The purpose of this study was to explore how individual students understand various aspects of quadratic functions such as quadratic growth, quadratic correspondence, quadratic graphs, vertex points, x-intercepts, y-intercept, line of symmetry, parameters of general quadratic functions, and quadratic equations, in order to provide detailed characterizations of the scope and depth of students’ understandings of these concepts. The study addressed the following research questions: What are students’ understandings of quadratic functions? How do individual students understand and organize various aspects and properties of quadratic functions? How are these understandings constituted within situations involving quadratic functions and their properties?

In the study, a qualitative multiple case study methodology was used.  Two semi-structured, video recorded, 75 minute long in-depth interviews with each of the three university students and one high school student, who either recently completed a formal pre-calculus course or were currently enrolled in a pre-calculus course, constituted the study’s primary data source. Students (as cases) were given a twelve problem instrument, with mostly self-generation tasks, and their problem solving activities were analyzed using cognitive constructivist theories in which the participants’ acts of understanding, bases of understanding (Sierpinska, 1994), and cognitive structures were explicated and modeled. Besides the interviews, other data sources included students’ written solutions to the tasks, their self-generated diagrams and mathematical expressions, their responses to an individual mathematics background survey and the researcher’s observations throughout their problem solving activity.

The construct of act of understanding helped interpret how and what the participating students were thinking about various aspects of quadratic functions while solving quadratic function tasks. Briefly, an act of understanding is defined by Ajdukiewicz (1974) as “an act of mentally relating the object of understanding to another object” (cf., Sierpinska, 1994, p. 28). Sierpinska (1994) further developed Ajdukiewicz’s (1974) theory, and defined the ‘another object’ in the above definition as the basis of understanding. According to Sierpinska (1994), there are four mental operations involved in an act of understanding that either determine the object of understanding or relate or link the object and basis of understanding: Identification, discrimination, generalization and synthesis. In terms of conditions of understanding and criteria for relating the object of understanding with its basis, Sierpinska argued that attention, intention and question are three necessary conditions of an act of understanding. Finally, according to Sierpinska’s (1994) theory of understanding that is espoused here, an act of understanding should be distinguished from a process of understanding. An act of understanding is an actual single experience occurring in an individual’s mind at a given time. On the other hand, process of understanding is “regarded as lattices of acts of understanding linked by reasoning [inference or deduction]” (p. 72).

The structure of the analysis of each case study consisted of: An opening vignette that includes general information about the participant, the participant’s past mathematical experiences, their overall approach to and performance on the quadratic function tasks presented in the interview instrument; selected quotes from the participant’s responses to multiple data sources that characterize their reasoning and thinking about quadratic functions; and data analysis and discussion of the findings about the case. For each case, data analysis included: description, direct interpretation, generation of categories or themes, category aggregation, and generation of patterns among categories (Creswell, 1998; Stake, 1995). Video recordings and written documents were examined to discern problem solving episodes that reveal salient aspects of students’ understanding of quadratic function concepts. The themes or categories of how students understand quadratic functions and act on problem situations involving quadratic functions, evolved through the phases of observation, transcription, in vivo coding, and identifying portions of interview data that seemed to represent participants’ own meanings. Video recordings enabled the researcher to not only infer students’ inert actions but also observe their overt reactions, expressions and gestures.

As initial findings of the study, the first participant (or case), pseudo named Ken, yielded an understanding of quadratic function as a unique type of equation where one “solves for y,” whereas the analysis of the second case, pseudo named Sarah, led to the emergence of an understanding of quadratic function as a unique type of graph where every value of x has only one y value on the parabola shaped graph. And, three of all four cases suggested a way of understanding quadratic functions as a collection of things that are compartmentalized (Vinner, 1992; Gerson, 2008) in multiple ways. In addition, all cases confirmed some of the major findings in the literature on students’ understanding of functions. All four cases were compatible with the action view of functions. These findings emerged through several cross analyses between and among the multiple cases of the study. Detailed characterizations of the participants' bases of understanding of quadratic functions are currently being developed.

With regard to the educational significance of the study, the following implications for teaching are suggested based on the initial findings of the study. It is imperative that teachers help their students make sense of quadratic function situations in terms of covarying quantities or variables with corresponding values that belong to different sets called domain and range. For example, none of the participants of the current study mentioned or used the notions of domain and range of a function in any of the tasks. In order for them to develop the desired ways of understanding, they need to experience acts of understanding in a very rich set of situations involving quadratic functions. Teachers who are equipped with the knowledge of students’ current understandings can better design those learning situations. Indeed, teachers can communicate their own understandings to their students, but it is the students themselves who will attend and identify certain objects (i.e., objects that they identify as objects of their understanding) and relate them to their existing bases of understanding. The current study explicates four students’ bases of understanding of quadratic function concepts.

References:

Ajdukiewicz, K. (1974). Pragmatic logic. Boston, MA: D. Reidel Publishing Co. 

Creswell, J. W. (1998). Qualitative inquiry and research design: Choosing among five traditions. Thousand Oaks, CA: Sage.

Gerson, H. (2008). David’s understanding of functions and periodicity. School Science and Mathematics, 108(1), 28-38.

Sierpinska, A. (1994). Understanding in mathematics. London: The Falmer Press Ltd.

Stake, R. (1995). The art of case study research. Thousand Oaks, CA: Sage.

Vinner S. (1992). The Function Concept as a Prototype for Problems in Mathematics Learning. In E. Dubinsky et al. (Ed) The Concept of Function: Epistemological and Pedagogical Aspects, 195-214. Washington DC: Mathematical Association of America.