Understanding Quadratic Functions and Solving Quadratic Equations: An Analysis of Student Thinking and Reasoning

Abstract

Understanding quadratic functions is critical to student success in high school mathematics and beyond, yet very little is known about what students understand about these functions. There is agreement in the field that quadratics are one of the most conceptually challenging subjects in the secondary mathematics curriculum. However, research on student learning in this area has focused on procedural aspects of solving equations, with very little known about student understanding of the behavior of quadratic functions. This study sought to learn what high school students who have completed an Algebra 2 or Precalculus class understand about quadratics. Specifically, what connections, if any, do they make between representations of quadratic functions? How do students approach solving quadratic equations, and how do they interpret the solutions? Lastly, what cognitive affordances support them in their learning and understanding of quadratic functions, and what cognitive obstacles do they encounter? This qualitative study employed cognitive interviews of 27 students in grades nine through eleven. The data included video and audio recordings as well as student work, captured on a smart pen pencast. The data was analyzed in four phases: (1) focusing on one student at a time, (2) focusing on individual problems, (3) focusing across students, and then (4) revisiting individual problems across students using a conceptual framework grounded in big ideas and essential understandings of quadratics and a children’s mathematical learning perspective. I found that students have a strong sense of the symmetry of the parent function, but are not consistently able to explain the cause of that symmetry. As students solved equations and graphed functions, they transitioned between equations set equal to constant values, expressions, and equations defining functions. At times this was a productive strategy, but for some students it reflected confusion about what they were solving. Lastly, I found that students apply their understandings from work with linear functions to solving and graphing quadratic equations. This study provides an initial framework for how students think about quadratic functions which may enable mathematics educators to better interpret how students’ prior learning influences their understanding of big ideas within the study of quadratic functions.