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CC BY-NC-SA V4.0, Open Access
Dissertation
A learning trajectory for intermediate algebra students transitioning from solving linear equations to factorable quadratic equations
Washington State University
Doctor of Philosophy (PhD), Washington State University
01/2021
DOI:
https://doi.org/10.7273/000002422
Handle:
https://hdl.handle.net/2376/120454
Abstract
Research on learning quadratic equations reports students’ difficulties with procedural fluency and conceptual understanding of standard methods for solving such equations. There is little research on how to support students’ mathematical development for factorable quadratic equations without using the concept of function and function notation. I investigated how students may develop connections between essential concepts for solving factorable quadratic equations starting from their current conception of solving linear equations. To achieve this, I conducted a design research study. Based on the pilot’s data analysis, I proposed key developmental understandings (KDUs, M. A. Simon’s construct) for students learning to solve factorable quadratic equations. These KDUs informed the two subsequent iterative cycles through which I developed a hypothetical learning trajectory (HLT) for supporting students’ understanding of this topic. In each cycle, I prepared a HLT (including goals, mathematical tasks, and hypotheses), conducted individual task-based interviews, and used qualitative methods to analyze participants’ engagement with and reasoning during the tasks. I interviewed 12 university students enrolled in an intermediate algebra course. The data analysis was based on comparing the anticipated and observed learning trajectories.
This study contributes a HLT and an explanatory framework for supporting students in developing a richer understanding of solving factorable quadratic equations. I incorporated two perspectives of solutions to a linear or quadratic equation: symbolically as numbers that satisfy an equation (e.g., ax^2+bx+c=0) and graphically as the x-coordinate(s) of the x-intercept(s) of the corresponding graph (e.g., y=ax^2+bx+c). The instructional tasks in this trajectory offer students opportunities for subtle but crucial conceptual transitions as they engage their prior knowledge of linear equations, develop an intuitive understanding of why the method of factoring works, understand how many solutions a linear or quadratic equation may have, notice the algebraic structure of a factored equation and understand how the zero-product property applies to solving factorable quadratic equations. The data analysis shows that the proposed HLT is viable. The account of how participants engaged with the tasks and interacted with the researcher illustrates how teachers may probe and guide students towards reflecting on their mathematical activity and understanding of this topic.
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Details
- Title
- A learning trajectory for intermediate algebra students transitioning from solving linear equations to factorable quadratic equations
- Creators
- Silvia Anel Madrid Jaramillo
- Contributors
- Sandra Cooper (Advisor)Sandra Cooper (Committee Member)William Hall (Advisor)William L Hall (Committee Member)Shiv Karunakaran (Committee Member)Nairanjana Dasgupta (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Mathematics and Statistics, Department of
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 417
- Identifiers
- 99900606754301842
- Language
- English
- Resource Type
- Dissertation