Exploring the mathematics of motion through construction and collaboration

In this paper we give a detailed account of the design principles and construction of activities designed for learning about the relationships between position, velocity and acceleration, and corresponding kinematics graphs. Our approach is model-based, that is, it focuses attention on the idea that students constructed their own models – in the form of programs – to formalise and thus extend their existing knowledge. In these activities, students controlled the movement of objects in a programming environment, recording the motion data and plotting corresponding position-time and velocity-time graphs. They shared their findings on a specially-designed web-based collaboration system, and posted cross-site challenges to which others could react. We present learning episodes that provide evidence of students making discoveries about the relationships between different representations of motion. We conjecture that these discoveries arose from their activity in building models of motion and their participation in classroom and online communities

Community Detection on Evolving Graphs

Clustering is a fundamental step in many information-retrieval and data-mining applications. Detecting clusters in graphs is also a key tool for finding the community structure in social and behavioral networks. In many of these applications, the input graph evolves over time in a continual and decentralized manner, and, to maintain a good clustering, the clustering algorithm needs to repeatedly probe the graph. Furthermore, there are often limitations on the frequency of such probes, either imposed explicitly by the online platform (e.g., in the case of crawling proprietary social networks like twitter) or implicitly because of resource limitations (e.g., in the case of crawling the web). In this paper, we study a model of clustering on evolving graphs that captures this aspect of the problem. Our model is based on the classical stochastic block model, which has been used to assess rigorously the quality of various static clustering methods. In our model, the algorithm is supposed to reconstruct the planted clustering, given the ability to query for small pieces of local information about the graph, at a limited rate. We design and analyze clustering algorithms that work in this model, and show asymptotically tight upper and lower bounds on their accuracy. Finally, we perform simulations, which demonstrate that our main asymptotic results hold true also in practice

Perspectives on Deepening Teachers’ Mathematics Content Knowledge: The Case of the Oregon Mathematics Leadership Institute

The Oregon Mathematics Leadership Institute (OMLI) project served 180 Oregon teachers, and 90 administrators, across the K-12 grades from ten partner districts. OMLI offered a residential, three-week summer institute. Over the course of three consecutive summers, teachers were immersed in a total of six mathematics content classes– Algebra, Data & Chance, Discrete Mathematics, Geometry, Measurement & Change, and Number & Operations—along with an annual collegial leadership course. Each content class was designed and taught by a team of expert faculty from universities, community colleges, and K-12 districts. Each team chose a few “big ideas” on which to focus the course. For example, the Algebra team focused on algebraic structure and properties of the concept of a group, while the Data & Chance team centered their activities on the exploration of ideas of central tendency and variation using statistics and data analysis software packages. The content in all of the courses was addressed through deep investigation of the mathematics of tasks that had been selected and adapted from resources for K-12 mathematics classrooms. In addition to mathematics content, the courses were designed with specific attention to socio-mathematical norms, issues of status differences among learners, and the selection and implementation of group-worthy tasks for group work. The faculty attended sessions grounded in the work of Elizabeth Cohen on strategies for working with heterogeneous groups of learners (Cohen, 1994; Cohen et al, 1999) which was central to the OMLI design and implementation. Institute faculty modeled these strategies in the Institute classrooms and made their moves as transparent as possible, so that the teachers would be able to grapple with these strategies during the Institute and plan for implementation in their own classrooms. The Data & Chance course also modeled uses of technology in instruction using Tinkerplots. Generalization and justification were emphasized as mathematical ways of learning and knowing, and institute faculty conducted classroom discussions that intentionally modeled pushing for generalization and justification

Big Data Visualization Tools

Data visualization is the presentation of data in a pictorial or graphical format, and a data visualization tool is the software that generates this presentation. Data visualization provides users with intuitive means to interactively explore and analyze data, enabling them to effectively identify interesting patterns, infer correlations and causalities, and supports sense-making activities.Comment: This article appears in Encyclopedia of Big Data Technologies, Springer, 201