## Abstract

A great deal of progress has been made in dealing with the multiplicity and diversity of theories in mathematics education. However, relatively little attention has been paid to the opportunities offered by conflicts, tensions, and paradoxes among accepted yet opposing theoretical perspectives for theory building and theory advancement. In this paper, four modes of dealing with opposing perspectives are outlined: (1) taking contrasting theoretical perspectives as incommensurable; (2) holding opposites not as conflicting but as complementary; (3) dissolving or surpassing oppositions by blending perspectives; and (4) preserving paradoxes by recognizing the interdependence of constitutive oppositions. These four modes are illustrated by application to the long-standing debate of knowledge-in-structures versus knowledge-in-pieces and further exemplified by turning to the research literature on students’ understanding of limit.

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## Notes

- 1.
Prediger et al. (2008) organized various networking strategies on a scale according to their degree of integration, with “ignoring other theories” and “unifying theories globally” as poles of the scale. Networking strategies are located between these two extreme positions, including “understanding others” and “making understandable,” “comparing and contrasting,” “combing and coordinating,” as well as “synthesizing” and “integrating locally” (for a detailed explanation of the strategies; see Prediger et al. 2008).

- 2.
The term

*blending*has its origin in the work of Fauconnier and Turner (2002) on “conceptual blending,” who built a framework of blending two knowledge domains from which novel elements result that are not evident in either domain on its own. - 3.
The both-and relationship in the interplay position differs from that of the complementarity position as oppositions are not viewed as independent, operating on different levels; instead, oppositions are seen as dynamically interacting and interdependent, and united—principles that are more aligned with Eastern philosophies than Western philosophies.

- 4.
The notion of “phenomenological primitives” means to imply that these knowledge pieces are usually evident in our everyday experience (thus the notion of “phenomenological”) and that individuals cannot, in general, analyze or justify them—partly because they are not encoded in language (thus the notion of “primitive”).

- 5.
In the interest of the purposes of this paper and due to space restrictions, the focus is on a selection of key references regarding research on students’ understandings of limit. It is not the objective to offer a comprehensive overview of the existing research literature, nor to give justice to the vast and diverse lines of research in this area. The examples given here focus on research that has been framed in terms of Tall and Vinner’s (1981) notion of concept image; however, similar arguments could be made for other lines of research, such as that on epistemological obstacles (see Cornu, 1991; Sierpinska, 1987).

- 6.
The left and right circles in Fig. 1 represent the two perspectives—the knowledge-in-structures perspective on the left and the knowledge-in-pieces perspective on the right—that serve as input spaces for blending. The top circle in Fig. 1 represents the generic space containing fundamental characteristics common to the two perspectives (i.e., the common understanding that a knowledge system is complex and evolving). The bottom circle represents the blended space with different and novel characteristics emerging from the two input spaces (e.g., the construal of a knowledge system as dynamically forming). The solid lines designate the cross-space mapping between the input spaces, and the dashed lines designate links between input spaces and either generic or blended spaces. The words highlighted in italics in the blended space designate the new characteristics (i.e., a knowledge system being dynamic and emerging) along with selected characteristics of each perspective (i.e., a knowledge system being complex, evolving, robust, and fragmented). For a detailed elaboration of the process of blending; see Fauconnier and Turner (2002).

- 7.
For instance, the experiencing of contrary, yet mutually dependent, forces existing as opposites in unity—in short, the art of balancing opposites—is well embedded in the ancient Chinese philosophical principle “Yin-Yang” (see Wang, 2012).

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## Acknowledgments

I want to express my gratitude to Andy diSessa for inducing me into the present undertaking through his writings and conversations and for his critical comments and helpful suggestions on an earlier version of this paper.

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Scheiner, T. Dealing with opposing theoretical perspectives: knowledge in structures or knowledge in pieces?.
*Educ Stud Math* **104, **127–145 (2020). https://doi.org/10.1007/s10649-020-09950-7

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### Keywords

- Conceptual change
- Limit concept
- Networking of theories
- Opposition
- Paradox
- Theory building