Understanding the Concept of a Parabola: A Literature Review of the Conceptual, Algebraic, and Graphical Foundations

Abstract

Understanding the parabola is often fragmented among the quadratic function, the canonical equation, and the graphical representation, making it difficult to recognize it as a geometric locus and to coordinate its properties. The objective of this article was to analyze the difficulties, the conceptual, algebraic, and graphical foundations, and the instructional strategies reported in recent literature. A descriptive and integrative literature review was conducted of 42 studies published between 2020 and 2026. The information was organized into a 12-field matrix and coded by year, methodological approach, thematic focus, difficulties, contributions, limitations, and recommendations. 57.1% of the studies were published in 2024–2025; empirical qualitative research (52.4%) and literature reviews (26.2%) predominated. Fifteen studies focused directly on the parabola or conic sections, and 27 provided complementary evidence from ethnomathematics and the contextualization of geometry. The findings reveal a predominance of algebraic approaches, difficulties in converting between semiotic representations, weaknesses in visualization and prior knowledge, and confusion between the parabola and quadratic functions. The most consistent proposals integrate the focus-directrix definition, geometric construction, algebraic formalization, dynamic visualization, and contextualized problem-solving using GeoGebra, Tracker, photography, the history of mathematics, and cultural contexts. It is concluded that technology promotes understanding when accompanied by teacher mediation, guiding questions, collaborative work, and a sequence that moves from the phenomenon to the concept and from the concept to its representations.