Have you ever held a class discussion you thought went well until your students made a claim that had you question how they were interpreting the mathematics at hand? When such a situation happened in our classroom, we used the interaction to change our teaching and research to address preemptively the particular inconsistencies we had noticed. The classroom situation occurred during an activity based on the Power Tower (fig. 1a), an amusement ride at Cedar Point in Sandusky, Ohio, that shoots riders up the tower, then lets them fall toward the ground before shooting them up again, repeating this process two more times. After showing a video of the ride (see http://www.youtube.com/watch?v=HrGpGMrkUrM), we asked our students to sketch a graph of a rider’s total distance traveled and the vertical distance from the ground, with the intention of having them investigate changes in the vertical distance from the ground along with changes in the total distance traveled (see Moore et al.  for more on this task). The lead instructor presented two student-generated graphs (re-created in fig. 1b and c), and the class discussed how each graph accurately represented the quantities changing together. When the instructor concluded that the two graphs were “the same,” however, several students objected, saying that one “graph is a function” and the other graph “is not.” Although these students understood that the two graphs accurately represented some aspects of how the quantities covaried, the fact that one graph passed the vertical line test and the other did not meant to them that the graphs represented different relationships. This interaction piqued our interest. We conjectured that in their previous school experiences, our students lacked opportunities to distinguish which features are mathematically critical to representing relationships (and functions) and which are merely a choice or convention (e.g., orientation of axes). Hence, we began to investigate the extent to which our students’ understandings of mathematical concepts either were rooted in reasoning about relationships between quantities or relied on what we perceived to be curricular conventions. In this article, we present students’ ways of thinking about rate of change and functions that are grounded in quantitative and covariational reasoning. Many researchers (e.g., Carlson et al. 2002; Ellis et al. 2012; Johnson 2013; Thompson et al. 1994; Thompson 2011, 2013) have argued that such reasoning supports students’ engagement in several of the Process Standards described in Principles and Standards for School Mathematics (NCTM 2000) and Standards for Mathematical Practice described in the Common Core State Standards (CCSSI 2010). We conclude with suggestions that teachers can use to help their students develop productive ways of thinking to differentiate aspects that are critical to a concept from those only intended to be curricular conventions. RATE OF CHANGE Rate of change is an idea that unifies middle school through postsecondary mathematics courses, including algebra, calculus, and differential equations. Understanding that two quantities A and B covary at a constant rate of change involves understanding that if quantity A’s value changes by any but no particular amount a, quantity B’s value changes by an amount equal to some constant, m, times a (i.e., m • a). Further, if quantity B’s value changes by any but no particular amount b, then quantity A’s value changes by an amount equal to the constant 1/m times b (i.e., (1/m) • b). A student with such an understanding can interpret a linear relationship displayed in the Cartesian coordinate system by imagining variations in the quantity on either the horizontal (fig. 2a) or the vertical (fig. 2b) axis first and then imagining how the other quantity changes in relation to those variations; the student understands that the constant rate of change associated with a linear relationship can be thought of as either the measure m or the measure 1/m, depending on which quantity he or she chooses first. Such an understanding of rate of change can be generalized to any representational system (e.g., polar graphs as in fig. 2c and d, tables, or equations). To illustrate the flexibility and productivity of this way of thinking, consider the following illustration. We asked Student 1 to determine the rate of change of the relationship represented in the graph shown in figure 3. After she claimed that the rate of change of the relationship is 4, we posed that a hypothetical student identified a rate of change of one-fourth; this allowed us to investigate if Student 1 would consider an unconventional interpretation of the rate of change, specifically the rate of change of x with respect to y. This query is one example of a type of question we often pose: the hypothetical student response is mathematically viable but breaks from curricular conventions. Student 1 understood that the rate of change of a relationship could be conveyed in more than one way depending on which quantity she chose to coordinate first. She could interpret the relationship as having a rate of change of 4 if she coordinated x first (y varies by an amount 4 times as large as the amount x varies) or 1/4 if she coordinated y first (x varies by an amount 1/4 times as large as the amount y varies). As a point of contrast, we present Student 2’s response when we provided the labeled graph shown in figure 4a. We stated that a hypothetical student, Xavier, constructed the graph when asked to graph the equation y = 3x,and we asked Student 2 to evaluate Xavier’s graph. We wanted to know if Student 2 would interpret the graph as accurately representing the relationship defined by y = 3x.Whereas Student 1 thought about rate of change in terms of coordinating quantities’ values, Student 2’s understanding of rate of change was dominated by perceptual features of the graph (see fig. 4). Student 2 rotated the graph to obtain x on the horizontal axis (fig. 4b) and associated the rotated graph with a “negative slope.” To Student 2, slope (or rate of change) was indicated by the perceived tilt of a line, rather than a relationship between the values of two quantities on the graph. Such tilt-slope associations enable a student to respond correctly when common curricular conventions of graphing are maintained, but the associations are not productive in unconventional situations, such as describing the rate of change of the relationship represented in figure 4b. FUNCTIONA function is a particular type of relation in which each element of a defined set, called an input, is mapped to a unique element of a second defined set, called an output. “Function” is not an inherent property of a relationship between two sets, but instead depends on how one decides to define a relation between those sets. To illustrate a productive understanding of functions, consider Student 3, who was discussing the graph in figure 1b. (Note: [. . .] indicates a break in the transcript.) Student 3: Well, it depends which axis is your input and which axis is your output. [. . .] If your inputs were actually your vertical axis, the outputs were your horizontal axis, then that would be a function ’cause you would have a [. . .] unique output for every input. [. . .] [Student 3 continued by discussing how figure 1b may not represent a function] if the x-axis is your input and y-axis is your output. In contrast to the students who argued that the graphs in figure 1b and c were different because one “graph is a function” and the other graph “is not,” Student 3 implied that it did not make sense to claim the graph “is” or “is not” a function. Rather, Student 3 understood that discussing whether the graphed relationship represents a function requires first defining input and output quantities and then determining uniqueness (e.g., total distance is not a function of vertical distance, but vertical distance is a function of total distance). As did the students in the opening vignette, Student 4 initially claimed that the graph in figure 5 (a graph similar to fig. 1b but without explicitly defined quantitative referents on each axis) did not represent a function. We then posed that a hypothetical student, Yolanda, argued that the graph represents “x is a function of y” and asked Student 4 to respond to Yolanda’s claim. Student 4: That x is a function of y? I mean x and y definitely correlate, there’s definitely a relationship between them. But, I’ve always been taught a function, it has an x-value it can only correspond to one y-value. The same x-value cannot have a different y-value and still be a function. In contrast to Student 3, Student 4’s understanding entailed maintaining the input quantity of a function along the x-axis; representing the input quantity of a function along the x-axis was not merely a convention but inherent to Student 4’s way of thinking. Thus, Student 4 did not interpret Yolanda’s statement that the graph represents “x is a function of y” as different than the statement “y is a function of x.” STUDENTS’ UNDERSTANDINGS A notable difference exists between the students’ responses above with respect to what we perceive to be curricular conventions. Students 1 and 3 held ways of thinking that entailed understanding particular conventions of representing certain ideas as exactly that, conventions: standard ways of doing things chosen from equally viable ways of doing things. For example, Student 1 understood that the rate of change of a relationship could be measured in multiple ways. Although she usually considers the rate of change of y with respect to x, it was equally viable to describe the rate of change of x with respect to y. In contrast, what we perceive to be conventions were instead inherent to how Students 2 and 4 thought about rate of change and function. We are not surprised by Student 2’s and Student 4’s understandings, because these understandings likely worked in addressing questions posed by textbooks and teachers throughout their K–grade 14 schooling experiences. When we as educators (implicitly or explicitly) impose and maintain conventions, as opposed to having conventions emerge from student activity and need, students can construct mathematical understandings dependent upon those conventions. In turn, students’ understandings are restricted to specific representations and constraints, often obscuring features critical to a mathematical concept (e.g., rate of change as a coordination of covarying quantities). For example, associations between rate of change and the direction of a line are helpful only as long as graphs are represented in the Cartesian coordinate system with conventionally defined axes. Similarly, the vertical line test works if the input of a function is represented on the horizontal axis. However, these ways of thinking become problematic when conventions are not maintained (e.g., as in fig. 4b) or when students are asked to work in a different coordinate system (e.g., polar coordinates). When encountering graphs that depart from these conventions, students who hold such ways of thinking are not prepared to notice or interpret these departures as viable representations of an idea. STUDENTS ADDRESS THEIR OWN ASSUMPTIONS We provide sample questions (fig. 6) to focus classroom discussions. Rather than merely asking students to identify perceptual changes in the graphs, we encourage students to address their own assumptions about their graphical representations and to consider other students’ ways of representing the same situation. Tasks involving a collection of relationships between covarying quantities in various graphical orientations have helped students distinguish between what is mathematically critical versus what is merely a convention for representation. The key to orchestrating a discussion around such tasks is to focus on the mathematical ideas underlying the different relationships and representations as well as to discuss what aspects of representational options are not critical. We present the Deep Sea Diver problem (fig. 7) as another example of such a task and describe some common student solutions (fig. 8). In the Deep Sea Diver problem, we allow students to choose their own axes orientation and labeling, which often leads to four or more different, yet mathematically correct, ways to represent the relationship (see fig. 8 for sample graphs). Some students represent the diver’s distance above sea level on the horizontal axis (fig. 8a,b), whereas other students represent this quantity’s values along the vertical axis (fig. 8c,d). Because the diver’s height above sea level has a negative value throughout most of the situation, some students represent negative values in non-canonical orientations along the axes (fig. 8b,d), whereas other students maintain the canonical orientations of directed values (fig. 8a,c). Students might also change the point of intersection of the referent axes from the canonical (0, 0) (fig. 8a,c), since the diver starts by experiencing 14.5 psi 5 ft. above sea level (fig. 8b,d). After students construct their graphs, we highlight the work of students who used axes with different orientations and labels. As our students consider their classmates’ representations, we give them opportunities to discuss similarities and differences among their graphs; we provide hypothetical student solutions if additional variety is needed. We make sure to raise questions such as these: • Are the different graphs representing the same change in height for a certain change in pressure? How can you tell? • Does it matter in which direction the negative values are represented? • Does it matter if the intersection of the horizontal and vertical axis is at (0, 0) or at (14.5, 5)? We also prompt the students to discuss whether the graphs represent functions, leading to conversations regarding the necessity of defining input and output quantities. The conversations that ensue allow students to discuss whether each graph accurately represents the intended relationship and if differences among the representations affect the “correctness” of each graph. These discussions afford students the opportunity to negotiate and establish representational conventions. After such conversations, students can conceive of conventions as useful tools for communicative purposes, often easing the burden of attending to different considerations involved when creating, interpreting, or comparing graphs. CONSEQUENTIAL DISTINCTIONSProviding our students with repeated opportunities to work in what we perceive to be conventional and unconventional representations—often through hypothetical student work that could be interpreted as creative sense-making—has supported their using notation and representations purposefully and meaningfully. Such situations encourage our students to establish and differentiate aspects specific to representational conventions from those aspects that are critical to mathematical concepts. Moreover, we have found that our students can then critically reflect on their own ways of thinking and establish representational conventions within their class. Teachers who determine that their students are not distinguishing between what is essential to a concept and what is to be considered a representational convention can give the students a chance to develop richer ways of thinking for these concepts by engaging them in a multitude of representations with attention to differences and similarities in those representations. ACKNOWLEDGMENTSThis material is based on work supported by the NSF under Grant No. DRL-1350342. Any opinions, findings, and conclusions or recommendations expressed are those of the authors. We thank Jason Silverman, Stacy Musgrave, and David Liss for their contributions to our work. We thank Pat Thompson, Department of Mathematics and Statistics, Arizona State University, for providing the idea behind the Power Tower task and influencing our work. REFERENCESCarlson, Marilyn P., Sally Jacobs, Edward Coe, Sean Larsen, and Eric Hsu. 2002. “Applying Covariational Reasoning while Modeling Dynamic Events: A Framework and a Study.” Journal for Research in Mathematics Education 33 (November): 352–78. Common Core State Standards Initiative (CCSSI). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf Ellis, Amy B., Zekiye Ozgur, Torrey Kulow, Caroline Williams, and Joel Amidon. 2012. “Quantifying Exponential Growth: The Case of the Jactus.” In Quantitative Reasoning and Mathematical Modeling: A Driver for STEM Integrated Education and Teaching in Context, edited by R. Mayes and L. L. Hatfield, pp. 93–112. Laramie, WY: University of Wyoming. Johnson, Heather L. 2013. “Reasoning about Quantities That Change Together.” Mathematics Teacher 106 (May): 704–8. Moore, Kevin C., Jason Silverman, Teo Paoletti, and Kevin R. LaForest. 2014. “Breaking Conventions to Support Quantitative Reasoning.” Mathematics Teacher Educator 2 (2): 141–57. National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM. National Oceanic and Atmospheric Administration (NOAA). 2016. “How Does Pressure Change with Ocean Depth?” U.S. Department of Commerce, NOAA, National Ocean Service. http://oceanservice.noaa.gov/facts/pressure.html Thompson, Alba G., Randolph A. Philipp, Patrick W. Thompson, and Barbara A. Boyd. 1994. “Calculational and Conceptual Orientations in Teaching Mathematics.” In Professional Development for Teachers of Mathematics:1994 Yearbook of the National Council of Teachers of Mathematics, edited by Douglas B. Aichele, pp. 79–92. Reston, VA: National Council of Teachers of Mathematics. Thompson, Patrick W. 2011. “Quantitative Reasoning and Mathematical Modeling.” In New Perspectives and Directions for Collaborative Research in Mathematics Education: Papers from a Planning Conference for WISDOMe, edited by S. Chamberlin, L. L. Hatfield, and S. Belbase, pp. 33–57. Laramie, WY: University of Wyoming. ———. 2013. “In the Absence of Meaning.” In Vital Directions for Research in Mathematics Education, edited by Keith Leatham, pp. 57–93. New York: Springer. TEO PAOLETTI, email@example.com, teaches at Montclair State University. He is interested in student learning in the context of middle and secondary school mathematics. IRMA E. STEVENS, firstname.lastname@example.org, is a graduate student in mathematics education at the University of Georgia. She is interested in students’ learning of secondary and post-secondary mathematics. KEVIN C. MOORE, email@example.com, teaches at the University of Georgia in Athens. He is interested in students’ and teachers’ quantitative reasoning and enjoys working with in-service and preservice teachers.