Let two plus two not always equal four: A critical pedagogical exploration of mathematical certainty and context

This mixed-methods study was designed to investigate how engagement with modular arithmetic influences senior high school students’ beliefs about mathematical certainty, creativity, and reflective thinking. Grounded in constructivist learning theory and critical mathematics education, the study conceptualizes mathematical truth as context-dependent, rule-governed, and open to epistemic inquiry rather than as an absolute given. Conducted with 50 grade 12 learners from Stirling Schools in Erbil and Kirkuk during the 2024-2025 academic year, the study employed a convergent design integrating pre- and post-Likert-scale surveys with reflective journals. The intervention challenged the universal arithmetic statement “2 + 2 = 4” by introducing congruence modulo n, highlighting that mathematical outcomes depend on specific axiomatic rules. Quantitative data analysis revealed significant improvements in beliefs about mathematical certainty, creativity, and critical thinking (p < .001), with no gender-based differences noted. Thematic analysis of qualitative reflections identified six interconnected themes: recognition of context-dependent mathematical truth, awareness of underlying structures, epistemic curiosity from cognitive conflict, emergence of creativity in reasoning, understanding that rules shape meaning, and strengthened mathematical identity. The findings suggest that exposure to alternative mathematical systems fosters conceptual flexibility, metacognitive engagement, and a humanistic view of mathematics. The study has important implications for 21st century mathematics education, advocating for critical-pedagogical approaches that enrich classroom inquiry, promote creativity, and challenge absolutist notions of truth. Limitations and recommendations for future research, including longitudinal designs and cross-cultural comparisons, are also discussed.