Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

https://doi.org/10.1016/j.jmathb.2019.100741Get rights and content

Highlights

  • Presentation of ways students can develop productive meanings for inverse relations and functions.

  • Arya, a pre-service teacher, began the study with limited inverse function meanings consistent with pre-service teachers reported on elsewhere.

  • Arya experienced several perturbations to her meanings for inverse functions, graphs, and variables throughout the study.

  • Arya reorganized her limited inverse function meanings, as well as her meanings for other related ideas, into productive meanings.

Abstract

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.

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,

References (73)

  • C. Brown et al.

    Delineating four conceptions of function: A case of composition and inverse

  • M.P. Carlson

    A cross-sectional investigation of the development of the function concept

  • M.P. Carlson et al.

    Applying covariational reasoning while modeling dynamic events: A framework and a study

    Journal for Research in Mathematics Education

    (2002)
  • M.P. Carlson et al.

    A study of students’ readiness to learn Calculus

    International Journal of Research in Undergraduate Mathematics Education

    (2015)
  • J. Clement

    Analysis of clinical interviews: Foundations and model viability

  • J. Confrey et al.

    Exponential functions, rates of change, and the multiplicative unit

    Educational Studies in Mathematics

    (1994)
  • T.J. Cooney et al.

    Conceptualizing belief structures of preservice secondary mathematics teachers

    Journal for Research in Mathematics Education

    (1998)
  • J. Dewey

    How we think

    (1910)
  • DiNapoli

    Supporting secondary students’ perseverance for solving challenging mathematical tasks

  • E. Dubinsky

    Reflective abstraction in advanced mathematical thinking

  • The nature of the process conception of function

  • A.B. Ellis et al.

    Quantifying exponential growth: The case of the jactus

  • N. Engelke et al.

    Composition of functions: Precalculus students’ understandings

  • R. Even

    The inverse function: Prospective teachers use of’ undoing’

    International Journal of Mathematical Education in Science and Technology

    (1992)
  • R. Even et al.

    Subject-matter knowledge and knowledge about students as sources of teacher presentations

    Educational Studies in Mathematics

    (1995)
  • B. Fowler

    An investigation of the teaching and learning of function inverse

    (2014)
  • G.A. Goldin

    A scientific perspective on structured, task-based interviews in mathematics education research

  • G. Harel

    Intellectual need

  • G. Harel et al.

    Advanced mathematical-thinking at any age: Its nature and its development

    Mathematical Thinking and Learning

    (2005)
  • G. Leinhardt et al.

    Functions, graphs, and graphing: Tasks, learning, and teaching

    Review of Educational Research

    (1990)
  • C.A. Lucus

    Composition of functions and inverse function of a function: Main ideas as perceived by teachers and preservice teachers

    (2005)
  • O. Marmur et al.

    Space of fuzziness: Avoidance of deterministic decisions in the case of the inverse function

    Educational Studies in Mathematics

    (2018)
  • S.B. Merriam et al.

    Qualitative research: A guide to design and implementation

    (2005)
  • K.C. Moore

    Making sense by measuring arcs: A teaching experiment in angle measure

    Educational Studies in Mathematics

    (2013)
  • K.C. Moore

    Quantitative reasoning and the sine function: The case of Zac

    Journal for Research in Mathematics Education

    (2014)
  • K.C. Moore et al.

    Bidirectionality and covariational reasoning

  • Cited by (0)

    View full text