## 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.

### Similar content being viewed by others

## References

Bain, K., Rodriguez, J. G., Moon, A., & Towns, M. H. (2018). The characterization of cognitive processes involved in chemical kinetics using a blended processing framework.

*Chemistry Education Research and Practice, 19*, 617–628.Becker, N., & Towns, M. (2012). Students’ understanding of mathematical expressions in physical chemistry contexts: An analysis using Sherin’s symbolic forms.

*Chemistry Education Research and Practice, 13*, 209–220.Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study.

*Journal of Research in Mathematics Education, 33*(5), 252–378.Carpenter, P. A., & Shah, P. (1998). A model of the perceptual and conceptual processes in graph comprehension.

*Journal of Experimental Psychology: Applied, 4*(2), 75–100.Chabay, R., & Sherwood, B. A. (2015). The momentum principle. In R. Chabay & B. A. Sherwood (Eds.),

*Matter & interactions: Modern mechanics*(4th ed., pp. 45–87). New York, NY: Wiley.Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices.

*Cognitive Science, 5*(2), 121–152.Chi, M. T. H., Glaser, R., & Rees, E. (1982). Expertise in problem solving. In R. J. Sternberg (Ed.),

*Advances in the Psychology of Human Intelligence*(pp. 7–75). Hillsdale, NJ: Erlbaum.Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions.

*Journal for Research in Mathematics Education, 26*(1), 66–86.Cooper, M. M. (2013). Chemistry and the next generation science standards.

*Journal of Chemical Education, 90*(6), 679–680.diSessa, A. A. (1993). Toward an epistemology of physics.

*Cognition and Instruction, 10*(2–3), 105–225.Dorko, A., & Speer, N. (2015). Calculus students’ understanding of area and volume units.

*Investigations in Mathematics Learning, 8*(1), 23–46.Dreyfus, B. W., Elby, A., Gupta, A., & Sohr, E. R. (2017). Mathematical sense-making in quantum mechanics: An initial peek.

*Physical Review Physics Education Research, 13*, 020141.Driver, R., Asoko, H., Leach, J., Mortimer, E., & Scott, P. (1994). Constructing scientific knowledge in the classroom.

*Educational Researcher, 23*(7), 5–12.Driver, R., Leach, J., Millar, R., & Scott, P. (1996). Why does understanding the nature of science matter? In R. Driver, J. Leach, R. Millar, & P. Scott (Eds.),

*Young people’s images of science*(pp. 8-23). Philadelphia, PA: Open University Press.Ellis, A., Ozgur, Z., Kulow, T., Dogan, M., & Amidon, J. (2016). An exponential growth learning trajectory: Students’ emerging understanding of exponential growth through covariation.

*Mathematical Thinking and Learning, 18*(3), 151–181.Even, R. (1990). Subject matter knowledge for teaching and the case of functions.

*Educational Studies in Mathematics, 21*, 521–544.Glazer, N. (2011). Challenges with graph interpretation: A review of the literature.

*Studies in Science Education, 47*(2), 183–210.Grassian, V. H., Meyer, G., Abruña, H., Coates, G. W., Achenie, L. E., Allison, T., . . . Wood-Black, F. (2007). Viewpoint: Chemistry for a sustainable future.

*Environmental Science & Technology, 41*(14), 4840–4846.Habre, S. (2012). Students’ challenges with polar functions: Covariational reasoning and plotting in the polar coordinate system.

*International Journal of Mathematical Education in Science and Technology, 48*(1), 48–66.Hammer, D., & Elby, A. (2002). On the form of a personal epistemology. In B. K. Hofer & P. R. Pintrich (Eds.),

*Personal epistemology: The psychology of beliefs about knowledge and knowing*(pp. 169–190). Mahwah, NJ: Erlbaum.Hammer, D., & Elby, A. (2003). Tapping epistemological resources for learning physics.

*Journal of the Learning Sciences, 12*(1), 53–90.Hammer, D., Elby, A., Scherr, R. E., & Redish, E. F. (2005). Resources, framing, and transfer. In J. P. Mestre (Ed.),

*Transfer of learning from a modern multidisciplinary perspective*(pp. 89–119). Greenwich, CT: IAP.Hu, D., & Rubello, N. (2013). Using conceptual blending to describe how students use mathematical integrals in physics.

*Physical Review Special Topics - Physics Education Research, 9*(2), 1–15.Ivanjeck, L., Susac, A., Planinic, M., Andrasevic, A., & Milin-Sipus, Z. (2016). Student reasoning about graphs in different contexts.

*Physical Review Physics Education Research, 12*(1), 010106.Izak, A. (2000). Inscribing the Winch: Mechanisms by which students develop knowledge structures for representing the physical world with algebra.

*Journal of the Learning Sciences, 9*(1), 31–74.Izak, A. (2004). Students’ coordination of knowledge when learning to model physical situations.

*Cognition and Instruction, 22*(1), 81–128.Jones, S. (2013). Understanding the integral: Students’ symbolic forms.

*The Journal of Mathematical Behavior, 32*(2), 122–141.Jones, S. (2015a). The prevalence of area-under-a-curve and anti-derivative conceptions over Riemann sum-based conceptions in students’ explanations of definite integrals.

*International Journal of Mathematics Education in Science and Technology, 46*(5), 721–736.Jones, S. (2015b). Areas, anti-derivatives, and adding up pieces: Definite integrals in pure mathematics and applied science contexts.

*The Journal of Mathematical Behavior, 38*, 9–28.Jones, S. (2017). An exploratory study on student understanding of derivatives in real-world, non-kinematics contexts.

*The Journal of Mathematical Behavior, 45*, 95–110.Kozma, R. B., & Russell, J. (1997). Multimedia and understanding: Expert and novice responses to different representations of chemical phenomena.

*Journal of Research in Science Teaching, 24*(9), 949–968.Kuo, E., Hull, M., Gupta, A., & Elby, A. (2013). How students blend conceptual and formal mathematical reasoning in solving physics problems.

*Science Education, 97*(1), 32–57.Lee, V. R., & Sherin, B. (2006). Beyond transparency: How students make representations meaningful. In S. Barab, K. Hay, & D. Hickey (Eds.),

*Proceedings of the 7th International Conference on Learning Sciences*(pp. 397-403). Bloomington, IN: International Society of the Learning Sciences.Lunsford, E., Melear, C. T., Roth, W.-M., Perkins, M., & Hickok, L. G. (2007). Proliferation of inscriptions and transformations among preservice science teachers engaged in authentic science.

*Journal of Research in Science Teaching, 44*(4), 538-564.Mahaffy, P. G., Holme, T. A., Martin-Visscher, L., Martin, B. E., Versprille, A., Kirchhoff, M., . . . Towns, M. (2017). Beyond “inert” ideas to teaching general chemistry from rich contexts: Visualizing the chemistry of climate change (VC3).

*Journal of Chemical Education, 94*(8), 1027–1035.Matlin, S. A., Mehta, G., Hopf, H., & Krief, A. (2016). One-world chemistry and systems thinking.

*Nature Chemistry, 8*(5), 393–398.Moore, K. C., & Thompson, P. W. (2015). Shape thinking and students’ graphing activity. In T. Fukawa-Connelly, N. Infante, K. Keene, & M. Zandieh (Eds.),

*Proceedings of the Eighteenth Annual Conference on Research in Undergraduate Mathematics Education*(pp. 782–789). Pittsburgh, PA.Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fenemma, & T. P. Carpenter (Eds.),

*Integrating research on the graphical representation of functions*(pp. 69–100). New York: Erlbaum.National Research Council. (2012).

*A framework for K-12 science education: Practices, crosscutting concepts, and core ideas*(p. 2012). Washington, DC: The National Academies of Press.Nemirovsky, R. (1996). Mathematical Narratives, Modeling, and Algebra. In N. Bednarz, C. Kiernan, & L. Lee (Eds.),

*Approaches to algebra: Perspectives for research and teaching*(pp. 197–223). Dordrecht, The Netherlands: Kluwer Academic Publishers.Orton, A. (1983). Students’ understanding of differentiation.

*Educational Studies in Mathematics, 14*(3), 235–250.Phage, I. B., Lemmer, M., & Hitage, M. (2017). Probing factors influencing students’ graph comprehension regarding four operations in kinematics graphs.

*African Journal of Research in Mathematics, Science, and Technology Education, 21*(2), 200–210.Planinic, M., Ivanjeck, L., Susac, A., & Millin-Sipus, Z. (2013). Comparison of university students’ understanding of graphs in different contexts.

*Physical Review Special Topics - Physics Education Research, 9*, 020103.Potgieter, M., Harding, A., & Engelbrecht, J. (2007). Transfer of algebraic and graphical thinking between mathematics and chemistry.

*Journal of Research in Science Teaching, 45*(2), 297–218.Rasmussen, C., Marrongelle, K., & Borba, M. C. (2014). Research on calculus: What do we know and where do we need to go?

*ZDM Mathematics Education, 46*, 507–515.Reed, J. J., & Holme, T. A. (2014). The role of non-content goals in the assessment of chemistry learning. In L. K. Kendhammer & K. L. Murphy (Eds.),

*Innovative uses of assessment for teaching and research*(pp. 147–160). Washington, DC: American Chemical Society.Rodriguez, J. G., Bain, K., Towns, M. H., Elmgren, M., & Ho, F. M. (2019). Covariational reasoning and mathematical narratives: Investigating students’ understanding of graphs in chemical kinetics.

*Chemistry Education Research and Practice, 20*, 107–119.Rodriguez, J. G., Santos-Diaz, S., Bain, K., & Towns, M. H. (2018). Using symbolic and graphical forms to analyze students’ mathematical reasoning in chemical kinetics.

*Journal of Chemical Education, 95*, 2114–2125.Roschelle, J. (1991).

*Students’ construction of qualitative physics knowledge: Learning about velocity and acceleration in a computer microworld*. University of California, Berkley: Unpublished doctoral dissertation.Rodriguez, J.-M. G., Bain, K., & Towns, M. H. (2019). Graphs as objects: Mathematical resources used by undergraduate biochemistry students to reason about enzyme kinetics. In M. H. Towns, K. Bain, & J.-M. G. Rodriguez (Eds.),

*It’s just math: Research on students’ understanding of chemistry and mathematics*(pp. 69–80). Washington, DC: American Chemical Society. https://doi.org/10.1021/bk-2019-1316.ch005Saldanha L. & Thompson P. (1998). Re-thinking covariation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berenson & W. N. Coulombe (Eds.),

*Proceedings of the Annual Meeting of the Psychology of Mathematics Education – North America*(Vol. 1, pp. 298-304). Raleigh, NC: North Carolina University.Schermerhorn, B., & Thompson, J. (2016).

*Students’ use of symbolic forms when constructing differential length elements*. Paper presented at the Physics Education Research Conference, Sacramento, CA.Schwartz, J., & Yerushalmy, M. (1992). Getting students to function in and with algebra. In G. Harel & E. Dubinsky (Eds.),

*The concept of function: Aspects of epistemology and pedagogy (MAA Notes)*(Vol. 25, pp. 261–289). Washington, DC: Mathematical Association of America.Sengupta, P., & Wilensky, U. (2009). Lowering the learning threshold: Multi-agent-based models and learning electricity. In M. Khine & I. Saleh (Eds.),

*Models and Modeling*. Netherlands: Springer.Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification – The case function. In G. Harel & E. Dubinsky (Eds.),

*The concept of function: Aspects of epistemology and pedagogy (MAA Notes)*(Vol. 25, pp. 59–84). Washington, DC: Mathematical Association of America.Sherin, B. L. (2001). How students understand physics equations.

*Cognition and Instruction, 19*(4), 479–541.Silberberg, M., & Amateis, P. (2018). Kinetics: Rates and mechanisms of chemical reactions. In M. Silberberg & P. Amateis (Eds.),

*Chemistry: The molecular nature of matter and change*(8th ed., pp. 690–731). New York, NY: McGraw-Hill Education.Stewart, J. (2016). Functions and models. In J. Stewart (Eds.),

*Calculus: Early transcendentals*(8th ed., pp. 9–76). Australia: Cengage Learning.Thompson, P. (1994). Images of rate and operational understanding of the fundamental theorem of calculus.

*Educational Studies in Mathematics, 26*(2-3), 229–274.Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.),

*Compendium for research in mathematics education*(pp. 421–456). Reston, VA: National Council of Teachers of Mathematics.Underwood, S., Posey, L., Herrington, D., Carmel, J., & Cooper, M. Adapting assessment tasks to support three-dimensional learning.

*Journal of Chemical Education, 95*, 2017–2217.Urry, L. A., Cain, M. L., Wasserman, S. A., Minorsky, P. V., & Reece, J. B. (2017). Population ecology and the distribution of organisms. In

*Campbell biology in focus*(8th ed., pp. 840–866). Pearson Education, Ltd.Von Korff, J., & Rubello, N. (2014). Distinguishing between “change” and “amount” infinitesimals in first-semester calculus-based physics.

*American Journal of Physics, 82*, 695–705.White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus.

*Journal for Research in Mathematics Education, 27*(1), 79–95.

## 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.

## Author information

### Authors and Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

The authors declare that they have no conflict of interest.

## Electronic supplementary material

### ESM 1

(DOCX 30 kb)

## Rights and permissions

## About this article

### Cite this article

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

Received:

Accepted:

Published:

Issue Date:

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