Minimalism as a Guiding Principle: Linking Mathematical Learning to Everyday Knowledge

Studies report that students often fail to consider familiar aspects of reality in solving mathematical word problems. This study explored how different features of mathematical problems influence the way that undergraduate students employ realistic considerations in mathematical problem solving. Incorporating familiar contents in the word problems was found to have only a limited impact. Instead, removing contextual constraints from the problem goal was found to motivate students to validate their problem solving in terms of their everyday experiences. Based on these findings, what determines the authenticity and relevance of a mathematical problem seems to be whether the problem allows students to freely reconstruct the problem situation by making use of their imagination and everyday experiences. In short, the basic principle seems to be “less is more”; that is, fewer constraints in problem goals could function to help students personally associate problem solving with their everyday experiences.     

Minimalism as a Guiding Principle: Linking Mathematical Learning to Everyday Knowledge

Studies report that students often fail to consider familiar aspects of reality in solving mathematical word problems. This study explored how different features of mathematical problems influence the way that undergraduate students employ realistic considerations in mathematical problem solving. Incorporating familiar contents in the word problems was found to have only a limited impact. Instead, removing contextual constraints from the problem goal was found to motivate students to validate their problem solving in terms of their everyday experiences. Based on these findings, what determines the authenticity and relevance of a mathematical problem seems to be whether the problem allows students to freely reconstruct the problem situation by making use of their imagination and everyday experiences. In short, the basic principle seems to be “less is more”; that is, fewer constraints in problem goals could function to help students personally associate problem solving with their everyday experiences.     

An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation

An elementary proof of the maximum principle for optimal control problems whose states are governed by Volterra integral equations is given. Our proof is motivated by the work of Michel (Ref. 7) and utilizes only elementary results from analysis and mathematical programming. By appealing to Pontryagin-type perturbations of the controls, the above optimal control problem is effectively reduced to a mathematical programming problem. The results are then obtained by appealing to well-known mathematical programming results.     

With a focus on ‘Grundvorstellungen’ Part 1: a theoretical integration into current concepts

Current comparative studies such as PISA assess individual achievement in an attempt to grasp the concept of competence. Working with mathematics is then put into concrete terms in the area of application. Thereby, mathematical work is understood as a process of modelling: At first, mathematical models are taken from a real problem; then the mathematical model is solved; finally the mathematical solution is interpreted with a view to reality and the original problem is validated by the solution. During this cycle the main focus is on the transition between reality and the mathematical level. Mental objects are necessary for this transition. These mental objects are described in the German didactic with the concept of Grundvorstellungen'. In the delimitation to related educational constructs, ‘Grundvorstellungen’ can be described as mental models of a mathematical concept.     

限制情况下装卸工问题的最优解

装卸工问题是从现代物流技术中提出的一个实际问题,这个问题的雏形早在上个世纪60年代中国科学院数学研究所就提出和研究过.现代物流技术迅速发展,促成和推动装卸工问题的提出和研究.装卸工问题是一个新的NP困难的组合优化问题,首先介绍装卸工问题及限制情况下装卸工问题的数学模型,然后分析限制情况下的装卸工问题的性质,最后给出该问题的所有最优解.     

The Cereal Box Problem Revisited

The Cereal Box problem is fascinating for students of ail levels. Rich in mathematical content, this problem offers students the opportunity to collect data, make conjectures, and derive mathematical models. Using Monte Carlo methods, the Cereal Box problem is investigated in this paper, using both an experimental and theoretical framework. This investigation extends previous considerations of the Cereal Box problem. Using empirical data, students can discover patterns and relationships that help them understand the origin of the theoretical solution to the problem. Building on experimental findings, a theoretical model is derived, showing that the expected solution of the Cereal Box problem is formed from the sum of successive geometric series.     

Mathematical problem solving: an evolving research and practice domain

Research programs in mathematical problem solving have evolved with the development and availability of computational tools. I review and discuss research programs that have influenced and shaped the development of mathematical education in Mexico and elsewhere. An overarching principle that distinguishes the problem solving approach to develop and learn mathematics is to conceptualize the discipline as a set of dilemmas or problems that need to be explored and solved in terms of mathematical resources and strategies. In this context, relevant questions that help structure and organize this paper include: What does it mean to learn mathematics in terms of problem solving? To what extent do research programs in problem solving orient curricular proposals? What types of instructional scenarios promote the students’ development of mathematical thinking based on problem solving? What type of reasoning do students develop as a result of using distinct computational tools in mathematical problem solving?     

Investigating written expressions of mathematical reasoning for students with learning disabilities

Assessment results from two open-construction response mathematical tasks involving fractions and decimals were used to investigate written expression of mathematical reasoning for students with learning disabilities. The solutions and written responses of 51 students with learning disabilities in fourth and fifth grade were analyzed on four primary dimensions: (a) accuracy, (b) five elements of mathematical reasoning, (c) five elements of mathematical writing, and (d) vocabulary use. Results indicate most students were not accurate in their problem solution and communicated minimal mathematical reasoning in their written expression. In addition, students tended to use general vocabulary rather than academic precise math vocabulary and students who provided a visual representation were more likely to answer accurately. To further clarify the students struggles with mathematical reasoning, error analysis indicated a variety of error patterns existed and tended to vary widely by problem type. Our findings call for more instruction and intervention focused on supporting students mathematical reasoning through written expression. Implications for research and practice are presented.     

Wind-excited vibrations of transmission lines: A comparison of different mathematical models

In this paper, the modelling of wind-excited vibrations of conductor cables in overhead transmission lines is discussed. Only oscillations in the frequency range from 10 Hz to 50 Hz due to vortex shedding are considered. Different mathematical models are proposed for the cables, for the aerodynamic forces and for the dampers. The models are compared and their relative merits are summarized. Finally, a relatively simple approach is proposed for the calculation of the bending strains in the cables of a transmission line with dampers. The problem is an example of how mathematical modelling at different levels combines mathematical tools with engineering insight into the physical problem.     

The impact of a conceptual model-based mathematics computer tutor on multiplicative reasoning and problem-solving of students with learning disabilities

The study explored the impact of Please Go Bring Me-COnceptual Model-based Problem Solving (PGBM-COMPS) computer tutoring system on multiplicative reasoning and problem solving of students with learning disabilities. The PGBM-COMPS program focused on enhancing the multiplicative reasoning and problem solving through nurturing fundamental mathematical ideas and moving students above and beyond the concrete level of operation. This is achieved by taking advantages of the constructivist approach from mathematics education and explicit conceptual model-based problem solving approach from special education. Participants were three elementary students with learning disabilities (LD). A mixed method design was employed to investigate the effect of the PGBM-COMPS program on enhancing students’ multiplicative reasoning and problem solving. It was found that the PGBM-COMPS program significantly improved participating students’ problem solving performance not only on researcher developed criterion tests but also on a norm-referenced standardized test. Qualitative and quantities data from this study indicate that, in addition to nurturing fundamental concept of composite units, it is necessary to help students to understand underlying problem structures and move toward mathematical model-based problem representation and solving for generalized problem solving skills.     

Understanding How Students Develop Mathematical Models

in this article, we discuss findings from a research study designed to characterize students' development of significant mathematical models by examining the shifts in their thinking that occur during problem investigations. These problem investigations were designed to elicit the development of mathematical models that can be used to describe and explain the relations, patterns, and structure found in data from experienced situations. We were particularly interested in a close examination of the student interactions that appear to foster the development of such mathematical models. This classroom-based qualitative case study was conducted with precalculus students enrolled in a moderate-sized private research university. We observed several groups of 3 students each as they worked together on 5 different modeling tasks. In each task, the students were asked to create a quantitative system that could describe and explain the patterns and structures in an experienced situation and that could be used to make predictions about the situation. Our analysis of the data revealed 4 sources of mismatches that were significant in bringing about the occurrence of shifts in student thinking: conjecturing, questioning, impasses to progress, and the use of technology-based representations. The shifts in thinking in turn led to the development of mathematical models. These results suggest that students would benefit from learning environments that provide them with ample opportunity to express their ideas, ask questions, make reasoned guesses, and work with technology while engaging in problem situations that elicit the development of significant mathematical models.     

Understanding How Students Develop Mathematical Models

in this article, we discuss findings from a research study designed to characterize students' development of significant mathematical models by examining the shifts in their thinking that occur during problem investigations. These problem investigations were designed to elicit the development of mathematical models that can be used to describe and explain the relations, patterns, and structure found in data from experienced situations. We were particularly interested in a close examination of the student interactions that appear to foster the development of such mathematical models. This classroom-based qualitative case study was conducted with precalculus students enrolled in a moderate-sized private research university. We observed several groups of 3 students each as they worked together on 5 different modeling tasks. In each task, the students were asked to create a quantitative system that could describe and explain the patterns and structures in an experienced situation and that could be used to make predictions about the situation. Our analysis of the data revealed 4 sources of mismatches that were significant in bringing about the occurrence of shifts in student thinking: conjecturing, questioning, impasses to progress, and the use of technology-based representations. The shifts in thinking in turn led to the development of mathematical models. These results suggest that students would benefit from learning environments that provide them with ample opportunity to express their ideas, ask questions, make reasoned guesses, and work with technology while engaging in problem situations that elicit the development of significant mathematical models.     

Saddle-Point Criteria in an η-Approximation Method for Nonlinear Mathematical Programming Problems Involving Invex Functions

In this paper, the η-approximation method introduced by Antczak (Ref. 1) for solving a nonlinear constrained mathematical programming problem involving invex functions with respect to the same function η is extended. In this method, a so-called η-approximated optimization problem associated with the original mathematical programming problems is constructed; moreover, an η-saddle point and an η-Lagrange function are defined. By the help of the constructed η-approximated optimization problem, saddle-point criteria are obtained for the original mathematical programming problem. The equivalence between an η-saddle point of the η-Lagrangian of the associated η-approximated optimization problem and an optimal solution in the original mathematical programming problem is established.     

Historical origins of the nine-point conic. The contribution of Eugenio Beltrami

In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadratic transformations. In this sense, the work of Beltrami appears similar to the recent (after the anticipations of Magnus and Steiner) results of Schiaparelli and Cremona. Moreover, Beltrami's methods are closely related to the study of birational transformations, which in the same period were becoming one of the main topics of algebraic geometry. Finally, our work emphasises the role played by the nine-point conic problem in the studies of young Beltrami who, under Cremona's guidance, was then developing his mathematical skills. To this end, we make considerable use of the unedited correspondence Beltrami – Cremona, preserved in the Istituto Mazziniano, Genoa.     

A new method of solving nonlinear mathematical programming problems involving r-invex functions

A new approach to a solution of a nonlinear constrained mathematical programming problem involving r-invex functions with respect to the same function η is introduced. An η-approximated problem associated with an original nonlinear mathematical programming problem is presented that involves η-approximated functions constituting the original problem. The equivalence between optima points for the original mathematical programming problem and its η-approximated optimization problem is established under r-invexity assumption.     

The Stefan Problem: Comments on its Present State

After almost a century in the history of the Stefan problem,we can present a survey of many works containing a wide rangeof ideas for the solution of pure mathematical, numerical, scientificor technical problems. I have no doubt that a survey devotedespecially to pure mathematical topics, or to numerical aspectsof the problem or, finally, to specifically physical or technicalproblems would be of interest, even if it only represented aneffort to systematize the existing material. However, with theaim of stimulating an exchange of ideas concerning the mostimportant unsolved problems in the field under consideration,I would like to formulate some questions which seem to me theoreticallyinteresting, important from the pure mathematical and from scientificor technical points of view, and which remain unsolved in spiteof their importance. Naturally, the estimation of the importanceof a problem strongly depends on the individual point of viewand therefore the choice of topics for discussion reflects vefysubjective decisions. Since the discussion mainly touches onunsolved problems, the ideas discussed here are of a speculativenature.     

Problem modification as a tool for detecting cognitive flexibility in school children

This paper presents the results of an experiment in which fourth to sixth graders with above-average mathematical abilities modified a given problem. The experiment found evidence of links between problem posing and cognitive flexibility. Emerging from organizational theory, cognitive flexibility is conceptualized through three primary constructs: cognitive variety, cognitive novelty, and changes in cognitive framing. Among these components, changes in cognitive framing could be effectively detected in problem-posing situations, giving a relevant indication of students’ creative potential. The students’ capacity to generate coherent and consistent problems in the context of problem modification may indicate the existence of a strategy of functional type for generalizations, which seems to be specific to mathematical creativity.     

Mathematical programming based heuristics for the 0–1 MIP: a survey

The 0–1 mixed integer programming problem is used for modeling many combinatorial problems, ranging from logical design to scheduling and routing as well as encompassing graph theory models for resource allocation and financial planning. This paper provides a survey of heuristics based on mathematical programming for solving 0–1 mixed integer programs (MIP). More precisely, we focus on the stand-alone heuristics for 0–1 MIP as well as those heuristics that use linear programming techniques or solve a series of linear programming models or reduced problems, deduced from the initial one, in order to produce a high quality solution of a considered problem. Our emphasis will be on how mathematical programming techniques can be used for approximate problem solving, rather than on comparing performances of heuristics.     

A new lot sizing and scheduling heuristic for multi-site biopharmaceutical production

Biopharmaceutical manufacturing requires high investments and long-term production planning. For large biopharmaceutical companies, planning typically involves multiple products and several production facilities. Production is usually done in batches with a substantial set-up cost and time for switching between products. The goal is to satisfy demand while minimising manufacturing, set-up and inventory costs. The resulting production planning problem is thus a variant of the capacitated lot-sizing and scheduling problem, and a complex combinatorial optimisation problem. Inspired by genetic algorithm approaches to job shop scheduling, this paper proposes a tailored construction heuristic that schedules demands of multiple products sequentially across several facilities to build a multi-year production plan (solution). The sequence in which the construction heuristic schedules the different demands is optimised by a genetic algorithm. We demonstrate the effectiveness of the approach on a biopharmaceutical lot sizing problem and compare it with a mathematical programming model from the literature. We show that the genetic algorithm can outperform the mathematical programming model for certain scenarios because the discretisation of time in mathematical programming artificially restricts the solution space.