Knurek, R. & Johnson, H. L. (2022). Linear or nonlinear? Relating college algebra students’ covariational reasoning and graph selection. In Lischka, A. E., Dyer, E. B., Jones, R. S., Lovett, J. N., Strayer, J., & Drown, S. (Eds.), Proceedings of the 44th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (p. 863-864).Middle Tennessee State University.


We investigate the following problem: How does students’ covariational reasoning relate to their graph selection on a fully online assessment. Specifically, we’re going to focus on students’ distinction between linear and nonlinear graphs that represent the same direction of change in variables.

To theorize covariational reasoning, we draw on the work of Thompson and Carlson and colleagues. In particular, we are looking at the levels of covariational reasoning focusing on the gross coordination of values. This refers to a loose connection between the direction of change in attributes. For example, a student reasoning at the level of gross covariation might say something like, “as the height increases, the diameter decreases”. We situation this multiple case study in a larger interview-based validation study of a fully online covariation assessment that’s part of a larger NSF-funded project. The assessment contains six items.

For each item, students start by watching a video animation. For example, this assessment item shows a cart on a Ferris wheel that’s moving counter-clockwise. Students select one of four graphs that represents a relationship between variables, in this case, distance traveled and height from the ground. Lastly, they explain why they selected a particular graph.

Here, we’re going to talk about two students, Maya and Emma. We picked both of them because they both spontaneously wondered whether or not graphs would be linear or nonlinear.

What we learned is that when students engage in gross variation or covariational reasoning, it allows them to narrow down these graph choices to two viable choices, one linear and one nonlinear, that both represent the gross change in variables. For these two students, when they narrow down their selection, both graphs seem to be good enough. To distinguish between the two graphs, each of the students went to motion or iconic interpretations of the situation. For example, Maya draws on gross covariation to narrow down the selection to these two graphs, then goes to an iconic interpretation to select an answer, saying “we’re dealing with a circular kind of model”.

What we’re finding is if students already have an expectation that a graph should be linear or nonlinear, then the gross covariation turns out to be sufficient enough to select a single graph, even if it’s one that’ rather unconventional. When students are unsure, these students rely on motion or iconic interpretations to select a single graph.

So what do we take from this? We think about the kinds of learning of learning experiences students should be having in schools if we want them to be able to advance in their covariational reasoning. It seems like they should have opportunities to ask questions related to linear or nonlinear graphs.


Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352-378.

Johnson, H. L., McClintock, E. D., & Gardner, A. (2020). Opportunities for Reasoning: Digital Task Design to Promote Students’ Conceptions of Graphs as Representing Relationships between Quantities. Digital Experiences in Mathematics Education, 6(3), 340-366.

Stevens, I. E., Hobson, N. L. F., Moore, K. C., Paoletti, T., LaForest, K. R., & Mauldin, K. D. (2015). Changing cones: Themes in students’ representations of a dynamic situation. In Bartell, T. G., Bieda, K. N., Putnam, R. T., Bradfield, K., Dominguez, H. (Ed.), Proceedings of the 37th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 363–370). East Lansing, MI. Michigan State University.

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.

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