Das Kardinalzahlkonzept

This case-study investigates different aspects of the concept of cardinality of an eighteen-year-old student with mental retardation. At the age of six she could not relate number words, finger and objects in counting. These errors still persist in the classroom situation. This investigation shows that nevertheless her concept of cardinality is fairly highly developed. She knows that in counting she must match number words and objects one to one, the number word sequence she uses is stable, and her insight into the irrelevance of order of enumeration when counting, which she finds by trial, is a sign of the robustness of her cardinal concept. She also understands the relationships of equivalence and order of sets, and she solves arithmetical problems by counting on or down, which means that she understands the number words as cardinal and at the same time as sequence numbers. Errors occur in complex situations, where several components have to be considered. But her concept of cardinality is also incomplete: she has special difficulties concerning counting out objects bundled in tens. The same problems occur when she uses multidigit numbers: she does not see a ten-unit as composed of ten single unit items, that is to say, she replaces the hierarchic structure of the number sequence by a concatenated one. These difficulties must be interpreted as a consequence of her special weakness concerning synthetic thinking and simultaneous performing, as similar patterns can be seen in her spatial perception and in her speech. In the syntactic structure of her utterances, too, the combination of simple entities to complicated units is replaced by a mere concatenation. This means that due to brain dysfunction her behavior is determined by a particular pattern which repeatedly appears intrapersonally, and which is characteristic of some mentally retarded persons though not of all of them. Evidently mathematical thinking is also not a determined system, but a variable one. Mentally retarded students may therefore have great difficulties concerning some areas and at the same time make better progress in others. In particular, difficulties in counting objects are no obstacle to knowledge of cardinality.     

To teach or not to teach algorithms

Second, third, and fourth graders in 12 classes were individually interviewed to investigate the effects of teaching computational algorithms such as those of “carrying.” Some of the children had been encouraged to invent their own procedures and had not been taught any algorithms in grades 1 and 2, or in grades 1–3. Others had been taught the conventional algorithms prescribed by textbooks. The children were asked to solve multidigit addition and multiplication problems and to explain how they got their answers. It was found that those who had not been taught any algorithms produced significantly more correct answers. If children made errors, the incorrect answers of those who had not been taught any algorithms were much more reasonable than those found in the “Algorithms” classes. It was concluded that algorithms “unteach” place value and hinder children's development of number sense.     

Princess Elizabeth of Bohemia and Descartes’ letters (1650–1665)

After Descartes’ death in 1650, Princess Elizabeth generously shared with others several letters she had received from the philosopher, which contained philosophically as well as mathematically exciting material. In this article I place the transmission of these copies in context, revealing that Elizabeth steadily became an intellectually inspiring figure, attracting international attention. In the 1650s she stayed at Heidelberg where she discussed Cartesian philosophy with professors and students alike, including the professor of philosophy and mathematics Johann von Leuneschlos. In the mid-1660s, an initiative was taken from the English side of the Channel (Pell, More) to obtain Descartes’ mathematical letters to Elizabeth that had not yet been published. One letter of Elizabeth herself on this very subject has been preserved. The letter, addressed to Theodore Haak, will be published here for the first time. It is of special interest, because the princess supplies a general outline of her solution to the mathematical problem Descartes gave her to solve in 1643. It substantiates the hypothesis regarding Elizabeth’s solution earlier proposed by Henk Bos.     

Reflections on “Multiplication as Original Sin”: The implications of using a case to help preservice teachers understand invented algorithms

This article describes the use of a case report, Multiplication as original sin (Corwin, R. B. (1989). Multiplication as original sin. Journal of Mathematical Behavior, 8, 223-225), as an assignment in a mathematics course for preservice elementary teachers. In this case study, Corwin described her experience as a 6th grader when she revealed an invented algorithm. Preservice teachers were asked to write reflections and describe why Corwin’s invented algorithm worked. The research purpose was: to learn about the preservice teachers’ understanding of Corwin’s invented multiplication algorithm (its validity); and, to identify thought-provoking issues raised by the preservice teachers. Rather than using mathematical properties to describe the validity of Corwin’s invented algorithm, a majority of them relied on procedural and memorized explanations. About 31% of the preservice teachers demonstrated some degree of conceptual understanding of mathematical properties. Preservice teachers also made personal connections to the case report, described Corwin using superlative adjectives, and were critical of her teacher.     

Characterizing the growth of one student’s mathematical understanding in a multi-representational learning environment

The purpose of this study was to characterize the growth of one student’s mathematical understanding and use of different representations about a geometric transformation, dilation. We accomplished this purpose by using the Pirie-Kieren model jointly with the Semiotic Representation Theory as a lens. Elif, a 10^th- grade student, was purposefully chosen as the case for this study because of the growth of mathematical understanding about dilation she exhibited over time. Elif participated in task-based interviews before, during and after participating in a variety of transformation lessons where she used multiple representations, including physical and virtual manipulatives. The results revealed that Elif was able to progress in her mathematical understanding from informal levels to the formal levels in the Pirie-Kieren model as she performed treatments and conversions, movements involving different registers of representations. The results also showed numerous examples of Elif’s mathematical understanding based on folding back activities, complementary aspects of acting and expressing, and interventions. Using the two theories together provides a powerful and holistic approach to a deeper understanding of mathematical learning by characterizing and articulating the growth of mathematical understanding and the way of mathematical thinking.     

A global MINLP approach to symbolic regression

Symbolic regression methods generate expression trees that simultaneously define the functional form of a regression model and the regression parameter values. As a result, the regression problem can search many nonlinear functional forms using only the specification of simple mathematical operators such as addition, subtraction, multiplication, and division, among others. Currently, state-of-the-art symbolic regression methods leverage genetic algorithms and adaptive programming techniques. Genetic algorithms lack optimality certifications and are typically stochastic in nature. In contrast, we propose an optimization formulation for the rigorous deterministic optimization of the symbolic regression problem. We present a mixed-integer nonlinear programming (MINLP) formulation to solve the symbolic regression problem as well as several alternative models to eliminate redundancies and symmetries. We demonstrate this symbolic regression technique using an array of experiments based upon literature instances. We then use a set of 24 MINLPs from symbolic regression to compare the performance of five local and five global MINLP solvers. Finally, we use larger instances to demonstrate that a portfolio of models provides an effective solution mechanism for problems of the size typically addressed in the symbolic regression literature.     

Inexact restoration method for minimization problems arising in electronic structure calculations

An inexact restoration (IR) approach is presented to solve a matricial optimization problem arising in electronic structure calculations. The solution of the problem is the closed-shell density matrix and the constraints are represented by a Grassmann manifold. One of the mathematical and computational challenges in this area is to develop methods for solving the problem not using eigenvalue calculations and having the possibility of preserving sparsity of iterates and gradients. The inexact restoration approach enjoys local quadratic convergence and global convergence to stationary points and does not use spectral matrix decompositions, so that, in principle, large-scale implementations may preserve sparsity. Numerical experiments show that IR algorithms are competitive with current algorithms for solving closed-shell Hartree-Fock equations and similar mathematical problems, thus being a promising alternative for problems where eigenvalue calculations are a limiting factor.     

Using a Hybrid Genetic-Algorithm/Branch and Bound Approach to Solve Feasibility and Optimization Integer Programming Problems

The satisfiability problem in forms such as maximum satisfiability (MAX-SAT) remains a hard problem. The most successful approaches for solving such problems use a form of systematic tree search. This paper describes the use of a hybrid algorithm, combining genetic algorithms and integer programming branch and bound approaches, to solve MAX-SAT problems. Such problems are formulated as integer programs and solved by a hybrid algorithm implemented within standard mathematical programming software. Computational testing of the algorithm, which mixes heuristic and exact approaches, is described.     

The role of ethnomathematics in mathematics education cases from the horn of Africa

The aim of this paper is to highlight the role that ethnomathematics may have in the mathematical curriculum in the Horn of Africa. It is also a first attempt to document some social pratices and native procedures that people living in this region use to manage their “daily mathematical problems”. Examples from the local culture, which could be used to introduce mathematical arguments in the classroom, are described. The paper finally deals with the possible ways these cultural events may be included in the mathematical syllabi, such as coining new mathematics terms in the local languages, or as preparing mathematics textbooks and classroom activities.     

A comparison of heuristic algorithms for flow shop scheduling problems with setup times and limited batch size

In this paper, we propose different heuristic algorithms for flow shop scheduling problems, where the jobs are partitioned into groups or families. Jobs of the same group can be processed together in a batch but the maximal number of jobs in a batch is limited. A setup is necessary before starting the processing of a batch, where the setup time depends on the group of the jobs. In this paper, we consider the case when the processing time of a batch is given by the maximum of the processing times of the operations contained in the batch. As objective function we consider the makespan as well as the weighted sum of completion times of the jobs. For these problems, we propose and compare various constructive and iterative algorithms. We derive suitable neighbourhood structures for such problems with batch setup times and describe iterative algorithms that are based on different types of local search algorithms. Except for standard metaheuristics, we also apply multilevel procedures which use different neighbourhoods within the search. The algorithms developed have been tested in detail on a large collection of problems with up to 120 jobs.     

基于乌鸦喝水的元启发式优化算法

启发式优化算法已成为求解复杂优化问题的一种有效方法,可用于解决传统的优化方法难以求解的问题.受乌鸦喝水寓言故事启发,提出一种新型元启发式优化算法—乌鸦喝水算法,首先建立了乌鸦喝水算法数学模型;其次,给出实现该算法的详细步骤;最后,将该算法用于基准函数优化,并将该算法与乌鸦搜索算法、粒子群优化算法、多元宇宙优化算法、花授...     

A comparison of some algorithms for the nonlinear least squares problem

The problem of minimizing a sum of squares of nonlinear functions is studied. To solve this problem one usually takes advantage of the fact that the objective function is of this special form. Doing this gives the Gauss-Newton method or modifications thereof. To study how these specialized methods compare with general purpose nonlinear optimization routines, test problems were generated where parameters determining the local behaviour of the algorithms could be controlled. The order of 1000 test problems were generated for testing three algorithms: the Gauss-Newton method, the Levenberg-Marquardt method and a quasi-Newton method.     

A Teacher's Journey with a New Generation Handheld: Decisions,Struggles, and Accomplishments

In this technology‐oriented age, teachers face daily decisions regarding the use of advanced digital technologies—graphing calculators, dynamic geometry software, blogs, wikis, podcasts and the like—to enhance student mathematical understanding in their classrooms. In this case study, the authors use the Technological, Pedagogical, and Content Knowledge (TPACK) model in conjunction with a five‐stage developmental model, which can be used to describe growth in TPACK to describe the initial attempts of a teacher, Jane, to develop TPACK as she learns and attempts to integrate an advanced teaching technology into her classroom, namely the TI‐Nspire graphing calculator. The study tracks her struggles to reconcile some traditional beliefs about how students learn with her desire to be responsive to what she perceives as affordances of advanced digital technologies. Main data collection methods were journal writing, observations, document analysis, and interviews. Using the five‐stage developmental model, we saw that this experience helped Jane to move among different stages. This study showed that the TPACK model with the five‐stage developmental model can be a beneficial tool for researchers to study teachers' professional growth and is also a valuable tool for teachers to reflect on their own growth.     

It's Not Too Late to Conceptualize: Constructing a Generalized Subtraction Schema by Abstracting and Connecting Procedures

In this study, children were encouraged to abstract mathematical principles by making connections between procedures. Children from 2 sixth-grade classes (N = 58) were asked to solve and explain subtraction examples in 3 different number domains (whole numbers, fractions, and decimals), solve subtraction word problems, compare the procedures, and discuss subtraction principles. Forty percent of the children could identify procedural similarities and could abstract general subtraction principles. The rest of the children received more instruction. One group received individual abstraction and mapping instruction that encouraged them to generalize procedural steps and connect procedures. Another group received domain-specific instruction without connections between domains. The results show that following mapping instruction, children whose original instruction was mostly procedural could make connections and abstract principles, that is, construct a general subtraction schema. These children improved in conceptual knowledge and in solving transfer problems.     

It's Not Too Late to Conceptualize: Constructing a Generalized Subtraction Schema by Abstracting and Connecting Procedures

In this study, children were encouraged to abstract mathematical principles by making connections between procedures. Children from 2 sixth-grade classes (N = 58) were asked to solve and explain subtraction examples in 3 different number domains (whole numbers, fractions, and decimals), solve subtraction word problems, compare the procedures, and discuss subtraction principles. Forty percent of the children could identify procedural similarities and could abstract general subtraction principles. The rest of the children received more instruction. One group received individual abstraction and mapping instruction that encouraged them to generalize procedural steps and connect procedures. Another group received domain-specific instruction without connections between domains. The results show that following mapping instruction, children whose original instruction was mostly procedural could make connections and abstract principles, that is, construct a general subtraction schema. These children improved in conceptual knowledge and in solving transfer problems.     

A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 Special Session on Real Parameter Optimization

In recent years, there has been a growing interest for the experimental analysis in the field of evolutionary algorithms. It is noticeable due to the existence of numerous papers which analyze and propose different types of problems, such as the basis for experimental comparisons of algorithms, proposals of different methodologies in comparison or proposals of use of different statistical techniques in algorithms’ comparison. In this paper, we focus our study on the use of statistical techniques in the analysis of evolutionary algorithms’ behaviour over optimization problems. A study about the required conditions for statistical analysis of the results is presented by using some models of evolutionary algorithms for real-coding optimization. This study is conducted in two ways: single-problem analysis and multiple-problem analysis. The results obtained state that a parametric statistical analysis could not be appropriate specially when we deal with multiple-problem results. In multiple-problem analysis, we propose the use of non-parametric statistical tests given that they are less restrictive than parametric ones and they can be used over small size samples of results. As a case study, we analyze the published results for the algorithms presented in the CEC’2005 Special Session on Real Parameter Optimization by using non-parametric test procedures.     

Extended duality for nonlinear programming

Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other high-level search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex problems, including discrete and mixed-integer problems where the feasible sets are generally nonconvex.     

A Teacher’s Conception of Definition and Use of Examples When Doing and Teaching Mathematics

To contribute to an understanding of the nature of teachers’ mathematical knowledge and its role in teaching, the case study reported in this article investigated a teacher’s conception of a metamathematical concept, definition, and her use of examples in doing and teaching mathematics. Using an enactivist perspective on mathematical knowledge, the authors give an account of the case of Lily, a prospective, then beginning, teacher who conceived of mathematical definition as an object with particular form and function and engaged in purposeful, specialized use of examples when doing and teaching mathematics. Lily’s case illustrates how a teacher’s interpretation of examples (as exemplifications or single instances) and conception of the form and function of definitions can influence her doing and teaching mathematics. An implication is that teacher preparation should foster teachers’ abilities to use examples purposefully to provide students with rich opportunities to engage in mathematical processes such as defining.     

Problems of mathematical physics with boundary conditions incompletely defined

The possibility to construct the analytic solutions of boundary-value problems of mathematical physics for noncanonical domains is important from the viewpoint of the development of efficient algorithms to quantitatively estimate the characteristics of fields under study. The use of the superposition method allows one to analyze a wide class of specific problems applying the introduced notion of the general solution of a boundary-value problem. However, in this case, some difficulties can arise in the construction of calculation algorithms, because the boundary conditions are incompletely defined on the intervals, where the functions appearing in the general solution are orthogonal to one another. We present examples of problems with such difficulties and study their nature and methods to overcome them. The quantitative estimates of the exactness of constructed solutions are given.