The parametric nature of two students’ covariational reasoning

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.     

Reasoning about variation in the intensity of change in covarying quantities involved in rate of change

This paper extends work in the area of quantitative reasoning related to rate of change by investigating numerical and nonnumerical reasoning about covarying quantities involved in rate of change via tasks involving multiple representations of covarying quantities. The findings suggest that by systematically varying one quantity, an individual could simultaneously attend to variation in the intensity of change in a quantity indicating a relationship between covarying quantities. The results document how a secondary student, prior to formal instruction in calculus, reasoned numerically and nonnumerically about covarying quantities involved in rate of change in a way that was mathematically powerful and yet not ratio-based. I discuss how coordinating covariational and transformational reasoning supports attending to variation in the intensity of change in quantities involved in rate of change.     

Students’ challenges with polar functions: covariational reasoning and plotting in the polar coordinate system

Covariational reasoning has been the focus of many studies but only a few looked into this reasoning in the polar coordinate system. In fact, research on student's familiarity with polar coordinates and graphing in the polar coordinate system is scarce. This paper examines the challenges that students face when plotting polar curves using the corresponding plot in the Cartesian plane. In particular, it examines how students coordinate the covariation in the polar coordinate system with the covariation in the Cartesian one. The research, which was conducted in a sophomore level Calculus class at an American university operating in Lebanon, investigates in addition the challenges when students synchronize the reasoning between the two coordinate systems. For this, the mental actions that students engage in when performing covariational tasks are examined. Results show that coordinating the value of one polar variable with changes in the other was well achieved. Coordinating the direction of change of one variable with changes in the other variable was more challenging for students especially when the radial distance r is negative.     

Backward transfer influences from quadratic functions instruction on students’ prior ways of covariational reasoning about linear functions

The study reported in this article examined the ways in which new mathematics learning influences students’ prior ways of reasoning. We conceptualize this kind of influence as a form of transfer of learning called backward transfer. The focus of our study was on students’ covariational reasoning about linear functions before and after they participated in a multi-lesson instructional unit on quadratic functions. The subjects were 57 students from two authentic algebra classrooms at two local high schools. Qualitative analysis suggested that quadratic functions instruction did influence students’ covariational reasoning in terms of the number of quantities and the level of covariational reasoning they reasoned with. These results further the field’s understanding of backward transfer and could inform how to better support students’ abilities to engage in covariational reasoning.     

Developing prospective teachers’ covariational reasoning through a model development sequence

Forming part of a wider research study, the current study investigated prospective middle school mathematics teachers’ ways of covariational reasoning on tasks involving simultaneously changing quantities. As the introductory theme of a larger unit on derivative, a model development sequence on covariational reasoning was designed and experimented with 20 participants in a mathematical modeling course offered to prospective teachers. The participants’ developing abilities of covariational reasoning were documented under three categories: (i) identifying the variables, (ii) ways of coordinating the variables, and (iii) ways of quantifying the rate of change. The results revealed significant improvement in the prospective teachers’ ways of identifying and coordinating the variables, and in quantifying the rate of change. Moreover, the results indicated that preference for a particular way of thinking in identifying and coordinating the variables determined the prospective teachers’ way of quantifying the rate of change and thereby their level of covariational reasoning.     

Covariational reasoning and invariance among coordinate systems

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.     

The progression of preservice teachers’ covariational reasoning as they model global warming

The study examines how the covariational reasoning of three preservice mathematics teachers (PSTs) advances, and what they learned about an important metric in climate science, as they examine the link between carbon dioxide (CO₂) pollution and global warming. The PSTs completed a mathematical task during an individual, task-based interview. Their responses were analyzed by complementing the Covariation Framework and the Change in Covarying Quantities Framework. The analysis revealed that the PSTs’ covariational reasoning increased in sophistication as they completed the task, advancing from describing direction of change to reasoning about the rate of change. Each level of sophistication either supported or constrained the PSTs’ ability to specify nonlinear growth, anticipate concavity, draw accurate graphs, and make viable claims about the rate of change. The PSTs also learned about important ideas related to the metric radiative forcing by CO₂, suggesting it is possible to learn mathematics while promoting climate change education.     

Middle school students’ construction of quantitative unknowns

Two iterative, after school design experiments with small groups of middle school students were conducted to investigate how students constructed quantitative unknowns, conceived of as values of fixed quantities that are not known but can be determined. Students solved problems about an unknown height or length measured in two different units. Of 13 students who participated, 6 structured quantities into three levels of units. These students constructed an unknown as a height consisting of an indeterminate number of length units, each of which consisted of smaller length units, and they symbolized these relationships in their equations. The other 7 students structured quantities into two levels of units. Five of these students symbolized only the relationships between the measurement units, with two students demonstrating more basic and advanced solutions. The study shows that grappling with unknowns as measured and indeterminate is beneficial for students’ construction of variable.     

Students’ images of problem contexts when solving applied problems

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.     

Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship

Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers’ thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way of promoting preservice teachers’ reasoning about the tangent relationship.     

The effect of quantitative reasoning on prospective mathematics teachers’ proof comprehension: The case of real numbers

We report a mixed-methods research study investigating the effect of quantitative reasoning on prospective mathematics teachers’ comprehension of a proof on real numbers. Nineteen prospective mathematics teachers engaged in quantitative reasoning while developing real numbers as rational number sequences in a series of instructional activities. All participants completed a proof comprehension assessment prior to and upon completion of the instruction. Six of the prospective mathematics teachers also participated in semi-structured interviews after the post-test. Results showed a significant difference in proof comprehension performance between the pre- and post-tests. Moreover, results from the interviews showed that prospective teachers reasoned quantitatively on the proof comprehension dimensions. Results suggest that engaging in quantitative reasoning during instruction may help to develop proof comprehension, particularly in situations involving the analysis of proofs entailing properties of the real number system. We recommend embedding quantitative reasoning in teacher education and professional development programs to facilitate mathematics teachers’ proof comprehension and proving activities.     

Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation

We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequality is derived, is proved in an elementary and direct way based on standard integration by parts.     

“Approximate” multiplicative relationships between quantitative unknowns

Three 18-session design experiments were conducted, each with 6–9 7th and 8th grade students, to investigate relationships between students’ rational number knowledge and algebraic reasoning. Students were to represent in drawings and equations two multiplicatively related unknown heights (e.g., one was 5 times another). Twelve of the 22 participating students operated with the second multiplicative concept, which meant they viewed known quantities as units of units, or two-levels-of-units structures, but not as three-levels-of-units structures. These students were challenged to represent multiplicative relationships between unknowns: They changed the given relationship, did not think of the relationship as multiplicative until after concerted work, and used numerical values in lieu of unknowns. Our account for these challenges is that students needed to simplify the involved units coordinations. Ultimately students abstracted the relationship as multiplicative, but the exact relationship was not certain or had to be constituted in activity. Implications for teaching are explored.     

调和单叶映射反函数调和的充要条件

给出复值，调和，单叶函数的反函数是调和单叶的充要条件；并且举例说明反函数一     

On the inverse spectral theory of Schrödinger and Dirac operators

We prove that under some conditions finitely many partially known spectra and partial information on the potential entirely determine the potential. This extends former results of Hochstadt, Lieberman, Gesztesy, Simon and others.

     

convergence of the reconstruction formula for the potential function

It is known that the potential function of the Sturm-Liouville problem can be reconstructed from the nodal data by a pointwise limit. We show that this convergence is in fact .

     

Slowly oscillating solutions of parabolic inverse problems

We first extend slowly oscillating functions to a more general setting and investigate their properties. Then we show the existence and uniqueness of slowly oscillating solutions of parabolic equations and parabolic inverse problems.     

Modeling preferences and conditional preferences on resource consumption and production in ASP

In this paper, we extend our previous work on Resourced ASP, or for short RASP, where we have introduced the possibility of defining and using resources in ASP. In RASP, one can define resources with their amounts, where available resources can be used for producing other resources and the remaining amount, if any, can be used in a different way. In this paper, we introduce P-RASP (RASP with Preferences) where it is possible to express preferences about which resources should be either consumed or produced. Moreover, conditional preferences, of different forms, allow one to express preferences according to certain conditions, that are to be evaluated “dynamically”, namely, with respect to the specific answer set at hand. The semantic of conditional preferences is given in terms of (non-conditional) preferences, though the translation is not straightforward and thus the new features are not syntactic sugar. Complexity of P-RASP is also discussed.