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More than half of the coronavirus disease 19 (COVID-19) related mortality rates in the United States and Europe are associated with long-term-care facilities (LTCFs) such as old-age organizations, nursing homes, and disability centers. These facilities are considered most vulnerable to spread of an pandemic like COVID-19 because of multiple reasons including high density of elderly population with a diverse range of medical requirements, limited resources, nursing activities/medications, and the role of external visitors. In this study, we aim to understand the role of visitor’s family members and specific interventions (such as use of face masks and restriction of visiting hours) on the dynamics of infection in a community using a mathematical model. The model considers two types of social contexts (community and LTCFs) with three different groups of interacting populations (non-mobile community individuals, mobile community individuals, and long-term facility residents). The goal of this work is to compare the outbreak burden between different centre of disease control (CDC) planning scenarios, which capture distinct types of intensity of diseases spread in LTCF observed during COVID-19 outbreak. The movement of community mobile members is captured via their average relative times in and out of the long-term facilities to understand the strategies that would work well in these facilities the CDC planning scenarios. Our results suggest that heterogeneous mixing worsens epidemic scenario as compared to homogeneous mixing and the epidemic burden is hundreds times greater for community spread than within the facility population.

This article details the design and implementation of a professional development model called Learning Trajectory-Based Lesson Study focused on issues of equity, identity, and agency. We developed the Vertical Articulation to Unpack the Learning Trajectory (VAULT) tool to orient teachers' instructional planning toward an asset-based view of students' mathematics competencies. We examined teachers' use of the VAULT to plan, implement, and debrief on student strategies for one spatial reasoning task in elementary, middle, and high school classrooms. The VAULT facilitated intentional planning for a progression of anticipated strategies and equitable access to instruction. Teachers demonstrated an asset-based view of all student thinking independent of grade-level expectations.

In this work, the dynamics of the spread of COVID-19 is considered in the presence of both human-to-human transmission as well as environment-to-human transmission. Specifically, we expand and modify traditional epidemiological model for COVID-19 by incorporating a compartment to study the dynamics of pathogen concentration in the environmental reservoir, for instance concentration of droplets in closed spaces. We perform a mathematical analysis for the model proposed including an endemic equilibrium analysis as well as a next-generation approach both of which help to derive the basic reproduction number. We also study the e˚cacy of wearing a facemask through this model. Another important contribution of this work is the introduction to physics informed deep learning methods (PINNs) to study the dynamics. We propose this as an alternative to traditional numerical methods for solving system of differential equations used to describe dynamics of infectious diseases. Our results show that the proposed PINNs approach is a reliable candidate for both solving such systems and for helping identify important parameters that control the disease dynamics.

With the increasing popularity of active teaching methods, universities have become more interested in changing instruction from direct lecture to interactive engagement. The George Mason University (Mason, Virginia, USA) Biology Department is achieving this goal through: 1) faculty participation in the SIMPLE project; 2) faculty and student involvement in Discipline-Based Education Research projects; 3) participating in the Learning Assistant program for undergraduate students as scholars; and 4) the Accelerator Program. These approaches are discussed in this paper. Also, interactive teaching strategies, which the biology faculty and graduate teaching assistants use at Mason, are documented in the form of design memos; these memos are introduced in this paper

The sandwich problem described in this article would likely be considered more appropriate for third grade or above. However, as members of the research team for a lesson study involving teachers from kindergarten through fourth grade, the authors saw kindergarten students rise to the challenge. The authors invite readers to review two scenarios, and imagine themselves observing each one. They ask readers to examine the student work and the dialogue provided and ask themselves the following questions: (1) What knowledge is revealed about the students? (2) How might you respond to groups 1 and 2? (3) What might be the learning goal? (4) What questions might you pose? and (5) How would you interpret and respond to student thinking?

As COVID-19 cases continue to rise globally, many researchers have developed mathematical models to help capture the dynamics of the spread of COVID-19. Specifically, the compartmental SEIR model and its variations have been widely employed. These models differ in the type of compartments included, nature of the transmission rates, seasonality, and several other factors. Yet, while the spread of COVID-19 is largely attributed to a wide range of social behaviors in the population, several of these SEIR models do not account for such behaviors. In this project, we consider novel SEIR-based models that incorporate various behaviors. We created a baseline model and explored incorporating both explicit and implicit behavioral changes. Furthermore, using the Next Generation Matrix method, we derive a basic reproduction number, which indicates the estimated number of secondary cases by a single infected individual. Numerical simulations for the various models we made were performed and user-friendly graphical user interfaces were created. In the future, we plan to expand our project to account for the use of face masks, age-based behaviors and transmission rates, and mixing patterns.

In this paper, we introduce novel instructional approaches to engage students in using modelling with data to motivate and teach differential equations. Specifically, we introduce a pedagogical framework that will execute instructional modules to teach different solution techniques for differential equations through repositories and notebook environments during real-time instruction. Each of these teaching modules employs a literate programming approach that uses the notebook environment to explain the concepts in a natural language, such as English, interspersed with snippets of macros and traditional source code on a web browser. The pedagogical approach employed is reproducible and leads to openaccess material for students to motivate and teach differential equations efficiently. We will share examples of this framework applied to teaching advanced concepts such as machine learning and neural network approaches for solving ordinary and partial differential equations as well as estimating parameters in these equations for given datasets. More details of the work can be accessed from . [ABSTRACT FROM AUTHOR]

In this article, we provide some useful perspectives and experiences in mentoring students in undergraduate research (UR) in mathematical modeling using differential equations. To engage students in this topic, we present a systematic approach to the creation of rich problems from real-world phenomena; present mathematical models that are derived from calculus, but can be suitably modified to solve more complex problems; and finally provide pointers on how to train students through the UR process that can help to impact student's future academic and non-academic careers.

In this work, we review and share some useful technology for mathematics educators that can help to enhance their pedagogical practices to motivate students to improve their further understanding of differential equations after an initial exposure to the topic in an undergraduate class. In particular, we present some innovative ways to employ technology tools to motivate the learning of students and their appreciation of the applications of differential equations to real-world problems. The objective is to engage the students in exploration to solve differential equations using technology. This work presents a benchmark example involving modelling epidemics as a real-world problem and how to engage students who have basic familiarity with differential equations through literate programming. Visualization of the results using graphical user interface is also presented. The readers are also exposed to a novel free network computing laboratory called NCLab which is applied to a variety of applications involving differential equations.

This study examines elementary- and middle-grade teachers' understanding of the mathematical learning progression as they participated in a 6-month professional learning project. Teachers participated in a professional development project that consisted of a 1-week summer content-focused institute with school-based follow-up Lesson Study cycles in the fall that focused on the vertical articulation of algebraic concepts across grade levels. The following research examines how a vertical team of teachers from multiple grades designed, taught and learned from the Lesson Study cycle. The video analysis from the research lessons and the teachers' reflections revealed teachers' developing vertical connections and representational fluency from their planning, teaching, observation and debriefs. In addition, Lesson Study afforded teachers opportunities to deepen their understanding of the mathematical learning progression through observation and analysis of students' thinking through a situated school-based professional development experience.