In this chapter, we discuss Piagetian notions of figurative thought and operative thought. Consistent with many of the ideas introduced by Piaget, characterizations of figurative and operative thought have evolved in different ways since their introduction. Specifically, mathematics educators have adapted Piaget’s ideas in order to develop models of students’ mathematics. Evolutions in the use of these constructs typically stemmed from researchers’ needs to adjust them—in ways faithful to particular aspects of the original distinctions—in order to yield more viable and generalizable models of students’ mathematics. Here, we provide a summary of these evolutions. In doing so, we draw from our own work to provide concrete examples of researchers’ uses of figurative and operative thought in order to illustrate distinguishable aspects of the two forms of thought in multiple settings. We also discuss methodological implications of figurative and operative thought including how the constructs can be used in task design during empirical studies and can inform researchers’ claims regarding students’ mathematical meanings. We close with suggestions for future research in the hopes this chapter can be a springboard for pursuits in constructing viable models of students’ mathematics. 

Generalizing is a critical aspect of mathematics learning, with researchers and policy documents highlighting generalizing as a core mathematical practice. It can also be challenging to foster in class settings, and teachers need access to better resources to teach generalizing, including an understanding of effective forms of instruction. This article proposes Classroom Supports for Generalizing (CSGs), investigating how multiple elements—such as tasks, teacher moves, student interactions, and representations—interact to meaningfully foster student generalizing. Drawing on class video data from a middle school teacher and two high school teachers, we present the CSG Framework, which identifies three categories of supports: Interactions for Generalizing, Structures for Generalizing, and Routines for Generalizing.

Over the most recent several decades, researchers have argued the importance of quantitative and covariational reasoning for students’ learning. These same researchers have illustrated the importance of these reasoning processes with respect to local and longitudinal development. In both grain sizes, researchers are detailed in their descriptions of the intended topics or reasoning processes. There is, however, a lack of specificity of generalized criteria for concept construction from a quantitative reasoning perspective. In this chapter, we introduce such criteria through the construct of an abstracted quantitative structure, which has its roots in quantitative reasoning, covariational reasoning, and various Piagetian notions. In introducing the construct, we focus on ideas informing its development and its criteria, and we use it to characterize examples of student actions. We close with comments regarding implications for both teaching and research.

The Relative Risk Tool web app allows students to compare risks relating to COVID-19 with other more familiar risks, to make multiplicative comparisons, and to interpret them.

Generalization is a critical component of mathematical reasoning, with researchers recommending that it be central to education at all grade levels. However, research on students’ generalizing reveals pervasive difficulties in creating and expressing general statements, which underscores the need to better understand the processes that can support more productive generalizations. In response, we report on results from 146 interviews with 93 participants in middle school through college in the domains of algebra, advanced algebra, trigonometry/pre-calculus, and combinatorics while solving complex problems. Our findings yielded the Relating-Forming-Extending (RFE) Framework, which distinguishes multiple related forms and types of generalizing. We also present two aspects of mental activity that promote generative generalizations: operative activity, and building and refining activity.

I structure the chapter as follows to accomplish several goals. I first provide two vignettes in order to introduce the two graphical shape thinking constructs and motivate a focus on particular aspects of transfer. A concise discussion of the theoretical underpinnings central to this chapter follows the opening vignettes. I subsequently describe the two graphical shape thinking constructs and, using accompanying student data, illustrate them in terms of students’ transfer processes. Generalizing from these cases, I introduce a way to frame concept construction in terms of theories of transfer and graphical shape thinking, and I provide a data example to illustrate the productive nature of such a framing. As part of this discussion, I provide suggestions for future research.

We investigate United States and South Korean citizens’ mathematical schemes and how these schemes supported or hindered their attempts to assess the severity of COVID-19. We selected web and media-based COVID-19 data representations that we  hypothesized citizens would interpret differently depending on  their mathematical schemes. We  included items that we conjectured would be easier or more difficult to interpret with schemes that prior research had reported were more or less productive, respectively. We used the representations during clinical interviews with 25 United States and seven South Korean citizens. We illustrate that citizens’ mathematical schemes (as  well as  their beliefs) impacted how they assessed the  severity of COVID-19. We present vignettes of citizens’ schemes that inhibited interpreting representations of COVID-19 in ways compatible with the displayed quantitative data, schemes that aided them in assessing the severity of COVID-19, and beliefs about the reliability of scientific data that over-rode their mathematical conclusions. 

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.

We report on semi-structured clinical interviews to describe U.S. pre-service secondary mathematics teachers’ graphing meanings. Our primary goal is to draw on Piagetian notions of figurative and operative thought to identify marked differences in the students’ meanings. Namely, we illustrate students’ meanings dominated by fragments of sensorimotor experience and com- pare those with students’ meanings dominated by the coordination of mental actions in the form of covarying quantities. Our findings suggest students’ meanings that foreground operative aspects of thought are more generative with respect to graphing. Our findings also indicate that students can encounter perturbations due to potential incompatibilities between figurative and operative aspects of thought.

In this paper, we use relevant literature and data to motivate a more detailed look into relationships between what we perceive to be conventions common to United States (U.S.) school mathematics and individuals’ meanings for graphs and related topics. Specifically, we draw on data from pre-service (PST) and in-service (IST) teachers to characterize such relationships. We use PSTs’ responses during clinical interviews to illustrate three themes: (a) some PSTs’ responses implied practices we perceive to be conventions of U.S. school mathematics were instead inherent aspects of PSTs’ meanings; (b) some PSTs’ responses implied they understood certain practices in U.S. school mathematics as customary choices not necessary to represent particular mathematical ideas; and (c) some PSTs’ responses exhibited what we or they perceived to be contradictory actions and claims. We then compare our PST findings to data collected with ISTs.

Quantitative reasoning is important in the development of K–16 mathematical ideas such as function and rate of change. Coordinate systems are used to coordinate sets of quantities by establishing frames of reference and constructing representational spaces in which sets of quantities are joined. Despite the critical role of coordinate systems in mathematics, much is left to understand about how students construct and reason within frames of reference and associated coordinate systems. In this report, we draw from a teaching experiment to discuss how an undergraduate student, Lydia, constructed and reasoned within frames of reference when graphing in non-canonical coordinate systems. We pay specific attention to distinctions between figurative and operative aspects of thought in her committing to reference points and directionality of measure comparison within frames of reference. In this regard, we present shifts in Lydia’s reasoning during the teaching experiment and consider implications and future research directions.

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.

Researchers have argued that covariational reasoning is foundational for learning a variety of mathematics topics. We extend prior research by examining two students’ covariational reasoning with attention to the extent they became consciously aware of the parametric nature of their reasoning. We first describe our theoretical background including different conceptions of covariation researchers have found useful when characterizing student reasoning. We then present two students’ activities during a teaching experiment in which they constructed and reasoned about covarying quantities. We highlight aspects of the students’ reasoning that we conjectured created an intellectual need that resulted in their constructing a parameter quantity or attribute, a need we explored in closing teaching episodes. We discuss implications of these results for perspectives on covariational reasoning, students’ understandings of graphs and parametric functions, and areas of future research.

Strategy can uncover students’ thinking about representational conventions.

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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. 

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Much of the research on mathematical thinking, reasoning, and learning has concerned how students and mathematicians cope with this abstraction—the meanings individuals develop for abstract mathematical concepts, as well as how individuals use abstract concepts to draw inferences and solve problems. This chapter describes recent developments in mathematical thinking and reasoning through three theoretical frames of abstraction: Mathematical reasoning as logical reasoning; mathematical reasoning as commonalities of effective expert reasoning across mathematical situations; and mathematical reasoning as experienced by learners of mathematics.

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 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 de- centering is related to other research constructs including teachers’ development and enactment of mathematical knowledge for teaching.

The Pathways Precalculus curriculum consists of 12 modules. Each module contains 6 to 12 in-class investigations that include problems and prompts that engage students in making meaning, building connections and understanding ideas. Students learn general approaches for solving problems that enable them to emerge as competent and confident problem solvers.

We discuss a teaching experiment that explored two pre-service secondary teachers’ meanings for the unit circle. Our analyses suggest that the participants’ initial unit circle meanings predominantly consisted of calculational strategies for relating a given circle to what they called "the unit circle." These strategies did not entail conceiving a circle’s radius as a unit of measure. In response, we implemented tasks designed to focus the participants’ attention on various measurement ideas including conceiving a circle’s radius as a unit magnitude. Against the backdrop of the participants’ actions on these tasks, we characterize shifts in the participants’ unit circle meanings and we briefly describe how these shifts influenced their ability to use the unit circle in trigonometric situations.

This chapter describes the processes involved in conceptualizing functions as processes that entail two quantities varying in tandem. We make use of examples to illustrate covariational reasoning in the context of using functions to model relationships between quantities in various situations. We also describe a promising approach that fosters students’ covariational reasoning abilities as a primary focus for developing the function conceptions needed to understand calculus and continue in mathematics and the sciences.

Epistemic Algebraic Students: Emerging Models of Students' Algebraic Knowing (Papers from an Invitational Conference) is the fourth volume in a monograph series published in support of the activities of participants in the Wyoming Institute for the Study and Development of Mathematics Education (WISDOM^e). 

Students’ representational activity remains an area of focus in mathematics education research. From a radical constructivist perspective, this area of research faces an inherent difficulty: a person’s knowledge, and hence what they are representing, is fundamentally unknowable to another individual. A consequence of this is that, as a researcher, we can only develop models of the mental operations that might lead a student to produce our interpretations of her or his actions and representations. In this article I draw on notions put forth by several researchers including von Glasersfeld and Piaget to make sense of students’ representational activity. Namely, I distinguish between representations as signals or as symbols, each stemming from different levels of abstraction. In the former, students’ representational activity and meanings are halted at the level of pseudo-empirical abstractions and are thus constrained to carrying out actions. In the latter, students’ representational activity and meanings involve reflective abstractions and are thus operative in that the representations become pointers to internalized processes that need not be carried out. I illustrate these perspectives, including their implications, by presenting student activity across several contexts.

Despite the importance of the polar coordinate system (PCS) to students’ study of mathematics and science, there is a limited body of research that explores students’ ways of thinking about the PCS. Research on students’ construction of the PCS is especially sparse. In this article, we highlight several issues that arose spontaneously during a teaching experiment that explored students’ construction of the PCS. We illustrate how students’ angle measure meanings influenced their construction of the PCS. We also discuss how the students’ ways of thinking about the Cartesian coordinate system (CCS) became problematic as they transitioned to the PCS. Collectively, we highlight that students’ ways of thinking about coordinate systems evolve when students reason within and across multiple coordinate systems.

Education researchers often explain student activity in terms of general thinking and learning processes including those identified by Cifarelli and Sevim. In this commentary, I refocus Cifarelli and Sevim’s discussion in order to hypothesize the organization of mental actions that comprise and support those learning processes.

A connected introduction of angle measure and the sine function entails quantitative reasoning.

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How do your 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 your students?

Quantitative reasoning is critical to developing understandings of function
that are important for sustained success in mathematics. Unfortunately, preservice teachers often do not receive sufficient quantitative reasoning experiences during their schooling. In this paper, we illustrate consequences of underdeveloped quantitative reasoning abilities against the backdrop of central function concepts. We also illustrate tasks that can perturb preservice teachers’ thinking in ways that produce opportunities for quantitative reasoning. By implementing strategically designed tasks, teacher educators can support preservice teachers—and students in general—in advancing their quantitative reasoning abilities and their understanding of secondary mathematics content.

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.

Imagine a bottle being filled with liquid. How do you think about the volume of the liquid as it changes relative to the height of the liquid in the bottle? One way is to envision sections along the height of the bottle and determine or estimate amounts of volume in each section. Alternatively, one might envision both the volume and the height changing together so that each is continually increasing. The former way of thinking could be likened to filling a bottle with successive cups of liquid: changes in volume and height occur in discrete chunks. The latter way of thinking could be likened to filling the bottle from a hose: changes in volume and height continually progress. These different ways of thinking indicate how students might draw on differing images of change when constructing relationships between changing quantities (in this case, the volume and height of liquid in a bottle).

Researchers continue to emphasize the importance of covariational reasoning in the context of students’ function concept, particularly when graphing in the Cartesian coordinate system (CCS). In this article, we extend the body of literature on function by characterizing two pre-service teachers’ thinking during a teaching experiment focused on graphing in the polar coordinate system (PCS). We illustrate how the participants engaged in covariational reasoning to make sense of graphing in the PCS and make connections with graphing in the CCS. By foregrounding covariational relationships, the students came to understand graphs in different coordinate systems as representative of the same relationship despite differences in the perceptual shapes of these graphs. In synthesizing the students’ activity, we provide remarks on instructional approaches to graphing and how the PCS forms a potential context for promoting covariational reasoning.

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.

Over the past five years I have sought to better understand student thinking and learning in the context of topics central to trigonometry, including angle measure, the unit circle, trigonometric functions, periodicity, and the polar coordinate system. While each study has provided unique insights into students’ learning of trigonometry, a common theme connects the studies’ findings: quantitative reasoning plays a central role in students’ trigonometric understandings. In this chapter, I first describe a coherent system of understandings for trigonometry that is grounded in quantitative reasoning. Against this backdrop, I compare students’ quantitative reasoning in the context of trigonometry in order to illustrate the role of quantitative reasoning in the learning of a particular mathematical topic.

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.

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.