Developing mathematical problem solving skills

Problem solving often requires a representation of given information in a structured form which can stimulate and support the problem solving process. FunctionLab is a computer based learning environment meeting this requirement and designed to assist problem solving and the development of problem solving skills in algebra word problems. A feature of the system is its interface design which incorporates tools for students to represent problem information as generic and dynamic process models which support investigatory learning and illustrate the structural characteristics of solutions. The design of FunctionLab is described together with an initial evaluation study of its effects on problem solving performances and methods of solution.     

A taxonomy for computer-based assessment of problem solving

Computer-based assessment of problem solving is motivated by the need for educational assessments that are valid and efficient. Based on a recent revision of Bloom's taxonomy (Anderson et al., 2001, A taxonomy for learning, teaching, and assessing: a revision of Bloom's taxonomy of educational objectives. New York: Longman), assessment items should require applying a particular cognitive process to a particular type of knowledge. There are 19 types of cognitive processes that can be classified into six major categories: remember, understand, apply, analyze, evaluate, and create. There are four major categories of knowledge: factual, conceptual, procedural, and metacognitive. Examples of computer-based assessments of problem-solving are provided based on the evaluation of the cognitive consequences of children's participation in an after-school computer club.     

Structured development of problem solving methods

Problem solving methods (PSMs) describe the reasoning components of knowledge-based systems as patterns of behavior that can be reused across applications. While the availability of extensive problem solving method libraries and the emerging consensus on problem solving method specification languages indicate the maturity of the field, a number of important research issues are still open. In particular, very little progress has been achieved on foundational and methodological issues. Hence, despite the number of libraries which have been developed, it is still not clear what organization principles should be adopted to construct truly comprehensive libraries, covering large numbers of applications and encompassing both task-specific and task-independent problem solving methods. In this paper, we address these "fundamental" issues and present a comprehensive and detailed framework for characterizing problem solving methods and their development process. In particular, we suggest that PSM development consists of introducing assumptions and commitments along a three-dimensional space defined in terms of problem-solving strategy, task commitments, and domain (knowledge) assumptions. Individual moves through this space can be formally described by means of adapters. In the paper, we illustrate our approach and argue that our architecture provides answers to three fundamental problems related to research in problem solving methods: 1) what is the epistemological structure and what are the modeling primitives of PSMs? 2) how can we model the PSM development process? and 3) how can we develop and organize truly comprehensive and manageable libraries of problem solving methods?     

Online assessment of problem solving skills

This overview evaluates the effectiveness of an alternative interactive scenario based assessment system (ISBAS). ISBAS addresses the limitations of traditional assessment methods, allowing students to troubleshoot complex scenarios, ask questions, and make diagnoses through an interactive web interface. The choices the student makes are tracked and scored based on predetermined, objective values. An analysis of student choices/decisions is used to assign a grade/proficiency level and provide the student feedback. ISBAS is efficient, flexible, and objectively evaluates the sophistication of complex knowledge and skills.     

改进的超视距空战态势评估方法

根据现代超视距空战的特点,结合目前超视距空战态势评估模型的不足,提出了一种改进的态势评估模型。该模型针对战机在超视距空域内寻找目标、互射导弹实施打击的特点,以载机速度、高度、空战能力、雷达探测性能,以及导弹攻击性能等作为主要研究对象,分析并建立了武器运用优势函数、飞行状态优势函数以及空战能力优势函数,用加权综合的方法得到双方的优势函数,模型更符合现代超视距空战实战特点。大量的数据仿真和实例应用结果验证了该评估算法的有效性。     

基于OpenMP的并行蚁群算法求解协同空战火力分配

根据动态火力分配中"动静结合"的思想,建立了一种带毁伤概率门限的火力分配模型。针对协同空战的第一阶段,在求得对目标机群最大毁伤效果的同时尽量节约导弹武器资源,以应对下一阶段的火力分配。根据粗粒度的并行策略,采用OpenMP并行优化技术对蚁群系统(ACS)中最耗时的循环迭代、循环赋值部分进行并行化处理,在此基础上,将优化后的蚁群算法应用到空战火力分配中,通过对各种规模的火力分配问题进行仿真实验,并验证所提出的火力分配模型的合理性和并行蚁群算法的有效性。     

Models and methods for solving the problem of network vulnerability

The problem of analysis of the vulnerability of single-commodity and multi-commodity networks taking into account complete breakdown of one or several arcs is considered. The guaranteed estimate of the loss of users of this network is chosen as the criterion of the network performance. A number of formal statements of the problem are proposed for vulnerability analysis. These statements of the problem are studied by flow, graph theoretical, probabilistic, and other methods. The cases when these statements of the problem are polynomially solvable are specified, and the corresponding algorithms are given. In the other cases, it is proposed to use approximation, combinatorial, and other methods.     

A generic library of problem solving methods for scheduling applications

In this paper, we propose a generic library of problem-solving methods for scheduling applications. Although some attempts have been made in the past at developing the libraries of scheduling problem-solvers, these only provide limited coverage. Many lack generality, as they subscribe to a particular scheduling domain. Others simply implement a particular problem-solving technique, which may be applicable only to a subset of the space of scheduling problems. In addition, most of these libraries fail to provide the required degree of depth and precision. In our approach, we subscribe to the task-method-domain-application knowledge modeling framework which provides a structured organization for the different components of the library. At the task level, we construct a generic scheduling task ontology to formalize the space of scheduling problems. At the method level, we construct a generic problem-solving model of scheduling that generalizes from the variety of approaches to scheduling problem-solving, which can be found in the literature. The generic nature of this model is demonstrated by constructing seven methods for scheduling as an alternative specialization of the model. Finally, we validated our library on a number of applications to demonstrate its generic nature and effective support for developing scheduling applications.     

大规模编队空战队形优化算法

大规模编队队形优化是空战战术研究的一个重要方面,但是对队形优化评估标准还没有建立一种有效的方法.在利用市场机制完成目标分配的基础上,提出使用编队目标函数作为队形优化评估标准.为了验证该评估标准的有效性,提出利用自适应遗传算法完成大规模编队协同空战队形优化.首先采用层级方法对大规模飞机编队队形进行编码和解码,然后利用编队目标函数构造出了适应度函数.最后通过3组例子阐明了队形优化算法的有效性.     

USSOR methods for solving the rank deficient linear least squares problem

In order to find the least squares solution of minimal norm to linear system \(Ax=b\) with \(A \in \mathcal{C}^{m \times n}\) being a matrix of rank \(r< n \le m\), \(b \in \mathcal{C}^{m}\), Zheng and Wang (Appl Math Comput 169:1305–1323, 2005) proposed a class of symmetric successive overrelaxation (SSOR) methods, which is based on augmenting system to a block \(4 \times 4\) consistent system. In this paper, we construct the unsymmetric successive overrelaxation (USSOR) method. The semiconvergence of the USSOR method is discussed. Numerical experiments illustrate that the number of iterations and CPU time for the USSOR method with the appropriate parameters is respectively less and faster than the SSOR method with optimal parameters.     

An assessment of higher-order isoparametric elements for solving an elastic problem

In this report isoparametric elements of various orders are used to solve an elastic problem in plane stress. The particular engineering structure has been selected to represent problems in which damage growth takes place at stress concentrators by the mechanisms of creep and fatigue. Since the inelastic solution of such problems may be treated as a succession of many elastic solutions, attention is focussed on the identification of the most appropriate element to obtain elastic solutions for a single geometry. The elastic solutions obtained for one geometry are compared with each other and with an exact solution in order to assess their relative performance in solving the problem. Both triangular and quadrilateral shaped elements are included in the trials. The element which gives the best results is the cubic quadrilateral, but the quadratic quadrilateral with reduced integration proved to be the most economical.     

Trust region methods for solving the optimal output feedback design problem

We consider the problem of designing feedback control laws when a complete set of state variables is not available. For linear autonomous control systems with quadratic performance criterion, the design problem consists of choosing an appropriate static output feedback (SOF) gain matrix according to a certain objective function. The corresponding non-linear matrix optimization problem can be interpreted as an equality constrained minimization problem. For solving this problem, we propose a constrained trust region (CTR) algorithm, which presents a new and efficient numerical approach for this problem class. On the other hand, based on the formulation used in the past, the SOF problem can be also interpreted as an unconstrained programming problem. Thus, based on this interpretation, we also develop an unconstrained trust region (UTR) method. Finally, several numerical examples for optimal SOF problems demonstrate the applicability of the considered algorithm.     

Tracing success: graphical methods for analysing successful collaborative problem solving

The aim of this paper is to evaluate the use of trace diagrams for analysing collaborative problem solving. The paper describes a study where trace diagrams were used to analyse joint navigation in a virtual environment. Ten pairs of undergraduates worked together on a distributed virtual task to collect five flowers using two bees with each participant controlling one of the bees. This task is used extensively in research on multi-robot systems. The joint navigation of the pairs was analysed using trace diagrams. They showed that more successful pairs divided the task up, showed very little overlap and very little backtracking. Whereas, the less successful pairs, showed no task division, there was significant overlap and extensive backtracking. From this analysis we developed numerical measures of task division, overlap and backtracking. Task division was significantly and negatively related with task performance. Backtracking and overlap were significantly and positively correlated.     

Large-scale convex optimization methods for air quality policy assessment

This paper presents the implementation of a large-scale systems analytic method in a model permitting the assessment of air quality policies in urban regions. This method is based on the use of a convex optimization technique called the analytic center cutting plane method. One realizes the coupling of a photochemical model (TAPOM-Lite), used to simulate ozone creation and dispersion in the region under study, and of a technoeconomic model (MARKAL-Lite) that represents the technology and energy uses by different economic sectors in the same region. Although the models correspond to different time and space scales, one realizes the coupling through a series of approximating convex reduced order optimization problems with constraints that are implicitly defined by the photochemical and the technoeconomic model, respectively. The implementation of the method in a case study corresponding to the region of Geneva, Switzerland is described.     

Cognitive process validation of an online problem solving assessment

In this study we tested a novel cognitive validation strategy that yoked participants' verbal protocols with their clickstream data using a problem solving assessment (IMMEX—Interactive Multimedia Exercises). Participants were presented with a scenario and provided with relevant and irrelevant information to solve the task. Participants could access the information in any order and attempt to solve the problem at any time. The most frequently occurring cognitive processes were paraphrasing text, accurate cause–effect inferences, and monitoring of problem solving behavior. Productive processes were related to success and consistent with scientific reasoning behavior on the task and unproductive processes were related to unsuccessful performance and consistent with poor reasoning. We found strong evidence of the cognitive validity of the IMMEX task. Components of an online process-based assessment testbed are identified.     

Training individuals for distributed teams: problem solving assessment for distributed mission research

In this paper we describe an effort investigating the feasibility and utility of cognitively diagnostic assessment of problem solving when training for distributed team tasks. We utilized computer-based knowledge elicitation methods to assess both relational problem solving, requiring the semantic integration of concepts, and dynamic problem solving, requiring the ability to integrate and apply these concepts. Additionally, we addressed how metacognitive processes interact with learning outcomes when training for complex synthetic task environments. We find first, that multiple methods of assessing problem solving performance are diagnostic of knowledge acquisition for a complex synthetic team task, and second, that general metacomprehension predisposition is related to metacomprehension accuracy in synthetic task environments.     

Chebyshev polynomial acceleration for block SOR methods for solving the rank-deficient least-squares problem

In this article, we give the acceleration of the block successive overrelaxation (SOR) method for solving the rank-deficient least-squares problem. Santos and Silva proposed the two-block SOR method and the three-block SOR method. Here, we consider the acceleration of the two-block SOR method and the three-block SOR method using the Chebyshev polynomial and derive what we term the C-2-block SOR method and the C-3-block SOR method. The advantage of our methods is that we can get good results with very small iteration number. The comparison between the C-2-block method and the C-3-block method is presented. Finally, numerical examples are given.     

Direct and indirect boundary element methods for solving the heat conduction problem

The boundary element method is used to solve the stationary heat conduction problem as a Dirichlet, a Neumann or as a mixed boundary value problem. Using singularities which are interpreted physically, a number of Fredholm integral equations of the first or second kind is derived by the indirect method. With the aid of Green's third identity and Kupradze's functional equation further direct integral equations are obtained for the given problem. Finally a numerical method is described for solving the integral equations using Hermitian polynomials for the boundary elements and constant, linear, quadratic or cubic polynomials for the unknown functions.     

A comparison of numerical methods for solving diffusion-reaction equations in air quality models

In this paper we analyze the efficiency of some stiff ode solvers, applied to the coupled solution of vertical turbulent diffusion and chemical kinetics in Air Quality Models. We consider four general-purpose solvers, based on bdf or Rosenbrock methods, and two special-purpose solvers, developed for odes from atmospheric chemistry, and compare their performance on three test problems using different chemical models. The general-purpose solvers have been modified to take advantage of the sparsity of the Jacobian matrices arising in the application of implicit methods. The obtained results show that general-purpose solvers, provided with suitable sparse matrix techniques, perform generally better than special-purpose ones. Our analysis extends to the diffusion-reaction equations the recent work of other research groups, comparing ode solvers on atmospheric chemical kinetics equations. Received: 28 January 1998 / Accepted: 26 April 1999     

Using XQuery for problem solving

XQuery is a recommendation by the World Wide Web Consortium as a standard XML query language. In addition to being a special‐purpose query language, XQuery has also features that support unexpected applications such as problem solving. We demonstrate and discuss these features by presenting several XQuery programs for solving recreational problems and puzzles. Copyright © 2011 John Wiley & Sons, Ltd.