Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya
Introduction
Demonstrated by its presence in both U.S. and international curricula (Bergeron, 2015), inverse function is an important concept in secondary and post-secondary mathematics. Whereas numerous researchers have investigated students’ function meanings (e.g., Blanton, Brizuela, Gardiner, Sawrey, & Newman-Owens, 2015; Breidenbach, Dubinsky, Hawks, & Nichols, 1992; Carlson, 1998; Leinhardt, Zaslavsky, & Stein, 1990; Oehrtman, Carlson, & Thompson, 2008; Thompson, 1994), fewer researchers have focused on individuals’ inverse function meanings. Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings (Brown & Reynolds, 2007; Engelke, Oehrtman, & Carlson, 2005; Even, 1992; Lucus, 2005; Marmur & Zazkis, 2018; Paoletti, Stevens, Hobson, Moore, & LaForest, 2018; Teusher, Palsky, & Palfreyman, 2018; Vidakovic, 1996), and there are few empirical examples presenting ways to support any of these populations in developing productive inverse function meanings.1 Particular to pre-service and in-service teachers, several researchers (e.g., Cooney, Shealy, & Arvold, 1998; Even & Tirosh, 1995; Thompson, 2013) have noted these populations maintain meanings that lack coherence, and there is a need to explore ways to support teachers in developing productive meanings they can carry into their classrooms.
In this report, I first synthesize the research on students’ and teachers’ inverse function meanings. I then describe my theoretical perspective, including a description of key terminology (e.g., understandings, meanings, accommodations) and constructs related to quantitative and covariational reasoning. Next, I present a conceptual analysis (Thompson, 2008) of a productive meaning for inverse relations and functions grounded in Thompson and Carlson’s (2017) proposed covariational meaning of function. In this analysis, I characterize limited meanings for inverse function and related ideas students and teachers often hold, which they may need to reorganize to develop productive meanings for inverse function. I then characterize one pre-service teacher’s activities as she reorganized her inverse function meanings in ways compatible with my conceptual analysis, providing one of the few empirical examples of a student or teacher developing productive meanings for inverse relations and functions. I draw implications from these results including how school-taught techniques for determining inverse functions give rise to didactic obstacles, barriers “one encounters in developing a way of thinking… [as a] result of narrow or faulty instruction” (Harel & Sowder, 2005, p. 34).
Section snippets
Research examining students’ and teachers’ inverse function meanings
Researchers using an APOS lens (Breidenbach et al., 1992; Dubinsky & Harel, 1992) have provided preliminary genetic decompositions for specific inverse function classes (Martinez-Planell & Cruz Delgado, 2016; Weber, 2002) and inverse function generally (Vidakovic, 1996). Dubinsky (1991) described the process of generating a preliminary genetic decomposition of a concept as a researcher’s description “based on empirical data, of the mathematics involved and how a subject might make the
Understandings, meanings, and reorganizations
Thompson and Harel (Harel, 2013; Thompson, 2016) have described a system of knowing built on Piagetian notions. Thompson (2016) described an understanding as a cognitive state resulting from assimilation, and a meaning as the space of implications available at the moment of understanding. This space of implications can consist of a collection of actions, objects, or schemes the person brings to mind when an understanding is achieved (Thompson, Carlson, Byerley, & Hatfield, 2014).
Thompson and
Conceptual analysis and covariational inverse relations and functions
Thompson (2008) offered one use of conceptual analysis is to construct “ways of understanding an idea that, if students had them, might be propitious for building more powerful ways to deal mathematically with their environments than they would build otherwise” (p. 45). Although this resembles the process of constructing a preliminary genetic decomposition, there are important differences. Thompson (2002) stated:
Conceptual analyses are given in terms grounded in conceptual experience—to make it
Research questions, subjects, and methods
In this report, I characterize one pre-service teacher’s (hereafter student’s) meanings as she engaged in an instructional approach I conjectured had the potential to support her in reorganizing her limited inverse function meanings into productive meanings. I address the research questions:
- 1
How can addressing activities intended to emphasize reasoning about relationships quantitatively, covariationally, and bidirectionally support a student in reorganizing her inverse function meanings in
Results
I structure the results section chronologically to provide insights into and characterize shifts in Arya’s meanings. Specifically, I present the following:
- 1
Arya’s activity in the pre-interview providing insights into her inverse function meanings and related ideas (i.e., graphing conventions).
- 2
Arya’s activity in the Sine sessions providing insights into her developing sine relation meanings.
- 3
Arya’s activity in the Inverse sessions in which she experienced perturbations that promoted shifts in her
Discussion
In this study, I examined the potential of supporting a pre-service teacher in reorganizing her limited inverse function meanings into productive meanings via reasoning quantitatively, covariationally, and bidirectionally. As such, the findings of this study contribute to several areas of related research including the research on students’ inverse function meanings, on students’ quantitative and covariational reasoning, and students’ meaning of graphs and related ideas. In the sections that
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant No. DRL-1350342. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or our respective universities. This article is based on dissertation research directed by Kevin C. Moore at the University of Georgia. I thank Dr. Kevin C. Moore, Dr. Leslie Steffe, Dr. Sybilla Beckmann,
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