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 theirpartitioning 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.

This paper introduces a new mode of variational and covariational reasoning, which we call scaling-continuous reasoning. Scaling-continuous reasoning entails (a) imagining a variable taking on all values on the continuum at any scale, (b) understanding that there is no scale at which the continuum becomes discrete, and (c) re-scaling to any arbitrarily small increment for x and coordinating that scaling with associated values for y. Based on the analysis of a 15-h teaching experiment with two 12-year-old pre-algebra students, we present evidence of scaling-continuous reasoning and identify two implications for students’ understanding of rates of change: seeing constant rate as an equivalence class of ratios, and viewing instantaneous rate of change as a potential rate. We argue that scaling-continuous reasoning can support a robust understanding of function and rates of change. © 2020, Springer Nature B.V.

This paper introduces a quadratic growth learning trajectory, a series of transitions in students’ ways of thinking (WoT) and ways of understanding (WoU) quadratic growth in response to instructional supports emphasizing change in linked quantities. We studied middle grade (ages 12–13) students’ conceptions during a small-scale teaching experiment aimed at fostering an understanding of quadratic growth as phenomenon of constantly-changing rate of change. We elaborate the duality, necessity, repeated reasoning framework, and methods of creating learning trajectories. We report five WoT: Variation, Early Coordinated Change, Explicitly Quantified Coordinated Change, Dependency Relations of Change, and Correspondence. We also articulate instructional supports that engendered transitions across these WoT: teacher moves, norms, and task design features. Our integration of instructional supports and transitions in students’ WoT extend current research on quadratic function. A visual metaphor is leveraged to discuss the role of learning trajectories research in unifying research on teaching and learning. © 2020 Elsevier Inc.

Asked to quantify the changes in area of growing rectangles, these students reasoned about multiplicative relationships in interesting new ways.