Bridging the Gap Between the Dense and the Discrete: The Number Line and the “Rubber Line” Bridging Analogy

In two experiments we explored the instructional value of a cross‐domain mapping between “number” and “line” in secondary school students' understanding of density. The first experiment investigated the hypothesis that density would be more accessible to students in a geometrical context (infinitely many points on a straight line segment) compared to a numerical context (infinitely many numbers in an interval). The participants were 229 seventh to eleventh graders. The results supported this hypothesis but also showed that students' conceptions of the line segment were far from that of a dense array of points. We then designed a text-based intervention that attempted to build the notion of density in a geometrical context, making explicit reference to the number-to-points correspondence and using the “rubber line” bridging analogy (the line as an imaginary unbreakable rubber band) to convey the no-successor principle. The participants were 149 eighth and tenth graders. The text intervention improved student performance in tasks regarding the infinity of numbers in an interval; the “rubber line” bridging analogy further improved performance successfully conveying the idea that these numbers can never be found one immediately next to the other.     

Localized Solutions of a Piecewise Linear Model of the Stationary Swift–Hohenberg Equation on the Line and on the Plane

In this paper we study a simplified model of the stationary Swift–Hohenberg equation, where the cubic nonlinearity is replaced by a piecewise linear function with similar properties. The main goal is to prove the existence of so-called localized solutions of this equation, i.e., solutions decaying to a homogeneous zero state with unbounded increase of the space variable. The following two cases of the space variable are considered: one-dimensional (on the whole line) and two-dimensional; in the latter case, radially symmetric solutions are studied. The existence of such solutions and increase of their number with change in the equation parameters are shown.     

“Everything is Math in the Whole World”: Integrating Critical and Community Knowledge in Authentic Mathematical Investigations with Elementary Latina/o Students

This critical ethnographic study of an after-school mathematics club for elementary-aged Latina/o youth focuses on connecting critical, community, and mathematical knowledge in the context of authentic, community-based investigations. We present cases of two extended projects to highlight tensions and dilemmas that emerged, particularly tensions related to ensuring rich mathematics in the contexts of projects that were personally and socially meaningful to the students. Our analysis offers insights into critical mathematics education with elementary aged students, and has the potential to counter dominant deficit perspectives of Latina/o youth. Additionally, the findings of this study inform critical approaches to teaching mathematics in schools attended by marginalized students in order to reverse prevalent trends of our educational system failing these students.     

A One–parameter Family of Bounded Solutions for a System of Nonlinear Integral Equations on the Whole Line

A system of nonlinear integral equations with a convolution type operator arising in the p–adic string theory for the scalar tachyons field is studied. The existence of a one–parameter family of monotone continuous and bounded solutions for this system is proved. The limits of the constructed solutions at ±∞ are calculated.     

Linear Equations for the Number of Intervals Which are Isomorphic with Boolean Lattices and the Dehn–Sommerville Equations

Let P be a finite poset. Let L: = J(P) denote the lattice of order ideals of P. Let b _i (L) denote the number of Boolean intervals of L of rank i.

We construct a simple graph G(P) from our poset P. Denote by f _i (P) the number of the cliques K _i+1, contained in the graph G(P).

Our main results are some linear equations connecting the numbers f _i (P) and b _i (L).

We reprove the Dehn–Sommerville equations for simplicial polytopes.

In our proof, we use free resolutions and the theory of Stanley–Reisner rings.     

Students’ mathematical work on absolute value: focusing on conceptions,errors and obstacles

ZDM – Mathematics Education - This study investigates students’ conceptions of absolute value (AV), their performance in various items on AV, their errors in these items and the...     

Backward Transfer: An Investigation of the Influence of Quadratic Functions Instruction on Students’ Prior Ways of Reasoning About Linear Functions

The transfer of learning has been the subject of much scientific inquiry in the social sciences. However, mathematics education research has given little attention to a subclass called backward transfer, which is when learning about new concepts influences learners’ ways of reasoning about previously encountered concepts. This study examined when and in what ways a quadratic functions instructional unit productively influenced middle school students’ ways of reasoning about linear functions. Results showed that students’ ways of reasoning about essential properties of linear functions were productively influenced. Furthermore, conceptual connections were identified linking changes in students’ ways of reasoning about linear functions to what they learned during the quadratics unit. These findings suggest that it is possible to productively influence learners’ ways of reasoning about previously learned-about concepts in significant respects while teaching them new material and that backward transfer offers promise as a new focus for mathematics education research.     

Bojanov–Naidenov Problem for Differentiable Functions on the Real Line and the Inequalities of Various Metrics

Ukrainian Mathematical Journal - For given r ∈ N, p, λ > 0 and any fixed interval [a, b] ? R, we solve the extreme problem $$ underset{a}{overset{b}{int...     

Integer comparisons across the grades: Students’ justifications and ways of reasoning

This study is an investigation of students’ reasoning about integer comparisons—a topic that is often counterintuitive for students because negative numbers of smaller absolute value are considered greater (e.g., −5 > − 6). We posed integer-comparison tasks to 40 students each in Grades 2, 4, and 7, as well as to 11th graders on a successful mathematics track. We coded for correctness and for students’ justifications, which we categorized in terms of 3 ways of reasoning: magnitude-based, order-based, and developmental/other. The 7th graders used order-based reasoning more often than did the younger students, and it more often led to correct answers; however, the college-track 11th graders, who responded correctly to almost every problem, used a more balanced distribution of order- and magnitude-based reasoning. We present a framework for students’ ways of reasoning about integer comparisons, report performance trends, rank integer-comparison tasks by relative difficulty, and discuss implications for integer instruction.     

Explicit Formulas for Green’s Functions on the Annulus and on the Möbius Strip

Given a fixed point free antianalytic involution k of a domain G in thecomplex plane, bounded by a finite number of analytic curves, k-invariant Greensfunctions are defined on G. The Lindelöfs principle is extended to k-invariantGreens functions. When G is the annulus, k-invariant Greens functions areobtained in the explicit form. Since the factorization of the annulus by the group kgenerated by k produces a Möbius strip, the respective result helped us to obtain explicitforms for Greens functions on the Möbius strip.     

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.     

A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2

Mathematical Notes - A particular class of estimates related to the Nelson–Erd?s–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ?1,...     

Constantin and Iyer’s Representation Formula for the Navier–Stokes Equations on Manifolds

The purpose of this paper is to establish a probabilistic representation formula for the Navier–Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case of ? ⁿ or of T ⁿ . On a Riemannian manifold, however, there are several different choices of Laplacian operators acting on vector fields. In this paper, we shall use the de Rham–Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy–Le Jan–Li’s idea to decompose it as a sum of the square of Lie derivatives.     

Inequalities and asymptotics for the Euler–Mascheroni constant based on DeTemple’s result

Let \(R_{n}={\sum }_{k=1}^{n}\frac {1}{k}-\ln \left (n+\frac {1}{2}\right )\). DeTemple proved the following inequality:     

Science Fairs: What Are the Sources of Help for Students and How Prevalent Is Cheating?

This study examined the sources and kinds of help that students who were required to participate in science fairs considered fair and reasonable and the kinds of help they actually received for their project. In addition, the possibility of cheating was explicitly probed. A previously reported gap between potential and actual sources and kinds of help was confirmed, and 5 of the 24 students whose participation was required in a science fair admitted to making up their data or results. Pressure of time was the most highly reported obstacle faced by all students. Although 5 students cheated, one demonstrated a strong sense of right and wrong, but all the students who cheated lacked or did not make use of adaptive strategies.     

Secondary Students’ Quantification of Ratio and Rate: A Framework for Reasoning about Change in Covarying Quantities

Contributing to a growing body of research addressing secondary students’ quantitative and covariational reasoning, the multiple case study reported in this article investigated secondary students’ quantification of ratio and rate. This article reports results from a study investigating students’ quantification of rate and ratio as relationships between quantities and presents the Change in Covarying Quantities Framework, which builds from Carlson, Jacobs, Coe, Larsen, and Hsu’s (2002) Covariation Framework. Each of the students in this study was consistent in terms of the quantitative operation he or she used (comparison or coordination) when quantifying both ratio and rate. Illustrating how students can engage in different quantitative operations when quantifying rate, the Change in Covarying Quantities Framework helps to explain why students classified as operating at a particular level of covariational reasoning appear to be using different mental actions. Implications of this research include recommendations for designing instructional tasks to foster students’ quantitative and covariational reasoning.