## Abstract

In this paper, we introduce and discuss a construct called *graphical forms*, an extension of Sherin’s symbolic forms. In its original conceptualization, symbolic forms characterize the ideas students associate with patterns in a mathematical expression. To expand symbolic forms beyond only characterizing mathematical equations, we use the general term *registration* to describe structural features attended to by individuals (parts of an equation or regions in a graph). When mathematical ideas are assigned to registrations in a graph, we characterize this as reasoning using graphical forms. As an analytic framework, graphical forms provide the language to discuss intuitive mathematical ideas associated with features in a graph, but we are also interested in engagement in modeling. Our approach to investigating graphical reasoning involves conceptualizing modeling as discussing mathematical narratives. This affords the language to describe reasoning about the process (or “story”) that could give rise to a graph; in practice, this occurs when mathematical reasoning (i.e. reasoning using graphical forms) is integrated with context-specific ideas. In this work we describe graphical forms as an extension of symbolic forms and emphasize its utility for analyzing graphical reasoning. In order to illustrate how the framework could be applied, we provide examples of interpretations of graphs across disciplines, using graphs selected from introductory biology, calculus, chemistry, and physics textbooks.

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## Acknowledgments

We wish to thank the individuals in our research group for their support and helpful comments on the manuscript.

## Funding

This work was supported by the National Science Foundation under Grant DUE-1504371. Any opinions, conclusions, or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Rodriguez, JM.G., Bain, K. & Towns, M.H. Graphical Forms: The Adaptation of Sherin’s Symbolic Forms for the Analysis of Graphical Reasoning Across Disciplines.
*Int J of Sci and Math Educ* **18**, 1547–1563 (2020). https://doi.org/10.1007/s10763-019-10025-0

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DOI: https://doi.org/10.1007/s10763-019-10025-0

### Keywords

- Framework
- Inter-disciplinary
- Mathematical/graphical reasoning
- Modeling