Conceptualizing Perseverance in Problem Solving as Collective Enterprise

Students are expected to learn mathematics such that when they encounter challenging problems they will persist. Creating opportunities for students to persist in problem solving is therefore argued as essential to effective teaching and to children developing positive dispositions in mathematical learning. This analysis takes a novel approach to perseverance by conceptualizing it as collective enterprise among learners in lieu of its more conventional treatment as an individual capacity. Drawing on video of elementary school children in two US classrooms (n = 52), this paper offers: (1) empirical examples that define perseverance as collective enterprise; (2) indicators of perseverance for teachers (and researchers) to support (and study) its emergence; and (3) evidence of how the task, peer dynamics, and student-teacher interactions afford or constrain its occurrence. The significance of perseverance as collective enterprise and as an object of design in developing effective learning communities, is discussed.     

Conceptual learning in a principled design problem solving environment

To what extent can instructional design be based on principles for instilling a culture of problem solving and conceptual learning? This is the main focus of the study described in this paper, in which third grade students participated in a one-year course designed to foster problem solving and mathematical reasoning. The design relied on five principles: (a) encouragement to produce multiple solutions; (b) creating collaborative situations; (c) socio-cognitive conflicts; (d) providing tools for checking hypotheses; and (e) inviting students to reflect on solutions. We describe how a problem solving task designed according to the above principles, promoted students' understanding of the area concept. We show that the design afforded the surfacing of multiple solutions and justifications in various modalities (including gestures) and initiated peer argumentation, leading to deep learning of the area concept.     

Effect of self-constructed concept map with peer feedback on chemistry learning outcome

Concept mapping remains widely used in education. However, little is known about how self-constructed concept maps and peer feedback can improve student learning outcomes in chemistry. We investigated the effects of peer feedback on concept mapping and how it improves students' learning performance in a large second-semester, introductory chemistry course. Three hundred and twenty students were randomly assigned to one of two concept mapping conditions: self-constructed concept map with peer feedback and self-constructed concept map without peer feedback. Each group constructed concept maps that depicted the relationship between concepts on the topic of intermolecular force. The results showed that students in the self-constructed concept map with peer feedback condition outperformed students in the no peer feedback condition in chemistry learning outcome. Overall, this study demonstrates that peer feedback enhances the effectiveness of learning with generative concept maps. The implications and future directions are discussed.     

问题驱动 多元表征的概念教学探究——对偶线性规划教学案例设计

数学与应用数学(师范)专业中的《运筹学》具有跨学科、实践性的课程特点,目标在于培养职前教师用数学方法解决实际问题的能力.结合义务教育阶段新课程标准中"四基"的提出这一背景,本文将以线性规划部分(运筹数学)对偶线性规划概念的引入这一知识模块为例,探讨通过问题串形式进行问题驱动、多元表征的概念教学过程.即遵循问题驱动—兴趣驱动—问题意识发展—提出和解决新问题,依据数学与外部联系、数学内部联系两条主线设计教学和学习,探索如何通过问题驱动、多元表征的结构化教学过程引导学生的学习方式发生改变,增强探究学习的动机,发展问题解决能力.课堂教学实践证明效果优于以往单一的讲授式教学法,一定程度上提高了学生的学业成绩、应用问题的兴趣和问题解决意识.     

Beliefs and engagement structures: behind the affective dimension of mathematical learning

Beliefs influencing students’ mathematical learning and problem solving are structured and intertwined with larger affective and cognitive structures. This theoretical article explores a psychological concept we term an engagement structure, with which beliefs are intertwined. Engagement structures are idealized, hypothetical constructs, analogous in many ways to cognitive structures. They describe complex “in the moment” affective and social interactions as students work on conceptually challenging mathematics. We present engagement structures in a self-contained way, paying special attention to their theoretical justification and relation to other constructs. We suggest how beliefs are characteristically woven into their fabric and influence their activation. The research is based on continuing studies of middle school students in inner-city classrooms in the USA.     

Learning to improve reliability during system development

Research, conducted in collaboration with a leading aerospace manufacturer, aimed to facilitate learning in order to improve the reliability of engineering systems during their development phase. In particular, the processes and mathematical models used during reliability growth testing were investigated to assess how they might be better used to support this improvement. This required both soft and hard OR approaches to be adopted. For example, information flows were mapped and reengineered in order to provide a basis for more effective data collection and feed-back to decision-makers. A new mathematical model that combines failure data with engineering judgement was developed to estimate reliability growth. The paper presents a case study describing the problem, the modelling conducted, the recommendations made and the actions implemented. The ways in which the researchers and the manufacturer learnt to improve both the modelling and the reliability growth testing process are reflected upon.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

A multiobjective decomposition-coordination framework with an application to vehicle layout and configuration design

This short paper addresses both researchers in multiobjective optimization as well as industrial practitioners and decision makers in need of solving optimization and decision problems with multiple criteria. To enhance the solution and decision process, a multiobjective decomposition-coordination framework is presented that initially decomposes the original problem into a collection of smaller-sized subproblems that can be solved for their individual solution sets. A common solution for all decomposed and, thus, the original problem is then achieved through a subsequent coordination mechanism that uses the concept of epsilon-efficiency to integrate decisions on the desired tradeoffs between these individual solutions. An application to a problem from vehicle configuration design is selected for further illustration of the results in this paper and suggests that the proposed method is an effective and promising new solution technique for multicriteria decision making and optimization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于改进的粒子群算法求解供应链网络均衡问题

传统的供应链求解方法为投影法,针对其要对投影进行计算,十分复杂的缺点,提出用改进的粒子群算法求解供应链均衡问题,利用动态异步调整学习因子来有效的提高了算法搜索能力与精度。本文介绍了供应链网络均衡问题转变为无约束优化问题的方法,然后用改进的粒子群优化算法进行求解。通过四个数值算例,将实验结果与标准粒子群算法、蜂群算法、学习因子同步变化的粒子群算法进行比较,验证了改进的粒子群优化算法在解决供应链网络均衡问题中的有效性与优越性,为供应链网络求解提供了一种新的方法。     

An empirically-based trajectory for fostering abstraction of equivalent-fraction concepts: A study of the Learning Through Activity research program

Promoting deep understanding of equivalent-fractions has proved problematic. Using a one-on-one teaching experiment, we investigated the development of an increasingly sophisticated, sequentially organized set of abstractions for equivalent fractions. The article describes the initial hypothetical learning trajectory (HLT) which built on the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction), analysis of the empirical study, conclusions, and the resulting revised HLT (based on the conclusions). Whereas recursive partitioning proved to provide a strong conceptual foundation, the analysis revealed a need for more effective ways of promoting reversibility of concepts. The revised HLT reflects an approach to promoting reversibility derived from the empirical and theoretical work of the researchers.     

How can self-regulated learning support the problem solving of third-grade students with mathematics anxiety?

The study compares 140 third-grade Israeli students (lower and higher achievers) who were either exposed to self-regulated learning (SRL) supported by metacognitive questioning (the MS group) or received no direct SRL support (the N_MS group). We investigated: (a) mathematical problem solving performance; (b) metacognitive strategy use in three phases of the problem-solving process; and (c) mathematics anxiety. Findings indicated that the MS students showed greater gains in mathematical problem solving performance than the N_MS students. They reported using metacognitive strategies more often, and showed a greater reduction in anxiety. In particular, the lower MS achievers showed these gains in the basic and complex tasks, in strategy use during the on-action phase of the problem solving process and a decrease in negative thoughts. The higher achievers showed greater improvement in transfer tasks and an increase in positive thoughts towards mathematics. Both the theoretical and practical implications of this study are discussed.     

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.     

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.     

Connecting multiplicative thinking and mathematical reasoning in the middle years

Mathematical reasoning and problem solving are recognised as essential 21st century skills. However, international assessments of mathematical literacy suggest these are areas of difficulty for many students. Evidenced-based learning trajectories that identify the key ideas and strategies needed to teach mathematics for understanding and support these important capacities over time are needed to support teachers and curriculum developers so that they do not have to rely solely on mathematics content knowledge. Given this goal and recent evidence to suggest a relationship between the development of multiplicative thinking and mathematical reasoning, this paper explores the processes involved in developing a single, integrated scale for multiplicative thinking and mathematical reasoning using data from a four-year design-based project to establish learning and assessment frameworks for algebraic, geometrical and statistical reasoning in the middle years of schooling.     

Case-based reasoning for repetitive combinatorial optimization problems, part I: Framework

This article presents a case-based reasoning approach for the development of learning heuristics for solving repetitive operations research problems. We first define the subset of problems we will consider in this work: repetitive combinatorial optimization problems. We then present several general forms that can be used to select previously solved problems that might aid in the solution of the current problem, and several different techniques for actually using this information to derive a solution for the current problem. We describe both fixed forms and forms that have the ability to change as we solve more problems. A simple example for the 0–1 knapsack problem is presented.     

An efficacious dynamic mathematical modelling approach for creation of best collaborative groups

In many disciplines, including business, publishing, management, health, sports, arts and education, there is a population of people which should be optimally divided into multiple groups based on certain attributes to collaboratively perform a particular task. The problem becomes more complex when some other requirements are also added. They might be importance degrees of grouping criteria, homogeneity, heterogeneity or a mixture of teams, amount of consideration to the preferences of individuals, variability or invariability of group size, having moderators, aggregation or distribution of persons, overlapping level of teams, and so forth. Several researchers have addressed the problem, but they suffered from failure to satisfy all the requirements and/or developed inexact solutions and/or had very long process times. This work reveals how these problems can be mathematically formulated through a binary integer programming approach to construct an effective model which is solvable by exact methods in an acceptable time. The suggested model was validated through data obtained from collaboration of a set of learners in an online learning discussion forum grouped by means of the provided method. The achieved outcomes confirmed that the new approach is satisfactory and promising.