Teacher time out as a site for studying mathematical knowledge for teaching

The special mathematical knowledge that is needed for teaching has been studied for decades but the methods for studying it have challenges. Some methods, such as measurement and cognitive interviews, are removed from the dynamics of teaching. Other methods, such as observation, are closer to practice but mostly involve an outsider perspective. Moreover, few methods tap into the tacit and often invisible demands that teachers encounter in teaching. This article develops an argument that teacher time outs in rehearsals and enactments might be a productive site for studying mathematical knowledge for teaching. Teacher time outs constitute a site for professional deliberation, which 1) preserves the complexity and gets inside the dynamics of teaching, where 2) tacit and implicit challenges and demands are made explicit, and where 3) insider and outsider perspectives are combined.     

Fast Bootstrap Confidence Intervals for Continuous Threshold Linear Regression

Continuous threshold regression is a common type of nonlinear regression that is attractive to many practitioners for its easy interpretability. More widespread adoption of threshold regression faces two challenges: (i) the computational complexity of fitting threshold regression models and (ii) obtaining correct coverage of confidence intervals under model misspecification. Both challenges result from the nonsmooth and nonconvex nature of the threshold regression model likelihood function. In this article we first show that these two issues together make the ideal approach for making model-robust inference in continuous threshold linear regression an impractical one. The need for a faster way of fitting continuous threshold linear models motivated us to develop a fast grid search method. The new method, based on the simple yet powerful dynamic programming principle, improves the performance by several orders of magnitude. Supplementary materials for this article are available online.     

Statistical Measures for the Computer-Aided Diagnosis of Mammographic Masses

Abstract

We propose statistical measures for finding masses in mammograms. The measures are based on fitting broken line regressions to local intensity plots of the images. The method is illustrated on a small database of mammograms that have been read by a radiologist and confirmed by operative data. This work illustrates some of the statistical challenges in working with large diagnostic images     

A branch-and-bound approach for maximum quasi-cliques

Detecting quasi-cliques in graphs is a useful tool for detecting dense clusters in graph-based data mining. Particularly in large-scale data sets that are error-prone, cliques are overly restrictive and impractical. Quasi-clique detection has been accomplished using heuristic approaches in various applications of graph-based data mining in protein interaction networks, gene co-expression networks, and telecommunication networks. Quasi-cliques are not hereditary, in the sense that every subset of a quasi-clique need not be a quasi-clique. This lack of heredity introduces interesting challenges in the development of exact algorithms to detect maximum cardinality quasi-cliques. The only exact approaches for this problem are limited to two mixed integer programming formulations that were recently proposed in the literature. The main contribution of this article is a new combinatorial branch-and-bound algorithm for the maximum quasi-clique problem.     

The Omegahat Environment: New Possibilities for Statistical Computing

Abstract

Statistical environments such as S, R, XLisp-Stat, and others have had a dramatic effect on the way we, statistics practitioners, think about data and statistical methodology. However, the possibilities and challenges introduced by recent technological developments and the general ways we use computing conflict with the computational model of these systems. This article explores some of these challenges and identifies the need to support easy integration of functionality from other domains, and to export statistical methodology to other audiences and applications, both statically and dynamically. Existing systems can be improved in these domains with some already implemented extensions (see Section 5). However, the development of a new environment and computational model that exploits modern tools designed to handle many general aspects of these challenges appears more promising as a long-term approach. We present the architecture for such a new model named Omegahat. It lends itself to entirely new statistical computing paradigms. It is highly extensible at both the user and programmer level, and also encourages the development of new environments for different user groups. The Omegahat interactive language offers a continuity between the different programming tasks and levels via optional type checking and seamless access between the interpreted user language and the implementation language, Java. Parallel and distributed computing, network and database access, interactive graphics, and many other aspects of statistical computing are directly accessible to the user as a consequence of this seamless access. We describe the benefits of using Java as the implementation language for the environment and several innovative features of the user-level language which promise to assist development of software that can be used in many contexts. We also outline how this architecture can be integrated with existing environments such as R and S.

The ideas are drawn from work within the Omega Project for Statistical Computing. The project provides open-source software for researching and developing next generation statistical computing tools.     

A linear condition determining local or global existence for nonlinear problems

Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This work is the second part of a program, the first part being [Neuberger J.W., How to distinguish local semigroups from global semigroups, Discrete Contin. Dyn. Syst. (in press), available at http://arxiv.org/abs/1109.2184]. Future work points to a distant goal for problems as in [Fefferman C.L., Existence and Smoothness of the Navier-Stokes Equation, In: The Millennium Prize Problems, Clay Mathematics Institute, Cambridge/American Mathematical Society, Providence, 2006, 57–67].     

Beyond mathematical content knowledge: A mathematician's knowledge needed for teaching an inquiry-oriented differential equations course

In this research report we examine knowledge other than content knowledge needed by a mathematician in his first use of an inquiry-oriented curriculum for teaching an undergraduate course in differential equations. Collaboratively, the mathematician and two mathematics education researchers identified the challenges faced by the mathematician as he began to adopt reform-minded teaching practices. Our analysis reveals that responding to those challenges entailed formulating and addressing particular instructional goals, previously unfamiliar to the instructor. From a cognitive analytical perspective, we argue that the instructor's knowledge — or lack of knowledge — influenced his ability to set and accomplish his instructional goals as he planned for, reflected on, and enacted instruction. By studying the teaching practices of a professional mathematician, we identify forms of knowledge apart from mathematical content knowledge that are essential to reform-oriented teaching, and we highlight how knowledge acquired through more traditional instructional practices may fail to support research-based forms of student-centered teaching.     

Algorithms for checking rational roots of b-functions and their applications

The Bernstein–Sato polynomial of a hypersurface is an important object with many applications. However, its computation is hard, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called checkRoot for optimized checking whether a given rational number is a root of Bernstein–Sato polynomial and in the affirmative case, computing its multiplicity. These algorithms are used in the new approach to compute the global or local Bernstein–Sato polynomial and b-function of a holonomic ideal with respect to a weight vector. They can be applied in numerous situations, where a multiple of the Bernstein–Sato polynomial can be established. Namely, a multiple can be obtained by means of embedded resolution, for topologically equivalent singularities or using the formula of A?Campo and spectral numbers. We also present approaches to the logarithmic comparison problem and the intersection homology D-module. Several applications are presented as well as solutions to some challenges which were intractable with the classical methods. One of the main applications is the computation of a stratification of affine space with the local b-function being constant on each stratum. Notably, the algorithm we propose does not employ primary decomposition. Our results can be also applied for the computation of Bernstein–Sato polynomials for varieties. The examples in the paper have been computed with our implementation of the methods described in Singular:Plural.     

A framework for using learning theories to inform ‘growth mindset’ activities

ABSTRACT

The social psychology theory of fixed and growth mindsets offers one reason for observed underachievement in science, technology, engineering and mathematics (STEM), particularly for students who have previously excelled in these disciplines. Fixed mindset beliefs are linked to behaviours that can lead to avoiding challenges and reduced learning, such as concealing a lack of understanding to retain an image of being ‘smart’. The potential impact of a growth mindset on STEM achievement, particularly for minority and low-household-income students, resulted in calls to develop interventions that encourage growth mindsets and discourage fixed mindsets. However, education interventions are influenced by the educator's understanding of how learning occurs. A framework to show how activities based on different learning theories may encourage growth mindsets or (unintentionally) encourage fixed mindsets can guide the developers of growth mindset interventions. We present such a framework in six tables relating to key areas associated with growth and fixed mindsets: dealing with challenges, persistence, effort, praise, the success of others and learning goals. Each table gives examples of learning activities that may encourage growth or fixed mindsets, fitting with each of four key learning theories: behaviourism, constructivism, communities of practice and connectivism.     

Online Variational Bayes Inference for High-Dimensional Correlated Data

High-dimensional data with hundreds of thousands of observations are becoming commonplace in many disciplines. The analysis of such data poses many computational challenges, especially when the observations are correlated over time and/or across space. In this article, we propose flexible hierarchical regression models for analyzing such data that accommodate serial and/or spatial correlation. We address the computational challenges involved in fitting these models by adopting an approximate inference framework. We develop an online variational Bayes algorithm that works by incrementally reading the data into memory one portion at a time. The performance of the method is assessed through simulation studies. The methodology is applied to analyze signal intensity in MRI images of subjects with knee osteoarthritis, using data from the Osteoarthritis Initiative. Supplementary materials for this article are available online.     

Asymptotic Theory for Relative-Risk Models with Missing Time-Dependent Covariates

Relative-risk models are often used to characterize the relationship between survival time and time-dependent covariates. When the covariates are observed, the estimation and asymptotic theory for parameters of interest are available; challenges remain when missingness occurs. A popular approach at hand is to jointly model survival data and longitudinal data. This seems efficient, in making use of more information, but the rigorous theoretical studies have long been ignored. For both additive risk models and relative-risk models, we consider the missing data nonignorable. Under general regularity conditions, we prove asymptotic normality for the nonparametric maximum likelihood estimators.     

Advanced routing for city logistics service providers based on time-dependent travel times

Increasing traffic demand, recurring congestion and sophisticated e-commerce business models lead to enormous challenges for routing in city logistics. We introduce a planning system for city logistics service providers, which faces those challenges by more realistic vehicle routing considering time-dependent travel times. Time-dependent travel times arise from telematics-based traffic data collection city-wide. Well-known heuristics for the Traveling Salesman Problem and for the Vehicle Routing Problem are adapted to time-dependent planning data. Computational experiments allow for insights into the efficiency of individual heuristics, their adaptability to time-dependent planning data sets, and the quality and structure of resulting delivery tours.     

Lessons from the trenches: A survey of approaches for large-scale projects

Technical documents are generally difficult to author, publish and maintain due to the complexity inherent in mathematical notation, diagrams, and tables. This article surveys approaches for addressing these challenges used by large-scale, Web-based projects involving ongoing workflows. Such workflows fall into four broad categories: the XML Repository model, the TeX Repository model, the Page Layout model, and exotic models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical model and a metaheuristic approach for a memory allocation problem

Memory allocation in embedded systems is one of the main challenges that electronic designers have to face. This part, rather difficult to handle is often left to the compiler with which automatic rules are applied. Nevertheless, an optimal allocation of data to memory banks may lead to great savings in terms of running time and energy consumption. This paper introduces an exact approach and a vns-based metaheuristic for addressing a memory allocation problem. Numerical experiments have been conducted on real instances from the electronic community and on dimacs instances expanded for our specific problem.     

Large-scale high-fidelity atomistic simulations for emerging materials and electronics devices

The quest for ever higher levels of detail and realism in nanoelectronics simulations presents formidable modeling and computational challenges. These simulations are expected, however, to benefit from the development of specific advanced mathematical techniques and high-performance parallel algorithms to achieve high-fidelity simulations of materials and transport problems. Here, we propose to investigate novel simulation strategies which aim to go beyond qualitative predictions by proposing quantitative simulations from the atomistic scale to the transistor macroscale. Results obtained on carbon nanotubes to compute the electron density using these advanced techniques and first-principle calculations, are presented here. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strategic capacity planning for light emitting diode (LED) supply chains across Taiwan and China

This paper describes a mixed-integer programming (MIP) model formulated for strategic capacity planning for light emitting diode (LED) makers of Taiwan, major companies in the global LED market. These firms have complex supply chains across Taiwan and China, and the region’s unique political and economic environment has created not only competitive advantages but also challenges in supply chain management: government regulations require that customer orders be accepted from Taiwan or China according to customer attributes; when conducting manufacturing, Taiwanese firms may need to transfer orders across national borders for reasons such as manufacturing technology (the required technology is available only at certain manufacturing facilities) or more efficient capacity utilization; and there are operations to be performed with specific processing requirements to follow, posing substantial challenges for planners. Motivated by the significance of these firms in the global market, we develop a MIP model with novel features to support their strategic capacity planning, covering demand and manufacturing-related decisions, including order acceptance and transfer, manufacturing starts, capacity expansion, and logistics. We illustrate the model’s performance using modified industry data in a numerical example; we also describe the potential impacts the model may create in industry applications.     

Quasi-Monte Carlo for Highly Structured Generalised Response Models

Highly structured generalised response models, such as generalised linear mixed models and generalised linear models for time series regression, have become an indispensable vehicle for data analysis and inference in many areas of application. However, their use in practice is hindered by high-dimensional intractable integrals. Quasi-Monte Carlo (QMC) is a dynamic research area in the general problem of high-dimensional numerical integration, although its potential for statistical applications is yet to be fully explored. We survey recent research in QMC, particularly lattice rules, and report on its application to highly structured generalised response models. New challenges for QMC are identified and new methodologies are developed. QMC methods are seen to provide significant improvements compared with ordinary Monte Carlo methods.      

A validation study of the use of mathematical knowledge for teaching measures in Ireland

Researchers who study mathematical knowledge for teaching (MKT) are interested in how teachers deploy their mathematical knowledge in the classroom to enhance instruction and student learning. However, little data exists on how teachers’ scores on the US-developed measures relate to classroom instruction in other countries. This article documents a validation study of Irish teachers’ scores on measures of MKT that were adapted for use in Ireland. A validity argument is made identifying elemental, structural and ecological assumptions. The argument is evaluated using qualitative and quantitative data to analyse inferences related to the three assumptions. The data confirmed the elemental assumption but confirming the structural and ecological assumptions was more difficult. Only a weak association was found between teachers’ MKT scores and the mathematical quality of instruction. Possible reasons for this are outlined and challenges in validating the use of measures are identified.     

On the challenges of healthcare modelling and a proposed project life cycle for successful implementation

OR methods and tools are being increasingly applied within the health services domain to support objective decision-making for the effective and efficient provision of resources. Healthcare modelling, however, is beset with many challenges and this paper examines a number of issues that define the primary challenges faced by the modeller in this field. It is fair to say that their resolution determines the likely success or failure of healthcare modelling in general. Working with a number of participating health service organizations, a proposed framework towards successful implementation has evolved and is presented in this paper