The Road To Digitopolis: Perils of Problem Solving

How students solve problems is a topic of central concern both to educational researchers and to math/science teachers: What is the nature of good and poor problem solving? How can students improve their problem-solving capacities? Teachers are in a unique position to witness problem solving in action, and to draw connections between the classroom experiences of their students and the findings of research. This article presents an instance of problem solving (drawn from a popular children's book) annotated with references to current research in cognition and education. The annotations explore issues such as the effect of performance anxiety on problem solving, how problem solvers handle the experience of confusion, and the role of self-monitoring and metacognition in problem solving.     

Journey toward Teaching Mathematics through Problem Solving

Teaching mathematics through problem solving is a challenge for teachers who learned mathematics by doing exercises. How do teachers develop their own problem solving abilities as well as their abilities to teach mathematics through problem solving? A group of teachers began the journey of learning to teach through problem solving while taking a Teaching Elementary School Mathematics graduate course. This course was designed to engage teachers in problem solving during class meetings and required them to do problem solving action research in their classrooms. Although challenged by the course problem solving work, teachers became more comfortable with the mathematics and recognized the importance of group work while problem solving. As they worked with their students, teachers were more confident in their students' abilities to be successful problem solvers. For some teachers, a strong problem solving foundation was established. For others, the foundation was more tentative.     

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

An Improvement of Multigrid Methods Using Multiple Grids on Each Layer for Parallel Computing

Multigrid methods are widely used and well studied for linear solvers and preconditioners of Krylov subspace methods. The multigrid method is one of the most powerful approaches for solving large scale linear systems;however, it may show low parallel efficiency on coarse grids. There are several kinds of research on this issue. In this paper, we intend to overcome this difficulty by proposing a novel multigrid algorithm that has multiple grids on each layer.Numerical results indicate that the proposed method shows a better convergence rate compared with the existing multigrid method.     

一类多商品设施选址问题的基于线性松弛解的启发式方法

多商品设施选址问题是众多设施选址问题中一类重要而困难的问题.在这一问题中,顾客的需求可能包含不止一种商品.对于大规模问题,成熟的商业求解器往往不能在满意的时间内找到高质量的可行解.研究了无容量限制的单货源多商品设施选址问题的一般形式,并给出了应用于此类问题的两个启发式方法.这两个方法基于原选址问题的线性规划松弛问题的最优解,分别通过求解紧问题和邻域搜索的方式给出了原问题的一个可行上界.理论分析指出所提方法可以实施于任意可行问题的实例.数值结果表明所提方法可以显著地提高求解器求解此类设施选址问题的求解效率.     

Mathematical problem solving in small groups: Exploring the interplay of students' metacognitive behaviors,perceptions, and ability levels

This exploratory study extends our earlier work that identified the importance of metacognitive behaviors in mathematical problem solving in a small-group setting. In that study 27 seventh-grade students of varying ability were observed working in six small groups. The current investigation examines the perceptions of those students about themselves as problem solvers and about working in a small group. Data were obtained through videotapes of the students working in small groups and audiotapes of stimulated-recall interviews of the individual students. The results provided insight regarding the ways that beliefs, emotions and attitudes of students of varying ability influenced their own and their peers' metacognitive behaviors within their respective groups. The findings suggest a number of implications for teachers regarding the modality, level and frequency of assessment of group problem solving.     

偏微分方程组的Lie群与高阶对称群的Taylor多项式逐步精化算法

本文基于生成函数的Ｔａｙｌｏｒ展开式及逐步简化步骤，提出了计算偏微分方程组的Ｌｉｅ群与高阶对称群的Ｔａｙｌｏｒ多项式算法，把标准算法中的求解超定偏微分方程组的问题转化为求解代数方程组的问题，降低了求解的难度，提高了计算效率，并且易用计算机代数系统在计算机上全过程实现，并得到重要的对称群     

Improving the robustness of descent-based methods for semismooth equations using proximal perturbations

A common difficulty encountered by descent-based equation solvers is convergence to a local (but not global) minimum of an underlying merit function. Such difficulties can be avoided by using a proximal perturbation strategy, which allows the iterates to escape the local minimum in a controlled fashion. This paper presents the proximal perturbation strategy for a general class of methods for solving semismooth equations and proves subsequential convergence to a solution based upon a pseudomonotonicity assumption. Based on this approach, two sample algorithms for solving mixed complementarity problems are presented. The first uses an extremely simple (but not very robust) basic algorithm enhanced by the proximal perturbation strategy. The second uses a more sophisticated basic algorithm based on the Fischer-Burmeister function. Test results on the MCPLIB and GAMSLIB complementarity problem libraries demonstrate the improvement in robustness realized by employing the proximal perturbation strategy. Received July 15, 1998 / Revised version received June 28, 1999?Published online November 9, 1999     

Fast Solvers for an Elliptic Problem from Dipole Localization

The model‐based reconstruction of electrical brain activity from electroencephalographic measurements is of constantly growing importance in the fields of Neurology and Neurosurgery. Algorithms for this task involve the solution of a 3D Poisson problem on a complicated geometry and with non‐continuous coefficients for a considerable number of different right hand sides. Thus efficient solvers for this subtask are required. We will report on our experiences with different iterative solvers for a discretization based on cell‐centered finite‐differences.     

Logical Processing for Integer Programming

This paper reviews tools which have great potential for reducing the difficulty of solving IP (and also MIP) problems, if well implemented in solvers. Recent experiments with Branch and Bound solvers, in connection with “Short Start Features”, have shown that implementations need and can still be improved. Concepts which are likely to be specially important for (0,1) MIP are pointed out.     

Exercising QS: quantitative skills in an exercise science course

This study seeks to bring the discipline of exercise science into the discussion of Quantitative Skills (QS) in science. The author's experiences of providing learning support to students and working with educators in the field are described, demonstrating the difficulty of encouraging students to address their skills deficit. A survey of students’ perceptions of their own QS and of that required for their course, demonstrates the difficulties faced by students who do not have the prescribed assumed knowledge for the course. Limited results from academics suggest that their perceptions of students’ QS deficits are even direr than those of the under-prepared students.     

An integer linear formulation for the file transfer scheduling problem

In this paper, we propose a new integer linear programming (ILP) formulation for solving a file transfer scheduling problem (FTSP), which is to minimize the overall time needed to transfer all files to their destinations for a given collection of various sized files in a computer network. Each computer in this network has a limited number of communication ports. The described problem is proved to be NP-hard in a general case. Our formulation enables solving the problem by standard ILP solvers like CPLEX or Gurobi. To prove the validity of the proposed ILP formulation, two new reformulations of FTSP are presented. The results obtained by CPLEX and Gurobi solvers, based on this formulation, are presented and discussed.     

The impact of integrated STEM professional development on teacher quality

This study of a state‐funded, 3‐year implementation of an integrated STEM professional development (PD) program for teachers from two middle schools in the midwestern U.S. examined if participants in the PD were enabled to transform their practice and perceptions of STEM. An integrated STEM approach includes a focus on the STEM disciplines, along with leveraging social studies/history and English/language arts as important context and tools for solving society’s biggest challenges. Findings in this study indicated that teachers implemented more effective STEM teaching strategies and had more positive perceptions regarding STEM overall. Further, participants became more aware of their personal needs for resources and support to teach through integrated STEM. Implications for research and practice are discussed.     

A CAPACITY EXPANSION PROBLEM WITH BUDGET CONSTRAINT AND BOTTLENECK LIMITATION

1 IntroductionIn [2] tl1e authors considered a type of coustrained maximim capacity path problem whichcan be described briefly as: kuowing the costs for expallding one unit of capacity along differentedge8 of a l1etwork aud the availabIe budget, how to iucrease the caparities of the edges so thattlle capasity between any pair of nodes in the lletwork can be raised unifornily to the maximumextent? As the total cost is a summation of the expansion costs on all edges, this problem i8related to mi…     

A competitive and cooperative approach to propositional satisfiability

Propositional satisfiability (SAT) has attracted considerable attention recently in both Computer Science and Artificial Intelligence, and a lot of algorithms have been developed for solving SAT. Each SAT solver has strength and weakness, and it is difficult to develop a universal SAT solver which can efficiently solve a wide range of SAT instances. We thus propose parallel execution of SAT solvers each of which individually solves the same SAT instance simultaneously. With this competitive approach, a variety of SAT instances can be solved efficiently in average. We then consider a cooperative method for solving SAT by exchanging lemmas derived by conflict analysis among different SAT solvers. To show the usefulness of our approach, we solve SATLIB benchmark problems, planning benchmark problems as well as the job-shop scheduling problem with good performance. The system has been implemented in Java with both systematic and stochastic solvers.     

Computing solutions to algebraic problems using a symbolic versus a schematic strategy

To improve access to algebraic word problems, primary aged students in Singapore are taught to utilise schematic models. Symbolic algebra is not taught until the secondary school years. To examine whether the two methods drew on different cognitive processes and imposed different cognitive demands, we used functional magnetic resonance imaging to examine patterns of brain activation whilst problem solvers were using the two methods. To improve our ability to detect differences attributable to the two methods, rather than participant’s abilities to use the two methods, we used adult problem solvers who had high levels of competency in both methods. In a previous study, we focused on the initial stages of problem solving: translating word problems into either schematic or symbolic representations (Lee et al. in Brain Res 1155:163–171, 2007). In this study, we focused on the later stages of problem solving: in computing numeric solutions from presented schematic or symbolic representations. Participants were asked to solve simple algebraic questions presented in either format. Greater activation in the symbolic method was found in the middle and medial frontal gyri, anterior cingulate, caudate, precuneus, and intraparietal sulcus. Greater activation in the model condition was found largely in the occipital areas. These findings suggest that generating and computing solutions from symbolic representations require greater general cognitive and numeric processing resources than do processes involving model representations. Differences between the two methods appear to be of both a quantitative and qualitative nature.     

Problem solving in mathematics education in Italy: dreams and reality

Drawing on personal experiences in in-service teacher training and curricular innovation in Italy, this paper addresses some questions relevant to mathematics education in the specific area of problem solving research. What are the effects of research results on national programs and curricula? To what extent, and how, are these assimilated in the school system? The analysis of some specific aspects of the evolution of the Italian situation in the last 30 years will suggest some possible answers suitable for comparison with other countries.     

On relaxations of the max k-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.     

On solving sparse algebraic equations over finite fields

A system of algebraic equations over a finite field is called sparse if each equation depends on a small number of variables. In this paper new deterministic algorithms for solving such equations are presented. The mathematical expectation of their running time is estimated. These estimates are at present the best theoretical bounds on the complexity of solving average instances of the above problem. In characteristic 2 the estimates are significantly lower the worst case bounds provided by SAT solvers.