The role of framing in productive classroom discussions: A case for teacher learning

In this paper, we contrast two mathematical arguments that occurred during an algebra lesson to illustrate the importance of relevant framings in the ensuing arguments. The lesson is taken from a graduate course for elementary teachers who are enrolled in a mathematics specialist program. We use constructs associated with enthnography of argumentation to characterize the framings for warrants and backings that supported the ensuing arguments. Our findings suggest that teachers fully participated in argumentations that were framed by problem situations that were familiar to them, ones that were couched in elementary, fundamental mathematical ideas, and that these types of argumentations were arguably more productive in terms of opportunities for learning.     

Effectively coaching middle school teachers: A case for teacher and student learning

In this paper we report findings from a two-year, large-scale research project that describes the work of middle school mathematics specialists (also referred to as mathematics coaches or instructional coaches) who served in 10 school districts. We use mixed methods to describe how mathematics specialists spent their time supporting teachers and how these supports contributed to meaningful changes that teachers made in their instructional practices. We also report results that correlate student achievement scores with whether or not teachers were highly engaged with the mathematics specialists. We coordinate these quantitative results with findings from several case studies to illustrate the qualitatively different ways that mathematics specialists supported teachers’ ongoing work with their students. We also account for why some teachers participated more fully than others. Importantly, because mathematics specialists’ work was situated in different school settings with different demands, resources and administrative supports, these constraints and affordances contributed in part to how they could effectively support teachers’ work with their students.     

Benefits of a teacher and coach collaboration: A case study

This paper describes a case study of a math teacher working with a math coach and the effects of their interaction. A guiding question was whether the coaching intervention had affected the teacher's classroom practices and, if so, in what way. The study utilized data from teacher/coach planning sessions, classroom lessons, follow-up debriefing sessions, and interviews with the teacher, coach and school principal. These data enabled the author to study the impact, if any, of the coaching on teacher beliefs and practices.     

Operational research and ethics: A literature review

Any human activity raises ethical questions, questions about ‘good’ and ‘right’ ways to act and to live; or to put it differently, questions of values and responsibility. From its inception operational research (OR) has engaged with such questions in terms of professional behavior, the handling of preferences in OR, the societal role of OR, the process of OR intervention and the content of OR analysis. As a result, analytical methods and processes have been developed to help clients explore the ethical dimension of their decisions. The paper reviews the literature published in selected OR journals (Management Science; Operations Research; Interfaces; the European Journal of Operational Research; the Journal of the Operations Research Society; Omega; International Transactions in Operational Research; the Journal of Multi-Criteria Decision Analysis), organizing it along the lines of OR’s core competences. The review identifies a number of significant research programmes that are well established and are being energetically pursued; the research findings are being applied to a wide range of important issues. Ethical questions lie at the heart of the great governmental and commercial issues of the day: economic growth and instability; inequality and injustice; environmental degradation and sustainability. They also lie at the heart of the more mundane decisions of day-to-day OR. ‘Ethics’ therefore provides a useful focus for OR both in terms of raising the awareness of all concerned and in providing a theme for research. As a result of the review some research questions are suggested. There is much of interest, much to do.     

Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking

Recent work by researchers has focused on synthesizing and elaborating knowledge of students’ thinking on particular concepts as core progressions called learning trajectories. Although useful at the level of curriculum development, assessment design, and the articulation of standards, evidence is only beginning to emerge to suggest how learning trajectories can be utilized in teacher education. Our paper reports on two studies investigating practicing and prospective elementary teachers’ uses of a learning trajectory to make sense of students’ thinking about a foundational idea of rational number reasoning. Findings suggest that a mathematics learning trajectory supports teachers in creating models of students’ thinking and in restructuring teachers’ own understandings of mathematics and students’ reasoning.     

A combined approach for evaluating papers, authors and scientific journals

An integrated model for ranking scientific publications together with authors and journals recently presented in [Bini, Del Corso, Romani, ETNA 2008] is closely analyzed. The model, which relies on certain adjacency matrices H,K and F obtained from the relations of citation, authorship and publication, provides the ranking by means of the Perron vector of a stochastic matrix obtained by combining H,K and F. Some perturbation theorems concerning the Perron vector previously introduced by the authors are extended to more general cases and a counterexample to a property previously addressed by the authors is presented. The theoretical results confirm the consistency and effectiveness of our model. Some paradigmatic examples are reported together with some results obtained on a real set of data.     

Detecting Students' Experiences of Discontinuities Between Middle School and High School Mathematics Programs: Learning During Boundary Crossing

Transitions from middle school to high school mathematics programs can be problematic for students due to potential differences between instructional approaches and curriculum materials. Given the minimal research on how students experience such differences, we report on the experiences of two students as they moved out of an integrated, problem-based mathematics program in their middle school into a high school mathematics program that emphasized procedural fluency. We conducted an average of two interviews per year for two and a half years with participants and engaged in participant-observation at their high school. In this study, we illustrate an analytic process for detecting discontinuities between settings from participants' perspectives. We determined that students experienced a discontinuity if they reported meaningful differences between settings (frequent mention, in detail, with emphasis terms) and concurrently reported a change in attitude. Additionally, these students' experiences illustrate two opportunities to learn during boundary-crossing experiences: identification and reflection.     

A stopping rule for choosing the best of three coins

A recently proposed stopping rule for choosing the best of three coins is presented as the first exit time from a certain polygonal region by a two-dimensional random walk. A diffusion approximation to the expected value of this exit time is developed, using techniques of conformal mapping. An algebraic method for calculating the exact value of this expectation is also described.     

A comparison of three metaheuristics for the workover rig routing problem

The workover rig routing problem (WRRP) is a variant of the Vehicle Routing Problem with Time Windows (VRPTW) and arises in the operations of onshore oil fields. In this problem, a set of workover rigs located at different positions must service oil wells requesting maintenance as soon as possible. When a well requires maintenance, its production is reduced or stopped for safety reasons and some workover rig must service it within a given deadline. It is therefore important to service the wells in a timely fashion in order to minimize the production loss. Whereas for classical VRPTWs the objective is to minimize route length, in the WRRP the objective is to minimize the total lost production, equal to the sum of arrival times at the wells, multiplied by production loss rates. The WRRP generalizes the Delivery Man Problem with Time Windows by considering multiple open vehicle routes and multiple depots. This paper compares three metaheuristics for the WRRP: an iterated local search, a clustering search, and an Adaptive Large Neighborhood Search (ALNS). All approaches, in particular ALNS, have yielded good solutions for instances derived from a real-life setting.     

A three level linearized compact difference scheme for the Cahn-Hilliard equation

This article is devoted to the study of high order accuracy difference methods for the Cahn-Hilliard equation.A three level linearized compact difference scheme is derived.The unique solvability and unconditional convergence of the difference solution are proved.The convergence order is O(τ 2 + h 4 ) in the maximum norm.The mass conservation and the non-increase of the total energy are also verified.Some numerical examples are given to demonstrate the theoretical results.     

A stochastic lagrangean picture for the three dimensional navier-stokes equation +

We derive a representation formula for the solutions of the navier-Stokes flow in three dimensions in absence of boundary in terms of stochastic current lines. This pictures generalizes an analogous one given in two dimensions and on the other hand the classical Lafrangian picture for the Euler flow. We prove a Vanishing viscosity limit for the whole structure     

A comparison of three linear rules for classification

This paper compares three different linear procedures for classification: the normal one, the canonical one and a distribution-free one recently described by Heuchenne. The study is mainly conducted using a simulation which makes it possible to compute the probabilities of correct allocation of the three methods in 3888 different cases. The normal rule looks slightly better than Heuchenne's, which looks clearly better than the canonical one. Finally, inference on Heuchenne's method is examined and conditions under which this method is optimal are given.     

A rigorous stability result for the Vlasov-Poisson system in three dimensions

It is proven that in a neutral two-component plasma with space homogeneous positively charged background, which is governed by the Vlasov-Poisson system and for which Poisson's equation is considered on a cube inR ³ with periodic boundary conditions, the space homogeneous stationary solutions g with energy gradient g/ 0 and compact support are (nonlinearly) stable in the L¹-norm with respect to weak solutions of the initial value problem.