Mapping algebraic and geometric thinking with the van Hiele model among college students in STEM programs

Judel V. Protacio; Judel V. Protacio

doi:10.3934/steme.2026026

STEM Education

2026, Volume 6, Issue 4: 632-658. doi: 10.3934/steme.2026026

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Research article

Mapping algebraic and geometric thinking with the van Hiele model among college students in STEM programs

Judel V. Protacio ^,

College of Education, Arts and Sciences, Capiz State University, Pontevedra, 5802 Capiz, Philippines

Academic Editor: Ana Barbosa

Received: 03 December 2025 Revised: 04 June 2026 Accepted: 15 June 2026 Published: 02 July 2026

Mathematical proficiency, grounded in the integration of disciplinary knowledge and fundamental mathematical cognitive skills, is essential for achieving the intended learning outcomes of STEM programs. Despite this curricular emphasis, persistent weaknesses in students' mathematical ability continue to be widely documented. To provide insights on this underperformance, this descriptive–associational study investigated the algebraic and geometric thinking within the van Hiele hierarchy among 167 college students enrolled in STEM programs in a Philippine state university. Data were gathered using the van Hiele geometry test and a researcher-developed algebraic thinking assessment to examine development across three strands of algebraic thinking—generalized arithmetic, functional thinking, and modeling language—within van Hiele hierarchy and analyze variations of algebraic and geometric thinking across programs. Most students remained at van Hiele's visualization and analysis stages and at the transition stage between generalized arithmetic and functional thinking, underscoring a limited progression toward advanced mathematical reasoning. Teacher education students demonstrated stronger geometric and algebraic performance than engineering and computer science students, reflecting differences in disciplinary learning trajectories. The significant and systematic associations between algebraic and geometric thinking confirm the cognitive connections between the two domains, implying that students in higher van Hiele levels are better positioned to achieve higher levels of proficiency in complex algebraic thinking strands. The study recommends an in-depth study on instructional approaches that simultaneously cultivate geometric and algebraic thinking through conceptually rich, developmental, and discipline-aligned learning tasks.
- algebraic thinking,
- functional thinking,
- geometric achievement,
- van Hiele geometry test,
- mathematics education
Citation: Judel V. Protacio. Mapping algebraic and geometric thinking with the van Hiele model among college students in STEM programs[J]. STEM Education, 2026, 6(4): 632-658. doi: 10.3934/steme.2026026

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Abstract

Mathematical proficiency, grounded in the integration of disciplinary knowledge and fundamental mathematical cognitive skills, is essential for achieving the intended learning outcomes of STEM programs. Despite this curricular emphasis, persistent weaknesses in students' mathematical ability continue to be widely documented. To provide insights on this underperformance, this descriptive–associational study investigated the algebraic and geometric thinking within the van Hiele hierarchy among 167 college students enrolled in STEM programs in a Philippine state university. Data were gathered using the van Hiele geometry test and a researcher-developed algebraic thinking assessment to examine development across three strands of algebraic thinking—generalized arithmetic, functional thinking, and modeling language—within van Hiele hierarchy and analyze variations of algebraic and geometric thinking across programs. Most students remained at van Hiele's visualization and analysis stages and at the transition stage between generalized arithmetic and functional thinking, underscoring a limited progression toward advanced mathematical reasoning. Teacher education students demonstrated stronger geometric and algebraic performance than engineering and computer science students, reflecting differences in disciplinary learning trajectories. The significant and systematic associations between algebraic and geometric thinking confirm the cognitive connections between the two domains, implying that students in higher van Hiele levels are better positioned to achieve higher levels of proficiency in complex algebraic thinking strands. The study recommends an in-depth study on instructional approaches that simultaneously cultivate geometric and algebraic thinking through conceptually rich, developmental, and discipline-aligned learning tasks.

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Author's biography Judel V. Protacio, PhD is a Professor at the College of Education, Arts and Sciences of Capiz State University, Pontevedra, Capiz, Philippines. His research interests include mathematics and pre-service teacher education

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