable to good. The factor structure itself carried a theoretical punch. Items that the researchers had expected to split across affective, skill-based, and participatory domains instead converged into a single participation factor, suggesting that students experience argumentation holistically, as one integrated social practice rather than a bundle of separable skills. This finding aligns with a sociocultural view of mathematics learning, in which knowing how to argue and actually arguing in front of peers are inseparable from the classroom environment a teacher creates. It also echoes prior research showing that classroom argumentation is nonlinear and messy, requiring students to coordinate warrants, rebuttals, and qualifiers in real time while responding to one another’s reasoning.

The descriptive statistics delivered the study’s headline result. Students rated the importance of mathematical reasoning at a mean of 4.18 on the five-point scale, one of the highest endorsements the instrument recorded. Yet their reported participation in argumentation averaged just 2.88, well below the midpoint of active engagement. In other words, students believe argumentation is central to mathematics but say they rarely get to do it. The researchers are careful to note that because these two factors come from different item sets, the gap should be read as an exploratory observation rather than a proven attitude-behavior discrepancy. Still, the pattern is consistent with a well-documented distinction in mathematics education between possessing component knowledge and deploying it in performance. Students reported moderate use of reasoning components, at a mean of 3.68, suggesting they have cognitive tools they do not consistently bring into classroom discourse. The authors point to motivational and affective barriers, including limited confidence and limited enjoyment of reasoning tasks, as plausible culprits, alongside classroom discussion norms that may simply not invite justification.

Group comparisons added further texture. Male students reported significantly higher participation than female students, with a small-to-moderate effect size, and also reported slightly more difficulty in geometric reasoning. No significant differences emerged between rural and urban students on any factor, a result the authors attribute in part to Vietnam’s standardized national curriculum, which imposes shared goals and content across regions. Grade level made no difference either: tenth, eleventh, and twelfth graders held broadly comparable perceptions. That null result is quietly one of the most consequential findings in the paper, because it implies that argumentation-related perceptions do not mature automatically as students advance. Higher-order competencies, the study argues, require deliberate and sustained pedagogical support rather than time alone.

Participation in mathematics enrichment courses, by contrast, mattered considerably. Students who had taken supplementary enrichment modules reported higher perceived importance of reasoning, higher participation, and, intriguingly, more reported difficulty. The participation difference was the largest in the study, with a medium effect size of 0.644. The authors caution that enrichment participation is likely confounded with prior motivation, achievement, and family resources through self-selection, so the enrichment experience itself cannot be isolated as the cause. Nevertheless, the pattern is consistent with the idea that structured exposure to argumentation activities builds both the habit of participating and the metacognitive awareness of where one’s reasoning breaks down.

The most counterintuitive result concerned achievement level. Across all four factors, students’ self-reports differed significantly by mathematics achievement, and the direction of the difficulty finding defied expectations: students rated Excellent reported more difficulty in geometric reasoning, at a mean of 2.966, than students rated Satisfactory, at 2.330. Higher achievers also reported significantly higher participation and perceived importance. The researchers offer three speculative interpretations, all flagged for future investigation. Higher-achieving students may be more metacognitively aware and more willing to acknowledge gaps in their understanding, a phenomenon consistent with research on active learners. They may also be tackling more demanding tasks that expose genuinely harder reasoning challenges. Or the difficulty items, which probe trouble recalling and misapplying theorems, may be tapping into productive struggle, the productive confusion that characterizes engaged learners pushing at the edge of their competence rather than incompetence. Any of these explanations, if confirmed, would complicate how teachers interpret students who say geometry is hard.

Correlation analyses reinforced the interlocking nature of the four dimensions. Perceived importance correlated moderately with use of reasoning components, and component use correlated moderately with participation, a chain suggesting that valuing argumentation, wielding its tools, and engaging in it tend to travel together. Notably, the correlation between valuing argumentation and actually participating in it was weak, at 0.282, confirming that attitude alone does not drive classroom behavior. Perhaps most suggestive was the modest positive correlation between participation and reported difficulty: students who argue more also report more struggles, hinting that engagement surfaces cognitive challenges rather than dissolving them.

The study’s limitations are candidly enumerated, and they matter for how far the findings travel. All data are self-reported and subject to social desirability bias. The sample was geographically concentrated, with 82 percent of participants from An Giang Province, and drawn through convenience sampling of intact classrooms. Two of the four factors rest on only two items each, and no confirmatory factor analysis or external content validation was performed. School-level clustering was non-negligible for the participation and difficulty factors, effectively shrinking the usable sample for those comparisons. The authors call for confirmatory validation, geographically diverse replication, additional items, and qualitative methods such as classroom observation and think-aloud protocols to unpack the counterintuitive difficulty finding. Even with those caveats, the practical message is clear. If students prize argumentation but rarely practice it, interventions should target the motivational barriers, the participatory structures, and the component reasoning skills that stand between conviction and classroom voice. Geometry, with its built-in demand for hypotheses, conclusions, and deductive bridges, remains the ideal arena for that work, provided teachers design discussions in which every student is expected to justify, and to be justified.

Subject of Research: High school students' self-reported perceptions of mathematical argumentation in geometry learning

Article Title: Exploring high school students’ perspectives on mathematical argumentation in geometry learning

Article References: Duc, N. H. T., Triet, L. V. M., Loc, N. P., & Kiet, B. A. (2026). Exploring high school students’ perspectives on mathematical argumentation in geometry learning. Discover Education, 5(1), Article 1044. https://doi.org/10.1007/s44217-026-02227-w

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02227-w

Keywords: mathematical argumentation, geometry education, Toulmin argumentation model, exploratory factor analysis, high school students, student perceptions, geometric reasoning, mathematics education, Vietnam, survey research, classroom participation, reasoning difficulties