The catwalk task: Reflections and synthesis: Part 1

In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional.     

Multipurpose Professional Growth Sequence: The catwalk problem as a paradigmatic example

An important concern in mathematics teacher education is how to create learning opportunities for prospective and practicing teachers that make a difference in their professional growth as educators. The first purpose of this article is to describe one way of working with prospective and practicing teachers in a graduate mathematics education course that holds promise for positively influencing the way teachers think about mathematics, about student learning, and about mathematics teaching. Specifically, I use the “catwalk” task as an example of how a single problem can serve as the basis for a coherent sequence of professional learning experiences. A second purpose of this article is to provide background information that contextualizes the subsequent two articles, each of which details the positive influence of the catwalk task sequence on the authors’ professional growth.     

数学建模能力与大学生综合素质的培养

首先论述了提高大学生综合素质是当今高校的重要任务,数学建模能力是大学生综合素质的重要标志.然后从数学建模课堂教学、数学建模课外活动阐述了数学建模能力的培养,并介绍了培养数学建模能力、提升综合素质的几点尝试和体会.     

Mathematical cognition: individual differences in resource allocation

Individuals scoring higher in tests of general cognitive abilities tend to perform better on novel and familiar mathematical tasks. It has been scarcely investigated how this superior mathematical performance relates to the amount of cognitive resources that is invested to solve a given task. In this study we propose that, on novel tasks, individuals with high cognitive abilities outperform less able individuals, because they allocate a higher amount of resources. On familiar tasks, however, individuals with higher abilities profit from more efficient processes compared to individuals of lower cognitive abilities. We tested this hypothesis by administering to 11th graders a geometric analogy task not practiced at school and an algebraic transformation task comprising operations that are routinely required during mathematical courses. General cognitive abilities were measured with Ravens Advanced Progressive matrices (fluid intelligence), the d2 (focused attention) and KAI-N (working memory capacity). Resource allocation was measured by assessing pupil diameter during the problem-solving process. Performance on both the analogy and the algebra task was correlated with general cognitive abilities, especially fluid intelligence. In line with our assumptions, a positive correlation between fluid intelligence and resource allocation was observed in the novel geometric analogy task, whereas the correlation was not significant in the more familiar algebra task.     

Group Theory,Computational Thinking,and Young Mathematicians

In this article, we investigate the artistic puzzle of designing mathematics experiences (MEs) to engage young children with ideas of group theory, using a combination of hands-on and computational thinking (CT) tools. We elaborate on: (1) group theory and why we chose it as a context for young mathematicians’ experiences with symmetry and transformations; (2) our ME design principles of agency, access, surprise and audience; (3) the affordances of CT that complement our design principles; and (4) three ME variations we tested in grades 3–6 classrooms. We then reflect on the ME variations based on our design principles and the affordances of CT, and consider how the MEs may be further adapted and improved.     

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.     

The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.     

Looking back to the roots of partially correct constructs: The case of the area model in probability

We use the notion Partially Correct Constructs (PaCCs) for students’ constructs that partially match the mathematical principles underlying the learning context. A frequent expression of partial construction of mathematical principles is that a student’s words or actions provide an inaccurate or misleading picture of the student’s knowledge. In this study, we analyze the learning process of a grade 8 student, who learns a topic in elementary probability. The student successfully accomplishes a sequence of several tasks without apparent difficulty. When working on a further task, which seems to require nothing beyond his proven competencies, he encounters difficulties. Using the epistemic actions of the RBC model for abstraction in context as tracers, we analyze his knowledge constructing processes while working on the previous tasks, and identify some of his constructs as PaCCs that are concealed in these processes and explain his later difficulties. In addition, our research points to the complexity of the knowledge structures students are expected to deal with in their attempts to learn an elementary mathematical topic with understanding.     

Handheld technology for mathematics education: flashback into the future

In the 1990s, handheld technology allowed overcoming infrastructural limitations that had hindered until then the integration of ICT in mathematics education. In this paper, we reflect on this integration of handheld technology from a personal perspective, as well as on the lessons to be learnt from it. The main lesson in our opinion concerns the growing awareness that students’ mathematical thinking is deeply affected by their work with technology in a complex and subtle way. Theories on instrumentation and orchestration make explicit this subtlety and help to design and realise technology-rich mathematics education. As a conclusion, extrapolation of these lessons to a future with mobile multi-functional handheld technology leads to the issues of connectivity and in- and out-of-school collaborative work as major issues for future research.     

“数学实验”教学的几个点体会

对近几年从事"数学实验"教学进行总结.根据教学实践中的体会,针对教学中出现的问题阐明我们的观点,陈述我们解决这些问题的办法.实践证明这些办法是行之有效的.     

Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference?

Due to uncertainty regarding the relationship between mathematical creativity, mathematical expertise and general giftedness, we have conducted a large-scale study that explores the relationship between mathematical creativity and mathematical ability. We distinguish between relative and absolute creativity in order to address personal creativity as a characteristic that can be developed in schoolchildren. This paper presents part of a study that focuses on the power of multiple solution tasks (MSTs) as a tool for the evaluation of relative creativity. We discuss relationships between mathematical creativity, mathematical ability and general giftedness as reflected in the present empirical study of senior high school students in Israel which implemented the MST tool. The study demonstrates that between-group differences are task dependent and are a function of mathematical insight as it is integrated in the mathematical task. Thus, we conclude that different types of MSTs can be used for different research purposes, which we discuss at the end of this paper.     

Part-whole number knowledge in preschool children

Ability to reflect on a number as an object of thought, and to isolate its constituent parts, is basic to a deep knowledge of arithmetic, as well as much practical and applied mathematical problem solving. Part-whole reasoning and counting are closely related in children’s numerical development. The mathematical behavior of young children in part-whole problem settings was examined by using a dynamic problem situation, in which a small set of items was partitioned such that one of the subsets remained perceptually inaccessible. Issues addressed include the problem solving strategies successful children used, adaptations children make in response to successive administrations of the task over time, and characterizations of children’s mathematical thinking based on their responses to the task.     

数学建模竞赛的发展与数学建模教师的作用

本文简要回顾数学建模竞赛活动的起源和发展,介绍国内外特别是国外学者关于提高数学建模教学质量的主要观点,并结合作者自身的经验和体会,探讨教师在数学建模教学中的作用。     

Lipschitz Continuity of Approximate Solution Mappings to Parametric Generalized Vector Equilibrium Problems

In this paper, we consider a class of multiobjective problems with equilibrium constraints. Our first task is to extend the existing constraint qualifications for mathematical problems with equilibrium constraints from the single-objective case to the multiobjective case, and our second task is to derive some stationarity conditions under the proper Pareto sense for the considered problem. After doing that, we devote ourselves to investigating the relationships among the extended constraint qualifications and the proper Pareto stationarity conditions.     

Computer Mediated Learning: An Example of an Approach

There are several possible approaches in which dynamic computerized environments play a significant, and possibly unique, role in supporting innovative learning trajectories in mathematics in general, and particularly in geometry. These approaches are influenced by the way one views mathematics and mathematical activity.In this paper we briefly describe an approach based on a problem situation and our experiences using it with students and teachers. This leads naturally to a discussion of some of the ways in which parts of the mathematics curriculum, classroom practice, and student learning may differ from the traditional approach.This revised version was published online in September 2005 with corrections to the Cover Date.     

Cooperative advertising models in supply chain management: A review

This paper reviews articles on cooperative advertising, a topic which has gained substantial interest in the recent years. Thereby, we first briefly distinguish five different definitions of cooperative advertising which can be found in operations research literature. After that, we concentrate on vertical cooperative advertising, which is the most common object of investigation and is understood as a financial agreement where a manufacturer offers to pay a certain share of his retailer’s advertising expenditures. In total, we identified 58 scientific papers considering mathematical modeling of vertical cooperative advertising. These articles are then analyzed with regard to their general model setting (e.g., the underlying supply chain structure and design of the cooperative advertising program). After that, we explain the different demand and cost functions that are employed, whereupon we distinguish between static and dynamic models. The last dimension of our review is dedicated to the game-theoretic concepts which are mostly used to reflect different forms of distribution of power within the channel.     

Modeling empowered by information and communication technologies

In this article, we describe our work with mathematical modeling (MM) at different educational levels and discuss how the use of information and communication technologies (ICTs) empowered such work. Characteristics of two trends in research which have influenced our work are presented: one is a Brazilian perspective of MM, and the other is the use of ICTs in mathematics classrooms seen through the lens of the theoretical construct “humans-with-media”. We introduce some key questions regarding the notion of mathematical model and the phases of the modeling process that were paramount for us. Finally, we describe and analyze two experiences using modeling in different educational contexts, and present some evidence of the empowering role of ICTs in such contexts.     

Living with influenza: Impacts of government imposed and voluntarily selected interventions

Focusing on mitigation strategies for global pandemic influenza, we use elementary mathematical models to evaluate the implementation and timing of non-pharmaceutical intervention strategies such as travel restrictions, social distancing and improved hygiene. A spreadsheet model of infection spread between several linked heterogeneous communities is based on analytical calculations and Monte Carlo simulations. Since human behavior will likely change during the course of a pandemic, thereby altering the dynamics of the disease, we incorporate a feedback parameter into our model to reflect altered behavior. Our results indicate that while a flu pandemic could be devastating; there are coping methods that when implemented quickly and correctly can significantly mitigate the severity of a global outbreak.     

Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

中国运筹学发展研究报告

运筹学是自20世纪三四十年代发展起来的一门新兴交叉学科,主要研究如何应用数学和计算的理论与方法对社会系统和工程系统做出最优或满意的决策。为了更好地推动中国运筹学及相关领域的研究和应用,学会组织国内运筹学领域的部分专家,并参考了国内外运筹学多个分支代表性人物对运筹学的起源和发展的回忆与评述,共同完成了此报告。报告概述了运筹学的主要特征和方法,简述了运筹学的发展历程,剖析了运筹学研究中的成功经验,综述了运筹学几个主要分支的发展状况,介绍了运筹学中十几个有代表性的难题,展望了运筹学未来发展的方向。希望此报告能引起读者进一步思考运筹学的本质,运筹学是如何成长和将如何发展,并在一定程度上推动中国运筹学更好地发展。