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We report partial analysis of a survey of instructors of undergraduate geometry courses for teachers, attending to how they described the nature of the mathematical work they engage students in and the opportunities to learn that students had. Analysis of latent construct correlations showed that engagement of students in inquiry into geometry was significantly associated with opportunity to learn about mathematical definitions and conjecturing and engagement of students in the study of geometry was significantly associated with opportunity to learn about axioms and about history of geometry. Latent variable means comparisons showed group differences in claimed opportunity to learn between instructors whose highest degree was in mathematics and those whose highest degree was in mathematics education. 

This paper explores the potential of online lesson visualization and annotation tools in fostering international lesson-centered teacher collaboration. In an era where teachers face diverse challenges and limited opportunities for peer-to-peer collaboration, leveraging digital tools for asynchronous exchanges emerges as a promising avenue for professional development. This paper will illustrate the potential of emerging technologies for supporting cross-cultural exchanges in which teachers can share insights, perspectives, and innovative practices in durable and archivable forms, thereby enriching the collective knowledge base for teaching. We share data from an ongoing project focused on engaging groups of secondary mathematics teachers in collectively refining a single storyboarded lesson. Through collaborative lesson development and iterative refinement, we illustrate how these tools transcend temporal and spatial constraints by sharing data gathered from three different groups involved in cross-cultural exchange (one situated in the western part of the U.S, one situated in the eastern part of the U.S., and one situated in Bulgaria) centered on storyboard representation of a lesson. In this way, we provide insights on how the lean graphics of the storyboard and the asynchronous nature of annotation can foster a culture of continuous improvement and mutual support among mathematics teachers spread over large geographic distances. Ultimately, we advocate for the widespread adoption of online multimedia authoring tools as integral components of contemporary approaches to cross-cultural collaboration on lessons for facilitating meaningful exchanges and promoting excellence in teaching and learning on a global scale 

In an instance of the weighted Nash Social Welfare problem, we are given a set of m indivisible items, G, and n agents, A, where each agent i in A has a valuation v_ij ≥ 0 for each item j in G. In addition, every agent i has a non-negative weight w_i such that the weights collectively sum up to 1. The goal is to find an assignment of items to players that maximizes the weighted geometric mean of the valuation received by the players. When all the weights are equal, the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present an approximation algorithm whose approximation depends on the KL-divergence between the weight distribution and the uniform distribution. We generalize the convex programming relaxations for the symmetric variant of Nash Social Welfare presented in [CDG+17, AGSS17] to two different mathematical programs. The first program is convex and is necessary for computational efficiency, while the second program is a non-convex relaxation that can be rounded efficiently. The approximation factor derives from the difference in the objective values of the convex and non-convex relaxation. 

For centuries, there has been a debate about the role of undergraduate education in society. Some have argued that universities should focus on practical skills and knowledge to prepare students for the workforce, while others have supported the idea that universities should prioritize providing a broad understanding of disciplinary knowledge and practices. In this paper, we leverage data collected from 32 interviews to explore how instructors of the undergraduate geometry course for teachers (GeT) talk about the various tensions they experience in their work. Three distinct ways of talking about tensions emerged from the data: the tension as a dilemma that needs to be managed, the tension as a place to take sides, the tension as an opportunity to reframe aspects of the work. In closing we draw connections between these patterns in the data and the two perspectives about the role of undergraduate mathematics courses in preparing PTs for the work of teaching. 

This theoretical contribution draws on earlier work by Herbst and Chazan (2012; also Chazan et al., 2016) in which they describe the position of a mathematics teacher in an educational institution as accountable to stakeholders who issue four types of professional obligations. We propose an application and adaptation of that framework intended to address the case of instructors who teach undergraduate mathematics courses to future teachers. Considerations of not only the academic but also the professional ends of these courses are key in our application of the theory of obligations. 

Abstract Technology-mediated simulations of teaching are used increasingly to represent practice in the context of professional development interventions and assessment. Some such simulations represent students as cartoon characters. An important question in this context is whether simplified cartoon representations of students can convey similar meanings as real facial expressions do. Here, we share results from an implementation and replication study designed to observe whether and how (1) cartoon-based representations of emotion using graphical facial expressions can be interpreted at similar levels of accuracy as photo representations of emotions using actors and (2) the inclusion of markers of student emotions in storyboard-based scenarios of secondary mathematics teaching affects teachers’ appropriateness rating of the actions taken by a teacher represented in the storyboard. We show graphical representations of emotions can evoke particular intended emotions and that markers of student emotions in representations of practice could cue mathematics teachers into particular judgments of action. The Impact Sheet to this article can be accessed at10.6084/m9.figshare.24219964 

The high-energy radiative output, from the X-ray to the ultraviolet, of exoplanet host stars drives photochemical reactions and mass loss in the upper regions of planetary atmospheres. In order to place constraints on the atmospheric properties of the three closest terrestrial exoplanets transiting M dwarfs, we observe the high-energy spectra of the host stars LTT 1445A and GJ 486 in the X-ray withXMM-NewtonandChandraand in the ultraviolet with HST/COS and STIS. We combine these observations with estimates of extreme-ultraviolet flux, reconstructions of the Lyαlines, and stellar models at optical and infrared wavelengths to produce panchromatic spectra from 1 Å to 20 µm for each star. While LTT 1445Ab, LTT 1445Ac, and GJ 486b do not possess primordial hydrogen-dominated atmospheres, we calculate that they are able to retain pure CO₂atmospheres if starting with 10, 15, and 50% of Earth’s total CO₂budget, respectively, in the presence of their host stars’ stellar wind. We use age-activity relationships to place lower limits of 2.2 and 6.6 Gyr on the ages of the host stars LTT 1445A and GJ 486. Despite both LTT 1445A and GJ 486 appearing inactive at optical wavelengths, we detect flares at ultraviolet and X-ray wavelengths for both stars. In particular, GJ 486 exhibits two far-ultraviolet flares with absolute energies of 10^29.5and 10^30.1erg (equivalent durations of 4357 ± 96 and 19 724 ± 169 s) occurring 3 h apart. Based on the timing of the observations, we suggest that these high-energy flares are related and indicative of heightened flaring activity that lasts for a period of days, but our interpretations are limited by sparse time-sampling. Consistent high-energy monitoring is needed to determine the duration and extent of high-energy activity on individual M dwarfs and the population as a whole. 

Determinant maximization problem gives a general framework that models problems arising in as diverse fields as statistics [Puk06], convex geometry [Kha96], fair allocations [AGSS16], combinatorics [AGV18], spectral graph theory [NST19a], network design, and random processes [KT12]. In an instance of a determinant maximization problem, we are given a collection of vectors U = {v1, . . . , vn} ⊂ Rd , and a goal is to pick a subset S ⊆ U of given vectors to maximize the determinant of the matrix ∑i∈S vivi^T. Often, the set S of picked vectors must satisfy additional combinatorial constraints such as cardinality constraint (|S| ≤ k) or matroid constraint (S is a basis of a matroid defined on the vectors). In this paper, we give a polynomial-time deterministic algorithm that returns a r O(r)-approximation for any matroid of rank r ≤ d. This improves previous results that give e O(r^2)-approximation algorithms relying on e^O(r)-approximate estimation algorithms [NS16, AG17,AGV18, MNST20] for any r ≤ d. All previous results use convex relaxations and their relationship to stable polynomials and strongly log-concave polynomials. In contrast, our algorithm builds on combinatorial algorithms for matroid intersection, which iteratively improve any solution by finding an alternating negative cycle in the exchange graph defined by the matroids. While the det(.) function is not linear, we show that taking appropriate linear approximations at each iteration suffice to give the improved approximation algorithm.