Designing hypothetical learning trajectories and instructional sequences related to quadratic functions

Quadratic functions are one of the nonlinear functions that students encounter first and a strong understanding of quadratic functions is foundational to be successful at many mathematical concepts. Along with the importance of acquiring the quadratic functions, the related literature indicates the difficulties faced by the students and the limitations regarding their understanding. The reasoning process about quantitative relations supports the students' conceptual learning related to the quadratic functions. However, in the literature of mathematics education, a research-based framework defining the students' understanding and reasoning of the quadratic functions was not designed. In this direction, the purpose of this PhD dissertation is to design an instructional sequence for students to learn the quadratic functions conceptually by triggering their cognitive actions, and to reveal how the students' understanding is shaped in the context of quantitative reasoning. As the main purpose of this dissertation is to develop an instructional model that will support students' conceptual understanding of quadratic functions, the study was conducted as a design-based research, following a cyclical process. The study consisted of three phases: the design, implementation, and analysis phases. During the design phase, a hypothetical learning trajectory was developed through a review of the related literature on quadratic functions, observations of a mathematics teacher's instruction of quadratic functions, a clinical interview conducted with an 11th grade student, and interviews with pre-service and in-service mathematics teachers. Then, based on real-life contexts, four logico-mathematical learning tasks were developed, with accompanying assessment tasks to evaluate the students' understanding. The implementation phase of this design-based research consisted of two consecutive cycles. The first cycle was carried out to evaluate the success of the instructional sequence in supporting student learning in a class. Following the implementation of the instructional sequence in the class, the instructional sequence was further revised based on the students' understanding of the concepts. In the second cycle of the implementation phase, two 10th grade students worked on the revised set of tasks in pair. Hence, the video recordings taken during the teaching experiments, the researchers' observation notes, the clinical interviews, and the students' reflective journals constituted the data sources of the study. In the data analysis process of the study, the constant comparison method simultaneously conducted with the data collection process was used. The data analysis process included two types of analysis. First, concurrent to the data collection, an ongoing analysis was conducted by using constant comparison method. Throughout the data analysis process, the focus was on the students' understanding in the context of the tasks, the quantities that were constructed or that were expected to be constructed, and the quantitative operations in the unstructured analyses. Second, a retrospective analysis of the teaching experiment data was conducted, with a focus on students' quantitative reasoning. As a result of the data analysis, the instructional sequence for quadratic functions was finalized. Because the tasks were grounded in real-life contexts involving dynamic situations, they contributed to the students' understanding of functional relations and helped them construct the idea of covarying change in the functions. The students continually performed multiplicative comparisons between the variables and the amounts of changes in the variables in the context of the tasks. In addition, the students constructed the quadratic function through multiplication of two linear functions by drawing on the idea of finding the area of a square. The students showed evidence of strong quantitative reasoning, as indicated by their recognition of and expression that there was a constant rate of change between ∆x and ∆y values in the area function. The students constructed the quantities such as vertex and axis of symmetry of a quadratic function with respect to the path a ball follows. Furthermore, the students were able to understand that the vertex is an ordered pair formed by matching the value of the x-axis with the y-value corresponding to the that x-value in the function, instead of viewing it as just the value of (-b)/2a. Hence, the instructional sequence designed in this study seemed to have supported the participant students' conceptual understanding of quadratic functions. Therefore, the instructional sequence may be a helpful resource for mathematics teachers in supporting their students' understanding of quadratic functions, provided that they revise the instructional sequence according to their own classroom context. In addition, mathematics teachers may also take into account the main principles of this study when designing a learning environment for teaching other mathematical concepts, as well. Design-based research studies contribute to the design, development and improvement of instructional products or an instructional program, as well as contributing to the participant teachers' professional development when they are creatively and carefully designed. Therefore, it is considered crucial for teachers to participate in this kind of design-based research studies.