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Researchers have emphasized the importance of characterizing students' abilities to coordinate changes in covarying quantities. In this paper, we characterize three undergraduate students' coordination of covarying quantities' amounts of change during a teaching experiment. We adopt Piagetian notions of figurative and operative thought to describe the extent their meanings for covariational relationships are constrained to or supported by their partitioning activity -- the mental and physical actions associated with constructing accruals in quantities' magnitudes. Our analysis suggests that students' construction of amounts of change is constrained by figurative "partitioning activity" that requires carrying out or emulating particular actions on perceptually available material. In contrast, operative partitioning activity supports the students' transformation and (anticipated) regeneration of partitioning activity in order to conceive equivalent covariational relationships among various situations and representational systems. We conclude by discussing how documenting these distinctive meanings contributes to extant literature on covariational reasoning and, more broadly, the theorization of mathematical concept construction.

Supporting students developing understandings of function has been a notoriously elusive task in mathematics education. We present Thompson and Carlson's (2017) description of a covariational meaning of function and provide an example of a student who maintains meanings compatible with this description. We use this student's activity to illustrate nuances in a covariational meaning of function and to highlight how such meanings can be powerful. We argue a student who has developed meanings compatible with a covariational meaning of function has the foundational meanings needed to understand a formal definition of function (i.e. the horse needed to pull the cart).

Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers' and students' inverse function meanings. Responding to this call, we characterize 25 pre-service teachers' inverse function meanings as inferred from our analysis of clinical interviews. After summarizing relevant research, we describe the methodology and theoretical framework we used to interpret the pre-service teachers' activities. We then present data highlighting the techniques the pre-service teachers used when responding to tasks that involved analytical and graphical representations of functions and inverse functions in both decontextualized and contextualized situations and discuss our inferences of their meanings based on their activities. We conclude with implications for the teaching and learning of inverse function and areas for future research.

Mathematics educators and writers of mathematics education policy documents continue to emphasize the importance of teachers focusing on and using student thinking to inform their instructional decisions and interactions with students. In this paper, we characterize the interactions between a teacher and student(s) that exhibit this focus. Specifically, we extend previous work in this area by utilizing Piaget's construct of decentering ("The language and thought of the child." Meridian Books, Cleveland, 1955) to explain teachers' actions relative to both their thinking and their students' thinking. In characterizing decentering with respect to a teacher's focus on student thinking, we use two illustrations that highlight the importance of decentering in making in-the-moment decisions that are based on student thinking. We also discuss the influence of teacher decentering actions on the quality of student-teacher interactions and their influence on student learning. We close by discussing various implications of decentering, including how decentering is related to other research constructs including teachers' development and enactment of mathematical knowledge for teaching.

How do students think about an angle measure of ninety degrees? How do they think about ratios and values on the unit circle? How might angle measure be used to connect right-triangle trigonometry and circular functions? And why might asking these questions be important when introducing trigonometric functions to students? When teaching trigonometric functions, most teachers would agree that students have difficulty using trigonometric functions to relate quantities in circle and right-triangle contexts. If students are to use trigonometric functions productively, they must understand angle measure, the unit circle, and right triangles in ways that let them see trigonometric functions as relations between two quantities. In addition, trigonometry instruction should adhere to the Common Core State Standards for Mathematics' emphasis on students' quantitative reasoning (CCSSI 2010). In this article, Kevin Moore and Kevin LaForest draw from recent research on student learning in trigonometry to illustrate a connected introduction of angle measure and the sine function that entails quantitative reasoning. To demonstrate these ideas in practice, they provide examples of precalculus students' solutions to various trigonometry problems. The ideas presented here also outline nuances involved in increasing student learning during a sequence of activities like that presented by Landers (2013). In particular, they draw attention to reasoning about varying quantities in the context of graphing and how issues of measurement help students understand the unit circle and ratios.

A growing body of literature has identified quantitative and covariational reasoning as critical for secondary and undergraduate student learning, particularly for topics that require students to make sense of relationships between quantities. The present study extends this body of literature by characterizing an undergraduate precalculus student's progress during a teaching experiment exploring angle measure and trigonometric functions. I illustrate that connecting angle measure to measuring arcs and conceiving the radius as a unit of measure can engender trigonometric meanings that encompass both unit circle and right triangle trigonometry contexts. The student's progress during the teaching experiment also indicates that a covariation meaning for the sine function supports using the sine function to represent emergent relationships between quantities in novel situations.

I discuss a teaching experiment that sought to characterize precalculus students' angle measure understandings. The study's findings indicate that the students initially conceived angle measures in terms of geometric objects. As the study progressed, the students formed more robust understandings of degree and radian measures by constructing an arc length image of angle measures; the students' quantification of angle measure entailed measuring arcs and conceiving multiplicative relationships between a subtended arc, a circle's circumference, and a circle's radius. The students leveraged these quantitative relationships to transition between units with a fixed magnitude (e.g., an arc length's measure in feet) and various angle measure units, while maintaining invariant meanings for angle measures in different units. These results suggest that quantifying angle measure, regardless of unit, through processes that involve measuring arc lengths can support coherent angle measure understandings.

Characterizing how quantities change (or vary) in tandem has been an important historical focus in mathematics that extends into the current teaching of mathematics. Thus, how students conceptualize quantities that change in tandem becomes critical to their mathematical development. In this paper, we propose two images of change: chunky and smooth. Chunky images of change are based in imagining change as occurring countable and completed amounts, whereas smooth images of change are based in imagining a continually changing experience. Using empirical examples of student work, we illustrate these images of change and their implications.

This article reports findings from an investigation of precalculus students' approaches to solving novel problems. We characterize the images that students constructed during their solution attempts and describe the degree to which they were successful in imagining how the quantities in a problem's context change together. Our analyses revealed that students who mentally constructed a robust structure of the related quantities were able to produce meaningful and correct solutions. In contrast, students who provided incorrect solutions consistently constructed an image of the problem's context that was misaligned with the intent of the problem. We also observed that students who caught errors in their solutions did so by refining their image of how the quantities in a problem's context are related. These findings suggest that it is critical that students first engage in mental activity to visualize a situation and construct relevant quantitative relationships prior to determining formulas or graphs. (Contains 7 tables and 3 figures.)

We introduce the sociomathematical norm of "speaking with meaning" and describe its emergence in a professional learning community (PLC) of secondary mathematics and science teachers. We use "speaking with meaning" to reference specific attributes of individual communication that have been revealed to improve the quality of discourse among individuals engaged in discourse in a PLC. An individual who is "speaking with meaning" provides conceptually based descriptions when communicating with others about solution approaches. The quantities and relationships between quantities in the problem context are described rather than only stating procedures or numerical calculations used to obtain an answer to a problem. Solution approaches are justified with logical and coherent arguments that have a conceptual rather than procedural basis. The data for this research was collected during a year-long study that investigated a PLC whose members were secondary mathematics and science teachers. Analysis of the data revealed that after one semester of participating in a PLC where "speaking with meaning" was emphasized, the PLC members began to establish their own criteria for an acceptable mathematical argument and what constituted "speaking with meaning." The group also emerged with common expectations that answers be accompanied by explanations and mathematical operations be explained conceptually (not just procedurally). The course and PLC design that supported the emergence of "speaking with meaning" by individuals participating in a PLC are described. (Contains 2 figures.)